Mathematics - Grade 12 CAPS

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Chapter 1: Sequences and series

Discuss and explain important terminology.

Be consistent with the use of “common” difference and “constant” ratio to avoid confusing learners.

Learners must understand the difference between arithmetic and geometric sequences.

Explain sigma notation carefully as many learners have difficulty with this concept.

Encourage learners to use the correct notation (for example etc.) when solving problems.

Use the investigation for the sum of an infinite series to introduce the concept of convergence and divergence.

In earlier grades we learnt about number patterns, which included linear sequences with a common difference and quadratic sequences with a common second difference. We also looked at completing a sequence and how to determine the general term of a sequence.

In this chapter we also look at geometric sequences, which have a constant ratio between consecutive terms. We will learn about arithmetic and geometric series, which are the summing of the terms in sequences.

1.1 Arithmetic sequences

An arithmetic sequence is a sequence where consecutive terms are calculated by adding a constant value (positive or negative) to the previous term. We call this constant value the common difference ().

For example,

This is an arithmetic sequence because we add to each term to get the next term:

First term

Second term

Third term

Fourth term

Fifth term

Video: 284G

Exercise 1.1

Find the common difference and write down the next terms of the sequence.

This is an example of a non-numeric arithmetic sequence.

The general term for an arithmetic sequence

For a general arithmetic sequence with first term and a common difference , we can generate the following terms:

Therefore, the general formula for the term of an arithmetic sequence is:

Arithmetic sequence

An arithmetic (or linear) sequence is an ordered set of numbers (called terms) in which each new term is calculated by adding a constant value to the previous term:

where

is the term;

is the position of the term in the sequence;

is the first term;

is the common difference.

Test for an arithmetic sequence

To test whether a sequence is an arithmetic sequence or not, check if the difference between any two consecutive terms is constant:

If this is not true, then the sequence is not an arithmetic sequence.

Worked example 1: Arithmetic sequence

Given the sequence .

Is this an arithmetic sequence?

Find the formula of the general term.

Determine the number of terms in the sequence.

Check if there is a common difference between successive terms

Determine the formula for the general term

Write down the formula and the known values:

A graph was not required for this question but it has been included to show that the points of the arithmetic sequence lie in a straight line.

Note: The numbers of the sequence are natural numbers and therefore we should not connect the plotted points. In the diagram above, a dotted line has been used to show that the graph of the sequence lies on a straight line.

Determine the number of terms in the sequence

Write the final answer

Therefore, there are terms in the sequence.

Arithmetic mean

The arithmetic mean between two numbers is the number half-way between the two numbers. In other words, it is the average of the two numbers. The arithmetic mean and the two terms form an arithmetic sequence.

For example, the arithmetic mean between and is calculated:

Plotting a graph of the terms of a sequence sometimes helps in determining the type of sequence involved. For an arithmetic sequence, plotting vs. results in the following graph:

If the sequence is arithmetic, the plotted points will lie in a straight line.

Arithmetic sequences are also called linear sequences, where the common difference () is the gradient of the straight line.

Exercise 1.2

Given the sequence

Find the next term in the sequence.

Determine the general term of the sequence.

Which term has a value of ?

Given the sequence

Is this an arithmetic sequence? Justify your answer by calculation.

Yes, this is an arithmetic sequence since there is a common difference of between consecutive terms.

Calculate .

Which term has a value of ?

Determine by calculation whether or not is a term in the sequence?

This value of is not a positive integer, therefore is not a term of this sequence.

An arithmetic sequence has the general term .

Calculate the second, third and tenth terms of the sequence.

Draw a diagram of the sequence for .

The first term of an arithmetic sequence is and . Find .

Calculate the common difference :

Now determine the general term for the sequence:

What are the important characteristics of an arithmetic sequence?

There is a common difference between any two successive terms in the sequence.

The graph of vs. is a straight line.

You are given the first four terms of an arithmetic sequence. Describe the method you would use to find the formula for the term of the sequence.

Use the given terms to calculate the common difference .

From given terms, we know that .

Substitute the values for and into the equation .

Simplify and gather like terms.

A single square is made from matchsticks. To make two squares in a row takes matchsticks, while three squares in a row takes matchsticks.

Write down the first four terms of the sequence.

What is the common difference?

Determine the formula for the general term.

How many matchsticks are in a row of squares?

If there are matchsticks, calculate the number of squares in the row.

A pattern of equilateral triangles decorates the border of a girl's skirt. Each triangle is made by three stitches, each having a length of .

Complete the table:

Figure no. |  |  |  |  |  |
No. of stitches |  |  |  |  |  |

The border of the skirt is in length. If the entire length of the border is decorated with the triangular pattern, how many stitches will there be?

The terms form an arithmetic sequence. Find and the term of the sequence.

[IEB, Nov 2011]

The arithmetic mean of and is . Determine the value of in terms of .

Insert seven arithmetic means between the terms and .

Let the arithmetic means be:

Therefore the sequence is:

Quadratic sequence

A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant.

The general formula for the term of a quadratic sequence is:

It is important to note that the first differences of a quadratic sequence form an arithmetic sequence. This sequence has a common difference of between consecutive terms. In other words, a linear sequence results from taking the first differences of a quadratic sequence.

Worked example 2: Quadratic sequence

Consider the pattern of white and blue blocks in the diagram below.

Determine the sequence formed by the white blocks .

Find the sequence formed by the blue blocks .

Pattern number |  |  |  |  |  |  |
No. of white blocks |  |  |  |  |  |  |
Common difference |  |  |  |  |  |  |

Pattern number |  |  |  |  |  |  |
No. of blue blocks |  |  |  |  |  |  |
Common difference |  |  |  |  |  |  |

Use the diagram to complete the table for the white blocks

Pattern number |  |  |  |  |  |  |
No. of white blocks |  |  |  |  |  |  |
Common difference |  |  |  |  |  |  |

We see that the next term in the sequence is obtained by adding to the previous term, therefore the sequence is linear and the common difference is .

The general term is:

Use the diagram to complete the table for the blue blocks

Pattern number |  |  |  |  |  |
No. of blue blocks |  |  |  |  |  |
Difference |  |  |  |  |  |

We notice that there is no common difference between successive terms. However, there is a pattern and on further investigation we see that this is in fact a quadratic sequence:

Pattern number |  |  |  |  |  |  |
No. of blue blocks |  |  |  |  |  |  |
First difference |  |  |  |  |  |  |
Second difference |  |  |  |  |  |  |
Pattern |  |  |  |  |  |  |

Draw a graph of vs. for each sequence

Since the numbers of the sequences are natural numbers , we should not connect the plotted points. In the diagram above, a dotted line has been used to show that the graph of the sequence formed by the white blocks is a straight line and the graph of the sequence formed by the blue blocks is a parabola.

Exercise 1.3

Determine whether each of the following sequences is:

a linear sequence;

a quadratic sequence;

or neither.

Quadratic sequence

Linear sequence

Quadratic sequence

Linear sequence

Quadratic sequence

Neither

Quadratic sequence

Quadratic sequence

A quadratic pattern is given by . Find the values of and if the sequence starts with the following terms:

Starting with the first term, we have and :

For the second term, we use and :

Now we must solve these equations simultaneously. We can do this by substitution, but here we will show the solution using the 'elmination' method (which means subtracting one equation from the other to cancel the 's).

Finally, calculate the value of . As usual for simultaneous equations, this means that we must substitute the into either of the equations we used above. Let's use the equation .

The final answers are and .

NOTE: Now we know that the general term of the sequence is . We can use this to check our answers. We know that . Substitute into the general formula to check:

are the first terms of a sequence.

Is the sequence linear or quadratic? Motivate your answer.

This is an arithmetic sequence since there is a common difference of between consecutive terms.

What is the next term in the sequence?

Calculate .

Given , determine the values of and if the sequence starts with the terms:

Starting with the first term, we have and :

For the second term, we use and :

Now we must solve these equations simultaneously. We can do this by substitution, but here we will show the solution using the 'elimination' method (which means subtracting one equation from the other to cancel the 's).

Finally, calculate the value of . As usual for simultaneous equations, this means that we must substitute the into either of the equations we used above. Let's use the equation . The final answers are and .

NOTE: Now we know that the general term of the sequence is . We can use this to check our answers. We know that . Substitute into the general formula to check:

The first term of a quadratic sequence is , the third term is and the common second difference is . Determine the first six terms in the sequence.

A quadratic sequence has a second term equal to , a third term equal to and a fourth term equal to .

Determine the second difference for this sequence.

Hence, or otherwise, calculate the first term of the pattern.

1.2 Geometric sequences

Geometric sequence

A geometric sequence is a sequence of numbers in which each new term (except for the first term) is calculated by multiplying the previous term by a constant value called the constant ratio ().

Video: 285K

This means that the ratio between consecutive numbers in a geometric sequence is a constant (positive or negative). We will explain what we mean by ratio after looking at the following example.

Example: A flu epidemic

Influenza (commonly called “flu”) is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severe illness that most of us get during winter time. The influenza virus is spread from person to person in respiratory droplets of coughs and sneezes. This is called “droplet spread”. This can happen when droplets from a cough or sneeze of an infected person are propelled through the air and deposited on the mouth or nose of people nearby. It is good practice to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu. Regular hand washing is an effective way to prevent the spread of infection and illness.

Assume that you have the flu virus, and you forgot to cover your mouth when two friends came to visit while you were sick in bed. They leave, and the next day they also have the flu. Let's assume that each friend in turn spreads the virus to two of their friends by the same droplet spread the following day. Assuming this pattern continues and each sick person infects 2 other friends, we can represent these events in the following manner:

Each person infects two more people with the flu virus.

We can tabulate the events and formulate an equation for the general case:

Day (n)

No. of newly-infected people

The above table represents the number of newly-infected people after days since you first infected your friends.

You sneeze and the virus is carried over to people who start the chain (). The next day, each one then infects of their friends. Now people are newly-infected. Each of them infects people the third day, and new people are infected, and so on. These events can be written as a geometric sequence:

Note the constant ratio () between the events. Recall from the linear arithmetic sequence how the common difference between terms was established. In the geometric sequence we can determine the constant ratio () from:

More generally,

Exercise 1.4

Determine the constant ratios for the following geometric sequences and write down the next three terms in each sequence:

The general term for a geometric sequence

From the flu example above we know that and , and we have seen from the table that the term is given by .

The general geometric sequence can be expressed as:

Therefore the general formula for a geometric sequence is:

where

is the first term in the sequence;

is the constant ratio.

Test for a geometric sequence

To test whether a sequence is a geometric sequence or not, check if the ratio between any two consecutive terms is constant:

If this condition does not hold, then the sequence is not a geometric sequence.

Exercise 1.5

Determine the general formula for the term of each of the following geometric sequences:

Worked example 3: Flu epidemic

We continue with the previous flu example, where is the number of newly-infected people after days:

Calculate how many newly-infected people there are on the tenth day.

On which day will people be newly-infected?

Write down the known values and the general formula

Use the general formula to calculate

Substitute into the general formula:

On the tenth day, there are newly-infected people.

Use the general formula to calculate

We know that and can use the general formula to calculate the corresponding value of :

There are newly-infected people on the day.

For this geometric sequence, plotting the number of newly-infected people () vs. the number of days () results in the following graph:

Day (n)

No. of newly-infected people

In this example we are only dealing with positive integers , therefore the graph is not continuous and we do not join the points with a curve (the dotted line has been drawn to indicate the shape of an exponential graph).

Geometric mean

The geometric mean between two numbers is the value that forms a geometric sequence together with the two numbers.

For example, the geometric mean between and is the number that has to be inserted between and to form the geometric sequence:

Important: remember to include both the positive and negative square root. The geometric mean generates two possible geometric sequences:

In general, the geometric mean () between two numbers and forms a geometric sequence with and :

Exercise 1.6

The term of a sequence is given by the formula .

Write down the first three terms of the sequence.

What type of sequence is this?

Therefore this is a geometric sequence with constant ratio .

Consider the following terms:

The first three terms form an arithmetic sequence and the last three terms form a geometric sequence. Determine the values of and if both are positive integers.

[IEB, Nov 2006]

First consider the arithmetic sequence:

Now consider the geometric sequence:

Therefore and giving the terms

Given a geometric sequence with second term and ninth term .

Determine the value of .

Find the value of .

Determine the general formula of the sequence.

The diagram shows four sets of values of consecutive terms of a geometric sequence with the general formula .

Determine and .

Find and .

Determine :

Determine :

Find the fifth term of the sequence.

Write down the next two terms for the following sequence:

Check if this is a geometric sequence:

This is a geometric sequence with . Therefore, and .

is an arithmetic sequence and is a geometric sequence. All terms in the sequences are integers. Calculate the values of and .

For the arithmetic sequence:

For the geometric sequence:

The two numbers and are given.

Write down the geometric mean between the two numbers in terms of and .

Note: in this case only the positive square root is valid.

Determine the constant ratio of the resulting sequence.

Insert three geometric means between and . Give all possible answers.

Let the geometric sequence be

Therefore possible geometric sequences are:

1.3 Series

It is often important and valuable to determine the sum of the terms of an arithmetic or geometric sequence. The sum of any sequence of numbers is called a series.

Finite series

We use the symbol for the sum of the first terms of a sequence :

If we sum only a finite number of terms, we get a finite series.

For example, consider the following sequence of numbers

We can calculate the sum of the first four terms:

This is an example of a finite series since we are only summing four terms.

Infinite series

If we sum infinitely many terms of a sequence, we get an infinite series:

Sigma notation

Sigma notation is a very useful and compact notation for writing the sum of a given number of terms of a sequence.

A sum may be written out using the summation symbol (Sigma), which is the capital letter “S” in the Greek alphabet. It indicates that you must sum the expression to the right of the summation symbol:

For example,

In general,

where

is the index of the sum;

is the lower bound (or start index), shown below the summation symbol;

is the upper bound (or end index), shown above the summation symbol;

is a term of a sequence;

the number of terms in the series .

The index increases from to by steps of .

Note that this is also sometimes written as:

When we write out all the terms in a sum, it is referred to as the expanded form.

If we are summing from (which implies summing from the first term in a sequence), then we can use either or notation:

Worked example 4: Sigma notation

Expand the sequence and find the value of the series:

Expand the formula and write down the first six terms of the sequence

This is a geometric sequence with a constant ratio of between consecutive terms.

Determine the sum of the first six terms of the sequence

Worked example 5: Sigma notation

Find the value of the series:

Expand the sequence and write down the five terms

Determine the sum of the five terms of the sequence

Worked example 6: Sigma notation

Write the following series in sigma notation:

Consider the series and determine if it is an arithmetic or geometric series

First test for an arithmetic series: is there a common difference?

We let:

We calculate:

There is a common difference of , therefore this is an arithmetic series.

Determine the general formula of the series

Be careful: brackets must be used when substituting into the general term. Otherwise the equation would be , which would be incorrect.

Determine the sum of the series and write in sigma notation

Rules for sigma notation

Given two sequences, and :

For any constant that is not dependent on the index :

Be accurate with the use of brackets:

Example 1:

Example 2:

Note: the series in the second example has the general term and the is added to the sum of the three terms. It is very important in sigma notation to use brackets correctly.

The values of :

start at ( is not always );

increase in steps of ;

and end at .

Exercise 1.7

Determine the value of the following:

Expand the series:

Calculate the value of :

Write the following in sigma notation:

Geometric series with , constant ratio and general formula .

Write the sum of the first terms of the series below in sigma notation:

Arithmetic series with , common difference and general formula .

Write the sum of the first natural, odd numbers in sigma notation.

Arithmetic series with , common difference and general formula .

1.4 Finite arithmetic series

An arithmetic sequence is a sequence of numbers, such that the difference between any term and the previous term is a constant number called the common difference ():

where

is the term of the sequence;

is the first term;

is the common difference.

When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series.

The sum of the first one hundred integers

A simple arithmetic sequence is when and , which is the sequence of positive integers:

If we wish to sum this sequence from to any positive integer, for example , we would write

This gives the answer to the sum of the first positive integers.

The mathematician, Karl Friedrich Gauss, discovered the following proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from to . Young Karl quickly realised how to do this and shocked the teacher with the correct answer, . This is the method that he used:

Write the numbers in ascending order.

Write the numbers in descending order.

Add the corresponding pairs of terms together.

Simplify the equation by making the subject of the equation.

General formula for a finite arithmetic series

If we sum an arithmetic sequence, it takes a long time to work it out term-by-term. We therefore derive the general formula for evaluating a finite arithmetic series. We start with the general formula for an arithmetic sequence of terms and sum it from the first term () to the last term in the sequence ():

This general formula is useful if the last term in the series is known.

We substitute into the above formula and simplify:

The general formula for determining the sum of an arithmetic series is given by:

or

For example, we can calculate the sum for the arithmetic sequence by summing all the individual terms:

or, more sensibly, we could use the general formula for determining an arithmetic series by substituting , and :

This example demonstrates how useful the general formula for determining an arithmetic series is, especially when the series has a large number of terms.

Video: 286N

Worked example 7: General formula for the sum of an arithmetic sequence

Find the sum of the first terms of an arithmetic series with by using the formula.

Use the general formula to generate terms of the sequence and write down the known variables

This gives the sequence:

Write down the general formula and substitute the known values

Write the final answer

Worked example 8: Sum of an arithmetic sequence if first and last terms are known

Find the sum of the series

Identify the type of series and write down the known variables

Determine the value of

Use the general formula to find the sum of the series

Write the final answer

Worked example 9: Finding given the sum of an arithmetic sequence

Given an arithmetic sequence with and , determine how many terms must be added together to give a sum of .

Write down the known variables

Use the general formula to determine the value of

but must be a positive integer, therefore .

We could have solved for using the quadratic formula but factorising by inspection is usually the quickest method.

Write the final answer

Worked example 10: Finding given the sum of an arithmetic sequence

The sum of the second and third terms of an arithmetic sequence is equal to zero and the sum of the first terms of the series is equal to . Find the first three terms in the series.

Write down the given information

Solve the two equations simultaneously

Write the final answer

The first three terms of the series are:

Calculating the value of a term given the sum of terms:

If the first term in a series is , then .

We also know the sum of the first two terms , which we rearrange to make the subject of the equation:

Similarly, we could determine the third and fourth term in a series:

and

Exercise 1.8

Determine the value of :

This is an arithmetic series with and :

The sum to terms of an arithmetic series is .

How many terms of the series must be added to give a sum of ?

but must be a positive integer, therefore .

Determine the sixth term of the series.

The common difference of an arithmetic series is . Calculate the values of for which the term of the series is , and the sum of the first terms is .

Explain why there are two possible answers.

There are two series that satisfy the given parameters:

The third term of an arithmetic sequence is and the seventh term is . Determine the sum of the first terms of the sequence.

Calculate the sum of the arithmetic series .

Evaluate without using a calculator:

Consider the numerator:

Consider the denominator:

Now consider the quotient of the two series:

The second term of an arithmetic sequence is and the sum of the first six terms of the series is .

Find the first term and the common difference.

Hence determine .

[IEB, Nov 2004]

Determine the value of the following:

Arithmetic series:

Arithmetic series:

Determine the value of .

Series:

but must be a positive integer, therefore .

Alternative method:

but must be a positive integer, therefore .

The sum of terms of an arithmetic series is for all values of . Determine the common difference.

The sum of an arithmetic series is times its first term, while the last term is times the first term. Calculate the number of terms in the series if the first term is not equal to zero.

1.5 Finite geometric series

When we sum a known number of terms in a geometric sequence, we get a finite geometric series. We generate a geometric sequence using the general form:

where

is the position of the sequence;

is the term of the sequence;

is the first term;

is the constant ratio.

General formula for a finite geometric series

The general formula for determining the sum of a geometric series is given by:

This formula is easier to use when .

Alternative formula:

The general formula for determining the sum of a geometric series is given by:

This formula is easier to use when .

Video: 2875

Worked example 11: Sum of a geometric series

Calculate:

Write down the first three terms of the series

We have generated the series

Determine the values of and

Use the general formula to find the sum of the series

Write the final answer

Worked example 12: Sum of a geometric series

Given a geometric series with and . Determine the values of and if .

Determine the values of and

Therefore the geometric series is Notice that the signs of the terms alternate because .

We write the general term for this series as .

Use the general formula for the sum of a geometric series to determine the value of

Write the final answer

Worked example 13: Sum of a geometric series

Use the general formula for the sum of a geometric series to determine if

Write down the first three terms of the series

We have generated the series

We can take out the common factor and write the series as:

Determine the values of and

Calculate the sum of the first eight terms of the geometric series

So then we can write:

Write the final answer

Exercise 1.9

Prove that and state any restrictions.

where .

Given the geometric sequence determine:

The eighth term of the sequence.

The sum of the first eight terms of the sequence.

Determine:

Find the sum of the first terms of the geometric series

Show that the sum of the first terms of the geometric series is given by .

The eighth term of a geometric sequence is . The third term is . Find the sum of the first terms.

Given:

Find the first three terms in the series.

Calculate the number of terms in the series if .

The ratio between the sum of the first three terms of a geometric series and the sum of the , and terms of the same series is . Determine the constant ratio and the first terms if the third term is .

1.6 Infinite series

So far we have been working only with finite sums, meaning that whenever we determined the sum of a series, we only considered the sum of the first terms. We now consider what happens when we add an infinite number of terms together. Surely if we sum infinitely many numbers, no matter how small they are, the answer goes to infinity? In some cases the answer does indeed go to infinity (like when we sum all the positive integers), but surprisingly there are some cases where the answer is a finite real number.

Sum of an infinite series

Cut a piece of string in length.

Now cut the piece of string in half and place one half on the desk.

Cut the other half in half again and put one of the pieces on the desk.

Repeat this process until the piece of string is too short to cut easily.

Draw a diagram to illustrate the sequence of lengths of the pieces of string.

Can this sequence be expressed mathematically? Hint: express the shorter lengths of string as a fraction of the original length of string.

What is the sum of the lengths of all the pieces of string?

Predict what would happen if these steps could be repeated infinitely many times.

Will the sum of the lengths of string ever be greater than ?

What can you conclude?

Worked example 14: Sum to infinity

Complete the table below for the geometric series and answer the questions that follow:

Terms

As more and more terms are added, what happens to the value of ?

As more more and more terms are added, what happens to the value of ?

Predict the maximum value of for the sum of infinitely many terms in the series.

Complete the table

Terms

Consider the value of and

As more terms in the series are added together, the value of increases:

However, by considering , we notice that the amount by which increases gets smaller and smaller as more terms are added:

We can therefore conclude that the value of is approaching a maximum value of ; it is converging to .

Write conclusion mathematically

We can conclude that the sum of the series

gets closer to 1 () as the number of terms approaches infinity (), therefore the series converges.

We express the sum of an infinite number of terms of a series as

Convergence and divergence

If the sum of a series gets closer and closer to a certain value as we increase the number of terms in the sum, we say that the series converges. In other words, there is a limit to the sum of a converging series. If a series does not converge, we say that it diverges. The sum of an infinite series usually tends to infinity, but there are some special cases where it does not.

Exercise 1.10

For each of the general terms below:

Determine if it forms an arithmetic or geometric series.

Calculate .

Determine if the series is convergent or divergent.

Therefore, this is a divergent series.

This is a divergent series.

This series is convergent (since the ) and converges to .

This is a divergent series.

Note the following:

An arithmetic series never converges: as tends to infinity, the series will always tend to positive or negative infinity.

Some geometric series converge (have a limit) and some diverge (as tends to infinity, the series does not tend to any limit or it tends to infinity).

Infinite geometric series

There is a simple test for determining whether a geometric series converges or diverges; if , then the infinite series will converge. If lies outside this interval, then the infinite series will diverge.

Test for convergence:

If , then the infinite geometric series converges.

If , then the infinite geometric series diverges.

We derive the formula for calculating the value to which a geometric series converges as follows:

Now consider the behaviour of for as becomes larger.

Let :

Since is in the range , we see that gets closer to as gets larger. Therefore gets closer to .

Therefore,

The sum of an infinite geometric series is given by the formula

where

is the first term of the series;

is the constant ratio.

Alternative notation:

In words: as the number of terms tends to infinity, the sum of a converging geometric series tends to the value .

Video: 287P

Worked example 15: Sum to infinity of a geometric series

Given the series . Find the sum to infinity if it exists.

Determine the value of

We need to know the value of to determine whether the series converges or diverges.

Since , we can conclude that this is a convergent geometric series.

Determine the sum to infinity

Write down the formula for the sum to infinity and substitute the known values:

As tends to infinity, the sum of this series tends to ; no matter how many terms are added together, the value of the sum will never be greater than .

Worked example 16: Using the sum to infinity to convert recurring decimals to fractions

Use two different methods to convert the recurring decimal to a proper fraction.

Convert the recurring decimal to a fraction using equations

Convert the recurring decimal to a fraction using the sum to infinity

This is a geometric series with . And since , we can conclude that the series is convergent.

Worked example 17: Sum to infinity

Determine the possible values of and if

Write down the sum to infinity formula and substitute known values

Apply the condition for convergence to determine possible values of

For a series to converge:

Write the final answer

For the series to converge, and .

Exercise 1.11

What value does approach as tends towards ?

Find the sum to infinity of the geometric series

Determine for which values of , the geometric series will converge.

For the series to converge, , therefore:

The sum to infinity of a geometric series with positive terms is and the sum of the first two terms is . Find , the first term, and , the constant ratio between consecutive terms.

Use the sum to infinity to show that .

Rewrite the recurring decimal:

This is a geometric series with and .

A shrub high is planted in a garden. At the end of the first year, the shrub is tall. Thereafter the growth of the shrub each year is half of it's growth in the previous year. Show that the height of the shrub will never exceed . Draw a graph of the relationship between time and growth.

[IEB, Nov 2003]

Write the annual growth of the shrub as a series:

This is a geometric series with and .

Therefore the growth of the shrub is limited to , and the maximum height of the shrub is therefore + = .

Note: we may join the points on the graph because the growth is continuous.

Find :

Write out the series on the RHS:

This is an arithmetic series with and .

Write out the series on the LHS:

This is a geometric series with and \quad ( for the series to converge).

1.7 Summary

Arithmetic sequence

common difference between any two consecutive terms:

general form:

general formula:

graph of the sequence lies on a straight line

Quadratic sequence

common second difference between any two consecutive terms

general formula:

graph of the sequence lies on a parabola

Geometric sequence

constant ratio between any two consecutive terms:

general form:

general formula:

graph of the sequence lies on an exponential curve

Sigma notation

Sigma notation is used to indicate the sum of the terms given by , starting from and ending at .

Series

the sum of certain numbers of terms in a sequence

arithmetic series:

geometric series:

if

if

Sum to infinity

A convergent geometric series, with , tends to a certain fixed number as the number of terms in the sum tends to infinity.

Exercise 1.12

Is an example of a finite series or an infinite series?

Infinite arithmetic series

A new soccer competition requires each of teams to play every other team once.

Calculate the total number of matches to be played in the competition.

If each of teams played each other once, determine a formula for the total number of matches in terms of .

Calculate:

Number of terms = end index - start index

The first three terms of a convergent geometric series are: ; ; .

Calculate the value of , ().

Sum to infinity of the series.

Write the sum of the first twenty terms of the following series in notation.

This is a geometric series:

Determine:

A man was injured in an accident at work. He receives a disability grant of in the first year. This grant increases with a fixed amount each year.

What is the annual increase if he received a total of over years?

This is an arithmetic series.

His initial annual expenditure is , which increases at a rate of per year. After how many years will his expenses exceed his income?

This is an arithmetic series.

Therefore his expenses will exceed his income after years.

The length of the side of a square is units. This square is divided into equal, smaller squares. One of the smaller squares is then divided into four equal, even smaller squares. One of the even smaller squares is divided into four, equal squares. This process is repeated indefinitely. Calculate the sum of the areas of all the squares.

This gives the geometric series:

Thembi worked part-time to buy a Mathematics book which costs . On 1 February she saved , and every day saves more than she saved the previous day. So, on the second day, she saved , and so on. After how many days did she have enough money to buy the book?

This is an arithmetic series.

since must be a positive integer.

A plant reaches a height of after one year under ideal conditions in a greenhouse. During the next year, the height increases by . In each successive year, the height increases by of the previous year's growth. Show that the plant will never reach a height of more than .

This is a geometric series.

Therefore the height of the plant is limited to + =

Calculate the value of if:

This is an arithmetic series.

since must be a positive integer.

Michael saved during the first month of his working life. In each subsequent month, he saved % more than what he had saved in the previous month.

How much did he save in the seventh working month?

This is a geometric series.

Therefore he saved in seventh month.

How much did he save all together in his first working months?

Therefore he saved in first months.

The Cape Town High School wants to build a school hall and is busy with fundraising. Mr. Manuel, an ex-learner of the school and a successful politician, offers to donate money to the school. Having enjoyed mathematics at school, he decides to donate an amount of money on the following basis. He sets a mathematical quiz with questions. For the correct answer to the first question (any learner may answer), the school will receive , for a correct answer to the second question, the school will receive , and so on. The donations form a geometric sequence. Calculate, to the nearest Rand:

The amount of money that the school will receive for the correct answer to the question.

This is a geometric series.

They receive for question .

The total amount of money that the school will receive if all questions are answered correctly.

They receive !

The first term of a geometric sequence is , and the ratio of the sum of the first eight terms to the sum of the first four terms is . Find the first three terms of the sequence, if it is given that all the terms are positive.

Given the geometric sequence:

Determine , ().

Show that the constant ratio is .

Determine the tenth term of this sequence correct to one decimal place.

The second and fourth terms of a convergent geometric series are and , respectively. Find the sum to infinity of this series, if all its terms are positive.

Evaluate:

represents the sum of the first terms of a particular series. Find the second term.

Determine whether the following series converges for the given values of . If it does converge, calculate the sum to infinity.

This is a geometric sequence.

This is a geometric sequence.

Calculate:

This is a geometric sequence.

The sum of the first terms of a sequence is . Find the tenth term.

Alternative method:

This is an arithmetic sequence.

The powers of are removed from the following set of positive integers

Find the sum of remaining integers.

Remove the powers of to make a separate series:

This is a geometric series.

Observe the pattern below:

If the pattern continues, find the number of letters in the column containing M's.

This is an arithmetic series.

If the total number of letters in the pattern is , which letter will be in the last column.

so the letter “s” will be in the last column.

Write as a proper fraction.

Rewrite the recurring decimal:

The part in the square brackets is a geometric series with and .

Given:

For which values of will converge?

This is a geometric series with and . For the series to converge:

Determine the value of .

From the definition of a geometric sequence, deduce a formula for calculating the sum of terms of the series

Calculate the tenth term of the series if .

Alternative method:

A theatre is filling up at a rate of people in the first minute, people in the second minute, and people in the third minute and so on. After minutes the theatre is half full. After how many minutes will the theatre be full?

[IEB, Nov 2001]

It takes minutes for the theatre to be full.

Chapter 2: Functions

Learners must be encouraged to check whether or not the inverse is a function.

It is very important that learners understand that notation can only be used if the inverse is a function.

Learners must not confuse the inverse function and the reciprocal .

Encourage learners to state restrictions, particularly for quadratic functions.

Learners must understand that has real solutions for .

Exercises on parabolic functions with horizontal and vertical shifts have been included for enrichment only and are clearly marked.

The logarithmic function is introduced as the inverse of the exponential function. Learners need to understand that the logarithmic function allows us to rewrite an exponential expression with the exponent as the subject of the formula.

It is very important that learners can change from exponential form to logarithmic form and vice versa. This skill is also important for finding the period of an investment or loan in the Finance chapter.

Learners should be encouraged to use the definition and change of base to solve problems. Manipulation involving the logarithmic laws is not examinable.

Learners should be encouraged to be familiar with the LOG function on their calculator and also to use their calculator to check answers.

Enrichment content is not examinable and is clearly marked.

2.1 Revision

In previous grades we learned about the characteristics of linear, quadratic, hyperbolic and exponential functions. In this chapter we will demonstrate the ability to work with various types of functions and relations including inverses. In particular, we will look at the graphs of the inverses of:

Linear functions: or

Quadratic functions:

Exponential functions:

Worked example 1: Linear function

Draw a graph of and determine the significant characteristics of this linear function.

Write the equation in standard form

Draw the straight line graph

To draw the straight line graph we can use the gradient-intercept method:

Alternative method: we can also determine and plot the - and -intercepts as follows:

For the -intercept, let :

This gives the point .

For the -intercept, let :

This gives the point .

Determine the characteristics

Gradient:

Intercepts:

Domain:

Range:

Decreasing function: as increases, decreases.

Worked example 2: Quadratic function

Write the quadratic function in standard form. Draw a graph of the function and state the significant characteristics.

Write the equation in standard form

Therefore, we see that:

Draw a graph of the parabola

For the -intercept, let :

This gives the point .

For the -intercept, let :

This gives the points and .

State the significant characteristics

Shape: , therefore the graph is a “smile”.

Intercepts:

Turning point:

Axes of symmetry:

Domain:

Range:

The function is decreasing for and increasing for .

Worked example 3: Exponential function

Draw the graphs of and on the same set of axes and compare the two functions.

Examine the functions and determine the information needed to draw the graphs

Consider the function:

Asymptotes: has a horizontal asymptote, the line , which is the -axis.

Consider the function:

Asymptotes: also has a horizontal asymptote at .

Draw the graphs of the exponential functions

State the significant characteristics

Symmetry: and are symmetrical about the -axis.

Domain of and :

Range of and :

The function decreases as increases and function increases as increases. The two graphs intersect at the point .

Exercise 2.1

Sketch the graphs on the same set of axes and determine the following for each function:

Intercepts

Turning point

Axes of symmetry

Domain and range

Maximum and minimum values

and

For :

For :

and

For :

For :

and

For :

For :

Given and . Determine the values of and if and passes through the point . Sketch both functions on the same system of axes.

Given and . Determine the - and -intercepts and sketch both graphs on the same system of axes.

For :

For :

Write the linear function in standard form:

Given and .

Sketch on the same system of axes.

For each of the functions, determine the intercepts, asymptotes, domain and range.

For :

For :

For :

2.2 Functions and relations

Relation

A rule which associates each element of set with at least one element in set .

Function

A rule which uniquely associates elements of one set with the elements of another set ; each element in set maps to only one element in set .

Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable. The domain is the set of values to which the rule is applied and the range is the set of values (also called the images or function values) determined by the rule.

Example of a one-to-one function:

Example of a many-to-one function:

However, some very common mathematical constructions are not functions. For example, consider the relation . This relation describes a circle of radius centred at the origin. If we let , we see that and thus either or . This is a one-to-many relation because a single -value relates to two different -values. Therefore is not a function.

Vertical line test

Given the graph of a relation, there is a simple test for whether or not the relation is a function. This test is called the vertical line test. If it is possible to draw any vertical line (a line of constant ) which crosses the graph of the relation more than once, then the relation is not a function. If more than one intersection point exists, then the intersections correspond to multiple values of for a single value of (one-to-many).

If any vertical line cuts the graph only once, then the relation is a function (one-to-one or many-to-one).

The red vertical line cuts the circle twice and therefore the circle is not a function.

The red vertical line only cuts the parabola once and therefore the parabola is a function.

Exercise 2.2

Consider the graphs given below and determine whether or not they are functions:

Yes

Yes

Yes

No

Yes

No

Yes

No

Sketch the following and determine whether or not they are functions:

The table below gives the average per capita income, , in a region of the country as a function of , the percentage of unemployed people. Write down an equation to show that the average income is a function of the percentage of unemployed people.

1

2

3

4

Per capita income is a measure of the average amount of money earned per person in a certain area.

We see that there is a constant difference of between the consecutive values of , therefore the relation is a linear function of the form :

is the independent variable and is the dependent variable.

Substitute any of the given set of values to solve for :

The function is:

Function notation

For the function , is the dependent variable, because the value of (output) depends on the value of (input). We say is the independent variable, since we can choose to be any number. Similarly, if , then is the independent variable and is the function name.

If and we need to determine when , then we solve for the value of such that:

If and we need to determine , then we calculate the value for when :

2.3 Inverse functions

An inverse function is a function which does the “reverse” of a given function. More formally, if is a function with domain , then is its inverse function if and only if for every .

A function must be a one-to-one relation if its inverse is to be a function. If a function has an inverse function , then is said to be invertible.

Given the function , we determine the inverse by:

interchanging and in the equation;

making the subject of the equation;

expressing the new equation in function notation.

Note: if the inverse is not a function then it cannot be written in function notation. For example, the inverse of cannot be written as as it is not a function. We write the inverse as and conclude that is not invertible.

If we represent the function and the inverse function graphically, the two graphs are reflected about the line . Any point on the line has - and -coordinates with the same numerical value, for example and . Therefore interchanging the - and -values makes no difference.

This diagram shows an exponential function (black graph) and its inverse (blue graph) reflected about the line (grey line).

Important: for , the superscript is not an exponent. It is the notation for indicating the inverse of a function. Do not confuse this with exponents, such as or .

Be careful not to confuse the inverse of a function and the reciprocal of a function:

Inverse | Reciprocal
and symmetrical about |
Example: | Example:

Video: 28B2

2.4 Linear functions

Inverse of the function

Worked example 4: Inverse of the function

Determine the inverse function of and sketch the graphs of and on the same system of axes.

Determine the inverse of the given function

Interchange and in the equation.

Make the subject of the new equation.

Express the new equation in function notation.

Therefore, .

Sketch the graphs of the same system of axes

The graph of is the reflection of about the line . This means that every point on the graph of has a mirror image on the graph of .

To determine the inverse function of :

Therefore the inverse of is . If a linear function is invertible, then its inverse will also be linear.

Worked example 5: Inverses - domain, range and intercepts

Determine and sketch the inverse of the function . State the domain, range and intercepts.

Determine the inverse of the given function

Interchange and in the equation.

Make the subject of the new equation.

Express the new equation in function notation.

Therefore, .

Sketch the graphs on the same system of axes

The graph of is the reflection of about the line .

Determine domain, range and intercepts

Notice that the intercepts of and are mirror images of each other. In other words, the - and -values have “swapped” positions. This is true of every point on the two graphs.

Domain and range

For a function of the form , the domain is and the range is . When a function is inverted the domain and range are interchanged. Therefore, the domain and range of the inverse of an invertible, linear function will be and respectively.

Intercepts

The general form of an invertible, linear function is and its inverse is .

The -intercept is obtained by letting :

This gives the point .

The -intercept is obtained by letting :

This gives the point .

It is interesting to note that if , then and the -intercept of is the -intercept of and the -intercept of is the -intercept of .

Video: 28B3

Exercise 2.3

Given , find .

Consider the relation .

Is the relation a function? Explain your answer.

It is a function. Every -value relates to only one -value, it is a one-to-one relation.

Identify the domain and range.

Determine .

Sketch the graph of the function and its inverse on the same system of axes. Indicate the intercepts and the axis of symmetry of the two graphs.

The intercepts are:

is a point on and is a point on . Determine the coordinates of if and are symmetrical.

Explain why the line is an axis of symmetry for a function and its inverse.

To reflect a function about the -axis, we replace every with . Similarly, to reflect a function about the -axis, we replace every with . To reflect a function about the line , we replace with and with , which is how we determine the inverse.

Will the line be an axis of symmetry for a function and its inverse?

No it will not.

Given , determine .

Calculate the intercepts of and .

Consider :

Consider :

Therefore the intercepts for are and and the intercepts for are and .

Determine the coordinates of , the point of intersection of and .

To find the point on intersection, we let :

This gives the point .

Sketch the graphs of and on the same system of axes. Indicate the intercepts and point on the graph.

Is an increasing or decreasing function?

Decreasing function. The function values decrease as increases.

2.5 Quadratic functions

Inverse of the function

Worked example 6: Inverse of the function

Determine the inverse of the quadratic function and sketch both graphs on the same system of axes.

Determine the inverse of the given function

Interchange and in the equation.

Make the subject of the new equation.

Sketch the graphs on the same system of axes

Notice that the inverse does not pass the vertical line test and therefore is not a function.

To determine the inverse function of :

The vertical line test shows that the inverse of a parabola is not a function. However, we can limit the domain of the parabola so that the inverse of the parabola is a function.

Domain and range

Consider the previous worked example and its inverse :

If we restrict the domain of so that , then passes the vertical line test and is a function.

If the restriction on the domain of is , then would also be a function.

The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

Similarly, a restriction on the domain of the function results in a restriction on the range of the inverse and vice versa.

Worked example 7: Inverses - domain, range and restrictions

Determine the inverse of and sketch both graphs on the same system of axes. Restrict the domain of so that the inverse is a function.

Examine the function and determine the inverse

Determine the inverse of the function:

Sketch both graphs on the same system of axes

Determine the restriction on the domain

Option : Restrict the domain of to so that the inverse will also be a function . The restriction on the domain of will restrict the range of such that .

or

Option : Restrict the domain of to so that the inverse will also be a function . The restriction on the domain of will restrict the range of such that .

Worked example 8: Inverses - domain, range and restrictions

Determine the inverse of .

Sketch both graphs on the same system of axes.

Restrict the domain of so that its inverse is a function.

Determine the inverse of the function

Note: is only defined if .

Sketch both graphs on the same system of axes

The inverse does not pass the vertical line test and is not a function.

Determine the restriction on the domain

If :

If :

Exercise 2.4

Determine the inverse for each of the following functions:

Given the function for .

Find the inverse of .

Draw and on the same set of axes.

Is a function? Explain your answer.

Yes. It passes the vertical line test and is a one-to-one relation.

State the domain and range for and .

Determine the coordinates of the point(s) of intersection of the function and its inverse.

To find the points of intersection, we equate and :

Therefore, we get the points or .

Given the graph of the parabola with and passing through the point .

Determine the equation of the parabola.

State the domain and range of .

Give the coordinates of the point on that is symmetrical to the point about the line .

Determine the equation of .

State the domain and range of .

Draw a graph of .

Determine the inverse of .

Sketch both graphs on the same system of axes.

Restrict the domain of so that the inverse is a function.

Option : Restrict the domain of to so that the inverse will also be a function . The restriction on the domain of will restrict the range of such that .

or

Option : Restrict the domain of to so that the inverse will also be a function . The restriction on the domain of will restrict the range of such that .

The diagram shows the graph of and . Both graphs pass through the point .

Determine the values of , and .

From the graph we see that the straight line passes through the origin, therefore .

Give the domain and range of and .

For which values of is ?

Determine .

Determine the coordinates of the point(s) of intersection of and intersect.

To determine the coordinates of the point(s) of intersection, we equate and :

Therefore, the two graphs intersect at and .

Worked example 9: Inverses - average gradient

Given:

Determine the inverse, .

Find the point where and intersect.

Sketch and on the same set of axes.

Use the sketch to determine if and are increasing or decreasing functions.

Calculate the average gradient of between the two points of intersection.

Determine the inverse of the function

Determine the point of intersection

Therefore, this gives the points A and .

Sketch both graphs on the same system of axes

Examine the graphs

From the graphs, we see that both and pass the vertical line test and therefore are functions.

Calculate the average gradient

Calculate the average gradient of between the points A and .

Note: this is also the average gradient of between the points and .

Exercise 2.5

Sketch the graph of and label a point other than the origin on the graph.

Find the equation of the inverse of .

Sketch the graph of the inverse on the same system of axes.

Is the inverse a function? Explain your answer.

No. For certain values of , the inverse cuts a vertical line in two places. Therefore, it is not function.

is a point on . Determine the coordinates of , the point on the graph of the inverse which is symmetrical to about the line .

Determine the average gradient between:

the origin and ;

the origin and .

Interpret the answers.

and .

Both gradients are positive, and they are also reciprocals of each other.

Given the function , , which passes through the point .

Find the value of .

State the domain and range of .

Find the equation of .

State the domain and range of .

Sketch the graphs of and on the same system of axes.

Is an increasing or decreasing function?

Decreasing function: as the value of increases, the function value decreases.

Given: .

Find .

Calculate the point(s) where and intersect.

Therefore, the points of intersection are and .

Sketch and on the same set of axes.

Use the sketch to determine if and are increasing or decreasing functions.

Calculate the average gradient of between the two points of intersection.

2.6 Exponential functions

Revision of exponents

An exponent indicates the number of times a certain number (the base) is multiplied by itself. The exponent, also called the index or power, indicates the number of times the multiplication is repeated. For example, .

Graphs of the exponential function

The value of affects the direction of the graph:

If , is an increasing function.

If , is a decreasing function.

If , is not defined.

Not defined

Inverse of the function

Determining the inverse

Function | Type of function | Inverse: | Inverse:
|  | interchange and | make the subject

Consider the exponential function

To determine the inverse of the exponential function, we interchange the - and -variables:

For straight line functions and parabolic functions, we could easily manipulate the inverse to make the subject of the formula. For the inverse of an exponential function, however, is the index and we do not know a method of solving for the index.

To resolve this problem, mathematicians defined the logarithmic function. The logarithmic function allows us to rewrite the expression with as the subject of the formula:

This means that is the same as and both are the inverse of the exponential function .

Logarithms

Logarithm

If , then , where , and .

Note that the brackets around the number are not compulsory, we use them to avoid confusion.

The logarithm of a number with a certain base is equal to the exponent , the value to which that certain base must be raised to equal the number .

For example, means the power of that will give . Since , we see that . Therefore the exponential form is and the logarithmic form is .

Restrictions on the definition of logarithms

Exponential and logarithmic form

Discuss the following statements and determine whether they are true or false:

is the inverse of .

is a one-to-one function, therefore is also a one-to-one function.

is the inverse of .

is the same as .

To determine the inverse function of :

Therefore, if we have the exponential function , then the inverse is the logarithmic function .

The “common logarithm” has a base and can be written as . In other words, the symbol written without a base is interpreted as the logarithm to base . For example, .

Video: 28BR

Worked example 10: Exponential form to logarithmic form

Write the following exponential expressions in logarithmic form and express each in words:

Determine the inverse of the given exponential expressions

Remember: is the same as .

Express in words

is the power to which must be raised to give the number .

is the power to which must be raised to give the decimal number .

is the power to which must be raised to give .

Worked example 11: Logarithmic form to exponential form

Write the following logarithmic expressions in exponential form:

Determine the inverse of the given logarithmic expressions

For , we can write .

Exercise 2.6

Write the following in logarithmic form:

.

.

.

.

Express each of the following logarithms in words and then write in exponential form:

The logarithm of to base is equal to .

The logarithm of to base is equal to .

The logarithm of to base is equal to .

The logarithm of to base is equal to .

The logarithm of to base is equal to .

The logarithm of to base is equal to .

The logarithm of to base is .

The logarithm of to base is .

Logarithm bases

From the definition of a logarithm we know that the base of a logarithm must be a positive number and it cannot be equal to . The value of the base influences the value of the logarithm. For example, is not the same as and is not the same as , ().

We often calculate the “common logarithm”, which has a base and can be written as . For example, .

The “natural logarithm”, which has a base (an irrational number between and ), can be written as . For example, .

Special logarithmic values

Laws of logarithms

In earlier grades, we used the following exponential laws for working with exponents:

where , and .

The logarithmic laws are based on the exponential laws and make working with logarithms much easier.

Logarithmic laws:

The last two logarithmic laws in the list above are not covered in this section. They are discussed at the end of the chapter and are included for enrichment only.

IMPORTANT: PROOFS ARE NOT REQUIRED FOR EXAMS

Logarithmic law:

In words: the logarithm of a number which is raised to a power is equal to the value of the power multiplied by the logarithm of the number.

Worked example 12: Applying the logarithmic law

Determine the value of .

Use the logarithmic law to simplify the expression

Write the final answer

Special case:

The following is a special case of the logarithmic law :

Video: 28CC

Exercise 2.7

Simplify the following:

Logarithmic law:

It is often necessary or convenient to convert a logarithm from one base to another base. This is referred to as a change of base.

Special applications:

Video: 28CT

Worked example 13: Applying the logarithmic law

Show:

Simplify the right-hand side of the equation

Simplify the left-hand side of the equation

Write the final answer

We have shown that .

Worked example 14: Applying the logarithmic law

If and , express the following in terms of and :

Use a change of base to simplify the expressions

Exercise 2.8

Convert the following:

to base

to base

to base

to base

to base

a logarithm is equal to the reciprocal of its inverse.

to base

Simplify the following using a change of base:

If and , determine (correct to decimal places):

Logarithms using a calculator

Calculating a logarithmic value

There are many different types and models of scientific calculators. It is very important to be familiar with your own calculator and the different function buttons. Some calculators only have two buttons for logarithms: one for calculating the common logarithm (base is equal to ) and another for calculating the natural log (base is equal to ). Newer models will have a third button which allows the user to calculate the logarithm of a number to a certain base.

Worked example 15: Using a calculator: logarithm function

Use a calculator to determine the following values (correct to decimal places):

Use the common logarithm function on your calculator

Make sure that you are familiar with the “LOG” function on your calculator. Notice that the base for each of the logarithms given above is .

Write the final answer

undefined

Worked example 16: Using a calculator: inverse logarithm function

Use a calculator to determine the following values (correct to decimal places):

Use the second function and common logarithm function on your calculator

For each of the logarithms given above we need to calculate the inverse of the logarithm (sometimes called the antilog). Make sure that you are familiar with the “nd F” button on your calculator.

Notice that by pressing the “nd F” button and then the “LOG” button, we are using the “” function on the calculator, which is correct since exponentials are the inverse of logarithms.

Write the final answer

Worked example 17: Using a calculator: change of base

Use a calculator to find correct to two decimal places.

Use a change of base to convert given logarithm to base

Use the common logarithm function on your calculator

Write the final answer

Important:

Do not write down an intermediate step when doing this type of calculation:

Perform the calculation in one step on your calculator:

Do not round off before the final answer as this can affect the accuracy of the answer.

Be sure that you determine the correct sequence and order of operations when using a calculator.

Exercise 2.9

Calculate the following (correct to three decimal places):

no value

Use a calculator to determine the value of (correct to two decimal places). Check your answer by changing to exponential form.

Exponential and logarithmic graphs

Worked example 18: Graphs of the inverse of

On the same system of axes, draw the graphs of and its inverse . Investigate the properties of and .

Determine the properties of

Function:

Shape: increasing graph

Intercept(s):

Asymptote(s): horizontal asymptote at -axis, line

Domain:

Range:

Draw the graphs

The graph of the inverse is the reflection of about the line .

Determine the properties of

Function:

Shape: increasing graph

Intercept(s):

Asymptote(s): vertical asymptote at -axis, line

Domain:

Range:

Notice that the inverse is a function: is a one-to-one function since every input value is associated with only one output value.

The exponential function and the logarithmic function are inverses of each other:

the domain of the function is equal to the range of the inverse

the range of the function is equal to the domain of the inverse

the -intercept of the function is equal to the -intercept of the inverse

the -intercept of the function is equal to the -intercept of the inverse

the asymptote for the function is and the asymptote for the inverse is

the graphs are reflected about the line

Worked example 19: Graphs of

Draw a sketch of .

Reflect the graph of about the -axis to give the graph .

Investigate the properties of .

Use and to suggest a general conclusion.

Sketch the graph of

Reflect about the -axis

An easy method for reflecting a graph about a certain line is to imagine folding the Cartesian plane along that line and the reflected graph is pressed onto the plane.

Investigate the properties of

Function: passes the vertical line test

Shape: decreasing graph

Intercept(s):

Asymptote(s): vertical asymptote at -axis, line

Domain:

Range:

Since is symmetrical to about the -axis, this means that every -value of corresponds to a -value of the opposite sign for .

General conclusion

From this example of and we see that:

Worked example 20: Graph of

Draw a sketch of .

Draw the graph of , the reflection of about the line .

Investigate the properties of .

Write down the new equation if is shifted unit upwards and units to the right.

Sketch the graph of

Reflect about the line

Investigate the properties of

Function: passes the vertical line test

Shape: decreasing graph

Intercept(s):

Asymptote(s): horizontal asymptote at -axis, line

Domain:

Range:

Since is symmetrical to about the line , this means that is the inverse of .

Therefore, is an exponential function of the form with . In words, the base is a positive fraction with a value between and .

Vertical and horizontal shifts

If is shifted unit upwards and units to the right, then the new equation will be:

The vertical asymptote is and the horizontal asymptote is .

Summary of graphs: and

| Exponential function | Logarithmic function | Axis of symmetry
Axis of symmetry | -axis, | -axis, |

Video: 28F3

Exercise 2.10

Given .

Sketch the graphs of and on the same system of axes. Label both graphs clearly.

State the intercept(s) for each graph.

and

Label , the point of intersection of and .

Notice that the function and its inverse intersect at a point that lies on the line .

State the domain, range and asymptote(s) of each function.

Function | Domain | Range | Asymptote
|  |  | -axis,
|  |  | -axis,

Given with a point on the graph of .

Determine the value of

Find the inverse of .

Use symmetry about the line to sketch the graphs of and on the same system of axes.

Point lies on the graph of and is symmetrical to point about the line . Determine the coordinates of .

Applications of logarithms

Logarithms have many different applications:

seismologists use logarithms to calculate the magnitude of earthquakes

financial institutions make use of logarithms to calculate the length of loan repayments

scientists use logarithms to determine the rate of radioactive decay

biologists use logarithms to calculate population growth rates

scientists use logarithms to determine pH levels

Video: 28F6

Worked example 21: Population growth

The population of a city grows by % every two years. How long will it take for the city's population to triple in size?

Write down a suitable formula and the known values

Let

The population triples in size, so

Growth rate

Growth rate is given for a year period, so we use

Substitute known values and solve for

Method : take the logarithm of both sides of the equation

Method : change from exponential form to logarithmic form

Write the final answer

It will take approximately years for the city's population to triple in size.

Exercise 2.11

The population of Upington grows % every years. How long will it take to triple in size?

Give your answer in years and round to the nearest integer.

This is a compound growth problem. You should first write down the initial size of the population, the final size of the population and the rate at which the population grows. Note that every period is years long.

Let the population at the start be We want to know how many periods it will take to triple in size. This means that the final population is

Let be the number of periods needed to grow to a size of . Note that every period is years long, therefore we must use .

The growth rate is

We use the formula for compound growth/interest:

We now use the definition of the logarithm to write the equation in terms of and then evaluate with a calculator:

It will take approximately years for the population to triple in size.

An ant population of ants doubles every month.

Determine a formula that describes the growth of the population.

Let the number of months be

where is the number of months.

Calculate how long it will take for the ant population to reach a quarter of a million ants.

There will be a quarter of a million ants after about months.

2.7 Summary

Function: a rule which uniquely associates elements of one set with the elements of another set ; each element in set maps to only one element in set .

Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable.

Vertical line test: if it is possible to draw any vertical line which crosses the graph of the relation more than once, then the relation is not a function.

Given the invertible function , we determine the inverse by:

replacing every with and with ;

making the subject of the equation;

expressing the new equation in function notation.

If we represent the function and the inverse function graphically, the two graphs are reflected about the line .

The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

The inverse function of a straight line is also a straight line. Vertical and horizontal lines are exceptions.

The inverse of a parabola is not a function. However, we can limit the domain of the parabola so that the inverse of the parabola is a function.

The inverse of the exponential function is the logarithmic function .

The “common logarithm” has a base and can be written as . The symbol written without a base means base .

Logarithmic laws:

| Straight line function | Quadratic function | Exponential function
Formula |  |  |
Inverse |  |  |
Inverse a function? | yes | no | yes
Graphs |  |  |

Exercise 2.12

Given the straight line with intercepts and .

Determine the equation of .

Find .

Draw both graphs on the same system of axes.

Calculate the coordinates of , the point of intersection of and .

This gives the point

State the property regarding the point of intersection that will always be true for a function and its inverse.

The value of the -coordinate and the -coordinate will always be the same since the point lies on the line .

The inverse of a function is .

Determine .

Draw and on the same set of axes. Label each graph clearly.

Is an increasing or decreasing function? Explain your answer.

Increasing. As increases, the function value increases. Alternative reason: gradient is positive, therefore function is increasing.

.

Draw the graph of and state its domain and range.

The domain is: and the range is: .

Determine the inverse and state its domain and range.

Domain: , Range: .

Given the function .

Sketch the graphs of and on the same system of axes.

Determine if the point lies on the graph of .

Yes, the point does lie on .

Write in the form

or

If the graphs of and intersect at , determine the value of .

, since the point lies on the line .

Give the equation of the new graph, , if the graph of is shifted units to the left.

Give the asymptote(s) of .

Vertical asymptote:

Consider the function .

Write down the inverse in the form

State the domain and range of .

Domain: and range: .

Sketch the graphs of and on the same system of axes, label all intercepts.

For which values of will ?

Consider the functions and .

Sketch the graphs of and on the same system of axes.

Determine whether or not and intersect at a point where .

The graphs do not intersect at .

How many solutions does the equation have?

Two

Below are three graphs and six equations. Write down the equation that best matches each of the graphs.

Graph :

Graph : or

Graph :

Given the graph of the function passing through the point .

Show that .

Determine the value of if lies on .

Write down the new equation if is shifted units upwards.

Write down the new equation if is shifted units to the right.

If the rhino population in South Africa starts to decrease at a rate of % per annum, determine how long it will take for the current rhino population to halve in size? Give your answer to the nearest integer.

It will take less than years for the current rhino population to halve in size.

Which of the following graphs best illustrates the rhino population's decline? Motivate your answer.

Important note: the graphs above have been drawn as a continuous curve to show a trend. Rhino population numbers are discrete values and should be plotted points.

Graph C

Currently the rhino population is . After years, it will have halved, . Note: the line in the graph indicates the trend, rhino population numbers are discrete values and should be plotted points.

At 8 a.m. a local celebrity tweets about his new music album to of his followers. Five minutes later, each of his followers retweet his message to two of their friends. Five minutes after that, each friend retweets the message to another two friends. Assume this process continues.

Determine a formula that describes this retweeting process.

This is a geometric sequence: and .

Therefore .

Calculate how many retweets of the celebrity's message are sent an hour after his original tweet.

hour minutes , therefore

.

retweets.

How long will it take for the total number of retweets to exceed million?

Therefore, hours.

Exercise 2.13

Given: , find the inverse of in the form

Draw the graph of .

Use symmetry to draw the graph of on the same set of axes.

Is a function?

Yes. It passes vertical line test.

Give the domain and range of .

Domain: , Range: .

The graph of the inverse of is shown below:

Find the equation of , given that is a parabola of the form .

First use the information provided in the graph of the inverse:

To get the turning point and intercepts of the function, we invert the given coordinates. Now we can use those coordinates to find the equation of the function:

Now find the equation of the function:

Will have a maximum or a minimum value?

Maximum value at

State the domain, range and axis of symmetry of .

Domain: and range: , Axis of symmetry: .

Given:

If lies on , determine the value(s) of .

This gives the points and .

Sketch the graph of , label the point(s) on the graph.

Find the equation of the inverse of in the form

Sketch and on the same system of axes.

Determine the coordinates of the point on the graph of the inverse that is symmetrical to about the line .

The sketch shows the graph of a parabola passing through the point .

Determine the equation of .

Determine and investigate the inverse.

Sketch the inverse and discuss the characteristics of the graph.

The inverse is a not a function. The turning point of the inverse is and -intercept is .

Given the function .

Determine the algebraic formula for the inverse of .

Draw graphs of and its inverse on the same system of axes. Indicate intercepts and turning points.

Determine the intercepts:

The intercepts are and .

Is the inverse a function? Give reasons.

No. The inverse does not pass the vertical line test. It is a one-to-many relation.

Show algebraically and graphically the effect of restricting the domain of to .

If the domain of is restricted to , then the inverse is .

The graph of cuts a vertical line only once at any one time and therefore passes the vertical line test.

2.8 Enrichment: more on logarithms

NOTE: THIS SECTION IS NOT PART OF THE CURRICULUM

Laws of logarithms

Logarithmic law:

Now we change from the exponential form back to logarithmic form:

In words: the logarithm of a product is equal to the sum of the logarithms of the factors.

Worked example 22: Applying the logarithmic law

Simplify:

Use the logarithmic law to simplify the expression

We combine the first two terms since the product of and is equal to , which is always useful when simplifying logarithms.

We expand the last term to simplify the expression further:

Write the final answer

Exercise 2.14

Simplify the following, if possible:

cannot be written as separate logarithms.

Note: . Do not confuse this with applying the distributive law: .

Write the following as a single term, if possible:

This cannot be written as one term because the bases are not the same.

This is already a single term:

Simplify the following:

Logarithmic law:

Now we change from the exponential form back to logarithmic form:

In words: the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.

Worked example 23: Applying the logarithmic law

Simplify:

Use the logarithmic law to simplify the expression

We combine the first two terms since both terms have the same base and the quotient of and is equal to :

We expand the last term to simplify the expression further:

Write the final answer

Exercise 2.15

Expand and simplify the following:

This cannot be simplified.

Write the following as a single term:

This cannot be simplified.

This cannot be simplified because the bases are not the same.

Simplify the following:

Vini and Dirk complete their mathematics homework and check each other's answers. Compare the two methods shown below and decide if they are correct or incorrect:

Question:

Simplify the following:

Vini's answer:

Dirk's answer:

Both methods are correct.

Useful summary:

Simplification of logarithms

Worked example 24: Simplification of logarithms

Simplify (without a calculator):

Apply the appropriate logarithmic laws to simplify the expression

Write the final answer

We cannot simplify any further, therefore .

Important: all the algebraic manipulation techniques , factorisation etc.) also apply for logarithmic expressions. Always be aware of the number of terms in an expression as this will help to determine how to simplify.

Exercise 2.16

Simplify the following without using a calculator:

Solving logarithmic equations

Worked example 25: Solving logarithmic equations

Solve for :

Make the subject of the equation

Change from logarithmic form to exponential form

Write the final answer

Worked example 26: Solving logarithmic equations

Solve for (correct to the nearest integer):

Change from exponential form to logarithmic form

Use a change of base to solve for

Write the final answer

Exercise 2.17

Determine the value of (correct to decimal places):

Alternative (longer) method:

Alternative (longer) method:

Alternative (longer) method:

Given .

Write down the equation of the inverse of in the form

If , determine the value of (correct to one decimal place).

Draw the graph of and its inverse. Plot the points and .

Summary

The logarithm of a number with a certain base is equal to the exponent , the value to which that certain base must be raised to equal the number .

If , then , where , and .

Logarithms and exponentials are inverses of each other.

and

Common logarithm: means

The “LOG” function on your calculator uses a base of .

Natural logarithm: uses a base of .

Special values:

Logarithmic laws:

Special reciprocal applications:

Exercise 2.18

State whether the following are true or false. If false, change the statement so that it is true.

False:

If , then

False:

True

False: cannot be simplified further.

True

False:

True

True

False: bases are different

True

False: cannot be simplified to single logarithm

False: do not confuse with or with the distributive law .

False:

False: do not confuse with . LHS cannot be simplified.

Simplify the following without using a calculator:

Given . Find the value of the following without using a calculator:

If , then .

Given .

Without using a calculator, show that .

Now solve for if .

Let . Draw the graph of and . Indicate the point on the graph.

Solve for if . Give answer correct to two decimal places.

Given and .

For which integer values of will .

Therefore, .

For which values of will . Give answer to the nearest integer.

Important: notice that the inequality sign changed direction when we divided both sides by , since it has a negative value.

Chapter 3: Finance

Discuss terminology.

Very important to emphasize not rounding off calculations until final answer as this affects accuracy.

Learners should do calculation in one step using the memory function on their calculators.

Draw timelines showing the different time periods, interest rates and any deposits/withdrawals.

Explain and discuss the difference between future and present value annuities.

Learners must always check how often a given interest rate is compounded.

Learners must be careful to calculate the correct number of payments for the term of the investment.

In earlier grades we studied simple interest and compound interest, together with the concept of depreciation. Nominal and effective interest rates were also described.

Simple interest:

Compound interest:

Simple depreciation:

Compound depreciation:

Nominal and effective annual interest rates:

In this chapter we look at different types of annuities, sinking funds and pyramid schemes. We also look at how to critically analyse investment and loan options and how to make informed financial decisions.

Financial planning is very important and it allows people to achieve certain goals, such as supporting a family, attending university, buying a house, and saving enough money for retirement. Prudent financial planning includes making a budget, opening a savings account, wisely investing savings and planning for retirement.

3.1 Calculating the period of an investment

For calculations using the simple interest formula, we solve for , the time period of an investment or loan, by simply rearranging the formula to make the subject. For compound interest calculations, where is an exponent in the formula, we need to use our knowledge of logarithms to determine the value of .

Solving for :

Worked example 1: Determining the value of

Thembile invests into a savings account which pays % per annum compounded yearly. After an unknown period of time his account is worth . For how long did Thembile invest his money?

Write down the compound interest formula and the known values

Substitute the values and solve for

Write final answer

The was invested for years.

Worked example 2: Duration of investments

Margo has to invest and needs the money to grow to at least to pay for her daughter's studies. If it is invested at a compound interest rate of % per annum, determine how long (in full years) her money must be invested?

Write down the compound interest formula and the known values

Substitute the values and solve for

Write the final answer

In this case we round up, because years will not yet deliver the required . Therefore the money must be invested for at least years.

Exercise 3.1

Nzuzo invests at an interest rate of % per annum compounded yearly. How long will it take for his investment to grow to ?

It will take just over years.

Sally invests at an interest rate of % per annum compounded quarterly. How long will it take for her investment to double?

It will take just over years and months.

When Banele was still in high school he deposited into a savings account with an interest rate of % per annum compounded yearly. How long ago did Banele open the account if the balance is now ? Write the answer as a combination of years and months.

This is a compound interest problem:

Now substitute the known values and solve for :

Change to logarithmic form:

Banele left the money in the account for about years.

However, we must give our answer in terms of years and months, not as a decimal number of years. years means years and some number of months; to figure out how many months, we need to convert years into months.

We know that there are months in a year.

To convert years into months, do the following:

Banele deposited the money into the account years and months ago.

The annual rate of depreciation of a vehicle is %. A new vehicle costs . After how many years will the vehicle be worth less than ?

The vehicle will be worth less than after about years.

Some time ago, a man opened a savings account at KMT South Bank and deposited an amount of . The balance of his account is now . If the account gets % compound interest p.a., determine how many years ago the man made the deposit.

We write the compound interest formula and the given information:

We know everything except for the value of . Substitute and then solve for .

Use the definition of a logarithm to solve for :

Use a calculator to evaluate the :

The man made the deposit years ago.

Mr. and Mrs. Dlamini want to save money for their son's university fees. They deposit in a savings account with a fixed interest rate of % per year compounded annually. How long will take for this deposit to double in value?

It will take years for their deposit to double in value.

A university lecturer retires at the age of . She has saved over the years.

She decides not to let her savings decrease at a rate faster than % per year. How old will she be when the value of her savings is less than ?

If she manages her money carefully, she will be years or older.

If she doesn't use her savings and invests all her money in an investment account that earns a fixed interest rate of % per annum, how long will it take for her investment to grow to ?

It will take less than years.

Simosethu puts into a bank account at the Bank of Upington. Simosethu's account pays interest at a rate of % p.a. compounded monthly. After how many years will the bank account have a balance of ?

Use the compound interest formula and determine the value of .

In this question the interest is payable every month. Therefore and . In this case, represents the number of years; the product represents the number of times the bank pays interest into the account.

At this point we must change the equation to logarithmic form:

To find the number of years, we solve for :

The money has been in the Simosethu's account for years.

3.2 Annuities

Annuity

A number of equal payments made at regular intervals for a certain amount of time. An annuity is subject to a rate of interest.

Future value annuity - regular equal deposits/payments are made into a savings account or investment fund to provide an accumulated amount at the end of the time period. The amount accumulating in the fund earns compound interest at a certain rate.

Present value annuity - regular equal payments/installments are made to pay back a loan or bond over a given time period. The reducing balance of the loan is usually charged compound interest at a certain rate.

For investment funds, pension funds, loan repayments, mortgage bonds (home loan) and other types of annuities, payments are typically made each month. To “default” on a payment means that a payment for a certain month was not paid. The period of an investment is also referred to as the term of an investment.

3.3 Future value annuities

For future value annuities, we regularly save the same amount of money into an account, which earns a certain rate of compound interest, so that we have money for the future.

Worked example 3: Future value annuities

At the end of each year for years, Kobus deposits into an investment account. If the interest rate on the account is % per annum compounded yearly, determine the value of his investment at the end of the years.

Write down the given information and the compound interest formula

Draw a timeline

The first deposit in the account earns the highest amount of interest (three interest payments) and the last deposit earns the least interest (no interest payments).

We can summarize this information in the table below:

| Deposit | No. of interest payments | Calculation | Accumulated amount
Year 1 |  |  |  |
Year 2 |  |  |  |
Year 3 |  |  |  |
Year 4 |  |  |  |
Total |  |  |  |

Deriving the formula

Note: for this section is it important to be familiar with the formulae for the sum of a geometric series (Chapter 1):

In the worked example above, the total value of Kobus' investment at the end of the four year period is calculated by summing the accumulated amount for each deposit:

We notice that this is a geometric series with a constant ratio .

Using the formula for the sum of a geometric series:

We can therefore use the formula for the sum of a geometric series to derive a formula for the future value () of a series of () regular payments of an amount () which are subject to an interest rate ():

Future value of payments:

If we are given the future value of a series of payments, then we can calculate the value of the payments by making the subject of the above formula.

Payment amount:

Worked example 4: Future value annuities

Ciza decides to start saving money for the future. At the end of each month she deposits into an account at Harringstone Mutual Bank, which earns % interest p.a. compounded monthly.

Determine the balance of Ciza's account after years.

How much money did Ciza deposit into her account over the year period?

Calculate how much interest she earned over the year period.

Write down the given information and the future value formula

Substitute the known values and use a calculator to determine

Remember: do not round off at any of the interim steps of a calculation as this will affect the accuracy of the final answer.

Calculate the total value of deposits into the account

Ciza deposited each month for years:

Calculate the total interest earned

Useful tips for solving problems:

Timelines are very useful for summarising the given information in a visual way.

When payments are made more than once per annum, we determine the total number of payments () by multiplying the number of years by :

Term |
yearly / annually |
half-yearly / bi-annually |
quarterly |
monthly |
weekly |
daily |

If a nominal interest rate is given, then use the following formula to convert it to an effective interest rate:

Worked example 5: Calculating the monthly payments

Kosma is planning a trip to Canada to visit her friend in two years' time. She makes an itinerary for her holiday and she expects that the trip will cost . How much must she save at the end of every month if her savings account earns an interest rate of % per annum compounded monthly?

Write down the given information and the future value formula

To determine the monthly payment amount, we make the subject of the formula:

Substitute the known values and calculate

Write the final answer

Kosma must save each month so that she can afford her holiday.

Worked example 6: Determining the value of an investment

Simon starts to save for his retirement. He opens an investment account and immediately deposits into the account, which earns % per annum compounded monthly. Thereafter, he deposits at the end of each month for years. What is the value of his retirement savings at the end of the year period?

Write down the given information and the future value formula

Note that we added one extra month to the years because Simon deposited immediately.

Substitute the known values and calculate

Write the final answer

Simon will have saved for his retirement.

Exercise 3.2

Shelly decides to start saving money for her son's future. At the end of each month she deposits into an account at Durban Trust Bank, which earns an interest rate of % per annum compounded quarterly.

Determine the balance of Shelly's account after years.

Write down the given information and the future value formula:

Substitute the known values and use a calculator to determine :

How much money did Shelly deposit into her account over the year period?

Calculate the total value of deposits into the account:

Shelly deposited each month for years:

Calculate how much interest she earned over the year period.

Calculate the total interest earned:

Gerald wants to buy a new guitar worth in a year's time. How much must he deposit at the end of each month into his savings account, which earns a interest rate of % p.a. compounded monthly?

Write down the given information and the future value formula:

To determine the monthly payment amount, we make the subject of the formula:

Substitute the known values and calculate :

Write the final answer:

Gerald must deposit each month so that he can afford his guitar.

A young woman named Grace has just started a new job, and wants to save money for the future. She decides to deposit into a savings account every month. Her money goes into an account at First Mutual Bank, and the account earns % interest p.a. compounded every month.

How much money will Grace have in her account after years?

After years, Grace will have in her account.

How much money did Grace deposit into her account by the end of the year period?

The total amount of money Grace saves each year is . From that we can determine the total amount she saves by multiplying by the number of years: .

After years, Grace deposited a total of into her account.

Ruth decides to save for her retirement so she opens a savings account and immediately deposits into the account. Her savings account earns % per annum compounded monthly. She then deposits at the end of each month for years. What is the value of her retirement savings at the end of the year period?

Write down the given information and the future value formula:

Substitute the known values and calculate :

Write the final answer:

Ruth will have saved for her retirement.

Musina MoneyLenders offer a savings account with an interest rate of % p.a. compounded monthly. Monique wants to save money so that she can buy a house when she retires. She decides to open an account and make regular monthly deposits. Her goal is to end up with in her account after years.

How much must Monique deposit into her account each month in order to reach her goal?

In order to save in years, Monique will need to save in her account every month.

How much money, to the nearest rand, did Monique deposit into her account by the end of the year period?

The final amount calculated in the question above includes the money Monique deposited into the account plus the interest paid by the bank. The total amount of money Monique put into her account during the year is the product of payments per year, years, and the payment amount itself:

After years, Monique deposited a total of into her account.

Lerato plans to buy a car in five and a half years' time. She has saved in a separate investment account which earns % per annum compound interest. If she doesn't want to spend more than on a vehicle and her savings account earns an interest rate of % p.a. compounded monthly, how much must she deposit into her savings account each month?

First calculate the accumulated amount for the in Lerato's investment account:

In five and a half years' time, Lerato needs to have saved .

Substitute the known values and calculate :

Write the final answer:

Lerato must deposit each month into her savings account.

Every Monday Harold puts into a savings account at the King Bank, which accrues interest of % p.a. compounded weekly. How long will it take Harold's account to reach a balance of . Give the answer as a number of years and days to the nearest integer.

To get to the final answer for this question, convert years into years and days.

days

Harold's investment takes years and days to reach the final value of .

How much interest will Harold receive from the bank during the period of his investment?

The total amount Harold invests is as follows:

Therefore, the total amount of interest paid by the bank: .

Sinking funds

Vehicles, equipment, machinery and other similar assets, all depreciate in value as a result of usage and age. Businesses often set aside money for replacing outdated equipment or old vehicles in accounts called sinking funds. Regular deposits, and sometimes lump sum deposits, are made into these accounts so that enough money will have accumulated by the time a new machine or vehicle needs to be purchased.

Worked example 7: Sinking funds

Wellington Courier Company buys a delivery truck for . The value of the truck depreciates on a reducing-balance basis at % per annum. The company plans to replace this truck in seven years' time and they expect the price of a new truck to increase annually by %.

Calculate the book value of the delivery truck in seven years' time.

Determine the minimum balance of the sinking fund in order for the company to afford a new truck in seven years' time.

Calculate the required monthly deposits if the sinking fund earns an interest rate of % per annum compounded monthly.

Determine the book value of the truck in seven years' time

Determine the minimum balance of the sinking fund

Calculate the price of a new truck in seven years' time:

Therefore, the balance of the sinking fund () must be greater than the cost of a new truck in seven years' time minus the money from the sale of the old truck:

Calculate the required monthly payment into the sinking fund

Substitute the values and calculate :

Therefore, the company must deposit each month.

Exercise 3.3

Mfethu owns his own delivery business and he will need to replace his truck in years' time. Mfethu deposits into a sinking fund each month, which earns % interest p.a. compounded monthly.

How much money will be in the fund in years' time, when Mfethu wants to buy the new truck?

Interest is compounded monthly: and .

After years, Mfethu will have in his sinking fund.

If a new truck costs in years' time, will Mfethu have enough money to buy it?

No, Mfethu does not have enough money in his account:

Atlantic Transport Company buys a van for . The value of the van depreciates on a reducing-balance basis at % per annum. The company plans to replace this van in five years' time and they expect the price of a new van to increase annually by %.

Calculate the book value of the van in five years' time.

Determine the amount of money needed in the sinking fund for the company to be able to afford a new van in five years' time.

Therefore, the balance of the sinking fund () must be greater than the cost of a new van in five years' time less the money from the sale of the old van:

Calculate the required monthly deposits if the sinking fund earns an interest rate of % per annum compounded monthly.

Calculate the required monthly payment into the sinking fund:

Substitute the values and calculate :

Therefore, the company must deposit each month.

Tonya owns Freeman Travel Company and she will need to replace her computer in years' time. Tonya creates a sinking fund so that she will be able to afford a new computer, which will cost . The sinking fund earns interest at a rate of % p.a. compounded each quarter.

How much money must Tonya save quarterly so that there will be enough money in the account to buy the new computer?

Interest is compounded per quarter, therefore and

Tonya must deposit into the sinking fund quarterly.

How much interest (to the nearest rand) does the bank pay into the account by the end of the year period?

Total savings:

Interest earned:

To the nearest rand, the bank paid into the account.

3.4 Present value annuities

For present value annuities, regular equal payments/installments are made to pay back a loan or bond over a given time period. The reducing balance of the loan is usually charged compound interest at a certain rate. In this section we learn how to determine the present value of a series of payments.

Consider the following example:

Kate needs to withdraw from her bank account every year for the next three years. How much must she deposit into her account, which earns % per annum, to be able to make these withdrawals in the future? We will assume that these are the only withdrawals and that there are no bank charges on her account.

To calculate Kate's deposit, we make the subject of the compound interest formula:

We determine how much Kate must deposit for the first withdrawal:

We repeat this calculation to determine how much must be deposited for the second and third withdrawals:

Notice that for each year's withdrawal, the deposit required gets smaller and smaller because it will be in the bank account for longer and therefore earn more interest. Therefore, the total amount is:

We can check these calculations by determining the accumulated amount in Kate's bank account after each withdrawal:

Calculation

Accumulated amount

Initial deposit

Amount after one year

=

=

Amount after first withdrawal

=

=

Amount after two years

=

=

Amount after second withdrawal

=

=

Amount after three years

=

=

Amount after third withdrawal

=

=

Completing this table for a three year period does not take too long. However, if Kate needed to make annual payments for years, then the calculation becomes very repetitive and time-consuming. Therefore, we need a more efficient method for performing these calculations.

Deriving the formula

In the example above, Kate needed to deposit:

We notice that this is a geometric series with a constant ratio .

Using the formula for the sum of a geometric series:

We can therefore use the formula for the sum of a geometric series to derive a formula for the present value () of a series of () regular payments of an amount () which are subject to an interest rate ():

Present value of a series of payments:

If we are given the present value of a series of payments, we can calculate the value of the payments by making the subject of the above formula.

Payment amount:

Worked example 8: Present value annuities

Andre takes out a student loan for his first year of civil engineering. The loan agreement states that the repayment period is equal to years for every year of financial assistance granted and that the loan is subject to an interest rate of % p.a. compounded monthly.

If Andre pays a monthly installment of , calculate the loan amount.

Determine how much interest Andre will have paid on his student loan at the end of the months.

Write down the given information and the present value formula

Substitute the known values and determine :

Therefore, Andre took out a student loan for .

Calculate the total amount of interest

At the end of the month period:

Worked example 9: Calculating the monthly payments

Hristo wants to buy a small wine farm worth . He plans to sell his current home for which he will use as a deposit for the purchase of the farm. He secures a loan with HBP Bank with a repayment period of years and an interest rate of % compounded monthly.

Calculate his monthly repayments.

Determine how much interest Hristo will have paid on his loan by the end of the years.

Write down the given information and the present value formula

To determine the monthly repayment amount, we make the subject of the formula:

Substitute the known values and calculate

Therefore, Hristo must pay per month to repay his loan over the year period.

Calculate the total amount of interest

At the end of the year period:

Prime lending rate

The prime lending rate is a benchmark rate at which private banks lend out money to the public. It is used as a reference rate for determining interest rates on many types of loans, including small business loans, home loans and personal loans. Some interest rates may be expressed as a percentage above or below prime rate. For calculations in this chapter, we will assume the prime lending rate is % per annum.

Worked example 10: Calculating the outstanding balance of a loan

A school sells its old bus and uses the proceeds as a % deposit for the purchase of the new bus, which costs . To finance the balance of the purchase, the school takes out a loan that is subject to an interest rate of compounded monthly. The repayment period of the loan is years.

Calculate the monthly repayments.

Determine the balance of the loan at the end of the first year, immediately after the payment.

Write down the given information and the present value formula

To determine the monthly repayment amount, we make the subject of the formula:

Substitute the known values and calculate

Therefore, the school must pay per month to repay the loan over the year period.

Calculate the balance of the loan at the end of the first year

We can calculate the balance of the loan at the end of the first year by determining the present value of the remaining payments:

Therefore, the school must still pay of the loan.

Alternative method: we can also calculate the balance of the loan at the end of the first year by determining the accumulated amount of the loan for the first year less the future value of the first payments:

Therefore, the school must still pay of the loan.

(Note the difference of cents due to rounding).

Exercise 3.4

A property costs . Calculate the monthly repayments if the interest rate is % p.a. compounded monthly and the loan must be paid off in years' time.

Write down the given information and the present value formula:

To determine the monthly repayment amount, we make the subject of the formula:

Substitute the known values and calculate :

A loan of is to be returned in two equal annual installments. If the rate of interest is % compounded annually, calculate the amount of each installment.

Write down the given information and the present value formula:

To determine the monthly repayment amount, we make the subject of the formula:

Substitute the known values and calculate :

Stefan and Marna want to buy a flat that costs R million. Their parents offer to put down a % payment towards the cost of the house. They need to get a mortgage for the balance. What is the monthly repayment amount if the term of the home loan is years and the interest is % p.a. compounded monthly?

Write down the given information and the present value formula:

To determine the monthly repayment amount, we make the subject of the formula:

Substitute the known values and calculate :

Ziyanda arranges a bond for from Langa Bank. If the bank charges % p.a. compounded monthly, determine Ziyanda's monthly repayment if she is to pay back the bond over years.

Ziyanda must pay each month.

What is the total cost of the bond?

Total cost:

Ziyanda paid the bank a total of .

Dullstroom Bank offers personal loans at an interest rate of % p.a. compounded twice a year. Lubabale borrows and must pay every six months until the loan is fully repaid.

How long will it take Lubabale to repay the loan?

It will take years.

How much interest will Lubabale pay for this loan?

The total cost of the loan: .

The total amount of interest Lubabale pays is:

Likengkeng has just started a new job and wants to buy a car that costs . She visits the Soweto Savings Bank, where she can arrange a loan with an interest rate of % p.a. compounded monthly. Likengkeng has enough money saved to pay a deposit of . She arranges a loan for the balance of the payment, which is to be paid over a period of years.

What is Likengkeng's monthly repayment on her loan?

The balance of the payment:

Therefore, Likengkeng takes out a loan for .

Likengkeng must pay each month.

How much will the car cost Likengkeng?

The total amount paid:

Therefore, Likengkeng paid a total of

Anathi is a wheat farmer and she needs to buy a new holding tank which costs . She bought her old tank years ago for . The value of the old grain tank has depreciated at a rate of % per year on a reducing balance, and she plans to trade it in for its current value. Anathi will then need to arrange a loan for the balance of the cost of the new grain tank.

Orsmond bank offers loans with an interest rate of % p.a. compounded monthly for any loan up to and % p.a. compounded monthly for a loan above that amount. The loan agreement allows Anathi a grace period for the first six months (no payments are made) and it states that the loan must be repaid over years.

Determine the monthly repayment amount.

Anathi will trade in the old grain tank to offset some of the cost of the new one. Therefore, we need to determine the current value of the old grain tank:

The value of the old grain tank for the trade in is .

Therefore, the loan amount for the new tank is:

Determine which interest rate to use: the loan amount is more than , therefore Anathi gets the lower interest rate of %.

Calculate how much the loan will be worth at the end of the grace period:

After the first six months of the loan period, the amount she owes increases to .

Now we can use the present value formula to solve for the value of . Remember that the time period for this calculation is years.

Therefore, Anathi must pay each month.

What is the total amount of interest Anathi will pay for the loan?

The total cost of the loan is:

Therefore, the amount of interest is:

The total amount of interest Anathi paid is .

How much money would Anathi have saved if she did not take the six month grace period?

For this calculation the loan amount is and the time period is years:

Therefore, Anathi would pay each month.

The total cost of the loan would be:

The amount of interest:

Anathi would have saved:

3.5 Analysing investment and loan options

In the next worked example we consider the effects of the duration of the repayment period on the total amount repaid (the amount borrowed plus the accrued interest) for a loan.

Worked example 11: Repayment periods

David and Julie take out a home loan of R million with an interest rate of % per annum compounded monthly.

Calculate the monthly repayments for a repayment period of years.

Calculate the interest paid on the loan at the end of the year period.

Determine the monthly repayments for a repayment period of years.

Determine the interest paid on the loan at the end of the year period.

What is the difference in the monthly repayment amounts?

Comment on the difference in the interest paid for the two different time periods.

Consider a year repayment period on the loan

Therefore, the monthly repayment is for a year period.

At the end of the years, David and Julie will have paid a total amount of:

The total amount of interest on the loan:

We notice that the interest on the loan is more than double the amount borrowed.

Consider a year repayment period on the loan

Therefore, the monthly repayment is for a year period.

At the end of the years, David and Julie will have paid a total amount of:

The total amount of interest on the loan:

We notice that the interest on the loan is about times the borrowed amount.

Consider the difference in the repayment and interest amounts

It is also very interesting to look at the difference in the total interest paid:

Therefore, by paying an extra each month over a shorter repayment period, David and Julie could save more than million on the repayment of their home loan.

When considering taking out a loan, it is advisable to investigate and compare a few options offered by financial institutions. It is very important to make informed decisions regarding personal finances and to make sure that the monthly repayment amount is serviceable (payable). A credit rating is an estimate of a person's ability to fulfill financial commitments based on their previous payment history. Defaulting on a loan can affect a person's credit rating and their chances of taking out another loan in the future.

Worked example 12: Analysing investment opportunities

Marlene wants to start saving for a deposit on a house. She can afford to invest between and each month and gets information from four different investment firms. Each firm quotes a different interest rate and a prescribed monthly installment amount. She plans to buy a house in years' time. Calculate which company offers the best investment opportunity for Marlene.

| Interest rate (compounded monthly) | Monthly payment
TBS Investments | % p.a. |
Taylor Anderson | % p.a. |
PHK | % p.a. |
Simfords Consulting | % p.a. |

Consider the different investment options

To compare the different investment options, we need to calculate the following for each option at the end of the seven year period:

The future value of the monthly payments.

The total amount paid into the investment fund.

The total interest earned.

TBS Investments:

Taylor Anderson:

PHK:

Simfords Consulting:

Draw a table of the results to compare the answers

TBS Investments |  |  |
Taylor Anderson |  |  |
PHK |  |  |
Simfords Consulting |  |  |

Make a conclusion

An investment with PHK would provide Marlene with the highest deposit () for her house at the end of the year period. However, we notice that an investment with Taylor Anderson would earn the highest amount of interest () and is therefore the better investment option.

Worked example 13: Analysing loan options

William wants to take out a loan of , so he approaches three different banks. He plans to start repaying the loan immediately and he calculates that he can afford a monthly repayment amount between and .

Calculate which of the three options would be best for William.

West Bank offers a repayment period of years and an interest rate of prime compounded monthly.

AcuBank offers a repayment period of years and an interest rate of prime compounded monthly.

FinTrust Bank offers a repayment period of years and an interest rate of prime compounded monthly.

Consider the different loan options

To compare the different loan options, we need to calculate the following for each option:

The monthly payment amount.

The total amount paid to repay the loan.

The amount of interest on the loan.

West Bank:

AcuBank:

FinTrust Bank:

Draw a table of the results to compare the answers

West Bank |  |  |
AcuBank |  |  |
FinTrust Bank |  |  |

Make a conclusion

A loan from FinTrust Bank would accumulate the lowest amount of interest but the monthly repayment amounts are not within William's budget. Although West Bank offers the lowest interest rate and monthly repayment amount, the interest earned on the loan is very high as a result of the longer repayment period. If we assume that William must repay the loan over the given time periods, then AcuBank offers the best option.

However, we know that William can afford to pay more than per month, and if the bank allows him to pay back the loan earlier, he should consider taking out a loan with West Bank and take advantage of the lower interest rate.

Exercise 3.5

Cokisa is years old and starting to plan for her future. She has been thinking about her retirement and wants to open an annuity so that she will have money when she retires. Her intention is to retire when she is 65 years old. Cokisa visits the Trader's Bank of Tembisa and learns that there are two investment options from which she can choose:

Option A: % p.a. compounded once every four months

Option B: % p.a. compounded half-yearly

Which is the better investment option for Cokisa if the amount she will deposit will always be the same?

There are two separate account options for Cokisa to consider; to determine which is the better option we need to compare the effective interest of each option. The higher the effective interest, the quicker the account will grow. The effective interest formula is:

Work out the effective interest for option A:

This calculation shows that option A an effective interest rate of about %.

Now work out the effective interest for option B:

For option B, the effective interest rate will be approximately %.

By comparing the two calculations above, we see that the option A is the better one.

NOTE: It may seem that we could answer this question by picking an amount for the regular payment and then using the future value formula for each of the two options to see which one produces more money. However, this will not work because the compounding periods are different. If we want to work it out this way, we MUST adjust the payment value we use according to the compounding period, for example if we choose a regular payment of once every four months for option A, then we will have to use semi-annually for option B (because is the same total amount per year as ).

Cokisa opens an account and starts saving every four months. How much money (to the nearest rand) will she have saved when she reaches her planned retirement?

Cokisa will have for her retirement.

Phoebe wants to take out a home loan of R million. She approaches three different banks for their loan options:

Bank A offers a repayment period of years and an interest rate of % per annum compounded monthly.

Bank B offers a repayment period of years and an interest rate of % per annum compounded monthly.

Bank C offers a repayment period of years and an interest rate of % per annum compounded monthly.

If Phoebe intends to start her monthly repayments immediately, calculate which of the three options would be best for her.

Bank A:

Bank B:

Bank C:

Bank A |  |  |
Bank B |  |  |
Bank C |  |  |

A loan from Bank A would have the lowest monthly repayments, however the interest paid is high as a result of the longer repayment period. Therefore, Phoebe should consider taking out a loan with Bank B, as this has the lowest total repayment amount.

Pyramid schemes

A pyramid scheme is a moneymaking scheme that promises investors unusually high returns on their investment. The concept of a pyramid scheme is quite simple and should be easy to identify, however, it is often cleverly disguised as a legitimate business. In some cases a product is offered and in others the scheme is marketed as a highly profitable investment opportunity. Unfortunately, many of these schemes have cost millions of people their savings. Pyramid schemes are illegal in South Africa.

"Ponzi schemes" are named after Charles Ponzi, a Italian businessman who lived in the U.S. and Canada. One of his investment schemes involved buying postal reply coupons in other countries and redeeming them for a higher value in the United States. He promised investors % profit within days of their investment. As the scheme grew, Ponzi paid off early investors with money from investors who joined the scheme at a later stage. Ponzi's fraudulent scheme was exposed and investors lost millions of dollars. He was sent to prison for a number of years.

South African Adriaan Nieuwoudt started a pyramid scheme commonly referred to as the "Kubus" scheme. Participants bought a biological substance called an activator that was supposed to be used in beauty products. The activator was used to grow cultures in milk that were then dried and ground up and resold to new participants. The scheme had thousands of investors and took in approximately million before it was declared an illegal lottery.

A pyramid scheme starts with one person, who finds other people to invest their money in the scheme. These people then enrol more people into the scheme and the base of the pyramid grows. The money invested by new investors goes to participants closer to the top of the pyramid. This is unsustainable because it requires that more and more people join the scheme and the growth must end at some point because there are a finite number of people. Therefore, most of the investors lose their money when the scheme collapses.

The South African Reserve Bank has launched a public awareness campaign, “Beware of oMashayana (crooks)”, to help educate people about pyramid schemes and how to identify them.

Beware of oMashayana

So, you think you’ve found the perfect investment? Regular high returns with no risk? Be careful. Don’t lose all your money. If it sounds too good to be true, it’s probably a Pyramid or Ponzi scheme.

What is a Ponzi scheme?

A Ponzi con-artist will only ask you to give them money that they say will be invested in a scheme or project, such as supposed property developments; bridging finance; foreign-exchange transactions; venture capital to other companies; or share units. The scheme operator promises to give you back much more money than you have given them initially, in a very short space of time.

What is a pyramid scheme?

A pyramid con-artist will offer you the chance to make quick money for yourself, often by selling something. You pay a joining fee, buy the product and then sell it. They tell you that the more people you get to sell for you, the more money you will make. It’s easy for the first people to introduce new members, but soon everyone is part of the scheme and it gets harder to find new members to join. For example, the con-artist recruits people who each pay a fee. Each of those people has to recruit people. There are now up to people. Now, those people each have to sign up people – this equals , and by the time you have to get to level , you have to get million people to join to keep the scheme going. This can never work.

What is the difference between a pyramid and a Ponzi scheme?

The main difference is that with a pyramid scheme you have to work or sell to recruit investors, while with a Ponzi scheme the con-artist will only ask you to invest in something (for example, property development). Both schemes are illegal.

What can I do to protect myself?

You are your own best protection. It is your responsibility to make sure that you never give your money to any company or person that is not registered as a deposit-taking institution in terms of the Banks Act.

Why is handing my money over so risky?

When you hand over your money (notes and coins) to another person who then loses the money, steals it or goes bankrupt, you only have an unsecured claim against that person or their estate and you might not get all your money back.

Why is it safer to hand my money to a bank?

Banks and investment companies have to be registered so that they can be regulated and supervised, to make sure that your money is safe. Unregulated and unsupervised persons and groups don’t follow these rules and your money is at great risk with them.

3.6 Summary

Always keep the rate of interest per time unit and the time period in the same units.

Simple interest:

Compound interest:

Simple depreciation:

Compound depreciation:

Nominal and effective annual interest rates:

Future value of payments:

Payment amount:

Present value of a series of payments:

Payment amount:

Exercise 3.6

Mpumelelo deposits into a savings account, which earns interest at % p.a. compounded quarterly. How long will it take for the savings account to have a balance of ?

In this question the interest is payable quarterly, so and . In this case, represents the number of years; the product represents the number of times the bank pays interest into the account.

At this point we must change the equation to logarithmic form:

To find the number of years, we solve for :

Therefore, it will take years.

How much interest will Gavin pay on a loan of for years at % per annum compounded monthly?

Monthly payments are .

Total loan:

Interest:

Wingfield school will need to replace a number of old classroom desks in years' time. The principal has calculated that the new desks will cost . The school establishes a sinking fund to pay for the new desks and immediately deposits an amount of into the fund, which accrues interest at a rate of % p.a. compounded monthly.

How much money should the school save every month so that the sinking fund will have enough money to cover the cost of the desks?

is deposited into the sinking fund immediately and will earn interest until the end of the years:

After years the deposit will be worth .

Balance required in the sinking fund:

We calculate the monthly payments that will give a future value of :

Therefore, Wingfield school must deposit into the sinking fund each month so that there will enough money in the account to buy the new desks.

How much interest does the fund earn over the period of years?

The total amount of money the school deposits into the account is .

Therefore, the amount of interest paid by the bank:

Determine how many years (to the nearest integer) it will take for the value of a motor vehicle to decrease to % of its original value if the rate of depreciation, based on the reducing-balance method, is % per annum.

Let the value of the vehicle be .

Therefore, it will take about years.

Angela has just started a new job, and wants to save money for her retirement. She decides to deposit into a savings account once each month. Her money goes into an account at Pinelands Mutual Bank, and the account receives % interest p.a. compounded once each month.

How much money will Angela have in her account after years?

After years, Angela will have in her account.

How much money did Angela deposit into her account after years?

The total amount of money Angela saves each year is . From that we can determine the total amount she saves by multiplying by the number of years: .

After years, Angela deposited a total of into her account.

Nicky has been working at Meyer and Associates for years and gets an increase in her salary. She opens a savings account at Langebaan Bank and begins making deposits of every month. The account earns % interest p.a., compounded monthly. Her plan is to continue saving on a monthly schedule until she retires. However, after years she stops making the monthly payments and leaves the account to continue growing.

How much money will Nicky have in her account years after she first opened it?

This calculation shows that the account will have a balance of after years, which is when Nicky stops making monthly deposits.

From this point on, the account grows from with compound interest only (no more monthly payments). This will continue for years.

The total amount of money in the account years after Nicky opens the account is .

Calculate the difference between the total deposits made into the account and the amount of interest paid by the bank.

The total deposits:

At the end of the period, the interest earned is:

Difference:

Every three months Louis puts into an annuity. His account earns an interest rate of % p.a. compounded quarterly. How long will it take Louis's account to reach a balance of ?

Now that the square bracket is alone, work inside of it: add one to both sides, and then change the equation to log-form to get the exponent.

It will take about years.

How much interest will Louis receive from his investment?

The total amount of deposits:

The total amount of interest:

A dairy farmer named Kayla needs to buy new equipment for her dairy farm which costs . She bought her old equipment years ago for . The value of the old equipment depreciates at a rate of % per year on a reducing balance. Kayla will need to arrange a bond for the remaining cost of the new equipment.

An agency which supports farmers offers bonds at a special interest rate of % p.a. compounded monthly for any loan up to and % p.a. compounded monthly for a loan above that amount. Kayla arranges a bond such that she will not need to make any payments on the loan in the first six months (called a 'grace period') and she must pay the loan back over years.

Determine the monthly payment.

Old equipment:

The value of the old equipment is .

Bond amount:

After the first six months of the loan period, during which she makes no payments, the amount she owes increases to .

Now we can use the present value formula to solve for the value of . Remember that the repayment period is years.

Kayla must pay each month.

What is the total amount of interest Kayla will pay for the bond?

Total cost of the loan:

Interest:

The total amount of interest Kayla paid is .

By what factor is the interest she pays greater than the value of the loan? Give the answer correct to one decimal place.

The interest is greater than the loan amount by a factor of .

Thabo invests in a special banking product which will pay % per annum for 1 month, then % per annum for the next months, then % per annum for the next 3 months, % per annum for the next months, and % for the rest of the year. If the bank charges him to open the account, how much can he expect to get back at the end of the year?

Subtract account fee from the investment amount: - =

Thabani and Lungelo are both using Harper Bank for their savings. Lungelo makes a deposit of at an interest rate of for six years. Three years after Lungelo made his first deposit, Thabani makes a deposit of at an interest rate of % per annum. If after years their investments are equal, calculate the value of (correct to three decimal places). If the sum of their investment is , determine how much Thabani earned in years.

Chapter 4: Trigonometry

Emphasize the value and importance of making sketches, where appropriate.

It is very important for learners to understand that it is incorrect to apply the distributive law to the trigonometric ratios of compound angles and that .

Emphasize that the area, sine and cosine rules do not require right-angled triangles.

Remind learners that angles in the Cartesian plane are always measured from the positive -axis.

Important to note that is not included in the curriculum.

Note that the co-function for tangent is also not included in the curriculum.

Remind learners to check that their answers are within the required interval.

For the general solution, determine the solution in the correct quadrants and within the required interval.

To prove that an identity is true, remind learners that they are only allowed to manipulate one side at a time.

To prove identities, we usually manipulate the more complicated expression so that it looks the same as the more simple expression.

4.1 Revision

Trigonometric ratios

We defined the basic trigonometric ratios using the lengths of the sides of a right-angled triangle.

Trigonometric ratios in the Cartesian plane

We also defined the trigonometric ratios with respect to any point in the Cartesian plane in terms of , and . Using the theorem of Pythagoras, .

CAST diagram

The sign of a trigonometric ratio depends on the signs of and :

Reduction formulae and co-functions:

The reduction formulae hold for any angle . For convenience, we assume is an acute angle ().

When determining function values of (), () and () the function does not change.

When determining function values of () and () the function changes to its co-function.

Negative angles

Special angle triangles

These values are useful when we need to solve a problem involving trigonometric functions without using a calculator. Remember that the lengths of the sides of a right-angled triangle obey the theorem of Pythagoras.

θ

°

°

°

°

°

undef

Trigonometric identities

Quotient identity:

Square identity:

It also follows that:

All these relationships and identities are very useful for simplifying trigonometric expressions.

Worked example 1: Revision

Determine the value of the expression, without using a calculator:

Use reduction formulae to express each trigonometric ratio in terms of an acute angle

Now use special angles to evaluate the simplified expression:

Worked example 2: Revision

Prove:

State restrictions where applicable.

Use trigonometric identities to simplify each side separately

Simplify the left-hand side of the identity:

Simplify the right-hand side of the identity so that it equals the left-hand side:

Alternative method: we could also have started with the left-hand side of the identity and substituted and simplified to get the right-hand side.

Restrictions

We need to determine the values of for which any of the terms in the identity will be undefined:

We must also consider the values of for which is undefined. Therefore, the identity is undefined for .

Useful tips:

It is sometimes useful to write in terms of and .

Never write a trigonometric ratio without an angle. For example, has no meaning.

For proving identities, only simplify one side of the identity at a time.

As seen in the worked example above, sometimes both sides of the identity need to be simplified.

Remember to write down restrictions:

the values for which any of the trigonometric ratios are not defined;

the values of the variable which make any of the denominators in the identity equal to zero.

Exercise 4.1

Given:

Write each of the following expressions in terms of :

Simplify to a single trigonometric ratio:

Simplify to a single trigonometric ratio:

Hence, determine:

If , express the following in terms of :

Evaluate the following without the use of a calculator:

Reduce the following to one trigonometric ratio:

Use reduction formulae and special angles to show that

can be simplified to .

Use reduction formulae and co-functions to simplify the expression

Without using a calculator, determine the value of .

Use special angles to determine the value of the expression

Troy's mathematics teacher asks the class to answer the following question.

Question:

Prove that .

Troy's answer:

Comment on Troy's answer and show the correct method for proving this identity.

The question requires that Troy prove the identity. However, by working with both sides of the identity at the same time, he accepted that it was true. The correct method for proving an identity is to work with only one side at a time and to show that one side equals the other. Sometimes it is necessary to first simplify one side of the identity, and then also to simplify the other side in order to show that they are equal. Troy also should have stated restrictions.

Correct method:

Restrictions: undefined where .

So then .

Therefore .

Prove the following identities:

(State any restricted values in the interval , where appropriate.)

Restrictions: undefined where is undefined.

Therefore .

Restrictions: undefined where and where is undefined.

Therefore .

Restrictions: undefined where .

Therefore .

Restrictions: undefined where and where is undefined.

Therefore .

Restrictions: undefined where .

Restrictions also include the values of for which is undefined.

Therefore .

Restrictions: where and where is undefined.

Therefore .

Determine whether the following statements are true or false.

If the statement is false, choose a suitable value between ° and ° to confirm your answer.

False

False

True

False

True

True

4.2 Compound angle identities

Derivation of

Compound angles

Danny is studying for a trigonometry test and completes the following question:

Question:

Evaluate the following:

Danny's solution:

Consider Danny's solution and determine why it is incorrect.

Use a calculator to check that Danny's answer is wrong.

Describe in words the mistake(s) in his solution.

Is the following statement true or false?

"A trigonometric ratio can be distributed to the angles that lie within the brackets."

From the investigation above, we know that . It is wrong to apply the distributive law to the trigonometric ratios of compound angles.

Using the distance formula and the cosine rule, we can derive the following identity for compound angles:

Consider the unit circle below. The two points and are shown on the circle.

We can express the coordinates of and in terms of the angles and :

We use the distance formula to determine :

Now we determine using the cosine rule for :

Equating the two expressions for , we have

Worked example 3: Derivation of

Derive an expression for in terms of the trigonometric ratios of and .

Use the compound angle formula for

We use the compound angle formula for and manipulate the sign of in so that it can be written as a difference of two angles:

Write the final answer

Worked example 4: Derivation of and

Derive the expanded formulae for and in terms of the trigonometric ratios of and .

Use the compound angle formula and co-functions to expand

Using co-functions, we know that , so we can write in terms of the cosine function as:

Apply the compound angle formula:

To derive the formula for , we use the compound formula for and manipulate the sign of :

Write the final answers

Compound angle formulae

Note: we can use the compound angle formulae to expand and simplify compound angles in trigonometric expressions (using the equations from left to right) or we can use the expanded form to determine the trigonometric ratio of a compound angle (using the equations from right to left).

Worked example 5: Compound angle formulae

Prove that without using a calculator.

Consider the given identity

We know the values of the trigonometric functions for the special angles ( °, °, °, etc.) and we can write .

Therefore, we can use the compound angle formula for to express in terms of known trigonometric function values.

Prove the left-hand side of the identity equals the right-hand side

When proving an identity is true, remember to only work with one side of the identity at a time.

Therefore, we have shown that .

Worked example 6: Compound angle formulae

Determine the value of the following expression without the use of a calculator:

Use co-functions to simplify the expression

We need to change two of the trigonometric functions from cosine to sine so that we can apply the compound angle formula.

We also need to make sure that the sum (or difference) of the two angles is equal to a special angle so that we can determine the value of the expression without using a calculator. Notice that .

Apply the compound angle formula and use special angles to evaluate the expression

Write the final answer

Checking answers: It is always good to check answers. The question stated that we could not use a calculator to find the answer, but we can use a calculator to check that the answer is correct:

Exercise 4.2

Given:

Draw a sketch and determine the following, without the use of a calculator:

Calculate the following without the use of a calculator (leave answers in surd form):

Prove:

Hence, evaluate without using a calculator.

We have shown that:

Use a calculator to check your answer.

For :

Simplify the following without using a calculator:

Prove:

Hence, calculate the value of without using a calculator.

We have shown that:

In the diagram below, points and lie on the circle with radius of units and centre at the origin.

Prove .

We can express the coordinates of and in terms of the angles and :

We use the distance formula to determine :

Now we determine using the cosine rule for :

Equating the two expressions for , we have

Note: earlier in this chapter we derived the compound angle identities using a unit circle (radius = unit) because it simplified the calculations. From the exercise above, we see that the compound angle identities can in fact be derived using a radius of any length.

4.3 Double angle identities

Derivation of

We have shown that . If we let , then we can write the formula as:

Derivation of

Similarly, we know that . If we let , then we have:

Using the square identity, , we can also derive the following formulae:

And

Double angle formulae

Worked example 7: Double angle identities

If is an acute angle and , determine the value of without using a calculator.

Draw a sketch

We convert to a fraction so that we can use the ratio to represent the sides of a triangle.

Use the double angle formula to determine the value of

Write the final answer

Check the answer using a calculator:

Worked example 8: Double angle identities

Prove that .

For which values of is the identity not valid?

Consider the given expressions

The right-hand side (RHS) of the identity cannot be simplified, so we simplify the left-hand side (LHS). We also notice that the trigonometric function on the RHS does not have a dependence, therefore we will need to use the double angle formulae to simplify and on the LHS.

Prove the left-hand side equals the right-hand side

Identify restricted values of

We know that is undefined for .

Note that division by zero on the LHS is not allowed, so the identity will also be undefined for:

for .

Exercise 4.3

Given and . Determine the value of the following, without a calculator:

Draw a sketch:

Given , determine (without a calculator):

Prove the identity:

Restrictions:

Hence, solve the equation for .

Without using a calculator, find the value of the following:

Check the answer using a calculator.

Check the answer using a calculator.

Prove the identity:

Restrictions:

Explain why the identity is undefined for

Consider the denominator on the LHS:

Consider the denominator on the RHS:

Therefore, the identity will be undefined because division by zero is not permitted.

4.4 Solving equations

The general solution

The periodicity of the trigonometric functions means that there are an infinite number of positive and negative angles that satisfy an equation. If we do not restrict the solution, then we need to determine the general solution to the equation. We know that the sine and cosine functions have a period of ° and the tangent function has a period of °.

Method for finding the solution:

Simplify the equation using algebraic methods and trigonometric identities.

Determine the reference angle (use a positive value).

Use the CAST diagram to determine where the function is positive or negative (depending on the given equation/information).

Restricted values: find the angles that lie within a specified interval by adding/subtracting multiples of the appropriate period.

General solution: find the angles in the interval that satisfy the equation and add multiples of the period to each answer.

Check answers using a calculator.

General solutions:

for .

Worked example 9: Finding the general solution

Determine the general solution for (correct to one decimal place).

Use a calculator to find the reference angle

Use CAST diagram to determine in which quadrants is positive

The CAST diagram indicates that is positive in the first and second quadrants.

Using reduction formulae, we know that .

In the first quadrant:

In the second quadrant:

where .

Check that the solution satisfies the original equation

We can select random values of to check that the answers satisfy the original equation.

Let :

This solution is correct.

Similarly, if we let :

This solution is also correct.

Write the final answer

or for .

Worked example 10: Trigonometric equations

Solve the following equation for , without using a calculator:

Simplify the equation

We first simplify the left-hand side of the equation using the double angle formulae. To solve this equation, we need to manipulate the given equation to be of the form:

Use a calculator to find the reference angle

Use CAST diagram to determine in which quadrants is negative

The CAST diagram indicates that is negative in the second and fourth quadrants.

Notice that for , , which is the angle in the fourth quadrant.

Check that the solution satisfies the original equation

Write the final answer

where .

Worked example 11: Trigonometric equations

Prove and hence solve (correct to one decimal place).

Prove the identity

Expand the right-hand side of the identity and show that it is equal to the left-hand side:

Solve the equation

Use a calculator to find the reference angle

Use CAST diagram to determine in which quadrants is positive

The CAST diagram indicates that is positive in the first and fourth quadrants.

In the first quadrant:

Important: remember to also divide by .

In the fourth quadrant:

where .

Check that the solution satisfies the original equation

Write the final answer

Worked example 12: Trigonometric equations

Find the general solution for .

Simplify the given equation

Do not divide both sides of the equation by :

part of the solution would be lost;

we would need to restrict the values of to those where (division by zero is not permitted).

Apply the zero product law and solve for

Write the final answer

Exercise 4.4

Find the general solution for each of the following equations (correct to two decimal places):

We work with the left-hand side of the equation since it contains , which is easier to simplify than the double angle on the right-hand side:

for .

Restrictions:

Restrictions:

Given:

Solve the equation for , without using a calculator.

Draw a graph and indicate the solution on the diagram.

The diagram below shows the graph of (grey) and (black).

Given:

Determine the general solution.

How many solutions does the given equation have in the interval ?

Therefore, there are ten solutions that lie within the interval .

Without using a calculator, solve in .

First we simplify the left-hand side of the equation:

Find the general solution for .

Use a graph to illustrate the solution for the interval .

The diagram below shows the graph of .

Explain why the general solution for the equation is and the general solution for is . Why are they different?

The period of the cosine function is °. This means that the function values repeat after °.

The period of the tangent function is °. This means that the function values repeat after °.

Solve for :

Check that both answers satisfy the original equation:

Therefore, .

4.5 Applications of trigonometric functions

Area, sine and cosine rule

Area rule

Sine rule

Cosine rule

area

area

area

How to determine which rule to use:

Area rule:

no perpendicular height is given

Sine rule:

no right angle is given

two sides and an angle are given (not the included angle)

two angles and a side are given

Cosine rule:

no right angle is given

two sides and the included angle are given

three sides are given

Problems in two dimensions

Worked example 13: Area, sine and cosine rule

Given with and .

Calculate the following:

Area

Use the cosine rule to determine the length of

Use the sine rule to determine

From the diagram, we see that , therefore .

Calculate the area

Worked example 14: Problem in two dimensions

In the figure below, and .

Show that .

Consider the given information

Use the given information to determine as many of the unknown angles as possible.

Determine the expression for

To derive the required expression, we need to write in terms of and .

In , we can use the cosine rule to determine :

Exercise 4.5

In the diagram below, is the centre of the semi-circle .

Find in terms of .

is a diameter of semi-circle . is the centre and therefore bisects .

Or

In , determine an expression for .

In , determine an expression for .

In , determine an expression for .

Use the results from the previous questions to show that .

is a diameter of the circle with centre and radius . , and .

Show that .

The figure below shows a cyclic quadrilateral with .

Show that the area of the cyclic quadrilateral is .

Write down two expressions for : one in terms of and one in terms of .

Show that .

Suppose that units, units, units and units. Calculate (correct to one decimal place).

Find the angle . Hence, calculate the area of (correct to one decimal place).

Two vertical towers and are and high, respectively. Point lies between the two towers. The angle of elevation from to is ° and from to is °. A cable is needed to connect and .

Determine the minimum length of cable needed to connect and (to the nearest metre).

How far apart are the bases of the two towers (to the nearest metre)?

Problems in three dimensions

Trigonometric formulae are useful for solving problems in two dimensions. However, in the real world all objects are three dimensional, so it is important that we extend the application of the area, sine and cosine formulae to three dimensional situations.

Drawing a three dimensional diagram is a crucial step in finding the solution to a problem. Interpreting the given information and sketching a three dimensional diagram are skills that need to be practised.

Worked example 15: Problems in three dimensions - height of a pole

is the top of a pole and its base, , is in the same horizontal plane as the points and . The angle of elevation measured from to is . , .

Use the given information to calculate , the height of the pole, to the nearest metre.

Draw a sketch

It is a challenge to analyze this situation when only a description is given. A diagram is very useful for representing three dimensional problems.

Draw a sketch using the given information. Indicate all right angles on the sketch, even those that do not look as though they are a ° angle (for example, ). It is also helpful to shade the horizontal plane (), as it adds depth to the diagram and gives a better visualization of the situation.

Consider the given information

There are two triangles to consider: and . We are required to find the length of , but the only information we have for is .

Important: notice that is a side of and it is also a side of - it is a link or bridge between the two triangles.

Therefore, to determine the height of the pole:

Write down all the given information on the sketch.

Use the information for to calculate .

Use and the information for to calculate .

Determine the length of

We notice that does not have a right-angle, so we use the sine rule to determine .

Determine the length of

In :

Note: do not round off answers in the intermediate steps as this will affect the accuracy of the final answer. Always try to do the calculation in one complete step and only round off the final answer.

Write the final answer

The height of the pole, , is .

The calculation in the above worked example is only applicable for the specific numbers given. However, if we derived a general formula for this contextual situation, then it could be applied to any suitable set of numerical values.

Worked example 16: Problems in three dimensions - height of a building

is the top of a building of height . The base of the building is at and lies on the ground (a horizontal plane). , , and .

Show that .

Consider the given information

We know that is right-angled and we are required to find a formula to calculate the length of . In , we are given two angles and a length of . We identify the side as the link between and .

Determine an expression for

does not have a right-angle but two angles and a side are given, so we use the sine rule to determine .

Determine an expression for

In :

Exercise 4.6

The line represents a tall tower, with at its base. The angle of elevation from to is . A man stands at such that and .

Find the height of the tower in terms of , and .

Find if we are given that , and .

is the top of a mast and its base, , is in the same horizontal plane as the points and . The angle of elevation measured from to is . , .

Use the given information to derive a general formula for , the height of the mast.

In :

If , , and , calculate (to the nearest metre).

is the height of a block of flats with at the base and at the top of the building. is a point in the same horizontal plane as points and . and .

Show that the height of the block of flats, , can be expressed as:

is the link between and .

If and , calculate (correct to one decimal place).

Assuming each level is high, estimate the number of levels in the block of flats.

Therefore, there are approximately levels.

Two ships at sea can see a lighthouse on the shore. The distance from the top of the lighthouse () to ship and to ship is . The angle of elevation from to is , and

Show that the distance between the two ships is given by .

Show that the area of the sea included in is given by area .

Calculate the triangular area of the sea if the angle of inclination from the ship to the top of the lighthouse is and the angle between the direct lines from the base of the lighthouse to each ship is .

and .

A triangular look-out platform () is attached to a bridge that extends over a deep gorge. The vertical depth of the gorge, the distance from the edge of the look-out to the bottom of the gorge , is . The angle of depression from to is ° and from to is °. The angle at the edge of the platform, is °.

Calculate the area of the look-out platform (to the nearest ).

If the platform is constructed so that the two angles of depression, and , are both equal to ° and the vertical depth of the gorge , and , show that .

If and , calculate (to the nearest integer).

4.6 Summary

Pythagorean Identities

Ratio Identities

Special angle triangles

θ

°

°

°

°

°

undef

CAST diagram and reduction formulae

Negative angles

Periodicity Identities

Cofunction Identities

Area Rule

Sine Rule

Cosine Rule

Compound Angle Identities

Double Angle Identities

Video: 28N2

Exercise 4.7

Determine the following without using a calculator:

Given . Using a diagram, find and .

Given for .

Determine the following (leave answers in surd form):

Draw a sketch.

If for , determine, without the use of a calculator:

Draw a sketch.

We are given that , therefore must lie in the second quadrant for the tangent function to be negative.

Simplify:

Solve the equation:

for .

Find the general solution, without a calculator, for the following equations:

Prove:

Hence, solve the equation for .

Prove the following identities:

Restrictions: .

Prove:

For which values of is the identity undefined?

The identity is undefined for the values of such that since division by zero is not permitted.

Restricted values are:

Given:

Find the general solution.

Find the solution for .

Draw a graph of for and indicate the solutions to the equation on the graph.

Use the graph to determine where .

Show that:

Use this result to solve for .

On the same system of axes, draw two graphs to solve graphically: for . Indicate the solutions on the graph using the letters etc.

Given: for

Solve for algebraically.

Verify the solution graphically by sketching two graphs on the same system of axes.

The following graphs are given below:

Explain why and .

From the graph we can see that the amplitude of the sine graph is , therefore .

For the cosine graph, the period is , therefore .

For how many -values in will ?

For:

For the interval , the diagram shows that the two graphs intersect at three places.

Use the graph to solve .

From the graph we have the following solution:

Solve for using trigonometric identities.

For which values of will for ?

In ,

Prove that .

Show that .

Given the graphs of and , determine the values of and .

and

is an acute angled triangle with . Show that area .

is a cyclic quadrilateral with .

Calculate .

Determine .

Find .

is a cyclic quadrilateral that lies in a horizontal plane. is a vertical pole with base . The angle of elevation from to is and . is an isosceles triangle with .

Show that .

In cyclic quadrilateral :

Show that

If , calculate the length of .

The first diagram shows a rectangular box with and . The lid of the box, , is opened to the position , as shown in the second diagram.

Write down the dimensions (length, breadth and diagonal) of the lid .

Calculate , the perpendicular height of above the base of the box.

Calculate the ratio .

is a vertical pole on a horizontal plane . is metres and the angle of elevation from to is . and .

Name the two right angles in the diagram.

Show that .

If it is given that , show that the height of the pole is given by .

Calculate the height of the pole if .

is a flagpole on top of a government building . units and is a point on the ground in the same horizontal plane as the base of the building, . The angle of elevation from to and is and , respectively.

Show that

Calculate the height of the flagpole (to the nearest metre) if the building is , and .

Chapter 5: Polynomials

Proofs of theorems are not examinable.

Be careful of spending too much time on revision. Learners should be familiar with these factorisation methods from Grade 10 and 11.

Explain terminology carefully. It is very important for learners to understand what a factor is and how it relates to the graphs of quadratic and cubic functions (covered in Differential Calculus chapter).

Long division and synthetic division are included as an introduction to the remainder theorem. Do not spend too much time on these techniques.

It is important for learners to understand that a remainder of zero means that divisor is a factor.

Encourage learners to practise factorising by inspection.

5.1 Revision

Identifying polynomials

Terminology:

Polynomial

An expression that involves one or more variables having different powers and coefficients.

Monomial

A polynomial with one term.

For example, .

Binomial

A polynomial that has two terms.

For example, .

Trinomial

A polynomial that has three terms.

For example, .

Degree/Order

The degree, also called the order, of a univariate polynomial is the value of the highest exponent in the polynomial.

For example, has a degree of .

It is important to notice that the definition of a polynomial states that all exponents of the variables must be elements of the set of natural numbers. If an expression contains terms with exponents that are not natural numbers, then it is not a polynomial.

The following examples are not polynomials:

More on polynomials

Discuss whether the following statements are true or false:

The expression is a trinomial of degree .

is a binomial of order .

is a constant polynomial of degree .

is a quadratic polynomial.

The expression is a monomial because it only has one term.

is a constant polynomial of undefined degree.

A cubic polynomial has three terms and all the exponents are natural numbers.

Exercise 5.1

Given , determine whether the following statements are true or false. If false, provide the correct statement.

is a trinomial.

True

The coefficient of the is zero.

True

.

is of degree .

True

The constant term is .

False:

will have real roots.

True

Given , determine the following:

the number of terms in .

the degree of .

the coefficient of the term.

the constant term.

Determine which of the following expressions are polynomials and which are not. For those that are not polynomials, give reasons.

Cubic polynomial

Quadratic polynomial

Not a polynomial; in the exponent is not a natural number.

Not a polynomial; in the exponent is not a natural number.

Cubic polynomial

Linear polynomial

Zero polynomial

Polynomial; degree

Cubic polynomial

Not a polynomial; in the exponent is not a natural number.

Peter's mathematics homework is shown below. Find and correct his mistakes.

Homework:

Given , answer the following questions:

Simplify the expression.

Is a polynomial?

What is the coefficient of the term?

Peter's answers:

Yes, because it can be simplified to have exponents that are all natural numbers. It is a quadratic binomial because the highest exponent is two and there are only two terms; and .

Before I simplified, the coefficient of the term was nothing and after I simplified it became .

Peter made the following mistakes:

This is not an equation, therefore Peter must not multiply through by . Peter did not factorise correctly, he made an error with the sign in the second bracket

.

The original expression is not a polynomial because the exponent of the second term is not a natural number. Peter's simplified expression is a quadratic trinomial because it has three terms and two factors.

In the original expression, the coefficient of the term is . After Peter wrongly simplified the expression, the coefficient of the term was and not . Always remember that the coefficient includes the sign and the number in front of the variable.

Quadratic polynomials

In earlier grades we learnt the following useful techniques and methods for factorising an expression:

taking out a common factor

factorising the difference of two squares

grouping in pairs

factorising the sum and difference of two cubes

We also looked at the different methods for factorising quadratic expressions:

factorising by inspection

completing the square

using the quadratic formula

making a suitable substitution

It is important to revise these methods; we use the quadratic formula for factorising cubic polynomials and we also use completing the square to find the equation of a circle in Chapter 7.

Terminology: |
Variable | A symbol used to represent an unknown numerical value. For example, .
Coefficient | The number or parameter that is multiplied by the variable of an expression.
Expression | A term or group of terms consisting of numbers, variables and the basic operators ().
Univariate expression | An expression containing only one variable.
Equation | A mathematical statement that asserts that two expressions are equal.
Identity | A mathematical relationship that equates one expression with another.
Solution | A value or set of values that satisfies the original problem statement.
Root/Zero | A root, also referred to as the “zero”, of an equation is the value of such that is satisfied.

Worked example 1: Solving quadratic equations using factorisation

Solve for :

Determine the restrictions

The restrictions are the values for that would result in the denominator being equal to , which would make the fraction undefined. Therefore and .

Determine the lowest common denominator

The lowest common denominator is .

Multiply each term in the equation by the lowest common denominator and simplify

Factorise and solve the equation

Check the solution by substituting both answers back into the original equation

Write the final answer

Therefore or .

Worked example 2: Using the quadratic formula

Find the roots of the function .

Finding the roots

To determine the roots of , we let .

Check whether the expression can be factorised

The expression cannot be factorised, so the general quadratic formula must be used.

Identify the coefficients to substitute into the formula

Apply the quadratic formula

Write the final answer

Therefore or

Worked example 3: Solving quadratic equations by completing the square

Solve by completing the square:

The equation is already in the form

Make sure the coefficient of the term is equal to

Take half the coefficient of the term and square it, then add and subtract it from the equation

The coefficient of the term is . Half of the coefficient of the term is and the square of it is . Therefore .

Write the trinomial as a perfect square

Method : Take square roots on both sides of the equation

Important: When taking a square root always remember that there is a positive and negative answer, since and .

Solve for

Therefore or .

Method : Factorise the expression as a difference of two squares

Simplify and solve for

Write the final answer

Notice that both methods produce the same answer. These roots are rational because is a perfect square.

Exercise 5.2

Solve the following quadratic equations by factorisation. Answers may be left in surd form, where applicable.

Solve the following equations by completing the square:

Solve the following using the quadratic formula.

Note: it would be easier and faster to solve by factorisation rather than using the quadratic formula. If the question does not specify which method to use, first try to solve by factorisation.

Factorise the following:

5.2 Cubic polynomials

Simple division

Consider the following and answer the questions below:

students are at a product promotion and there are free gifts to be given away. Each student must receive the same number of gifts.

Determine how many gifts each student will get?

How many gifts will be left over?

Use the following variables to express the above situation as a mathematical equation:

total number of gifts

total number of students

number of gifts for each student

number of gifts left over

A group of students go to dinner together at a restaurant and the total bill is . Each student contributes towards the bill. They count the money and find that they are still short.

Assign variables to the known and unknown values.

Write a mathematical equation to describe the situation.

Use this equation to determine how many students went to dinner.

We know that divided by gives an answer of with a remainder .

This means that:

We can write a general expression for the rule of division; if an integer is divided by an integer , then the answer is with a remainder .

where and .

This rule can be extended to include the division of polynomials; if a polynomial is divided by a polynomial , then the answer is with a remainder .

where .

In words: the dividend is equal to the divisor multiplied by the quotient, plus the remainder.

A cubic polynomial is an expression with the highest power equal to ; we say that the degree of the polynomial is .

The general form of a cubic polynomial is

where .

In Grade 10 we learnt how to factorise the sum and difference of two cubes by first finding a factor (the first bracket) and then using inspection (the second bracket). For example,

In this section we focus on factorising cubic polynomials with one variable (univariate) where and are not zero.

We use the following methods for factorising cubic polynomials:

long division

synthetic division

inspection

Worked example 4: Long division

Use the long division method to determine the quotient and the remainder if is divided by . Write your answer in the form .

Write down the known and unknown expressions

Use long division method to determine and

Make sure that and are written in descending order of the exponents. If a term of a certain degree is missing from , then write the term with a coefficient of .

Write the final answer

Video: 28S4

Synthetic division is a simpler and more efficient method for dividing polynomials. It allows us to determine the quotient and the remainder by considering the coefficients of the terms in each of the polynomials without needing to rewrite the variable and exponent for each term. If a term of a certain degree is missing from , then write the term with a coefficient of . For example, should be written as .

Notice that for synthetic division:

the coefficients of the dividend () are written below the horizontal line.

the coefficients of the quotient () are written above the horizontal line.

we add coefficients instead of subtracting as is the case with long division

we use the opposite sign of the divisor (); the divisor is and we use .

the coefficient of the term in the divisor is , so .

Worked example 5: Synthetic division

Use the synthetic division method to determine the quotient and the remainder if is divided by . Write your answer in the form .

Write down the known and unknown expressions

Use synthetic division to determine and

Write the final answer

The quotient will be one degree lower than the dividend if we divide by a linear expression, therefore we have:

General method: given the dividend and the divisor , we determine the quotient and the remainder using:

We determine the coefficients of the quotient by calculating:

Important note: is a function and are coefficients.

Video: 28S5

Worked example 6: Synthetic division

Use the synthetic division method to determine the quotient and the remainder if is divided by .

Write down the known and unknown expressions

Use synthetic division to determine and

Make the leading coefficient of the divisor equal to :

Write the final answer

Video: 28S6

Exercise 5.3

Factorise the following:

For each of the following:

Use long division to determine the quotient and the remainder .

Write in the form

Check your answer by expanding the brackets to get back to the original cubic polynomial.

is divided by

Check:

and

and

Check:

and

The remainder is equal to zero, therefore is a factor of .

Check:

and

Check:

is divided into

Check:

Check:

is divided by

Check:

and

Check:

Use synthetic division to determine the quotient and the remainder when is divided by :

5.3 Remainder theorem

Remainder theorem

Given the following functions:

Determine and .

Write your answers in the general form: .

Determine and .

What do you notice?

Consider the degree of the quotient and the remainder - is there a rule?

What conclusions can you draw?

Write a mathematical equation to describe your conclusions.

Complete the following sentence: a cubic function divided by a linear polynomial gives a quotient with a degree of and a remainder with a degree of , which is called a constant.

The Remainder theorem

A polynomial divided by gives a remainder of .

In words: the value of the remainder is obtained by substituting into the polynomial .

NOTE: PROOF NOT FOR EXAMS

Let the quotient be and let the remainder be . Therefore we can write:

Worked example 7: Finding the remainder

Use the remainder theorem to determine the remainder when is divided by the following linear polynomials:

Determine the remainder for each linear divisor

The remainder theorem states that any polynomial that is divided by gives a remainder of :

Worked example 8: Using the remainder to solve for an unknown variable

Given that divided by gives a remainder of , use the remainder theorem to determine the value of .

Use the remainder theorem to determine the unknown variable

From the remainder theorem we know that and we can therefore solve for :

Write the final answer

Therefore and .

Exercise 5.4

Use the remainder theorem to determine the remainder when is divided by :

Determine the value of if divided by gives a remainder of .

Calculate the value of if is divided by and gives a remainder of .

If is divided by and the remainder is , find the value of .

Determine the value of if is divided by and gives a remainder of .

If is divided by and the remainder is , calculate the value of .

Calculate the value of if is divided by and gives a remainder of .

If is divided by and the remainder is , express in terms of .

When the polynomial is divided by or , the remainder is . Determine the values of and .

5.4 Factor theorem

If an integer is divided by an integer , and the answer is with the remainder , then we know that is a factor of .

This is also true of polynomials; if a polynomial is divided by a polynomial , and the answer is with the remainder , then we know that is a factor of .

The factor theorem describes the relationship between the root of a polynomial and a factor of the polynomial.

The Factor theorem

If the polynomial is divided by and the remainder, given by is equal to zero, then is a factor of .

Converse: if is a factor of , then .

Worked example 9: Factor theorem

Using the factor theorem, show that is a factor of .

Determine how to approach the problem

For to be a factor, must be equal to .

Calculate

Conclusion

Since , is a factor of .

In general, to factorise a cubic polynomial we need to do the following:

Find one factor by trial and error: consider the coefficients of the given cubic polynomial and guess a possible root ().

Use the factor theorem to confirm that is a root; show that = 0.

Divide by the factor () to obtain a quadratic polynomial (remember to be careful with the signs).

Apply the standard methods of factorisation to determine the two factors of the quadratic polynomial.

Worked example 10: Factor theorem

Use the factor theorem to determine if is a factor of .

Determine how to approach the problem

For to be a factor, must be equal to .

Calculate

Conclusion

Since , is not a factor of .

Worked example 11: Factorising cubic polynomials

Factorise completely:

Find a factor by trial and error

Try

Therefore is not a factor.

We consider the coefficients of the given polynomial and try:

Therefore is a factor, because .

Factorise by inspection

Now divide by using inspection:

Write

The first term in the second bracket must be to give and make the polynomial a cubic.

The last term in the second bracket must be because .

So we have

Now, we must find the coefficient of the middle term:

gives the in the original polynomial. So, the coefficient of the -term must be .

.

Write the final answer

We can factorise the last bracket as a difference of two squares:

Worked example 12: Factorising cubic polynomials

Use the factor theorem to factorise .

Find a factor by trial and error

Try

Therefore is a factor.

Factorise by inspection

The first term in the second bracket must be to give if we work backwards.

The last term in the second bracket must be because .

So we have

Now, we must find the coefficient of the middle term:

gives . So, the coefficient of the -term in the second bracket must be to give another so that overall we have .

So .

Make sure that the expression has been factorised correctly by checking that the coefficient of the -term also works out: , which is correct.

Write the final answer

We can factorise the last bracket as:

Exercise 5.5

Find the remainder when is divided by .

Use the factor theorem to factorise completely.

Find .

Factorise completely.

Use the factor theorem to determine all the factors of the following expression:

Complete: If is a polynomial and p is a number such that , then is...

a factor of

5.5 Solving cubic equations

Now that we know how to factorise cubic polynomials, it is also easy to solve cubic equations of the form .

Worked example 13: Solving cubic equations

Solve:

Find one factor using the factor theorem

Let

Try

Therefore is not a factor.

Try

Therefore is a factor.

Factorise by inspection

Factorise fully

Solve the equation

Sometimes it is not possible to factorise a quadratic expression using inspection, in which case we use the quadratic formula to fully factorise and solve the cubic equation.

Worked example 14: Solving cubic equations

Solve for :

Use the factor theorem to determine a factor

Let

Try

Therefore is not a factor.

Try

Therefore is not a factor.

Therefore is a factor.

Factorise by inspection

cannot be factorised any further and we are left with

Solve the equation

Apply the quadratic formula for the second bracket

Always write down the formula first and then substitute the values of and .

Final solutions

or

Exercise 5.6

Solve the following cubic equations:

5.6 Summary

Terminology:

Expression
A term or group of terms consisting of numbers, variables and the basic operators ().

Univariate expression
An expression containing only one variable.

Root/Zero
A root, also referred to as the “zero”, of an equation is the value of such that is satisfied.

Polynomial

An expression that involves one or more variables having different powers and coefficients.

Monomial

A polynomial with one term.

For example, .

Binomial

A polynomial that has two terms.

For example, .

Trinomial

A polynomial that has three terms.

For example, .

Degree/Order

The degree, also called the order, of a univariate polynomial is the value of the highest exponent in the polynomial.

For example, has a degree of .

Quadratic formula:

Remainder theorem: a polynomial divided by gives a remainder of .

Factor theorem: if the polynomial is divided by and the remainder, , is equal to zero, then is a factor of .

Converse of the factor theorem: if is a factor of , then .

Synthetic division:

We determine the coefficients of the quotient by calculating:

Exercise 5.7

Solve for :

Solve for :

Solve for :

Solve for :

Solve for if .

Solve for :

Show that is a factor of .

Hence, by factorising completely, solve the equation:

for all values of . What is the value of ?

Use the factor theorem to solve the following equation for :

Hence, or otherwise, solve for :

Find the value of if is a factor of and divided by gives a remainder of .

Determine the values of for which the function

leaves a remainder of when it is divided by .

Alternative (long) method:

We first take out the factor using long division:

We take the remainder and set it equal to 9:

Calculate and if .

Chapter 6: Differential calculus

Explain terminology carefully as there are many new terms in this chapter.

Encourage learners to draw a sketch, as this is a very helpful tool in solving problems.

It is very important that learners understand that a limit is the value that the function tends towards (gets closer and closer to) and it is not an exact answer.

Explain to learners that the function does not have to be defined for a given value of for the limit to exist.

Learners must understand that the derivative of a function gives the gradient of the function at a certain point AND the gradient of the tangent at that point.

Learners must practise differentiating from first principles. This often asked in examinations.

Learners must be familiar with the different notations for differentiation.

6.1 Limits

Calculus is one of the central branches of mathematics and was developed from algebra and geometry. It is built on the concept of limits, which will be discussed in this chapter. Calculus consists of two related ideas: differential calculus and integral calculus. We will only be dealing with differential calculus in this chapter and will explore how it can be used to solve optimisation problems and finding rates of change.

The tale of Achilles and the tortoise

Zeno (born about 490 BC) was a philosopher of southern Italy who is famous for his paradoxes (a “paradox” is a statement that seems contradictory and yet may be true).

One of Zeno's paradoxes can be summarised as:

Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles is very fast. So, the tortoise asks Achilles for a head start. Achilles agrees to give the tortoise a head start. Does Achilles overtake the tortoise?

To solve this problem, we start by writing:

where

is the distance covered by Achilles

is Achilles' speed

is the time taken by Achilles to overtake the tortoise

is the distance covered by the tortoise

is the tortoise's speed

Achilles will overtake the tortoise when both of them have covered the same distance. If we assume that Achilles runs at and the tortoise runs at , then this means that Achilles will overtake the tortoise at a time calculated as:

However, Zeno looked at it as follows: Achilles takes to travel the head start that he gave the tortoise. However, in these , the tortoise has travelled a further .

Achilles then takes another to travel the . In these , the tortoise travels a further .

Zeno saw that Achilles would always get closer and closer but wouldn't actually overtake the tortoise.

So what does Zeno, Achilles and the tortoise have to do with calculus? Consider our earlier studies of sequences and series:

We know that the sequence can be defined by the expression and that the terms get closer to as gets larger.

Similarly, the sequence can be defined by the expression and the terms get closer to as gets larger.

We have also seen that an infinite geometric series can have a finite sum.

where is the first term of the series and is the common ratio.

We see that there are some functions where the value of the function gets close to or approaches a certain value as the number of terms increases.

Limits

Consider the function:

The numerator of the function can be factorised as:

Then we can cancel the from numerator and denominator and we are left with:

However, we are only able to cancel the term if . If , then the denominator becomes and the function is not defined. This means that the domain of the function does not include . But we can examine what happens to the values for as gets closer to . The list of values shows that as gets closer to , gets closer and closer to .

The graph of this function is shown below. The graph is a straight line with slope and -intercept , but with a hole at . As approaches from the left, the -value approaches and as approaches from the right, the -value approaches . Since the function approaches the same -value from the left and from the right, the limit exists.

Notation

We can now introduce new notation. For the function , we can write:

This is read: the limit of as tends to (from both the left and the right) is equal to .

Limits

If , determine:

What do you notice about the value of as gets closer and closer to ?

Worked example 1: Limit notation

Write the following using limit notation: as gets close to , the value of the function approaches .

This is written as:

This is illustrated in the diagram below:

We can also have the situation where a function tends to a different limit depending on whether approaches from the left or the right.

As from the left, approaches . As from the right, approaches .

The limit for approaching from the left is:

and for approaching from the right:

where means approaches zero from the left and means approaches zero from the right.

Therefore, since does not approach the same value from both sides, we can conclude that the limit as tends to zero does not exist.

As tends to from the left, the function approaches and as tends to from the right, the function approaches . Since the function approaches the same value from both sides, the limit as tends to exists and is equal to .

Video: 28VG

Worked example 2: Limits

Determine:

Illustrate answers graphically.

Simplify the expression and cancel all common terms

We cannot simplify further and there are no terms to cancel.

Calculate the limit

Worked example 3: Limits

Determine the following and illustrate the answer graphically:

Simplify the expression

Factorise the numerator:

As , the denominator , therefore the expression is not defined for since division by zero is not permitted.

Cancel all common terms

Calculate the limit

Draw the graph

Video: 28YM

Exercise 6.1

Determine the following limits and draw a rough sketch to illustrate:

Determine the following limits (if they exist):

Important: notice that even though the function is not defined at , the limit as tends to does exist and is equal to .

Even though the function is not defined at , the limit as tends to does exist and is equal to .

Even though the function is not defined at , the limit as tends to does exist and is equal to .

Note that the function is not defined at , but the limit as tends to does exist and is equal to .

Gradient at a point

Average gradient

In Grade 11 we learnt that the average gradient between any two points on a curve is given by the gradient of the straight line that passes through both points. We also looked at the gradient at a single point on a curve and saw that it was the gradient of the tangent to the curve at the given point. In this section we learn how to determine the gradient of the tangent.

Let us consider finding the gradient of a tangent to a curve with equation at a given point .

We know how to calculate the average gradient between two points on a curve, but we need two points. The problem now is that we only have one point, namely . To get around the problem we first consider a secant (a straight line that intersects a curve at two or more points) to the curve that passes through point and another point on the curve , where is an arbitrary distance from .

We can determine the average gradient of the curve between the two points:

If we let the -coordinate of be , then the -coordinate is . Similarly, if the -coordinate of is (), then the -coordinate is .

We can now calculate the average gradient as:

Gradient at a point

Imagine that moves along the curve, getting closer and closer to . The secant line approaches the tangent line as its limiting position. This means that the average gradient of the secant approaches the gradient of the tangent to the curve at .

We see that as point approaches point , gets closer to . If point lies on point , then and the formula for average gradient is undefined. We use our knowledge of limits to let tend towards to determine the gradient of the tangent to the curve at point :

Worked example 4: Gradient at a point

Given , determine the gradient of the curve at the point .

Write down the formula for the gradient at a point

Determine and

We need to find the gradient of the curve at , therefore we let :

Substitute into the formula and simplify

Notice that we only take the limit once we have removed from the denominator.

Write the final answer

The gradient of the curve at is .

Worked example 5: Gradient at a point

Given the function , determine the gradient of the tangent to the curve at the point .

Write down the formula for the gradient at a point

Determine and

We need to find the gradient of the tangent to the curve at , therefore we let :

Substitute into the formula and simplify

Write the final answer

The gradient of the tangent to the curve at is .

Worked example 6: Gradient at a point

Determine the gradient of at the point .

Write down the formula for the gradient at a point

Determine and

Let :

Substitute into the formula and simplify

Write the final answer

The gradient of at is .

Exercise 6.2

Given:

Find the average gradient of function , between and .

Illustrate this with a graph.

Find the gradient of at the point and illustrate this on your graph.

Determine the gradient of the tangent to if .

To simplify the two fractions in the numerator, we get a common denominator :

Determine the equation of the tangent to .

The tangent meets the graph of at

Therefore tangent goes through .

6.2 Differentiation from first principles

We know that the gradient of the tangent to a curve with equation at can be determine using the formula:

We can use this formula to determine an expression that describes the gradient of the graph (or the gradient of the tangent to the graph) at any point on the graph. This expression (or gradient function) is called the derivative.

Derivative

The derivative of a function is written as and is defined by:

Differentiation

The process of determining the derivative of a given function.

This method is called differentiation from first principles or using the definition.

Worked example 7: Differentiation from first principles

Calculate the derivative of from first principles.

Write down the formula for finding the derivative using first principles

Determine

Substitute into the formula and simplify

Write the final answer

The derivative .

Notation

There are a few different notations used to refer to derivatives. It is very important that you learn to identify these different ways of denoting the derivative and that you are consistent in your usage of them when answering questions.

If we use the common notation , where the dependent variable is and the independent variable is , then some alternative notations for the derivative are as follows:

The symbols and are called differential operators because they indicate the operation of differentiation.

means differentiated with respect to . Similarly, means differentiated with respect to .

Important: is not a fraction and does not mean .

Video: 28W2

Worked example 8: Differentiation from first principles

Find the derivative of from first principles.

Determine and interpret the answer.

Write down the formula for finding the derivative from first principles

Substitute into the formula and simplify

Calculate and interpret the answer

The derivative of at is .

The gradient of the function at is equal to .

The gradient of the tangent to at is equal to .

Worked example 9: Differentiation from first principles

Calculate from first principles if .

Write down the formula for finding the derivative using first principles

Substitute into the formula and simplify

It is sometimes easier to write the right-hand side of the equation as:

Notice: even though remains in the denominator, we can take the limit since it does not result in division by .

Write the final answer

Worked example 10: Differentiation from first principles

Differentiate from first principles and interpret the answer.

Write down the formula for finding the derivative from first principles

Substitute into the formula and simplify

Interpret the answer

The gradient of is equal to at any point on the graph. The derivative of this constant function is equal to .

Exercise 6.3

Given:

Determine .

Hence, determine .

Explain the meaning of your answer in (b).

The derivative of is . The gradient of the function is given by the expression . The gradient of the graph depends on the value of .

Find the derivative of using first principles.

Determine the derivative of using first principles.

Determine from first principles if .

Therefore:

If , determine using first principles.

Find the derivative of using first principles.

Differentiate using first principles.

(Hint: Use Pascal's triangle)

Step 1: Use first principles to find the derivative.

Expand using the pattern of the coefficients given by Pascal's Triangle:

All the terms, except , contain .

This is a very valuable general rule for finding the derivative of a function.

6.3 Rules for differentiation

Determining the derivative of a function from first principles requires a long calculation and it is easy to make mistakes. However, we can use this method of finding the derivative from first principles to obtain rules which make finding the derivative of a function much simpler.

Rules for differentiation

Differentiate the following from first principles:

Complete the table:

Can you identify a pattern for determining the derivative?

Rules for differentiation

General rule for differentiation:

The derivative of a constant is equal to zero.

The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

The derivative of a sum is equal to the sum of the derivatives.

The derivative of a difference is equal to the difference of the derivatives.

Video: 28WD

Worked example 11: Rules of differentiation

Use the rules of differentiation to find the derivative of each of the following:

Apply the appropriate rules to determine the derivative

Worked example 12: Rules of differentiation

Differentiate the following with respect to :

Expand the expression and apply the rules of differentiation

We have not learnt a rule for differentiating a product, therefore we must expand the brackets and simplify before we can determine the derivative:

Expand the expression and apply the rules of differentiation

We have not learnt a rule for differentiating a quotient, therefore we must first simplify the expression and then we can differentiate:

Important: always write the final answer with positive exponents.

When to use the rules for differentiation:

If the question does not specify how we must determine the derivative, then we use the rules for differentiation.

When to differentiate using first principles:

If the question specifically states to use first principles.

If we are required to differentiate using the definition of a derivative, then we use first principles.

Exercise 6.4

Differentiate the following:

Find if .

Find if .

Find if .

Determine if .

Determine the derivative of .

Find .

Find if .

Make the subject of the formula in order to differentiate with respect to .

Determine if .

Find if .

6.4 Equation of a tangent to a curve

At a given point on a curve, the gradient of the curve is equal to the gradient of the tangent to the curve.

The derivative (or gradient function) describes the gradient of a curve at any point on the curve. Similarly, it also describes the gradient of a tangent to a curve at any point on the curve.

To determine the equation of a tangent to a curve:

Find the derivative using the rules of differentiation.

Substitute the -coordinate of the given point into the derivative to calculate the gradient of the tangent.

Substitute the gradient of the tangent and the coordinates of the given point into an appropriate form of the straight line equation.

Make the subject of the formula.

The normal to a curve is the line perpendicular to the tangent to the curve at a given point.

Worked example 13: Finding the equation of a tangent to a curve

Find the equation of the tangent to the curve at the point . Sketch the curve and the tangent.

Find the derivative

Use the rules of differentiation:

Calculate the gradient of the tangent

To determine the gradient of the tangent at the point , we substitute the -value into the equation for the derivative.

Determine the equation of the tangent

Substitute the gradient of the tangent and the coordinates of the given point into the gradient-point form of the straight line equation.

Sketch the curve and the tangent

Worked example 14: Finding the equation of a tangent to a curve

Given , determine the equation of the tangent to the curve at .

Determine the -coordinate of the point

Therefore the tangent to the curve passes through the point .

Expand and simplify the given function

Find the derivative

Calculate the gradient of the tangent

Substitute into the equation for :

Determine the equation of the tangent

Substitute the gradient of the tangent and the coordinates of the point into the gradient-point form of the straight line equation.

Worked example 15: Finding the equation of a normal to a curve

Determine the equation of the normal to the curve at .

Draw a rough sketch.

Find the derivative

Make the subject of the formula and differentiate with respect to :

Calculate the gradient of the normal at

First determine the gradient of the tangent at the given point:

Use the gradient of the tangent to calculate the gradient of the normal:

Find the equation of the normal

Substitute the gradient of the normal and the coordinates of the given point into the gradient-point form of the straight line equation.

Draw a rough sketch

Exercise 6.5

Determine the equation of the tangent to the curve defined by at .

where is the -intercept.

Tangent meets at

Determine the point where the gradient of the tangent to the curve:

is equal to .

is equal to .

Determine the point(s) on the curve where the tangent is:

parallel to the line .

Therefore, the tangent is parallel to the given line at the point .

perpendicular to the line .

Therefore, the tangent is perpendicular to the given line at the point .

Given the function : .

Draw a graph of , indicating all intercepts and turning points.

Complete the square:

Find the equations of the tangents to at:

the -intercept of .

the turning point of .

the point where .

Draw the three tangents above on your graph of .

Write down all observations about the three tangents to .

Tangent at (blue line): gradient is positive, the function is increasing at this point.

Tangent at turning point (green line): gradient is zero, tangent is a horizontal line, parallel to -axis.

Tangent at (purple line): gradient is negative, the function is decreasing at this point.

6.5 Second derivative

The second derivative of a function is the derivative of the first derivative and it indicates the change in gradient of the original function. The sign of the second derivative tells us if the gradient of the original function is increasing, decreasing or remaining constant.

To determine the second derivative of the function , we differentiate using the rules for differentiation.

We also use the following notation for determining the second derivative of :

Worked example 16: Finding the second derivative

Calculate the second derivative for each of the following:

Exercise 6.6

Calculate the second derivative for each of the following:

Find the first and second derivatives of .

Find .

Given the function .

Determine and .

What type of function is:

Find the value of .

What do you observe about the degree (highest power) of each of the derived functions?

Each derivative function is one degree lower than the previous one.

6.6 Sketching graphs

Functions of the form

The effects of on a cubic function

Complete the table below and plot the graphs of and on the same system of axes.

Be careful to choose a suitable scale for the -axis.

The effects of on a cubic graph

Intercepts

Number of intercepts

Complete the table:

Degree of function |  |  |
Type of function |  |  |
Factorised form |  |  |
No. of -intercepts |  |  |
No. of -intercepts |  |  |

Worked example 17: Determining the intercepts

Given , find the - and -intercepts.

Determine the -intercept

The -intercept is obtained by letting :

This gives the point .

Use the factor theorem to factorise the expression

We use the factor theorem to find a factor of by trial and error:

Factorise further by inspection:

The -intercepts are obtained by letting :

This gives the points , and .

Exercise 6.7

Given the function .

Determine the - and -intercepts of .

For the -intercept, let :

This gives the point .

We use the factor theorem to find a factor of by trial and error:

Factorise further by inspection:

For the -intercept, let :

This gives the points , and .

Draw a rough sketch of the graph.

Is the function increasing or decreasing at ?

Increasing

Determine the - and -intercepts for each of the following:

For the -intercept, let :

This gives the point .

We use the factor theorem to find a factor by trial and error:

Factorise further by inspection:

For the -intercept, let :

This gives the points , and .

For the -intercept, let :

This gives the point .

We use the factor theorem to find a factor by trial and error:

Factorise further by inspection:

For the -intercept, let :

This gives the points , , and .

For the -intercept, let :

This gives the point .

We use the factor theorem to find a factor by trial and error:

Factorise further by inspection:

For the -intercept, let :

This gives the points , and .

For the -intercept, let :

This gives the point .

We take out a common factor of and then factorise the difference of two squares:

For the -intercept, let :

This gives the points , and .

For the -intercept, let :

This gives the point .

We use the factor theorem to find a factor by trial and error:

Factorise further by inspection:

For the -intercept, let :

This gives the points , and .

Determine all intercepts for and draw a rough sketch of the graph.

For the -intercept, let :

This gives the point .

For the -intercept, let :

This gives the points , and .

Stationary points

Complete the table below for the quadratic function :

-value |  |  |  |  |
Gradient of |  |  |  |  |
Sign of gradient |  |  |  |  |

Use the table to draw a rough sketch of the graph of .

Solve for if .

Indicate solutions to on the graph.

Complete the table below for the cubic function :

-value |  |  |  |  |
Gradient of |  |  |  |  |
Sign of gradient |  |  |  |  |

Use the table to draw a rough sketch of the graph of .

Solve for if .

Indicate solutions to on the graph.

Complete the following sentence:

The derivative describes the of a tangent to a curve at a given point and we have seen that the of a curve at its stationary point(s) is equal to . Therefore, we can use as a tool for finding the stationary points of the graphs of quadratic and cubic functions.

To determine the coordinates of the stationary point(s) of :

Determine the derivative .

Let and solve for the -coordinate(s) of the stationary point(s).

Substitute value(s) of into to calculate the -coordinate(s) of the stationary point(s).

Worked example 18: Finding stationary points

Calculate the stationary points of the graph of .

Determine the derivative of

Using the rules of differentiation we get:

Let and solve for

Therefore, the -coordinates of the turning points are and .

Substitute the -values into

We use the -coordinates to calculate the corresponding -coordinates of the stationary points.

Write final answer

The turning points of the graph of are and .

Local maximum and local minimum

We have seen that the graph of a quadratic function can have either a minimum turning point (“smile”) or a maximum turning point (“frown”).

For cubic functions, we refer to the turning (or stationary) points of the graph as local minimum or local maximum turning points. The diagram below shows local minimum turning point and local maximum turning point . These points are described as a local (or relative) minimum and a local maximum because there are other points on the graph with lower and higher function values.

Exercise 6.8

Use differentiation to determine the stationary point(s) for .

Determine the -values of the stationary points for .

Find the coordinates of the stationary points of the following functions using the rules of differentiation:

This is a straight line with a constant gradient, therefore there is no stationary point.

Sketching cubic graphs

General method for sketching cubic graphs:

Consider the sign of and determine the general shape of the graph.

Determine the -intercept by letting .

Determine the -intercepts by factorising and solving for .

Find the -coordinates of the turning points of the function by letting and solving for .

Determine the -coordinates of the turning points by substituting the -values into .

Plot the points and draw a smooth curve.

Worked example 19: Sketching cubic graphs

Sketch the graph of .

Determine the shape of the graph

The coefficient of the term is greater than zero, therefore the graph will have the following shape:

Determine the intercepts

The -intercept is obtained by letting :

This gives the point .

The -intercept is obtained by letting and solving for :

This gives the points , and .

Calculate the stationary points

Find the -coordinates of the stationary points by setting :

Substitute these -coordinates into to determine the corresponding -coordinates:

Therefore, the stationary points are and .

Draw a neat sketch

Concavity

Concavity indicates whether the gradient of a curve is increasing, decreasing or stationary.

Concave up: the gradient of the curve increases as increases.

Concave down: the gradient of the curve decreases as increases.

Zero concavity: the gradient of the curve is constant.

The diagram below shows the graph of the cubic function . The first derivative of is a quadratic function, and the second derivative is a linear function, .

Notice the following:

, the graph is concave up.

, the graph is concave down.

, change in concavity (point of inflection).

Points of inflection

This is the point where the concavity of a curve changes, as shown in the diagram below. If , then the concavity changes from concave up (purple) to concave down (grey) and if , concavity changes from concave down (blue) to concave up (green). Unlike a turning point, the gradient of the curve on the left-hand side of an inflection point ( and ) has the same sign as the gradient of the curve on the right-hand side.

A graph has a horizontal point of inflection where the derivative is zero but the sign of the gradient of the curve does not change. That means the graph (shown below) will continue to increase or decrease after the stationary point.

In the example above, the equation indicates that the gradient of this curve will always be positive (except where ). Therefore, the stationary point is a point of inflection.

: cubic function | : quadratic function | : linear function
(blue graph) | (green graph) | (red graph)
turning points | -intercepts |
point of inflection | turning point | -intercept

Exercise 6.9

Complete the following for each function:

Determine and discuss the change in gradient of the function.

Determine the concavity of the graph.

Find the inflection point.

Draw a sketch of the graph.

Gradient:

for : function decreasing

for : function stationary

for : function increasing

Concavity:

for : concave up

for : point of inflection

for : concave down

Point of inflection:

Gradient:

Gradient always (since ) for all except where the gradient will be .

Concavity:

for : concave down

for : point of inflection

for : concave up

Point of inflection:

Intercepts:

Gradient:

Gradient always for all except where the gradient will be .

Stationary point:

Concavity:

for : concave down

for : point of inflection

for : concave up

Point of inflection:

Intercepts:

Interpreting graphs

Worked example 20: Interpreting graphs

Consider the graph of the derivative of .

For which values of is decreasing?

Determine the -coordinate(s) of the turning point(s) of .

Given that , calculate , and .

Examine the parabolic graph and interpret the given information

We know that describes the gradient of . To determine where the cubic function is decreasing, we must find the values of for which :

Determine the -coordinate(s) of the turning point(s)

To determine the turning points of a cubic function, we let and solve for the -values. These -values are the -intercepts of the parabola and are indicated on the given graph:

Determine the equation of

From the graph, we see that the -intercept of is .

Video: 28VW

Exercise 6.10

Given .

Show that is a factor of and hence factorise .

First substitute in to check if is a factor:

Therefore is a factor of .

Determine the coordinates of the intercepts and the turning points.

For the -intercept, let : , which gives the point .

For the -intercepts, let :

Therefore, the -intercepts are: and .

Find the turning points:

To find the -values, we substitute these two values for into the original equation. We already know that when , . Substituting in the other value gives:

Therefore the turning points are: and

Sketch the graph.

Sketch the graph of . Show all the turning points and intercepts with the axes.

We find the -intercept by letting :

The -intercept is:

We find the -intercepts by letting .

We use the factor theorem to check if is a factor.

Therefore, is not a factor.

We now use the factor theorem to check if is a factor.

Therefore, is not a factor.

We now try :

Therefore, is a factor.

The -intercepts are: .

For the turning points, let .

The -coordinates of the turning points are: and .

The -coordinates of the turning points are calculated as:

and

Therefore, the turning points are: and .

Given , sketch the graph of without any further calculations. Describe the method for drawing the graph.

It is the mirror image of in the -axis. In other words, we reflect the graph of about the -axis.

The sketch shows the graph of a cubic function, , with a turning point at , going through and .

Find the equation of .

-intercepts are , and .

From the graph, is .

Find the coordinates of turning point .

The turning points are where :

Substitute into the equation of to calculate corresponding -value:

Find the intercepts and stationary point(s) of and draw a sketch of the graph.

This gives the point .

This gives the point .

There is only one stationary point at . From , we know that the gradient of the function is always negative, therefore this is a point of inflection.

For which values of will:

Motivate each answer.

For , the function values are negative and this is true for .

For , the gradient of is negative, so where , and this is true for , .

where is concave down and this is true where .

Use the information below to sketch a graph of each cubic function (do not find the equations of the functions).

The sketch below shows the curve of with turning points at and . is parallel to the -axis.

Determine the following:

length

length

length

length

Find by writing in expanded form:

coordinates of and

and are the turning points.

To find the -coordinates of and , find :

Substitute to find -coordinates of the turning points:

length

has same -coordinate as , . Therefore at .

Solve for to find the -coordinate of : .

We know that is a solution of this equation, therefore is a factor.

average gradient between and

the equation of the tangent to the graph at

Therefore the equation of the tangent is .

Given the graph of a cubic function with the stationary point , sketch the graph of the derivative function if it is also given that the gradient of the graph is at .

Derivative will be of the second degree: parabola. Gradient of function is negative throughout, except at , where gradient . has a maximum value of where . Therefore .

The sketch below shows the graph of with -intercepts at and .

Draw a sketch graph of if and .

is a cubic function because is a parabola. has two turning points because has two -intercepts.

The -values of the turning points of are the -intercepts of , where .

So will have turning points at : minimum turning point (where gradient changes from negative to positive) and at : maximum turning point (where gradient changes from positive to negative).

6.7 Applications of differential calculus

Optimisation problems

We have seen that differential calculus can be used to determine the stationary points of functions, in order to sketch their graphs. Calculating stationary points also lends itself to the solving of problems that require some variable to be maximised or minimised. These are referred to as optimisation problems.

The fuel used by a car is defined by , where is the travelling speed in .

What is the most economical speed of the car? In other words, determine the speed of the car which uses the least amount of fuel.

If we draw the graph of this function we find that the graph has a minimum. The speed at the minimum would then give the most economical speed.

We have seen that the coordinates of the turning point can be calculated by differentiating the function and finding the -coordinate (speed in the case of the example) for which the derivative is .

If we set we can calculate the speed that corresponds to the turning point:

This means that the most economical speed is .

Finding the optimum point:

Let and solve for to find the optimum point.

To check whether the optimum point at is a local minimum or a local maximum, we find :

If , then the point is a local maximum.

If , then the point is a local minimum.

Worked example 21: Optimisation problems

The sum of two positive numbers is . One of the numbers is multiplied by the square of the other. If each number is greater than , find the numbers that make this product a maximum.

Draw a graph to illustrate the answer.

Analyse the problem and formulate the equations that are required

Let the two numbers be and and the product be .

Make the subject of equation () and substitute into equation ():

Differentiate with respect to

Determine the stationary points by letting

We find the value of which makes a maximum:

Substitute into the equation () to solve for :

Determine the second derivative

We check that the point is a local maximum by showing that :

Write the final answer

The product is maximised when the two numbers are and .

Draw a graph

To draw a rough sketch of the graph we need to calculate where the graph intersects with the axes and the maximum and minimum function values of the turning points:

Intercepts:

Turning points:

Maximum and minimum function values:

Note: the above diagram is not drawn to scale.

Worked example 22: Optimisation problems

Michael wants to start a vegetable garden, which he decides to fence off in the shape of a rectangle from the rest of the garden. Michael has only of fencing, so he decides to use a wall as one border of the vegetable garden. Calculate the width and length of the garden that corresponds to the largest possible area that Michael can fence off.

Examine the problem and formulate the equations that are required

The important pieces of information given are related to the area and modified perimeter of the garden. We know that the area of the garden is given by the formula:

The fencing is only required for sides and the three sides must add up to .

Rearrange the formula to make the subject of the formula:

Substitute the expression for into the formula for the area of the garden. Notice that this formula now contains only one unknown variable.

Differentiate with respect to

We are interested in maximising the area of the garden, so we differentiate to get the following:

Calculate the stationary point

To find the stationary point, we set and solve for the value(s) of that maximises the area:

Therefore, the length of the garden is .

Substitute to solve for the width:

Therefore, the width of the garden is .

Determine the second derivative

We can check that this gives a maximum area by showing that :

Write the final answer

A width of and a length of will give the maximum area for the garden.

Important note:

The quantity that is to be minimised or maximised must be expressed in terms of only one variable. To find the optimised solution we need to determine the derivative and we only know how to differentiate with respect to one variable (more complex rules for differentiation are studied at university level).

Exercise 6.11

The sum of two positive numbers is . One of the numbers is multiplied by the square of the other. Find the numbers that make this product a maximum.

Let the first number be and the second number be and let the product be . We get the following two equations:

Rearranging the first equation and substituting into the second gives:

Differentiating and setting to gives:

Therefore, or .

If then and the product is a minimum, not a maximum.

Therefore, and .

Therefore the two numbers are and (approximating to the nearest integer gives and ).

A wooden block is made as shown in the diagram. The ends are right-angled triangles having sides , and . The length of the block is . The total surface area of the block is .

Show that .

We start by finding the surface area of the prism:

Solving for gives:

Find the value of for which the block will have a maximum volume.

(Volume = area of base height)

Start by finding an expression for volume in terms of :

Now take the derivative and set it equal to :

Since the length can only be positive,

Determine the shortest vertical distance between the curves of and if it is given that:

To minimise the distance between the curves, let

Therefore, the shortest distance:

The diagram shows the plan for a verandah which is to be built on the corner of a cottage. A railing is to be constructed around the four edges of the verandah.

If and , and the length of the railing must be , find the values of and for which the verandah will have a maximum area.

We need to determine an expression for the area in terms of only one variable.

The perimeter is:

The area is:

We use the expression for perimeter to eliminate the variable so that we have an expression for area in terms of only:

To find the maximum, we need to take the derivative and set it equal to :

Therefore, and substituting this value back into the formula for perimeter gives .

A rectangular juice container, made from cardboard, has a square base and holds of juice. The container has a specially designed top that folds to close the container. The cardboard needed to fold the top of the container is twice the cardboard needed for the base, which only needs a single layer of cardboard.

If the length of the sides of the base is cm, show that the total area of the cardboard needed for one container is given by:

Determine the dimensions of the container so that the area of the cardboard used is minimised.

Rates of change

It is very useful to determine how fast (the rate at which) things are changing. Mathematically we can represent change in different ways. For example we can use algebraic formulae or graphs.

Graphs give a visual representation of the rate at which the function values change as the independent (input) variable changes. This rate of change is described by the gradient of the graph and can therefore be determined by calculating the derivative.

We have learnt how to determine the average gradient of a curve and how to determine the gradient of a curve at a given point. These concepts are also referred to as the average rate of change and the instantaneous rate of change.

When we mention rate of change, the instantaneous rate of change (the derivative) is implied. When average rate of change is required, it will be specifically referred to as average rate of change.

Velocity is one of the most common forms of rate of change:

Velocity refers to the change in distance () for a corresponding change in time ().

Acceleration is the change in velocity for a corresponding change in time. Therefore, acceleration is the derivative of velocity

This implies that acceleration is the second derivative of the distance.

Worked example 23: Rate of change

The height (in metres) of a golf ball seconds after it has been hit into the air, is given by . Determine the following:

The average vertical velocity of the ball during the first two seconds.

The vertical velocity of the ball after .

The time at which the vertical velocity is zero.

The vertical velocity with which the ball hits the ground.

The acceleration of the ball.

Determine the average vertical velocity during the first two seconds

Calculate the instantaneous vertical velocity

Velocity after :

Determine the time at which the vertical velocity is zero

Therefore, the velocity is zero after

Find the vertical velocity with which the ball hits the ground

The ball hits the ground when

The ball hits the ground after . The velocity after will be:

The ball hits the ground at a speed of . Notice that the sign of the velocity is negative which means that the ball is moving downward (a positive velocity is used for upwards motion).

Acceleration

Just because gravity is constant does not mean we should necessarily think of acceleration as a constant. We should still consider it a function.

Exercise 6.12

A pump is connected to a water reservoir. The volume of the water is controlled by the pump and is given by the formula:

Determine the rate of change of the volume of the reservoir with respect to time after days.

Is the volume of the water increasing or decreasing at the end of days. Explain your answer.

The rate of change is negative, so the function is decreasing.

After how many days will the reservoir be empty?

When will the amount of water be at a maximum?

Maximum at turning point.

Calculate the maximum volume.

Maximum at turning point.

Draw a graph of .

A soccer ball is kicked vertically into the air and its motion is represented by the equation:

Determine the initial height of the ball at the moment it is being kicked.

Find the initial velocity of the ball.

Determine the velocity of the ball after .

Calculate the maximum height of the ball.

Maximum height is at the turning point.

Determine the acceleration of the ball after second and explain the meaning of the answer.

Interpretation: the velocity is decreasing by metres per second per second.

Calculate the average velocity of the ball during the third second.

Average velocity during third second:

Determine the velocity of the ball after seconds and interpret the answer.

Interpretation: this is the stationary point, where the derivative is zero. The ball has stopped going up and is about to begin its descent.

How long will it take for the ball to hit the ground?

The ball hits the ground at (time cannot be negative).

Determine the velocity of the ball when it hits the ground.

If the displacement (in metres) of a particle at time (in seconds) is governed by the equation , find its acceleration after seconds.

We know that velocity is the rate of change of displacement. This means that :

We also know that acceleration is the rate of change of velocity. This means that :

Substituting gives .

During an experiment the temperature (in degrees Celsius) varies with time (in hours) according to the formula: .

Determine an expression for the rate of change of temperature with time.

We find the rate of change of temperature with time by differentiating:

During which time interval was the temperature dropping?

We set the derivative equal to : We look at the coefficient of the term to decide whether this is a minimum or maximum point. The coefficient is negative and therefore the function must have a maximum value. The interval in which the temperature is increasing is . The interval in which the temperature is dropping is . We can check this by drawing the graph or by substituting in the values for into the original equation.

6.8 Summary

The limit of a function exists and is equal to if the values of get closer to from both sides as gets closer to .

Average gradient or average rate of change:

Gradient at a point or instantaneous rate of change:

Notation

Differentiating from first principles:

Rules for differentiation:

General rule for differentiation:

The derivative of a constant is equal to zero.

The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

The derivative of a sum is equal to the sum of the derivatives.

The derivative of a difference is equal to the difference of the derivatives.

Second derivative:

Sketching graphs:

The gradient of the curve and the tangent to the curve at stationary points is zero.

Finding the stationary points: let and solve for .

A stationary point can either be a local maximum, a local minimum or a point of inflection.

Optimisation problems:

Use the given information to formulate an expression that contains only one variable.

Differentiate the expression, let the derivative equal zero and solve the equation.

Exercise 6.13

Determine from first principles if .

Given , find using the definition of the derivative.

Calculate:

Determine if:

Given:

Use the definition of the derivative to calculate .

Hence, calculate the coordinates of the point at which the gradient of the tangent to the graph of is .

The value is: The coordinates of the point are

If , calculate .

Given:

Find .

Solve .

Find the derivative for each of the following:

If , determine .

Given:

Determine .

Differentiate with respect to .

Find .

To differentiate with respect to , express in terms of :

Show that .

Given:

Calculate .

Hence, solve .

We know that is a factor of because . We factorise further using inspection:

Determine .

Sketch the graph of , showing the coordinates of the turning points and the intercepts on both axes.

The -intercept is .

The -intercepts are .

To find the turning points we let the derivative equal . . The -values of the turning points are: and .

Therefore, the turning points are and .

Determine the coordinates of the points on the graph of where the gradient is .

Therefore, at and .

Draw the graph of on the same system of axes.

Determine and use this to make conclusions about the concavity of .

Given .

If , determine the -intercepts of .

The -intercepts of are , and .

Determine the coordinates of the turning points of .

Draw a sketch graph of . Clearly indicate the coordinates of the turning points and the intercepts with the axes.

For which value(s) of will the equation have three real roots of which two are equal?

To find the point when the cubic function has two real roots, we need to find the points where one of the turning points lies on the -axis. We look at the -values of the turning points to determine this. For the maximal turning point we lower the graph and for the minimal one we raise the graph. This gives: or

Determine the equation of the tangent to the graph of at the point where .

Find the -value at : Determine the value of substituting into the derivative:

The equation of the tangent is:

Given the function with -intercept , -intercept and a point of inflection at .

Show by calculation that , and .

Find the -coordinate of the point of inflection.

Draw the graph of .

Discuss the gradient of .

Gradient always positive except at where the gradient is at its minimum value, zero.

Discuss the concavity of .

Concavity changes at inflection point :

The sketch shows the graph of .

Identify the stationary points of the cubic function, .

Turning points are at and :

For : this is a local maximum turning point (the gradient of changes from positive to negative).

For : this is a local minimum turning point (the gradient of changes from negative to positive).

What is the gradient of function

where .

At , the gradient of the function is equal to . We can write this as .

If it is further given that has only two real roots, draw a rough sketch of

. Intercept values do not need to be shown.

Given that is a linear function with and , find the equation of .

is a linear function and is therefore of the form . The derivative of a linear function is , therefore from , we know that .

Therefore, .

The graphs of and and the following points are given below:

Use the graphs and determine the values of for which:

is a decreasing function.

.

and are both negative.

and

or (both and are negative)

where the gradient of is negative. Therefore, or .

And for or .

Therefore, and for or .

Given , determine the equation of the tangent to at the point .

Equation of tangent to at :

To calculate the value of at , substitute into :

Substitute into to determine :

Find the coordinates of the point(s) where the tangent in the question above meets the graph of again.

Equate the equation of the tangent and :

Therefore, tangent meets graph of at the turning point .

Without any calculations, give the -intercepts of the graph of . Explain reasoning.

-intercepts of are at and , the turning points of (points and ). At these two points the gradient of the graph is equal to zero, where cuts the -axis.

Sketch the graph of , show all intercepts with the axes and turning points.

We find the -intercept by finding the value for .

The -intercept is:

We find the -intercepts by finding the values for which the function .

We use the factor theorem to check if is a factor.

Therefore, is not a factor.

We now use the factor theorem to check if is a factor.

Therefore, is not a factor.

We now try :

Therefore, is a factor.

If we divide by we are left with:

This has factors:

The -intercepts are: .

Find the turning points by setting .

If we use the rules of differentiation we get:

The -coordinates of the turning points are: and .

The -coordinates of the turning points are calculated as: and

Therefore the turning points are: and .

Find the equation of the tangent to at .

We know that at , (this is a turning point of the graph).

We substitute into the derivative of the function to get :

Substituting this and the coordinates of the point into: gives:

The equation of the tangent to the graph at is .

Determine the point of inflection and discuss the concavity of .

Point of inflection:

This gives the point .

Concavity:

for : concave down

for : point of inflection

for : concave up

Determine the minimum value of the sum of a positive number and its reciprocal.

Let the number be and the reciprocal be . The sum of these two numbers is . To find the minimum value we must differentiate the expression for the sum and set it equal to zero: Therefore, the minimum value for the sum of a positive number and it's reciprocal is .

minutes after a kettle starts to boil, the height of the water in the kettle is given by , where is measured in millimetres.

Calculate the height of the water level in the kettle just before it starts to boil.

The kettle starts to boil at :

As the water boils, the water level in the kettle decreases. Determine the rate at which the water level is decreasing when .

To find the rate at which the water level is decreasing we take the derivative of the depth: This rate is negative since the water level is decreasing.

How many minutes after the kettle starts to boil will the water level be decreasing at a rate of per minute?

To find the number of minutes when the rate will be (water level is decreasing) we set the derivative equal to this and solve for : After minutes the water level will be decreasing at .

The displacement of a moving object is represented by the equation:

Calculate the acceleration of the object after seconds.

After seconds, the acceleration is .

In the figure is the diameter of the semi-circle . The sum of the lengths of and is . Calculate the perimeter of when covers the maximum area in the semi-circle. Leave the answer in simplified surd form.

Let be units and be units.

To find maximum area, let :

Therefore, units and units.

Perimeter of the triangle is units.

The capacity of a cylindrical water tank is . Let the height be and the radius be . The material used for the bottom of the tank is twice as thick and also twice as expensive as the material used for the curved part of the tank and the top of the tank.

Remember:

Express in terms of .

Show that the cost of the material for the tank can be expressed as:

Determine the diameter of the tank that gives the minimum cost of the materials.

[IEB, 2006]

Let the cost of the material be .

The diameter of an icecream cone is and the vertical height is . The sum of the diameter and the height of the cone is .

Determine the volume of the cone in terms of and .

(Volume of a cone: )

Determine the radius and height of the cone for the volume to be a maximum.

Let the volume of the cone be .

Calculate the maximum volume of the cone.

A water reservoir has both an inlet and an outlet pipe to regulate the depth of the water in the reservoir. The depth is given by the function:

Determine the rate at which the depth of the water is changing at 10h00.

Therefore, rate of change of the depth of the water is per hour at 10h00.

Is the depth of the water increasing or decreasing?

Decreasing (rate is negative).

At what time will the inflow of water be the same as the outflow?

[IEB, 2006]

We can convert this to minutes: minutes. Therefore, at 06h49

Chapter 7: Analytical geometry

7.1 Revision

Integrate Euclidean Geometry knowledge with Analytical Geometry.

Emphasize the value and importance of making sketches.

Emphasize the importance of writing coordinates consistently for the distance formula and gradient.

Learners must revise the method of completing the square for finding the general form of the equation of a circle with centre .

Remind learners that the tangent to a circle is perpendicular to the radius (and the diameter).

Straight line equations

Theorem of Pythagoras: |
Distance formula: |
Gradient: |
Mid-point of a line segment: |
Points on a straight line: |

Horizontal lines: |  |
Vertical lines: |  |

Two-point form: |  |
Gradient-point form: |  |
Gradient-intercept form: |  |

Worked example 1: Revision

Given quadrilateral with vertices and .

Determine the equation of the lines and .

Show that .

Calculate the lengths of and .

Determine the equation of the diagonal .

What type of quadrilateral is ?

Draw a sketch

Use the given information to determine the equation of lines and

Determine the equation of the line using the two point form of the straight line equation:

Determine the equation of the line using the gradient-intercept form of the straight line equation:

There is often more than one method for determining the equation of a line. The different forms of the straight line equation are used, depending on the information provided in the problem.

Show that line and line have equal gradients

Use the distance formula to determine the lengths of and

Determine the equation of the diagonal

Determine the gradient of the line:

Use gradient and the point to determine the equation of the line :

Examine the properties of quadrilateral

We have shown that and , therefore quadrilateral is a parallelogram (one pair of opposite sides equal and parallel).

Exercise 7.1

Determine the following for the line segment between the given points:

length

mid-point

gradient

equation

and

and

and

and

The line joining and has the mid-point . Determine the values of and .

Given and length units, determine the value(s) of .

There are two possible values of such that the length units:

Determine the equation of the straight line:

passing through the point and .

passing through the points and .

passing through the point and with .

parallel to the -axis and passing through the point .

passing through the point and with .

perpendicular to the -axis and passing through the point .

with undefined gradient and passing through the point .

with and passing through the point .

which cuts the -axis at and with .

Inclination of a line

The diagram shows a straight line which forms an acute angle with the positive -axis. This is called the angle of inclination of a straight line.

The gradient of a straight line is equal to the tangent of the angle formed between the line and the positive direction of the -axis.

Lines with positive gradients

A line with a positive gradient () has an acute angle of inclination ().

For example, we can determine the angle of inclination of a line with :

Lines with negative gradients

If we are calculating the angle of inclination of a line with a negative gradient (), then we add to change the negative angle to an obtuse angle ().

For example, we can determine the angle of inclination for a line with :

Worked example 2: Inclination of a straight line

Determine the acute angle (correct to decimal place) between the line passing through the points and and the straight line .

Draw a sketch

Draw the line through points and and the line on a suitable system of axes. Label and , the angles of inclination of the two lines. Label , the acute angle between the two straight lines.

Notice that and are acute angles and is an obtuse angle.

Use the gradient to determine the angle of inclination

From the equation we see that , therefore is an obtuse angle.

Determine the gradient and angle of inclination of the line through and

Determine the gradient

Determine the angle of inclination

Write the final answer

The acute angle between the two straight lines is °.

Parallel and perpendicular lines

Parallel lines |  |  |
Perpendicular lines |  |  |

Worked example 3: Parallel lines

Line passes through the point and has an angle of inclination of °.

Determine the equation of line which passes through the point and is parallel to .

Determine the equation of line , which passes through the origin and is perpendicular to both and .

Sketch lines , and on the same system of axes.

Use two different methods to determine the angle of inclination of .

Draw a rough sketch and use the angle of inclination to determine the equation of

Since we are given ,

Substitute the given point :

Determine the equation of

is perpendicular to , therefore the product of their gradients is equal to :

We know line passes through , therefore the equation of the line is:

Determine the angle of inclination of

Let the angle of inclination of be .

Method :

Method :

Draw a sketch

Exercise 7.2

Determine the angle of inclination (correct to decimal place) for each of the following:

a line with

the line passes through the points and

Horizontal line

a line with a gradient of

Find the angle between the line and the line passing through points and V.

Let the angle of inclination for the line be and let the angle of inclination for the other line be . Let the angle between the two lines be .

Determine the equation of the straight line that passes through the point and is parallel to the line .

Determine the equation of the straight line that passes through the point and is parallel to the line with angle of inclination .

Determine the equation of the straight line that passes through the point and is perpendicular to the line .

Determine the equation of the straight line that passes through the point and is perpendicular to the line with angle of inclination .

, and are the vertices of in the Cartesian plane. intersects the -axis at . Draw a sketch and determine the following:

the equation of line

Draw a sketch:

the coordinates of point

the angle of inclination of

the gradient of line

because

the equation of the line perpendicular to and passing through the origin

the mid-point of

the equation of the line parallel to and passing through point

Points , and are given.

Plot the points on the Cartesian plane.

Determine the coordinates of if is a parallelogram.

Prove that is a rhombus.

Given points , and .

Show that the equation of the line is .

Draw a sketch:

Determine the size of .

Let the angle of inclination of the line be .

Let the angle of inclination of the line be .

(exterior of sum on opp. interior 's)

7.2 Equation of a circle

Equation of a circle with centre at the origin

Draw a system of axes with a scale of unit on the -axis and on the -axis.

Draw the lines and .

Plot the following points:

What object do the points form?

Measure the following distances and complete the table below:

Line segment
Distance

Use the distance formula to check the results of the above table:

Line segment
Distance

What do you notice about the length of each line segment?

What is the general term given to this type of line segment?

If the point lies on the circle, use the distance formula to determine an expression for the length of .

Can you deduce a general equation for a circle with centre at the origin?

A circle is the set of all points that are an equal distance (radius) from a given point (centre). In other words, every point on the circumference of a circle is equidistant from its centre.

The radius of a circle is the distance from the centre of a circle to any point on the circumference.

A diameter of a circle is any line passing through the centre of the circle which connects two points on the circle. The diameter is also the name given to the maximum distance between two points on a circle.

Consider a point on the circumference of a circle of radius with centre at .

In :

Equation of a circle with centre at the origin:

If is a point on a circle with centre and radius , then the equation of the circle is:

Circle symmetry

A circle with centre is symmetrical about the origin: for every point on the circumference of a circle, there is also the point .

A circle centred on the origin is also symmetrical about the - and -axis. Is a circle centred on the origin symmetrical about the lines and ? How many lines of symmetry does a circle have?

Worked example 4: Equation of a circle with centre at the origin

Given: circle with centre and a radius of units.

Sketch the circle on the Cartesian plane.

Determine the equation of the circle.

Show that the point lies on the circle.

Draw a sketch

Determine the equation of the circle

Write down the general form of the equation of a circle with centre :

Show that point lies on the circle

Substitute the -coordinate and the -coordinate into the left-hand side of the equation and show that it is equal to the right-hand side:

Therefore, lies on the circle .

Worked example 5: Equation of a circle with centre at the origin

A circle with centre passes through the points and .

Plot the points and draw a rough sketch of the circle.

Determine the equation of the circle.

Calculate the length of .

Explain why is a diameter of the circle.

Draw a sketch

Determine the equation of the circle

Write down the general form of the equation of a circle with centre and substitute :

Therefore, the equation of the circle passing through and is .

Calculate the length

Use the distance formula to determine the distance between the two points.

Determine if is a diameter of the circle

Since connects two points, and , on the circle and is a line of length units, is a diameter of the circle.

Alternative method : Use symmetry to show that is a diameter of the circle.

The diameter is the name given to the maximum distance between two points on a circle; this means that the two points must lie opposite each other with respect to the centre.

Using symmetry about the origin, we know that lies opposite on the circle and vice versa:

lies opposite , which are the coordinates of

lies opposite , which are the coordinates of

Therefore, is a diameter of the circle.

Alternative method : Show that passes through the centre.

Determine the equation of and show that it passes through the origin.

lines on the line .

Therefore, passes through the centre and is a diameter of the circle.

Worked example 6: Equation of a circle with centre at the origin

Given a circle with centre and a radius of units. Determine the possible coordinates of the point(s) on the circle which have an -value that is twice the -value.

Determine the equation of the circle

Determine the coordinates of the points on the circle

To calculate the possible coordinates of the point(s) on the circle which have an -value that is twice the -value, we substitute into the equation of the circle:

This gives the points and .

Note: we can check that both points lie on the circle by substituting the coordinates into the equation of the circle:

Exercise 7.3

Complete the following for each circle given below:

Determine the radius.

Draw a sketch.

Calculate the coordinates of two points on the circle.

This gives the points and .

This gives the points and .

This gives the points and .

This gives the points and .

This gives the points and .

This gives the points and .

Determine the equation of the circle:

with centre at the origin and a radius of units.

with centre at and units.

passing through the point and with centre .

centred at the origin and units.

with centre at the origin and a diameter of units.

passing through the point and with centre at the origin.

Determine whether or not the following equations represent a circle:

Yes:

No, cannot be written in the form .

No, cannot be written in the form .

Yes:

Yes:

No, cannot be written in the form .

Yes:

Determine the value(s) of if is a point on the circle .

This gives the points and .

is a point on the circle with centre at the origin and a diameter of .

Determine the possible coordinates of if the value of is triple the value of .

Therefore, or

Determine the possible coordinates of if the value of is half the value of .

Therefore, or

lies on a circle with centre at .

Determine the equation of the circle.

Sketch the circle and label point .

If is a diameter of the circle, determine the coordinates of .

If is a diameter of the circle, then point must lie opposite point on the circumference of the circle. Using symmetry about the origin, the coordinates of point are .

Calculate the length of .

Alternative: Use the distance formula

Determine the equation of the line .

passes through the origin, therefore .

Determine the equation of the line perpendicular to and passing through the point .

For perpendicular line:

Equation of a circle with centre at

Shifting the centre of a circle

Complete the following for each equation in the table below:

Write down the shifted equation (do not simplify).

Draw a rough sketch to illustrate the shift(s).

Vertical shift: the graph is shifted unit up.

Horizontal shift: the graph is shifted units to the right.

Combined shifts: the graph is shifted unit up and units to the right.

The first example has been completed.

Equation
Vertical shift
Horizontal shift
Combined shift

Use the table to answer the following questions:

Write down the general equation of a circle with centre .

Write down the general equation of a circle with centre .

Write down the general equation of a circle with centre .

Write down the general equation of a circle with centre .

Consider a circle in the Cartesian plane with centre at and with a radius of units. If is any point on the circumference of the circle, we can use the distance formula to calculate the distance between the two points:

The distance is equal to the radius () of the circle:

If the coordinates of the centre of the circle are , then the equation of a circle not centred on the origin is:

Equation of a circle with centre at :

If is a point on a circle with centre and radius , then the equation of the circle is:

A circle with centre is a special case of the general equation:

Worked example 7: Equation of a circle with centre at

is a point on the circle with centre .

Draw a rough sketch of the circle and label .

Determine the equation of the circle.

Does the point lie on the circle?

Does the circle cut the -axis? Motivate your answer.

Draw a sketch

Determine the equation of the circle

Write down the general equation of a circle with centre and substitute the coordinates :

Substitute the coordinates of to determine the value of :

Determine whether or not lies on the circle

If lies on the circle, then it will satisfy the equation of the circle:

Therefore does not lie on the circle.

Determine the -intercept(s)

To determine the -intercept(s), we let :

The circle cuts the -axis at .

Worked example 8: Equation of a circle with centre at

Determine the coordinates of the centre of the circle and the length of the radius for .

Make the coefficient of the term and the term equal to

The coefficient of the and term must be , so we take out as a common factor:

Complete the square

Take half the coefficient of the term, square it; then add and subtract it from the equation.

The coefficient of the term is , so then .

Take half the coefficient of the term, square it; then add and subtract it from the equation.

The coefficient of the term is , so then .

The centre of the circle is and the radius is units.

Video: 293G

Worked example 9: Equation of a circle with centre at

Given and on the Cartesian plane.

The points and are symmetrical about the origin to and respectively. Determine the coordinates of and .

Determine the mid-point of .

Write down the equation of the circle .

Is ? Give reasons.

Determine the equation of the line perpendicular to and passing through point .

Determine the coordinates of and

For symmetry about the origin, every point is symmetrical to . So is symmetrical to and is symmetrical to .

Determine the mid-point of

The mid-point of the line is the origin.

Determine the equation of the circle

passes through the centre of the circle and is therefore a diameter. From Euclidean geometry, we know that the diameter of the circle subtends a right-angle at the circumference, therefore (angle in semi-circle).

Determine the equation of the line perpendicular to at point

Determine the gradient of :

Let the gradient of the line perpendicular to be :

Worked example 10: Equation of a circle with centre at

Given: a circle with centre and a radius of units.

If the circle is shifted units down and unit to the right, write down the equation of the shifted circle.

Sketch the original circle and the shifted circle on the same system of axes.

The shifted circle is reflected about the line . Sketch the reflected circle on the same system of axes as the question above.

Write down the equation of the reflected circle.

Write down the equation of the circle

Determine the equation of the shifted circle

Vertical shift: units down: is replaced with

Horizontal shift: unit to the right: is replaced with

Therefore the equation of the shifted circle is with centre at and radius of units.

Draw a sketch of the circles

The shifted circle is reflected about the line . The and variables are interchanged to give the circle with equation and centre at .

Worked example 11: Equation of a circle with centre at

A circle with centre on the line passes through the points and . Determine the equation of the circle.

Draw a rough sketch

Write down the general equation of a circle

Consider the line . Any point on this line will have the coordinates . Since the centre of the circle lies on the line , we can write the equation of the circle as

Solve for the unknown variables and

We need two equations to solve for the two unknown variables. We substitute the two given points, and and solve for and simultaneously:

Write the final answer

The equation of the circle is .

Circles with the same centre.

Circles touch internally at one point: the distance between the centres is equal to the difference between the radii.

Circles intersect at two points: the distance between the centres is less than the sum of the radii.

Circles touch externally at one point: the distance between the centres is equal to the sum of the radii.

Circles do not touch or intersect: the distance between the centres is greater than the sum of the radii.

Exercise 7.4

Determine whether or not each of the following equations represents a circle. If not, give a reason.

Yes

Yes

No, coefficients of term and term are different.

No, cannot be written in general form

Yes

No, must be greater than zero.

Write down the equation of the circle:

with centre and a radius of units.

such that and the centre is the origin.

with centre and passing through the point .

with centre and .

with and centre and

with centre and passing through the origin.

Determine the centre and the length of the radius for the following circles:

Centre: , units

Centre: , units

Centre: , units

Centre: , units

Centre: , units

Centre: , units

Centre: , units

Centre: , units

A circle cuts the -axis at and . If units, determine the possible equation(s) of the circle. Draw a sketch.

The equation of the circle passing through points and is or .

and are points on a circle such that is a diameter. Determine the equation of the circle.

Use the distance formula to determine the length of the diameter.

Given is a diameter of the circle, then the centre is the mid-point of PG:

Therefore the centre is and the equation of the circle is .

A circle with centre passes through the points and .

Determine the equation of the circle.

The equation of the circle is .

Determine the coordinates of , the mid-point of .

Show that .

If is a point on the circle such that is a diameter, determine the coordinates of .

Use the mid-point formula to calculate the coordinates of :

Determine the equation of the line .

A circle passes through the point and . If its centre lies on the line , determine the equation of the circle.

Given that the centre of the circle lies on the line , we can write the coordinates of the circle as and the equation of the circle becomes:

The equation of the circle is .

A circle with centre passes through the point .

Determine the equation of the circle.

If the circle is shifted units to the right and units down, determine the new equation of the circle.

Draw a sketch of the original circle and the shifted circle on the same system of axes.

On the same system of axes as the previous question, draw a sketch of the shifted circle reflected about the -axis. State the coordinates of the centre of this circle.

Centre of the shifted circle:

Determine whether the circle cuts, touches or does not intersect the -axis and the -axis.

The radius of the circle is units. The distance from the centre to the -axis is units, therefore the circle will touch the -axis. The distance from the centre to the -axis is units, therefore the circle will not intersect with the -axis.

7.3 Equation of a tangent to a circle

On a suitable system of axes, draw the circle with centre at .

Plot the point .

Plot the point . Draw and extend the line so that is cuts the positive -axis.

Measure .

Determine the gradient of the radius .

Determine the gradient of .

Prove that .

Plot the point and join .

Draw a tangent to the circle at .

Measure the angle between and the tangent line at .

Make a conjecture about the angle between the radius and the tangent to a circle at a point on the circle.

Complete the sentence: the product of the of the radius and the gradient of the is equal to .

A circle with centre and a radius of units is shown in the diagram above. is a point on the circumference and the equation of the circle is:

A tangent is a straight line that touches the circumference of a circle at only one place.

The tangent line touches the circle at .

The radius of the circle is perpendicular to the tangent at the point of contact .

The product of the gradient of the radius and the gradient of the tangent line is equal to .

How to determine the equation of a tangent:

Determine the equation of the circle and write it in the form

From the equation, determine the coordinates of the centre of the circle .

Determine the gradient of the radius:

The radius is perpendicular to the tangent of the circle at a point so:

Write down the gradient-point form of a straight line equation and substitute and the coordinates of . Make the subject of the equation.

Worked example 12: Equation of a tangent to a circle

Determine the equation of the tangent to the circle at the point .

Write the equation of the circle in the form

Use the method of completing the square:

Draw a sketch

The centre of the circle is and the radius is units.

Determine the gradient of the radius

Determine the gradient of the tangent

Let the gradient of the tangent line be .

Determine the equation of the tangent to the circle

Write down the gradient-point form of a straight line equation and substitute and .

Write the final answer

The equation of the tangent to the circle at is .

Worked example 13: Equation of a tangent to a circle

The straight line cuts the circle at and .

Calculate the coordinates of and .

Sketch the circle and the straight line on the same system of axes. Label points and .

Determine the coordinates of , the mid-point of chord .

If is the centre of the circle, show that .

Determine the equations of the tangents to the circle at and .

Determine the coordinates of , the point where the two tangents intersect.

Show that , and are on a straight line.

Determine the coordinates of and

Substitute the straight line into the equation of the circle and solve for :

This gives the points and .

Draw a sketch

Determine the coordinates of the mid-point

Show that is perpendicular to

We need to show that the product of the two gradients is equal to . From the given equation of , we know that .

Determine the equations of the tangents at and

Tangent at :

Determine the gradient of the radius :

The tangent of a circle is perpendicular to the radius, therefore we can write:

Substitute and into the equation of a straight line.

Tangent at :

Determine the gradient of the radius :

The tangent of a circle is perpendicular to the radius, therefore we can write:

Substitute and into the equation of a straight line.

The equations of the tangents are and .

Determine the coordinates of

Equate the two linear equations and solve for :

This gives the point .

Show that , and are on a straight line

We need to show that there is a constant gradient between any two of the three points. We have already shown that is perpendicular to , so we expect the gradient of the line through , and to be .

Therefore , and all lie on the line .

Worked example 14: Equation of a tangent to a circle

Determine the equations of the tangents to the circle , given that both are parallel to the line .

Draw a sketch

The tangents to the circle, parallel to the line , must have a gradient of . From the sketch we see that there are two possible tangents.

Determine the coordinates of and

To determine the coordinates of and , we must find the equation of the line perpendicular to and passing through the centre of the circle. This perpendicular line will cut the circle at and .

Notice that the line passes through the centre of the circle.

To determine the coordinates of and , we substitute the straight line into the equation of the circle and solve for :

This gives the points and .

Determine the equations of the tangents to the circle

Tangent at :

Tangent at :

The equation of the tangent at point is and the equation of the tangent at point is .

Worked example 15: Equation of a tangent to a circle

Determine the equations of the tangents to the circle , from the point outside the circle.

Draw a sketch

Consider where the two tangents will touch the circle

Let the two tangents from touch the circle at and .

Consider and apply the theorem of Pythagoras:

Note: from the sketch we see that must have a negative -coordinate, therefore we take the negative of the square root. Similarly, must have a positive -coordinate, therefore we take the positive of the square root.

This gives the points and .

Tangent at :

Tangent at :

Write the final answer

The equations of the tangents to the circle are and .

Video: 293H

Exercise 7.5

A circle with centre and the point on the circle are given. Determine the gradient of the radius.

Given:

the centre of the circle

a point on the circumference of the circle

Required:

the gradient of the radius,

The gradient of the radius is .

Determine the gradient of the tangent to the circle at the point .

The tangent to the circle at the point is perpendicular to the radius of the circle to that same point:

The gradient for the tangent is .

Given the equation of the circle:

Find the gradient of the radius at the point on the circle.

Given:

the equation for the circle

a point on the circumference of the circle

Required:

the gradient of the radius,

The coordinates of the centre of the circle are .

Draw a rough sketch:

The gradient for this radius is .

Determine the gradient of the tangent to the circle at the point .

Given:

The tangent to the circle at the point is perpendicular to the radius, so

The gradient for the tangent is .

Given a circle with the central coordinates . Determine the equation of the tangent to the circle at the point .

The tangent is perpendicular to the radius, therefore .

Write down the equation of a straight line and substitute and .

The equation of the tangent to the circle is .

Given the diagram below:

Determine the equation of the tangent to the circle with centre at point .

Given:

the centre of the circle

a point on the circumference of the circle

Required:

the equation for the tangent to the circle in the form

Calculate the gradient of the radius:

Equation of the tangent:

The equation for the tangent to the circle at the point is:

Given the point on the circle . Find the equation of the tangent at .

Given:

the equation for the circle

a point on the circumference of the circle

Required:

the equation of the tangent in the form

The coordinates of the centre of the circle are .

The gradient of the radius:

Equation of the tangent:

The equation of the tangent to the circle is

is the centre of the circle passing through and .

Determine the equation of the circle.

Use the distance formula to determine the length of the radius:

Write down the general equation of a circle and substitute and :

The equation of the circle is .

Determine the value of .

Substitute the and solve for the value.

The solution shows that or . From the graph we see that the -coordinate of must be positive, therefore .

Determine the equation of the tangent to the circle at point .

Calculate the gradient of the radius:

The radius is perpendicular to the tangent, so .

The equation for the tangent to the circle at the point is:

The straight line cuts the circle at and .

Calculate the coordinates of and .

Substitute the straight line into the equation of the circle and solve for :

This gives the points and .

Determine the length of .

Determine the coordinates of , the mid-point of chord .

If is the centre of the circle, show that .

Determine the equations of the tangents to the circle at and .

Tangent at :

Determine the gradient of the radius :

Let the gradient of the tangent at be . The tangent of a circle is perpendicular to the radius, therefore we can write:

Substitute and into the equation of a straight line.

Tangent at :

Determine the gradient of the radius :

Let the gradient of the tangent at be . The tangent of a circle is perpendicular to the radius, therefore we can write:

Substitute and into the equation of a straight line.

Therefore the equations of the tangents to the circle are and .

Determine the coordinates of , the point where the two tangents intersect.

Equate the two linear equations and solve for :

This gives the point .

Show that .

Determine the equations of the two tangents to the circle, both parallel to the line .

The tangent at , , is parallel to . To find the equation of the second parallel tangent:

7.4 Summary

Video: 294R

Theorem of Pythagoras: |
Distance formula: |
Gradient: |
Mid-point of a line segment: |
Points on a straight line: |

Straight line equations | Formulae
Two-point form: |
Gradient-point form: |
Gradient-intercept form: |
Horizontal lines: |
Vertical lines |

Parallel lines |  |  |
Perpendicular lines |  |  |

Inclination of a straight line: the gradient of a straight line is equal to the tangent of the angle formed between the line and the positive direction of the -axis.

Equation of a circle with centre at the origin:

If is a point on a circle with centre and radius , then the equation of the circle is:

General equation of a circle with centre at :

If is a point on a circle with centre and radius , then the equation of the circle is:

A tangent is a straight line that touches the circumference of a circle at only one point.

The radius of a circle is perpendicular to the tangent at the point of contact.

Exercise 7.6

Find the equation of the circle:

with centre and radius

with centre and radius

with centre and radius

with centre and diameter

with centre and radius

Find the equation of the circle with centre which passes through .

Where does it cut the line ?

Find the equation of the circle with centre which passes through .

Find the equation of the circle with centre which passes through .

Find the centre and radius of the following circles:

, units

, unit

, units

, units

, units

Find the and intercepts of the following graphs:

Find the centre and radius of the following circles:

The centre of the circle is and units.

The centre of the circle is and units.

The centre of the circle is and units.

The centre of the circle is and units.

The centre of the circle is and units.

The centre of the circle is and units.

Find the gradient of the radius between the point on the circle and its centre .

Given:

the centre of the circle

a point on the circumference of the circle

Required:

the gradient of the radius

The gradient for this radius is .

Find the gradient line tangent to the circle at the point .

The tangent to the circle at the point is perpendicular to the radius of the circle to that same point:

The gradient for the tangent is .

Given , determine the value(s) of if lies on the circle.

The points and lie on the circle.

Find the gradient of the tangent to the circle at the point .

The gradient of the tangent is .

Given a circle with the central coordinates . Determine the equation of the tangent line of the circle at the point .

The radius is perpendicular to the tangent, therefore :

Substitute and to determine :

The equation of the tangent to the circle at the point is .

Consider the diagram below:

Find the equation of the tangent to the circle at point .

Determine the -intercept () of the line by substituting the point .

The equation of the tangent to the circle at is

is a point on the circle . Determine the equation of the tangent at .

Complete the square:

Therefore the centre of the circle is and units.

Calculate the gradient of the radius:

Determine the -intercept of the line by substituting the point .

The equation for the tangent to the circle at the point is

is the centre of the circle passing through and .

Find the equation of the circle given.

The equation of the circle is .

Determine the value of .

From the graph we see that the correct -value is .

The coordinates for point is

Determine the equation of the tangent to the circle at point .

.

Determine the -intercept of the line by substituting the point .

The equation of the tangent to the circle at is .

Find the equation of the tangent to each circle:

at the point

The centre of the circle is and units.

The equation of the tangent to the circle is .

at the point

The centre of the circle is and units.

The equation of the tangent to the circle is .

at the point

The centre of the circle is and units.

The equation of the tangent to the circle is .

at the point

The centre of the circle is and units.

The equation of the tangent to the circle is .

Determine the equations of the tangents to the circle , given that both lines have an angle of inclination of .

The centre of the circle is and units.

Gradient of the tangents:

The line perpendicular to the tangents and passing through the centre of the circle is . Substitute into the equation of the circle and solve for :

This gives the points and .

Tangent at :

Tangent at :

The equations of the tangents to the circle are and .

The circle with centre has a tangent at point . The equation of is and lies on the -axis.

Determine the equation of .

Determine the coordinates of .

Equate the two equations and solve for :

Determine the equation of the circle.

Describe in words how the circle must be shifted so that is at the origin.

The circle must be shifted units down and units to the left.

If the length of is tripled and the circle is shifted units to the right and unit up, determine the equation of the new circle.

The equation of a circle with centre is and the equation of a circle with centre is . Prove that the two circles touch each other.

Therefore the two circles touch each other.

Chapter 8: Euclidean geometry

Sketches are valuable and important tools. Encourage learners to draw accurate diagrams to solve problems.

It is important to stress to learners that proportion gives no indication of actual length. It only indicates the ratio between lengths.

To prove triangles are similar, we need to show that two angles (AAA) are equal OR three sides in proportion (SSS).

Theorems are examinable and are often asked in examinations. It is also important that learners remember the correct construction required for each proof.

Notation - emphasize to learners the importance of the correct ordering of letters, as this indicates which angles are equal and which sides are in the same proportion.

If a length has to be calculated from a proportion, it helps to re-write the proportion with the unknown length in the top left position.

8.1 Revision

Types of triangles

Name

Diagram

Properties

Scalene

All sides and angles are different.

Isosceles

Two sides are equal in length. The angles opposite the equal sides are also equal.

Equilateral

All three sides are equal in length and all three angles are equal.

Acute-angled

Each of the three interior angles is less than °.

Obtuse-angled

One interior angle is greater than °.

Right-angled

One interior angle is °.

Congruent triangles

Condition

Diagram

SSS

(side, side, side)

SAS

(side, incl. angle, side)

AAS

(angle, angle, side)

RHS

(°, hypotenuse, side)

Similar triangles

Condition

Diagram

AAA

(angle, angle, angle)

SSS

(sides in prop.)

Circle geometry

If is the centre and , then .

If is the centre and , then .

If and , then passes through centre .

If an arc subtends an angle at the centre of a circle and at the circumference, then the angle at the centre is twice the size of the angle at the circumference.

Angles at the circumference subtended by arcs of equal length (or by the same arc) are equal.

Cyclic quadrilaterals

If the four sides of a quadrilateral are the chords of a circle with centre , then:

Reason: (opp. s cyclic quad. supp.)

Reason: (opp. s cyclic quad. supp.)

Reason: (ext. cyclic quad. = int. opp )

Reason: (vert. opp. s, ext. cyclic quad.)

Proving a quadrilateral is cyclic:

If or , then is a cyclic quadrilateral.

If or , then is a cyclic quadrilateral.

If or , then is a cyclic quadrilateral.

Tangents to a circle

A tangent is perpendicular to the radius (), drawn to the point of contact with the circle.

If and are tangents to a circle with centre , then:

(tangent radius)

(tangent radius)

(tangents from same point are equal)

If is a tangent, then and .

If or , then is a tangent touching at .

The mid-point theorem

The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

Given: and , we can conclude that and .

Exercise 8.1

in a circle with centre . and . Calculate the value of , giving reasons.

is the centre of the circle with and chord .

Determine the lengths of:

is a diameter of the circle with centre . bisects and .

Write down two other angles that are also equal to .

Calculate in terms of , giving reasons.

Prove that is a perpendicular bisector of .

is a diameter of the circle with centre . and .

Calculate the value of the following angles, giving reasons:

is the centre of the circle with diameter . at and chord intersects at .

Prove the following:

in the circle with centre is extended to so that . Express in terms of .

In the circle with centre , , and . Determine the length of .

is a diameter of the circle with centre . is extended to and is a tangent to the circle. and is a straight line.

Prove the following:

is a cyclic quadrilateral

and are tangents to the circle with centre . is a point on the circumference and .

Express the following in terms of , giving reasons:

8.2 Ratio and proportion

Ratio

A ratio describes the relationship between two quantities which have the same units. We can use ratios to compare weights, heights, lengths, currencies, etc. A ratio is a comparison between two quantities of the same kind and has no units.

Example: if the length of a rectangle is and the width is , then we can express the ratio between the length and width of the rectangle as:

The ratio of describes the length of the rectangle relative to its width.

A ratio written as a fraction is usually given in its simplest form.

A ratio gives no indication of actual length. For example,

Do not convert a ratio to a decimal (even though and have the same numerical value).

Proportion

Predicting heights

A record of heights is given.

If the ratio is to , complete the table below:

Name | Height at two years | Height as an adult
Hendrik |  |
Kagiso |  |
Linda |  |
Mandisa |  |
Prashna |  |

Consider the diagram below:

If two or more ratios are equal to each other, then we say that they are in the same proportion. Proportionality describes the equality of ratios.

If , then and are in the same proportion as and .

Given

The line segments and are in the same proportion as and . The following statements are also true:

Proportion
Reciprocal proportion
Cross multiplication

We can also substitute , , , and to show algebraically that the statements are true.

For example,

Exercise 8.2

Solve for :

A packet of sweets contains red, blue and yellow sweets in the ratio of respectively. Determine how many sweets of each colour there are in the packet.

A mixture contains parts of substance for every parts of substance . If the total weight of the mixture is , determine how much of substance is in the mixture (correct to decimal places).

Given the diagram below.

Show that:

Consider the line segment shown below.

Express the following in terms of and :

is a parallelogram with , and .

Calculate the ratio .

The area of a parallelogram base height:

The perimeter of a parallelogram .

To find the length of , we use and the theorem of Pythagoras.

and divides in the ratio . Determine and .

If and the ratio of is , calculate and .

Luke's biology notebook is long and wide. The dimensions of his desk are in the same proportion as the dimensions of his notebook.

If the desk is wide, calculate the area of the top of the desk.

Luke covers each corner of his desk with an isosceles triangle of cardboard, as shown in the diagram:

Calculate the new perimeter and area of the visible part of the top of his desk.

Use this new area to calculate the dimensions of a square desk with the same desk top area.

8.3 Polygons

Proportionality in polygons

A plane, closed shape consisting of three or more line segments.

In previous grades we studied the properties of the following polygons:

Triangle |  | Area
Parallelogram |  | Area
Rectangle |  | Area
Rhombus |  | Area

Square |  | Area
Trapezium |  | Area
Kite |  | Area

Worked example 1: Properties of polygons

is a rhombus with and .

Calculate the following (correct to two decimal places) and provide reasons:

Length of .

Length of .

Area of .

Use the ratio to determine the length of

Calculate the length of

We use the properties of a rhombus and the theorem of Pythagoras to find .

Determine the area of rhombus

Exercise 8.3

is a rectangle with and .

Calculate (correct to decimal places)

.

Calculate the area of (correct to decimal places).

Consider the trapezium shown below. If and area , calculate .

is a rhombus with sides of length millimetres. The diagonals intersect at and length millimetres. Express the area of in terms of .

In the diagram below, is a kite with , and .

Determine (correct to the nearest integer).

Calculate area .

is a rhombus. is the mid-point of and is the mid-point of . Prove that is also a rhombus.

8.4 Triangles

Proportionality of triangles

In the diagram below, and have the same height since both triangles are between the same parallel lines.

Triangles with equal heights have areas which are proportional to their bases.

and have the same base and the same height since both triangles lie between the same parallel lines.

Triangles with equal bases and between the same parallel lines are equal in area.

and have the same base and are equal in area.

Triangles on the same side of the same base and equal in area, lie between parallel lines.

Worked example 2: Proportionality of triangles

Given parallelogram with produced to . , and .

Calculate .

If , show that .

Determine the length of

We use the formula for area of a parallelogram to calculate .

Use proportionality to show that

We are given the ratio .

Exercise 8.4

The diagram below shows three parallel lines cut by two transversals and such that .

Determine:

We are given that , which we can write as a fraction and simplify:

The lengths of and , if it is given that .

We cannot determine the length of since we do not know the lengths of , or . We only know that .

In right-angled , is drawn parallel to , with the mid-point of . and . Determine and .

Use the theorem of Pythagoras to determine .

Given trapezium with and .

Prove that bisects .

Prove that the two diagonals are equal in length.

If , show that area .

In the diagram below, is given with .

Show that area .

In Grade we proved the mid-point theorem using congruent triangles.

Complete the following statement of the mid-point theorem:

"The line joining of a triangle is to the third side and equal to "

"The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side."

In , and are the mid-points of and respectively. Prove .

Hint: make a construction by drawing and .

Write down the converse of the mid-point theorem.

Converse: a line through the mid-point of one side of a triangle, parallel to a second side, bisects the third side.

Theorem: Proportion theorem

A line drawn parallel to one side of a triangle divides the other two sides of the triangle in the same proportion.

(Reason: line to one side )

with line

Draw from perpendicular to , and from perpendicular to .

Draw and .

Similarly, we use the same method to show that:

Converse: proportion theorem

If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side.

(Reason: line divides sides in prop.)

Worked example 3: Proportion theorem

In , , and . If , and , determine the values of and , giving reasons.

Use the proportion theorem to determine the values of and

Consider :

Consider :

Special case: the mid-point theorem

A corollary of the proportion theorem is the mid-point theorem: the line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

If and , then and .

We also know that .

Converse: the mid-point theorem

The line drawn from the mid-point of one side of a triangle parallel to another side, bisects the third side of the triangle.

If and , then .

Exercise 8.5

In , and .

, .

Calculate .

Calculate .

Use the given diagram to prove the Proportion Theorem.

Given:

Required to prove:

Construction: Draw and

Proof: use area of 's

In the diagram below,

If , calculate (correct to two decimal places).

Write down the converse of the proportion theorem and illustrate with a diagram.

Converse of the proportion theorem: a line cutting two sides of a triangle proportionally will be parallel to the third side.

In , is a point on . is the mid-point of , is the mid-point of and is the mid-point of .

Prove that is a parallelogram.

Consider

Prove that .

Consider

8.5 Similarity

Similar polygons

Polygons are similar if they are the same shape but differ in size. In other words, one polygon is an enlargement of the other. Congruent polygons are also similar since they are the same shape and the same size, only their position or orientation is different.

Two polygons with the same number of sides are similar when:

All pairs of corresponding angles are equal, and

All pairs of corresponding sides are in the same proportion.

In the diagram above, polygon is similar to polygon if:

Condition 1:

AND

Condition 2:

Important:

Both conditions must be true for two polygons to be similar.

If we are given two polygons that are similar, then we know that both conditions are true.

Worked example 4: Similar polygons

Polygons and are similar. Determine the value of .

Identify pairs of corresponding sides

Since the two polygons are similar,

Exercise 8.6

Determine whether or not the following polygons are similar, giving reasons.

Similar, all angles and sides are in the same proportion.

Not enough information is given. Corresponding angles are equal.

Not similar, sides are not in the same proportion.

Are the following statements true or false? If false, state reasons or draw an appropriate diagram.

All squares are similar.

True

All rectangles are similar.

False. Pairs of corresponding sides not necessarily in the same proportion.

All rhombi are similar.

False. Corresponding angles not necessarily equal.

All congruent polygons are similar.

True

All similar polygons are congruent.

False. Corresponding angles are equal but not necessarily true that corresponding sides are equal.

All congruent triangles are similar.

True

Isosceles triangles are similar.

False. Corresponding angles not necessarily equal and pairs of corresponding sides not necessarily in the same proportion.

Equilateral triangles are similar.

True

Similarity of triangles

To prove two polygons are similar, we need to show that two conditions are true: (a) all pairs of corresponding angles are equal and (b) all pairs of corresponding sides are in the same proportion.

To prove two triangles are similar, we need only show that one of the conditions is true. If one of the conditions is true for two triangles, then the other condition is also true.

Theorem: Equiangular triangles are similar

Equiangular triangles are similar.

(Reason: equiangular s)

and with ; ;

Construction: mark on so that , and mark on so that .

In and

Similarly, by constructing on so that , and mark on so that .

Therefore equiangular triangles are similar.

Notation

The symbol for congruency is .

The symbol for similarity is .

Be careful to label similar triangles correctly.

For example, if , and , then .

Do not write .

also indicates which sides of the triangles are in the same proportion:

Proving equiangular triangles are similar:

The sum of the interior angles of any triangle is °. If we know that two pairs of angles are equal, then the remaining angle in each triangle must also be equal. Therefore the two triangles are similar.

In other words, to prove that two triangles are equiangular, we need only show that two pairs of angles are equal.

Worked example 5: Similarity of triangles

Prove that is similar to .

Calculate the unknown angles in each triangle

Prove that the two triangles are similar

Exercise 8.7

In the diagram below, .

Prove .

In

If , calculate the length of (correct to one decimal place).

In

In circle ,

Prove .

In :

Prove .

If , calculate the radius of circle (correct to one decimal place).

Given the following figure with the following lengths, find , and .

Theorem: Triangles with sides in proportion are similar

If the corresponding sides of two triangles are in proportion, then the two triangles are similar.

(Reason: sides of s in prop.)

and with

, and

Construction: draw such that and .

Theorem: Triangles with two sides in proportion and equal included angles, are similar (PROOF NOT FOR EXAMS)

If two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, then the two triangles are similar.

(Reason: s with sides in prop. and equal incl. s)

and with and

and

Construction: draw such that and .

In and :

Proving triangles with sides in proportion are similar:

To prove that the sides of two triangles are in the same proportion, we must show that all three pairs of corresponding sides are in the same proportion.

Worked example 6: Similarity of triangles

In the diagram below, is ?

Show calculations.

Consider the diagram given below. Determine whether or not .

Show calculations.

(Diagrams are not drawn to scale)

Investigate proportionality of and

For similar triangles, the order in which the triangles are labelled indicates which pairs of sides correspond.

Investigate proportionality of and

Worked example 7: Similarity of triangles

is a trapezium, with . Prove that .

Identify triangles

We are required to prove that , which we can also write as the ratio .

To determine which two triangles we must consider, examine the lengths given in the ratio.

Prove triangles are equiangular

Use proportionality

Exercise 8.8

Consider the diagram given below. units and units. Is ? Give reasons.

Yes, .

is a diameter of the circle . is drawn parallel to and meets in .

If the radius is and , calculate the length of .

[NCS, Paper 3, November 2011]

OR

i

OR

are on the circumference of the circle.

Prove, with reasons, that:

In :

In :

is a parallelogram with diagonals intersecting at . is drawn parallel to . is produced to such that and is produced to such that .

[NCS, Paper 3, November 2011]

Show that is the midpoint of .

Prove .

OR

OR

If is , calculate the length of .

In and :

OR

In and :

8.6 Pythagorean theorem

Many different methods of proving the theorem of Pythagoras have been formulated over the years. Similarity of triangles is one method that provides a neat proof of this important theorem.

Theorem: Pythagorean theorem

The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

(Reason: Pythagoras or )

with

Construction: draw .

Converse: theorem of Pythagoras

If the square of one side of a triangle is equal to the sum of the squares of the other two sides of the triangle, then the angle included by these two sides is a right angle.

Worked example 8: Theorem of Pythagoras

In , and . If and , determine and (correct to the nearest integer).

Use the theorem of Pythagoras to determine

Use proportionality to determine and

Write the final answer

and

For any right-angled , if is drawn perpendicular to , then:

Exercise 8.9

is a point on circle with centre . and is the midpoint of radius .

If the diameter of the circle is , find .

Leave answer in simplified surd form.

In , and . units, units, units and units. Determine and , giving reasons.

Chord of circle with centre cuts at right angles at point .

Why is ?

In and :

If units and units, calculate the radius of the circle.

In :

In the diagram below, and are tangents to the circle with centre and .

Show that .

Prove that .

8.7 Summary

A ratio describes the relationship between two quantities which have the same units.

If two or more ratios are equal to each other , then and are in the same proportion as and .

A polygon is a plane, closed shape consisting of three or more line segments.

Triangles with equal heights have areas which are proportional to their bases.

Triangles with equal bases and between the same parallel lines are equal in area.

Triangles on the same side of the same base and equal in area lie between parallel lines.

A line drawn parallel to one side of a triangle divides the other two sides of the triangle in the same proportion.

Converse: proportion theorem

If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side.

Special case: the mid-point theorem

The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

If and , then and .

Converse: the mid-point theorem

The line drawn from the mid-point of one side of a triangle parallel to another side, bisects the third side of the triangle.

If and , then .

Polygons are similar if they are the same shape but differ in size. One polygon is an enlargement of the other.

Two polygons with the same number of sides are similar when:

All pairs of corresponding angles are equal, and

All pairs of corresponding sides are in the same proportion.

If two triangles are equiangular, then the triangles similar.

Triangles with sides in proportion are equiangular and therefore similar.

The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Converse: theorem of Pythagoras

If the square of one side of a triangle is equal to the sum of the squares of the other two sides of the triangle, then the angle included by these two sides is a right angle.

Exercise 8.10

Calculate

. Find .

Using the following figure and lengths, find and (correct to one decimal place).

and .

Find in the following figure.

In , , and . Determine .

, , , .

Calculate the lengths of , , , and , and find the ratio .

In , and . If and , determine and (correct to two decimal places).

Use the theorem of Pythagoras to determine :

Use proportionality to determine and :

Given the following figure with the following lengths, find , and .

cm, cm, cm, and cm.

is a parallelogram with on .

produced meets produced at . intercepts at .

Prove that .

In and :

Prove .

In :

If and , calculate .

is a diameter of circle . is produced to so that .

is a tangent to the circle and . produced meets at .

Prove that:

is a cyclic quadrilateral

In and :

In and :

is a diameter of circle . .

.

Prove that .

Hence, deduce that .

is a diameter of the circle with centre . Chord is produced to . Chord produced meets the tangent at . .

[NCS, Paper 3, November 2011]

Calculate the size of .

Find .

OR

Determine the size of .

Prove that .

OR

OR

OR

OR

In and

is a cyclic quadrilateral and

is a tangent to the circle at . and are straight lines.

Prove:

In and :

is a tangent to circle at . Chord is produced to . Chord cuts chord in and chord in . and . Let and .

[NCS, Paper 3, November 2011]

Determine THREE other angles that are each equal to .

Prove that .

In and

Hence, or otherwise, prove that .

Chapter 9: Statistics

It is easy for learners to be intimidated by the formulae in this chapter. Slowly go through the meaning of each term in the formulae and ensure that learners are familiar with summation notation.

Use by hand calculations to aid the understanding of key concepts.

Calculator skills are very important in this chapter. Methods for SHARP and CASIO calculators are shown but practical demonstration may be required.

The formula for population variance and population standard deviation are used and not sample variance and sample standard deviation.

Calculation of the correlation coefficient requires the use of the sample standard deviation, but the concepts of sample and population are beyond the scope of CAPS. The formula used for the correlation coefficient in this chapter is so as not to confuse learners. The actual formula is , however as this is a ratio, the difference in denominators between the population standard deviation and sample deviation will cancel. Note that there are other formulae for the correlation coefficient but these should be avoided as usage of the population standard deviation will result in erroneous results.

Discuss the misuse of statistics in the real world and encourage awareness.

9.1 Revision

Terminology

Measures of central tendency:

Provide information on the data values at the centre of the data set.

The mean is the 'average' value of a data set. It is calculated as where the are the data and is the number of data entries. We read as “x bar”.

The median is the middle value of an ordered data set. To find the median, we first sort the data in ascending or descending order and then pick out the value in the middle of the sorted list. If the middle is in between two values, the median is the average of those two values.

Measures of dispersion:

Tell us how spread out a data set is. If a measure of dispersion is small, the data are clustered in a small region. If a measure of dispersion is large, the data are spread out over a large region.

The range is the difference between the maximum and minimum values in the data set.

The inter-quartile range is the difference between the first and third quartiles of the data set. The quartiles are computed in a similar way to the median. The median is halfway into the ordered data set and is sometimes also called the second quartile. The first quartile is one quarter of the way into the ordered data set, whereas the third quartile is three quarters of the way into the ordered data set.

If you begin numbering your ordered data set with the number 1, the formulae for the location of each quartile are as follows:

The variance of the data is the average squared distance between the mean and each data value.

The variance of the data is in a population of elements, , with a mean of .

The standard deviation measures how spread out the values in a data set are around the mean. More precisely, it is a measure of the average distance between the values of the data in the set and the mean.

The standard deviation of the data is in a population of elements, , with a mean of .

The five number summary combines a measure of central tendency, the median, with measures of dispersion, namely the range and the inter-quartile range. More precisely, the five number summary is written in the following order:

minimum;

first quartile;

median;

third quartile;

maximum.

The five number summary is often presented visually using a box and whisker diagram, illustrated below.

Video: 29BS

Worked example 1: Five number summary

Draw a box and whisker diagram for the following data set:

Determine the minimum and maximum

Since the data set is already ordered, we can read off the minimum as the first value () and the maximum as the last value ().

Determine the quartiles

There are values in the data set. We can use the figure below or the formulae to determine where the quartiles are located.

Using the figure above we can see that the median is between the sixth and seventh values. We can confirm this using the formula: Therefore, the value of the median is

The first quartile lies between the third and fourth values. We can confirm this using the formula: Therefore, the value of the first quartile is

The third quartile lies between the ninth and tenth values. We can confirm this using the formula: Therefore, the value of the third quartile is

Draw the box and whisker diagram

We now have the five number summary as (; ; ; ; ). The box and whisker diagram representing the five number summary is given below.

Video: 29BT

Worked example 2: Variance and standard deviation

You flip a coin times and it lands on heads times. You then use the same coin and do another flips. This time in lands on heads times. You repeat this experiment a total of times and get the following results for the number of heads.

For the data set above:

Calculate the mean.

Calculate the variance and standard deviation using a table.

Confirm your answer for the variance and standard deviation using a calculator.

Calculate the mean

The formula for the mean is

In this case, we sum the data and divide by to get .

Calculate the variance using a table

The formula for the variance is

We first subtract the mean from each data point and then square the result.

The variance is the sum of the last row in this table divided by , so .

Calculate the variance using a calculator

Using the SHARP EL-531VH calculator:

Using your calculator, change the mode from normal to “Stat ”. Do this by pressing [2ndF] and then 1. This mode enables you to type in univariate data.

Key in the data, row by row:

Enter:

Press:

See:

44

DATA

n =

49

DATA

n =

52

DATA

n =

62

DATA

n =

53

DATA

n =

48

DATA

n =

54

DATA

n =

49

DATA

n =

46

DATA

n =

51

DATA

n =

Note: The [DATA] button is the same as the [M+] button.

Get the value for :

Press:

Press:

See:

RCL

Using the CASIO -82ZA PLUS calculator:

Switch on the calculator. Press [MODE] and then select STAT by pressing [2]. The following screen will appear:

1

2

3

4

5

6

7

8

Now press [1] for variance and standard deviation. Your screen should look something like this:

1

2

3

Press [44] and then [] to enter the first -value under . Then enter the other values in the same way.

1

44

2

49

3

52

Then press [AC]. The screen clears but the data remains stored.

Now press [SHIFT][1] to get the stats computations screen shown below.

1:

Type

2:

Data

3:

Sum

4:

Var

5:

MinMax

Choose variance by pressing [4].

1:

n

2:

3:

4:

s

Get the value for :

Press [3] and [=] to get the value of

Symmetric and skewed data

Last year you learnt about three shapes of data distribution: symmetric, left skewed and right skewed.

A symmetric distribution is one where the left and right hand sides of the distribution are roughly equally balanced around the mean. The histogram below shows a typical symmetric distribution.

For symmetric distributions, the mean is approximately equal to the median and the left and right tails are equally balanced, meaning that they have about the same length.

If large numbers of data are collected from a population, the graph will often have a bell shape. If the data was, say, examination results, a few learners usually get very high marks, a few very low marks and most get a mark in the middle range. This is a common type of symmetric data known as a normal distribution. We say a distribution is normal if

the mean, median and mode are equal.

it is symmetric around the mean.

% of the sample lies within one standard deviation of the mean, % within two standard deviations and % within three standard deviations of the mean.

Video: 29BV

What happens if the test was very easy or very difficult? Then the distribution may not be symmetrical. If extremely high or extremely low scores are added to a distribution, then the mean and median tend to shift towards these scores and the curve becomes skewed.

If the test was very difficult, the mean and median scores are shifted to the left. In this case, we say the distribution is positively skewed, or skewed right.

A distribution that is skewed right has the following characteristics:

the mean is typically more than the median;

the tail of the distribution is longer on the right hand side than on the left hand side; and

the median is closer to the first quartile than to the third quartile.

If the test was very easy, then many learners would get high scores, and the mean and median of the distribution would be shifted to the right. We say the distribution is negatively skewed, or skewed left.

A distribution that is skewed right has the following characteristics:

the mean is typically less than the median;

the tail of the distribution is longer on the left hand side than on the right hand side; and

the median is closer to the third quartile than to the first quartile.

The table below summarises the different categories visually.

Skewed right (positive) | Symmetric | Skewed left (negative)
mean median | mean median | mean less than median

Worked example 3: Skewed and symmetric data

Three Matric classes wrote a Mathematics test. The test is out of marks and each class has learners. The results of the test are shown in the table below:

Gr. 12A
Gr. 12B
Gr. 12C

For each class, determine the five number summary and draw a box and whisker diagram on the same set of axes using an appropriate scale.

Determine the mean and standard deviation for each class.

Comparing the mean and median values for each class, comment on the distribution of the test marks for each class.

First, we order the data from smallest to largest. This has already been done for us. Then, we divide our data into quartiles:

The minimum of each data set is . The maximum of each data set is .

Since there are values in the data set, the median lies on the eleventh mark, making it equal to for Gr. 12A, for Gr. 12B and for Gr. 12C.

The first quartile lies between the fifth and sixth values, making it equal to for Gr. 12A, for Gr. 12B and for Gr. 12C.

The third quartile lies between the and values, making it equal to for Gr. 12A and Gr. 12B, and for Gr. 12C.

Therefore, we are able to formulate the following five number summaries and subsequent box and whisker plots:

Gr. 12A

Gr. 12B

Gr. 12C

Gr. 12A: Gr. 12B: Gr. 12C:

If the mean is greater than the median, the data is typically positively skewed and if the mean is less than the median, the data is typically negatively skewed.

Gr. 12A: . The marks for 12A are therefore positively skewed, meaning that there were many low marks in the class with the high marks being more spread out.

Gr. 12B: . The marks for 12B are therefore negatively skewed, meaning that there were many high marks in the class with the low marks being more spread out.

Gr. 12C: . The marks for 12C are therefore normally distributed, meaning that there are as many low marks in the class as there are high marks.

Exercise 9.1

State whether each of the following data sets are symmetric, skewed right or skewed left.

A data set with this distribution:

skewed right

A data set with this box and whisker plot:

symmetric

A data set with this histogram:

skewed left

A data set with this frequency polygon:

skewed right

A data set with this distribution:

skewed left

The following data set:

; ; ; ; ; ; ; ; ;

The statistics of the data set are

mean: ;

first quartile: ;

median: ;

third quartile: .

Note that we get contradicting indications from the different ways of determining whether the data is skewed right or left.

The mean is slightly greater than the median. This would indicate that the data set is skewed right.

The median is slightly closer to the third quartile than the first quartile. This would indicate that the data set is skewed left.

Since these differences are so small and since they contradict each other, we conclude that the data set is symmetric.

For the following data sets:

Determine the mean and five number summary.

Draw the box and whisker plot.

Determine the skewness of the data.

; ; ; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the diagram below (or the formulae) to find at, or between, which values the quartiles lie.

The minimum is 3 and the maximum is 45.

For the first quartile, the position is between the third and fourth values. Since both these values are equal to , the first quartile is .

For the median (second quartile), the position is halfway between the sixth and seventh values. The sixth value is and the seventh value is , which means that the median is .

For the third quartile, the position is between the ninth and tenth values. Therefore the third quartile is .

Therefore, the five number summary is (3; 7; ; 33; 45)

Mean median, therefore the data is skewed right.

; ; ; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the formulae (or a diagram) to find at, or between, which values the quartiles lie.

The minimum is 8 and the maximum is 100.

For the first quartile, the position is between the third and fourth values. Since both these values are equal to , the first quartile is .

For the median (second quartile), the position is halfway between the sixth and seventh values. The sixth value and seventh value are , which means that the median is .

For the third quartile, the position is between the ninth and tenth values. Therefore the third quartile is .

Therefore, the five number summary is (8; 21; ; ; 100)

Mean median, therefore the data is skewed right.

; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the formulae (or a diagram) to find at, or between, which values the quartiles lie.

The minimum is 11 and the maximum is 92.

For the first quartile, the position is between the third and fourth values. Since the third is 57 and the fourth is 58, the first quartile is .

For the median (second quartile), the position is halfway between the fifth and sixth values. The sixth value is 65 and the seventh value is , therefore the median is .

For the third quartile, the position is between the seventh and eighth values. Therefore the third quartile is .

Therefore, the five number summary is (11; ; ; ; 92)

Mean

; ; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the formulae (or a diagram) to find at, or between, which values the quartiles lie.

The minimum is 1 and the maximum is 99.

For the first quartile, the position is between the third and fourth values. The third value is 33 and the fourth value is 41, therefore the first quartile is .

The second quartile (median) is at the sixth position. The sixth value is .

For the third quartile, the position is between the eighth and ninth values. Therefore the third quartile is .

Therefore, the five number summary is (1; 37; ; ; 99)

Mean

; ; ; ; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the formulae (or a diagram) to find at, or between, which values the quartiles lie.

The minimum is and the maximum is 3.

The first quartile is at the fourth position. Therefore, the first quartile is .

The median (second quartile) is at the seventh position. The seventh value is .

The third quartile is at the tenth position. Therefore the third quartile is .

Therefore, the five number summary is (; ; ; ; 3)

Mean

; ; ; ; ; ; ; ; ; ; ;

The mean is .

To determine the five number summary, we order the data:

We use the formulae (or a diagram) to find at, or between, which values the quartiles lie.

The minimum is 25 and the maximum is 97.

For the first quartile, the position is between the third and fourth values. The third value is 44 and the fourth value is , which means that the first quartile is .

For the median (second quartile), the position is halfway between the sixth and seventh values. The sixth value is 60 and the seventh value is , which means that the median is .

For the third quartile, the position is between the ninth and tenth values. Therefore the third quartile is .

Therefore, the five number summary is (25; 45; ; ; 97)

Mean median, therefore the data is symmetric.

For the following data sets:

Determine the mean.

Use a table to determine the variance and the standard deviation.

Determine the percentage of data points within one standard deviation of the mean. Round your answer to the nearest percentage point.

The formula for the mean is

The formula for the variance is

We first subtract the mean from each data point and then square the result.

The variance is the sum of the last row in this table divided by , so . The standard deviation is the square root of the variance, therefore .

The interval containing all values that are one standard deviation from the mean is . We are asked how many values are within than one standard deviation from the mean, meaning inside the interval. There are values from the data set within the interval, which is of the data points.

\{9; 5; 1; 3; 3; 5; 7; 4; 10; 8\}

The formula for the mean is

The formula for the variance is

We first subtract the mean from each data point and then square the result.

The variance is the sum of the last row in this table divided by , so . The standard deviation is the square root of the variance, therefore .

The interval containing all values that are one standard deviation from the mean is . We are asked how many values are within than one standard deviation from the mean, meaning inside the interval. There are values from the data set within the interval, which is of the data points.

\{81; 22; 63; 12; 100; 28; 54; 26; 50; 44; 4; 32\}

The formula for the mean is

The formula for the variance is

We first subtract the mean from each data point and then square the result.

The variance is the sum of the last row in this table divided by , so . The standard deviation is the square root of the variance, therefore .

The interval containing all values that are one standard deviation from the mean is . We are asked how many values are within than one standard deviation from the mean, meaning inside the interval. There are values from the data set within the interval, which is of the data points.

Use a calculator to determine the

mean,

variance,

and standard deviation

of the following data sets:

; ; ; ; ; ; ; ; ;

Mean =

; ; ; ; ; ; ; ; ; ; ;

Mean =

; ; ; ; ; ; ; ; ;

Mean =

; ; ; ; ; ; ; ; ;

Mean =

; ; ; ; ; ; ; ; ; ; ; ; ;

Mean =

Xolani surveyed the price of a loaf of white bread at two different supermarkets. The data, in rands, are given below.

Supermarket A |  |  |  |  |  |
Supermarket B |  |  |  |  |  |

Find the mean price at each supermarket and then state which supermarket has the lower mean.

Supermarket A: . Supermarket B: . Supermarket B has the lower mean.

Find the standard deviation of each supermarket's prices.

Standard deviation:

For Supermarket A:

For Supermarket B:

Which supermarket has the more consistently priced white bread? Give reasons for your answer.

The standard deviation of Supermarket A's prices is lower than that of Supermarket B's. That means that Supermarket A has more consistent (less variable) prices than Supermarket B.

The times for the athletes who swam the freestyle final at the 2012 London Olympic Games are shown below. All times are in seconds.

; ; ; ; ; ; ;

Calculate the mean time.

Calculate the standard deviation for the data.

How many of the athletes' times are more than one standard deviation away from the mean?

The mean is and the standard deviation is . Therefore the interval containing all values that are one standard deviation from the mean is . We are asked how many values are further than one standard deviation from the mean, meaning outside the interval. There are values from the data set outside the interval.

The following data set has a mean of and a variance of .

Calculate the values of and .

From the formula of the mean we have

From the formula of the variance we have

Substitute into this equation to get Therefore or .

Since we have or . So, the two unknown values in the data set are and .

We do not know which of these is and which is since the mean and variance tell us nothing about the order of the data.

The height of each learner in a class was measured and it was found that the mean height of the class was . At the time, three learners were absent. However, when the heights of the learners who were absent were included in the data for the class, the mean height did not change.

If the heights of two of the learners who were absent are and , calculate the height of the third learner who was absent. [NSC Paper 3 Feb-March 2013]

Let the number of learners who were first measured be .

The total measure of all heights is .

Let the height of the last learner be .

There are 184 students taking Mathematics in a first-year university class. The marks, out of 100, in the half-yearly examination are normally distributed with a mean of 72 and a standard deviation of 9. [NSC Paper 3 Feb-March 2013]

What percentage of students scored between 72 and 90 marks?

Therefore 90 lies at 2 standard deviations to the right of the mean.

Hence, % of students scored between 72 and 90 marks.

Approximately how many students scored between 45 and 63 marks?

9.2 Curve fitting

Intuitive curve fitting

In Grade 11, we used various means, such as histograms, frequency polygons and ogives, to visualise our data. These are very useful tools to depict univariate data, i.e. data with only one variable such as the height of learners in a class.

Last year we also learnt about a visual tool called scatter plots. Scatter plots are a common way to visualise bivariate data, i.e. data with two variables. This allows us to identify the direction and strength of a relationship between two variables.

We identify the nature of a relationship between two variables by examining if the points on the scatter plot conform to a linear, exponential, quadratic or some other function. The process of fitting functions to data is known as curve fitting.

The strength of a relationship can be described as strong if the data points conform closely to a function or weak if they are further away.

In the case of linear functions, the direction of a relationship is positive if high values of one variable occur with high values of the other or negative if high values of one variable occur with low values of the other.

The table below summarises the different relationships:

Strong, positive linear relationship | Strong, negative linear relationship
Weak, positive linear relationship | Exponential relationship
Quadratic relationship | No relationship

Worked example 4: Intuitive curve fitting

Examine the scatter plot below of data collected from a new shop:

What are the two variables being compared?

What type of function best fits the data?

Is the relationship between the two variables strong or weak?

Is the relationship between the two variables positive or negative?

Using your answers above, describe the relationship between the two variables in one sentence.

The variables being compared are average daily number of customers and time in months.

The data fit an exponential function.

The data points appear to fit the curve close to perfectly, so the relationship can be described as very strong.

As time increases, the number of customers increases, so the relationship can be described as positive.

There is a very strong, positive, exponential relationship between average daily customers and time in the new shop.

In the worked example above, by plotting the average daily customers and time data of a new shop on a scatter plot, we were able to identify the relationship between the two variables. Once we know the relationship between two variables, we are able to do another very useful thing - we are able to predict values where no data exist.

Interpolation and extrapolation

When we predict values that fall within the range of our data, this is known as interpolation. When we predict the values of a variable beyond the range of our data, this is known as extrapolation.

Extrapolation must be done with caution unless it is known that the observed relationship continues beyond the range of our data. For example, an exponential function may look linear if we only have the first few data points available but if we extrapolate far enough beyond the initial data points, our predictions will be inaccurate.

In order to interpolate or extrapolate values, we need to find the equation of the function which best fits the data. For linear data, we draw a straight line through the data which best approximates the available data points. This line is known as the line of best fit or trend line. Let us try our hand at this in the following example.

Worked example 5: Fitting by hand

Use the data below to draw a scatter plot and line of best fit.

Write down the equation of the line that best seems to fit the data.

Use your equation to calculate the estimated value for if .

Use your equation to calculate the estimated value for if .

Draw the graph

Choose a suitable scale for the axes.

Draw the axes.

Plot the points.

Drawing the line of best fit

The next step is to draw a straight line which goes as close to as many points as possible. It is generally best to have as many points above the line as below the line.

Calculating the equation of the line

The equation of the line is

From the graph we have drawn, we estimate the y-intercept to be . We estimate that when . So we have that points and lie on the line. The gradient of the line, m, is given by

So we finally have that the equation of the line of best fit is

Calculate the unknown values

The equation of the line is so in order to find the unknown values, we insert the known values into our equation.

For :

Since this -value is within the data range, this is interpolation.

For :

Since this -value is outside the data range, this is extrapolation.

Exercise 9.2

Identify the function (linear, exponential or quadratic) which would best fit the data in each of the scatter plots below:

quadratic

exponential

linear

linear

exponential

quadratic

Given the scatter plot below, answer the questions that follow.

What type of function fits the data best? Comment on the fit of the function in terms of strength and direction.

The data fit a strong, positive linear function.

Draw a line of best fit through the data and determine the equation for your line.

NB: The answer to this question is learner dependent. The method is more important than the final answer. Pay special attention to the -intercept of the line of best fit. Learners often draw their line through the origin, even when this is not appropriate. Below is an illustration of how the learner should go about finding the solution to this problem. The learner's answer does not have to look exactly like the model answer, but should at least be a good approximation.

The -intercept is approximately 1. The -value at is approximately 10. Therefore,

The equation for the line of best fit:

Using your equation, determine the estimated -value where .

Answer will depend on the learner's previous answer.

Using your equation, determine the estimated -value where .

Answer will depend on the learner's previous answer.

Tuberculosis (TB) is a disease of the lungs caused by bacteria which are spread through the air when an infected person coughs or sneezes. Drug-resistant TB arises when patients do not take their medication properly. Andile is a scientist studying a new treatment for drug-resistant TB. For his research, he needs to grow the TB bacterium. He takes two bacteria and puts them on a plate with nutrients for their growth. He monitors how the number of bacteria increases over time. Look at his data in the scatter plot below and answer the questions that follow.

What type of function do you think fits the data best?

Exponential

The equation for bacterial growth is where is the initial number of bacteria, is the growth rate per unit time as a proportion of 1, is time in hours, and is the number of bacteria at time, . Determine the number of bacteria grown by Andile after 24 hours if the number of bacteria doubles every hour (i.e. the growth rate is % per hour).

We are told that :

Marelize is a researcher at the Department of Agriculture. She has noticed that farmers across the country have very different crop yields depending on the region. She thinks that this has to do with the different climate in each region. In order to test her idea, she collected data on crop yield and average summer temperatures from a number of farmers. Examine her data below and answer the questions that follow.

Identify what type of function would fit the data best.

Quadratic

Marelize determines that the equation for the function which fits the data best is . Determine the optimal temperature to grow wheat and the respective crop yield. Round your answer to two decimal places.

This question requires us to find the turning point of the function. There are a number of ways to do this; two are shown below:

The first method is using the formula :

The first step is to write the equation in the form: . Our equation is already in this form, so we can immediately substitute the values into the formula for .

To find , we substitute our -value into the quadratic equation:

Another method is using differentiation:

The first step is to write the equation in the form: . Our equation is already in this form, so we can immediately differentiate the equation.

At the turning point, , therefore we can now solve for :

The -value can now be substituted into the quadratic equation to find :

Therefore the optimal temperature to grow wheat is and the respective crop yield is .

Dr Dandara is a scientist trying to find a cure for a disease which has an % mortality rate, i.e. % of people who get the disease will die. He knows of a plant which is used in traditional medicine to treat the disease. He extracts the active ingredient from the plant and tests different dosages (measured in milligrams) on different groups of patients. Examine his data below and complete the questions that follow.

Dosage (mg)

Mortality rate (\%)

Draw a scatter plot of the data

Which function would best fit the data? Describe the fit in terms of strength and direction.

The data show a strong, negative linear relationship.

Draw a line of best fit through the data and determine the equation of your line.

The -intercept is approximately 80. The -intercept is approximately 210. Therefore,

The equation for the line of best fit:

Use your equation to estimate the dosage required for a % mortality rate.

Dr Dandara decided to administer the estimated dosage required for a % mortality rate to a group of infected patients. However, he still found a mortality rate of %. Name the statistical technique Dr Dandara used to estimate a mortality rate of % and explain why his equation did not accurately predict his experimental results.

Dr Dandara used extrapolation to calculate the dosage where the mortality rate . Extrapolation can result in incorrect estimates if the trend observed within the available data range does not continue outside of the range. In this case, it appears that at dosages greater than , the equation of the line of best fit no longer fits the data, therefore extrapolation produced a false estimate.

In the previous worked example and exercises, you drew the line of best fit by hand. This can give us a reasonable approximation of which function best fits the data when the data points are close together. However, you and your classmates may have found that you obtained slightly different answers from one another. In the next section, we will learn about a more precise way of fitting a linear function to data.

Linear regression

Linear regression analysis is a statistical technique for finding out exactly which linear function best fits a given set of data. We can find out the equation of the regression line by using an algebraic method called the least squares method, available on most scientific calculators. The linear regression equation is written (we say y-hat) or . Of course these are both variations of the more familiar equation .

The least squares method is very simple. Suppose we guess a line of best fit, then at every data point, we find the distance between the data point and the line. If the line fitted the data perfectly, this distance would be zero for all the data points. The worse the fit, the larger the differences. We then square each of these distances, and add them all together.

The best-fit line is then the line that minimises the sum of the squared distances.

Video: 29D5

Suppose we have a data set of points . We also have a line that we are trying to fit to the data. The distance between the first data point and the line, for example, is

We now square each of these distances and add them together. Lets call this sum . Then we have that

Thus our problem is to find the value of m and c such that is minimised. Let us call these minimising values and respectively. Then the line of best-fit is . We can find and using calculus, but it is tricky, and we will just give you the result, which is that

Worked example 6: Method of least squares by hand

In the table below, we have the records of the maintenance costs in rands compared with the age of the appliance in months. We have data for five appliances. Determine the equation for the least squares regression line by hand.

Appliance

1

2

3

4

5

Age ()

Cost ()

Appliance

1

2

3

4

5

Total

80

Worked example 7: Using the SHARP EL-531VH calculator

Using a calculator, find the equation of the least squares regression line for the following data:

Days ()

1

2

3

4

5

Growth in m ()

NB. If you have a CASIO calculator, do the next worked example first. Come back to this worked example once you are done and see if you get the same answer on your calculator.

Getting your calculator ready

Using your calculator, change the mode from normal to “Stat ”. Do this by pressing [2ndF] and then 2. This mode enables you to type in bivariate data.

Entering the data

Key in the data row by row:

Enter:

Press:

Enter:

Press:

See:

1

1

DATA

n =

2

DATA

n =

3

DATA

n =

4

DATA

n =

5

DATA

n =

Note: The [()] button is the same as the [STO] button and the [DATA] button is the same as the [M+] button.

Getting regression results from the calculator

Ask for the values of the regression coefficients and .

Press:

Press:

See:

RCL

RCL

Worked example 8: Using the CASIO -82ZA PLUS calculator

Using a calculator determine the least squares line of best fit for the following data set.

Learner

1

2

3

4

5

Chemistry (\%)

Accounting (\%)

For a Chemistry mark of %, what mark does the least squares line predict for Accounting?

NB. If you have a SHARP calculator, ensure that you have done the previous worked example first. Once you have completed the previous worked example, attempt this example using your calculator and see if you get the same answer.

Getting your calculator ready

Switch on the calculator. Press [MODE] and then select STAT by pressing [2]. The following screen will appear:

1

2

3

4

5

6

7

8

Now press [2] for linear regression. Your screen should look something like this:

1

2

3

Entering the data

Press [52] and then [] to enter the first mark under . Then enter the other values, in the same way, for the -variable (the Chemistry marks) in the order in which they are given in the data set. Then move the cursor across and up and enter 48 under y opposite 52 in the -column. Continue to enter the other -values (the Accounting marks) in order so that they pair off correctly with the corresponding -values.

1

52

2

55

3

Then press [AC]. The screen clears but the data remains stored.

Now press [SHIFT][1] to get the stats computations screen shown below.

1:

Type

2:

Data

3:

Edit

4:

Sum

5:

Var

6:

MinMax

7:

Reg

Choose Regression by pressing [7].

1:

A

2:

B

3:

r

4:

5:

Getting regression results from the calculator

Press [1] and [=] to get the value of the -intercept, (to two decimal places)

Finally, to get the slope, use the following key sequence: [SHIFT][1][7][2][]. The calculator gives (to two decimal places)

The equation of the line of regression is thus:

Press [AC][65][SHIFT][1][7][5][]

This gives a (predicted) Accounting mark of

Exercise 9.3

Determine the equation of the least-squares regression line using a table for the data sets below. Round and to two decimal places.

Use your calculator to determine the equation of the least squares regression line for the following sets of data:

3

Determine the equation of the least squares regression line given each set of data values below. Round and to two decimal places in your final answer.

(Hint: multiply the numerator and denominator of the formula for by )

The table below shows the average maintenance cost in rands of a certain model of car compared to the age of the car in years.

Age () |  |  |  |  |  |  |
Cost () |  |  |  |  |  |  |

Draw a scatter plot of the data.

Complete the table below, filling in the totals of each column in the final row:

Age () | Cost () |  |
1 |  |  |
3 |  |  |
5 |  |  |
6 |  |  |
8 |  |  |
9 |  |  |
10 |  |  |

Age () | Cost () |  |
1 |  |  | 1
3 |  |  | 9
5 |  |  | 25
6 |  |  | 36
8 |  |  | 64
9 |  |  | 81
10 |  |  | 100

Use your table to determine the equation of the least squares regression line. Round and to two decimal places.

Use your equation to estimate what it would cost to maintain this model of car in its year.

Use your equation to estimate the age of the car in the year where the maintenance cost totals over for the first time.

Therefore the maintenance cost will exceed for the first time when the car is aged 13.

Miss Colly has always maintained that there is a relationship between a learner's ability to understand the language of instruction and their marks in Mathematics. Since she teaches Mathematics through the medium of English, she decides to compare the Mathematics and English marks of her learners in order to investigate the relationship between the two marks. A sample of her data is shown in the table below:

English \% () | 28 | 33 | 30 | 45 | 45 | 55 | 55 | 65 | 70 | 76 | 65 | 85 | 90
Mathematics \% () | 35 | 36 | 34 | 45 | 50 | 40 | 60 | 50 | 65 | 85 | 70 | 80 | 90

Complete the table below, filling in the totals of each column in the final row:

English \% ()
Mathematics \% ()

English \% () | Mathematics \% () |  |
45 | 45 |  |
45 | 50 |  |
55 | 40 |  |
65 | 50 |  |
70 | 65 |  |
76 | 85 |  |
65 | 70 |  |
85 | 80 |  |
90 | 90 |  |

Use your table to determine the equation of the least squares regression line. Round and to two decimal places.

Use your equation to estimate the Mathematics mark of a learner who obtained % for English, correct to two decimal places.

Use your equation to estimate the English mark of a learner who obtained % for Mathematics, correct to two decimal places.

Foot lengths and heights of ten students are given in the table below.

Height (cm)

Foot length (cm)

Using foot length as your -variable, draw a scatter plot of the data.

Identify and describe any trends shown in the scatter plot.

Strong (or fairly strong), positive, linear trend

Find the equation of the least squares line using the formulae and draw the line on your graph. Round and to two decimal places in your final answer.

Foot length () | Height () |  |
27 |  |  | 729
23 |  |  | 529
20 |  |  | 400
28 |  |  | 784
22 |  |  | 484
20 |  |  | 400
24 |  |  | 576
26 |  |  | 676
29 |  |  | 841
22 |  |  | 484

Confirm your calculations above by finding the least squares regression line using a calculator.

Answer should be the same as c).

Use your equation to predict the height of a student with a foot length of .

Use your equation to predict the foot length of a student tall, correct to two decimal places.

Now that we have a precise technique for finding the line of best fit, we still do not know how well our line of best fit really fits our data. We can fit a least squares regression line to any bivariate data, even if the two variables do not show a linear relationship. If the fit is not “good”, our assumption of the and values in might be incorrect. Next, we will learn of a quantitative measure to determine how well our line really fits our data.

9.3 Correlation

The linear correlation coefficient, , is a measure which tells us the strength and direction of a relationship between two variables. The correlation coefficient . When , there is perfect negative correlation, when , there is no correlation and when there is perfect positive correlation.

Positive, strong
Positive, fairly strong
Positive, weak

Negative, fairly strong
Negative, weak
No correlation

NB. See the fifth bullet at the beginning of the chapter regarding the formula for the correlation coefficient.

The linear correlation coefficient can be calculated using the formula

where is the gradient of the least squares regression line,

is the standard deviation of the -values and

is the standard deviation of the -values.

This is known as the Pearson's product moment correlation coefficient. It is much easier to do on a calculator where you simply follow the procedure for the regression equation, and go on to find .

In general:

Positive | Strength | Negative
| no correlation |
| very weak |
| weak |
| moderate |
| strong |
| very strong |
| perfect correlation |

Correlation does not imply causation! Just because two variables are correlated does not mean that they are causally linked, i.e. if A and B are correlated, that does not mean A causes B, or vice versa. This is a common mistake made by many people, especially journalists looking for their next juicy story.

For example, ice cream sales and shark attacks are correlated. This does not mean that the sale of ice cream is somehow causing more shark attacks. Instead, a simpler explanation is that the warmer it is, the more likely people are to buy ice cream and the more likely people are to go to the beach as well, thus increasing the likelihood of a shark attack.

Video: 29DV

Worked example 9: The correlation coefficient

A cardiologist wanted to test the relationship between resting heart rate and the peak heart rate during exercise. Heart rate is measured in beats per minute (bpm). The following set of data was generated from 12 study participants after they had run on a treadmill at for 10 minutes.

Resting heart rate | 48 | 56 | 90 | 65 | 75 | 78 | 80 | 72 | 82 | 76 | 68 | 62
Peak heart rate | 138 | 136 | 180 | 150 | 151 | 161 | 155 | 154 | 175 | 158 | 145 | 155

Draw a scatter plot of the data. Use resting heart rate as your -variable.

Use your calculator to determine the equation of the line of best fit.

Estimate what the heart rate of a person with a resting heart rate of will be after exercise.

Without using your calculator, find the correlation coefficient, . Confirm your answer using your calculator.

What can you conclude regarding the relationship between resting heart rate and the heart rate after exercise?

Draw the scatter plot

Choose a suitable scale for the axes.

Draw the axes.

Plot the points.

Calculate the equation of the line of best fit

As you learnt previously, use your calculator to determine the values for and .

Therefore, the equation for the line of best fit is

Calculate the estimated value for

If , using our equation, the estimated value for is:

Calculate the correlation co-efficient

The formula for is:

We already know the value of and you know how to calculate by hand from worked example 5, so we are just left to determine the value for and . The formula for standard deviation is:

First, you need to determine and and then complete a table like the one below.

Resting heart rate () | Peak heart rate () |  |
48 | 138 | 529 |
56 | 136 | 225 |
90 | 180 | 361 |
65 | 150 | 36 |
75 | 151 | 16 |
78 | 161 | 49 |
80 | 155 | 81 |
72 | 154 | 1 |
82 | 175 | 121 |
76 | 158 | 25 |
68 | 145 | 9 |
62 | 155 | 81 |

Confirm your answer using your calculator

Once you know the method for finding the equation of the best line of fit on your calculator, finding the value for is trivial. After you have entered all your and values into your calculator, in STAT mode:

on a SHARP calculator: press [RCL] then [r] (the same key as [])

on a CASIO calculator: press [SHIFT] then [STAT], [5], [3] then []

Comment on the correlation coefficient

Therefore, there is a strong, positive, linear relationship between resting heart rate and peak heart rate during exercise. This means that the higher your resting heart rate, the higher your peak heart rate during exercise is likely to be.

Exercise 9.4

Determine the correlation coefficient by hand for the following data sets and comment on the strength and direction of the correlation. Round your answers to two decimal places.

Therefore, the correlation between and is negative but weak.

Therefore, the correlation between and is positive and moderate.

Therefore, the correlation between and is positive and very strong.

Using your calculator, determine the value of the correlation coefficient to two decimal places for the following data sets and describe the strength and direction of the correlation.

, negative, very strong.

, negative, weak.

, no correlation

, positive, very weak.

, positive, strong.

Calculate and describe the direction and strength of for each of the sets of data values below. Round all -values to two decimal places.

The geography teacher, Mr Chadwick, gave the data set below to his class to illustrate the concept that average temperature depends on how far a place is from the equator (known as the latitude). There are 90 degrees between the equator and the North Pole. The equator is defined as 0 degrees. Examine the data set below and answer the questions that follow.

City | Degrees N () | Average temp. () |  |  |  |
Cairo | 43 | 22 |  |  |  |
Berlin | 53 | 19 |  |  |  |
London | 40 | 18 |  |  |  |
Lagos | 6 | 32 |  |  |  |
Jerusalem | 31 | 23 |  |  |  |
Madrid | 40 | 28 |  |  |  |
Brussels | 51 | 18 |  |  |  |
Istanbul | 39 | 23 |  |  |  |
Boston | 43 | 23 |  |  |  |
Montreal | 45 | 22 |  |  |  |
Total: |  |  |  |  |

Copy and complete the table.

City | Degrees N () | Average temp. () |  |  |  |
Cairo | 43 | 22 | 946 |  |  |
Berlin | 53 | 19 |  |  |  |
London | 40 | 18 |  |  |  |
Lagos | 6 | 32 | 192 | 36 |  |
Jerusalem | 31 | 23 | 713 | 961 |  |
Madrid | 40 | 28 |  |  |  |
Brussels | 51 | 18 | 918 |  |  |
Istanbul | 39 | 23 | 897 |  |  |
Boston | 43 | 23 | 989 |  |  |
Montreal | 45 | 22 | 990 |  |  |
Total: | 391 | 228 |  |  |  |

Using your table, determine the equation of the least squares regression line. Round and to two decimal places in your final answer.

Use your calculator to confirm your equation for the least squares regression line.

Answer should be as above.

Using your table, determine the value of the correlation coefficient to two decimal places.

What can you deduce about the relationship between how far north a city is and its average temperature?

There is a strong, negative, linear correlation between how far north a city is (latitude) and average temperature.

Estimate the latitude of Paris if it has an average temperature of

A taxi driver recorded the number of kilometres his taxi travelled per trip and his fuel cost per kilometre in Rands. Examine the table of his data below and answer the questions that follow.

Distance () | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 20 | 25 | 30
Cost () |  |  |  |  |  |  |  |  |  |  | 2

Draw a scatter plot of the data.

Use your calculator to determine the equation of the least squares regression line and draw this line on your scatter plot. Round and to two decimal places in your final answer.

Using your calculator, determine the correlation coefficient to two decimal places.

Describe the relationship between the distance travelled per trip and the fuel cost per kilometre.

There is a very strong, negative, linear relationship between distance travelled per trip and the fuel cost per kilometre.

Predict the distance travelled if the cost per kilometre is .

The time taken, in seconds, to complete a task and the number of errors made on the task were recorded for a sample of 10 primary school learners. The data is shown in the table below. [Adapted from NSC Paper 3 Feb-March 2013]

Time taken to complete task (in seconds) | 23 | 21 | 19 | 9 | 15 | 22 | 17 | 14 | 21 | 18
Number of errors made |  |  |  |  |  |  |  |  |  |

Draw a scatter plot of the data.

What is the influence of more time taken to complete the task on the number of errors made?

When more time is taken to complete the task, the learners make fewer errors.

OR

When less time is taken to complete the task, the learners make more errors.

Determine the equation of the least squares regression line and draw this line on your scatter plot. Round and to two decimal places in your final answer.

Determine the correlation coefficient to two decimal places.

Predict the number of errors that will be made by a learner who takes 13 seconds to complete this task.

Comment on the strength of the relationship between the variables.

There is a strong negative relationship between the variables.

A recording company investigates the relationship between the number of times a CD is played by a national radio station and the national sales of the same CD in the following week. The data below was collected for a random sample of 10 CDs. The sales figures are rounded to the nearest 50. [NSC Paper 3 November 2012]

Number of times CD is played | 47 | 34 | 40 | 34 | 33 | 50 | 28 | 53 | 25 | 46
Weekly sales of the CD |  |  |  |  |  |  |  |  |  |

Draw a scatter plot of the data.

Determine the equation of the least squares regression line.

Calculate the correlation coefficient.

Predict, correct to the nearest 50, the weekly sales for a CD that was played 45 times by the radio station in the previous week.

Comment on the strength of the relationship between the variables.

There is a very strong positive relationship between the number of times that a CD was played and the sales of that CD in the following week.

9.4 Summary

Curve fitting is the process of fitting functions to data.

Intuitive curve fitting is performed by visually interpreting if the points on the scatter plot conform to a linear, exponential, quadratic or some other function.

The line of best fit or trend line is a straight line through the data which best approximates the available data points. This allows for the estimation of missing data values.

Interpolation is the technique used to predict values that fall within the range of the available data.

Extrapolation is the technique used to predict the value of variables beyond the range of the available data.

Linear regression analysis is a statistical technique of finding out exactly which linear function best fits a given set of data.

The least squares method is an algebraic method of finding the linear regression equation. The linear regression equation is written , where

The linear correlation coefficient, , is a measure which tells us the strength and direction of a relationship between two variables, determined using the equation:

The correlation coefficient . When , there is perfect negative correlation, when , there is no correlation and when , there is perfect positive correlation.

Exercise 9.5

The number of SMS messages sent by a group of teenagers was recorded over a period of a week. The data was found to be normally distributed with a mean of 140 messages and a standard deviation of 12 messages. [NSC Paper 3 Feb-March 2012]

Answer the following questions with reference to the information provided in the graph:

What percentage of teenagers sent less than 128 messages?

128 is 1 standard deviation to the left of the mean, therefore the percentage of teenagers who sent less than 128 messages is:

What percentage of teenagers sent between 116 and 152 messages?

116 minutes is 2 standard deviations from the mean, therefore %

152 minutes is 1 standard deviation from the mean, therefore %

Percentage of the teenagers who sent between 116 and 152 messages

A company produces sweets using a machine which runs for a few hours per day. The number of hours running the machine and the number of sweets produced are recorded.

Machine hours
Sweets produced

Find the linear regression equation for the data, and estimate the machine hours needed to make sweets.

Using a calculator, the equation is:

Therefore, the estimated number of machine hours needed to make 300 sweets is:

The profits of a new shop are recorded over the first 6 months. The owner wants to predict his future sales. The profits by month so far have been ; ; ; ; ; .

Calculate the linear regression function for the data, using profit as your -variable. Round and to two decimal places.

Give an estimate of the profits for the next two months.

The owner wants a profit of . Estimate how many months this will take.

It will take 13 months to reach a profit of .

A fast food company produces hamburgers. The number of hamburgers made and the costs are recorded over a week.

Hamburgers made
Costs

Find the linear regression function that best fits the data. Use hamburgers made as your -variable and round and to two decimal places.

Calculate the value of the correlation coefficient, correct to two decimal places, and comment on the strength and direction of the correlation.

There is a strong, positive, linear correlation.

If the total cost in a day is , estimate the number of hamburgers produced. Round your answer down to the nearest whole number.

Therefore 528 burgers are produced.

What is the cost of hamburgers?

A collection of data related to an investigation into biceps length and height of students was recorded in the table below. Answer the questions to follow.

Length of right biceps (cm)
Height (cm)

Draw a scatter plot of the data set.

Calculate equation of the line of regression.

Draw the regression line onto the graph.

Calculate the correlation coefficient

What conclusion can you reach, regarding the relationship between the length of the right biceps and height of the students in the data set?

The length of the right biceps and the height of the students have a strong, positive linear relationship.

A class wrote two tests, and the marks for each were recorded in the table below. Full marks in the first test was , and the second test was out of .

Learner | Test 1 | Test 2
| (Full marks: ) | (Full marks: )

Is there a strong correlation between the marks for the first and second test? Show why or why not.

Using a calculator, which is a very strong, positive, linear correlation between the marks of the first and the second test.

One of the learners (in Row 18) did not write the second test. Given her mark for the first test, calculate an expected mark for the second test. Round the mark up to the nearest whole number.

Using a calculator, the least squares regression line equation is:

Therefore, the expected mark for the second test for the learner in Row 18 is:

Therefore the expected mark for the learner in row 18 for the second test is 11 out of 30.

Lindiwe works for Eskom, the South African power distributor. She knows that on hot days more electricity than average is used to cool houses. In order to accurately predict how much more electricity needs to be produced, she wants to determine the precise nature of the relationship between temperature and electricity usage.

The data below shows the peak temperature in degrees Celsius on ten consecutive days during summer and the average number of units of electricity used by a number of households. Examine her data and answer the questions that follow.

Peak temp. () | 32 | 40 | 30 | 28 | 25 | 38 | 36 | 20 | 24 | 26
Average no. of units () | 37 | 45 | 35 | 30 | 20 | 40 | 38 | 15 | 20 | 22

Draw a scatter plot of the data.

Using the formulae for and , determine the equation of the least squares line.

Average no. of units () | Peak temp. () |  |
37 | 32 |  |
45 | 40 |  |
35 | 30 |  | 1225
30 | 28 | 840 | 900
20 | 25 | 500 | 400
40 | 38 |  |
38 | 36 |  |
15 | 20 |  |
20 | 24 |  |
22 | 26 |  |

Determine the value of the correlation coefficient, , by hand.

We have already calculated the value of by hand in the question above, so we are left to determine and .

Average no. of units () | Peak temp. () |  |
32 | 37 |  |
40 | 45 |  |
30 | 35 |  |
28 | 30 |  |
25 | 20 |  |
38 | 40 |  |
36 | 38 |  |
20 | 15 |  |
24 | 20 |  |
26 | 22 |  |

What can Lindiwe conclude about the relationship between peak temperature and the number of electricity units used?

There is a very strong, positive, linear correlation between peak temperature and the average number of electricity units a household uses.

Predict the average number of units of electricity used by a household on a day with a peak temperature of . Give your answer correct to the nearest unit and identify what this type of prediction is called.

The value we were asked to predict is outside the range of the available data. This is known as extrapolation.

Lindiwe suspected that the relationship between temperature and electricity consumption was not linear for all temperatures. She then decided to collect data for peak temperatures down to . Examine the graph of her data below and identify which type of function would best fit the data and describe the nature of the relationship between temperature and electricity for the newly available data.

A quadratic function would best fit the data. At about average household electricity usage is at its minimum. As the peak temperature gets colder or warmer than this point, electrcity usage increases.

Lindiwe is asked by her superiors to determine which day is best to perform maintenance on one of their power plants. She determined that the equation best fit her data. Use her equation to estimate the peak temperature and average no. of units used on the day when the least amount of electricity generation is required.

This question requires us to find the minimum value of the quadratic equation. There are a number of ways to do this, two are shown below:

The first method is using the formula :

The first step is to write the equation in the form: . Our equation is already in this form, so we can immediately substitute the values into the formula for .

To find , we substitute our -value into the quadratic equation:

Another method is using differentiation:

The first step is to write the equation in the form: . Our equation is already in this form, so we can immediately differentiate the equation.

At the turning point, , therefore we can now solve for :

The -value can now be substituted into the quadratic equation to find :

Therefore the peak temperature when electricity demand is at its lowest is and the respective average household electricity usage is .

Below is a list of data concerning 12 countries and their respective carbon dioxide emission levels per person per annum (measured in tonnes) and the gross domestic product (GDP is a measure of products produced and services delivered within a country in a year) per person (in US dollars). Data sourced from the World Bank and the US Department of Energy's Carbon Dioxide Information Analysis Center.

| emmissions per capita (x) | GDP per capita (y)
South Africa |  |
Thailand |  |
Italy |  |
Australia |  |
China |  |
India |  |
Canada |  |
United Kingdom |  |
United States |  |
Saudi Arabia |  |
Iran |  |
Indonesia |  |

Draw a scatter plot of the data set.

Draw your estimate of the line of best fit on your scatter plot and determine the equation of your line of best fit.

The -intercept is approximately 1000. At , is approximately . Therefore,

The equation for the line of best fit:

Use your calculator to determine the equation for the least squares regression line. Round and to two decimal places in your final answer.

and , therefore

Use your calculator to determine the correlation coefficient, . Round your answer to two decimal places.

What conclusion can you reach regarding the relationship between emissions per annum and GDP per capita for the countries in the data set?

There is a strong, positive, linear correlation between emissions per annum and GDP per capita for the countries in the data set.

Kenya has a GDP per capita of . Use your equation of the least squares regression line to estimate the annual emissions of Kenya correct to two decimal places.

A group of students attended a course in Statistics on Saturdays over a period of 10 months. The number of Saturdays on which a student was absent was recorded against the final mark the student obtained. The information is shown in the table below. [Adapted from NSC Paper 3 Feb-March 2012]

Number of Saturdays absent | 0 | 1 | 2 | 2 | 3 | 3 | 5 | 6 | 7
Final mark (as ) |  |  |  |  |  |  |  |  |

Draw a scatter plot of the data.

Determine the equation of the least squares line and draw it on your scatter plot.

Calculate the correlation coefficient.

Comment on the trend of the data.

The greater the number of Saturdays absent, the lower the mark.

Predict the final mark of a student who was absent for four Saturdays.

Grant and Christie are training for a half-marathon together in 8 weeks time. Christie is much fitter than Grant but she has challenged him to beat her time at the race. Grant has begun a rigid training programme to try and improve his time.

Time taken to complete a half marathon was recorded each Sunday. The first recorded Sunday is denoted as week 1. The half-marathon takes place on the eighth Sunday, i.e. week 8. Examine the data set in the table below and answer the questions the follow.

Week | 1 | 2 | 3 | 4 | 5 | 6
Grant's time (HH:MM) | 02:01 | 01:59 | 01:55 | 01:53 | 01:47 | 01:42
Christie's time (HH:MM) | 01:40 | 01:42 | 01:38 | 01:39 | 01:37 | 01:35

Draw a scatter plot of the data sets. Include Grant and Christie's data on the same set of axes. Use a to denote Grant's data points and to denote Christie's data points. Convert all times to minutes.

Comment on and compare any trends that you observe in the data.

Both data sets show negative, linear trends. The trend in Grant's data appears to be more rapidly decreasing than the trend in Christie's data.

Determine the equations of the least squares regression lines for Grant's data and Christie's data. Draw these lines on your scatter plot. Use a different colour for each.

Calculate the correlation coefficient and comment on the fit for each data set.

Assuming the observed trends continue, will Grant beat Christie in the race?

Grant will beat Christie when . To find where the trends intersect, we equate each .

The race takes place in week 8. , therefore, Grant will be unable to beat Christie's time when the race takes place.

Assuming the observed trends continue, extrapolate the week in which Grant will be able to run a half-marathon in less time than Christie.

See answer to e). Grant will be able to beat Christie's time in the ninth week.

Chapter 10: Probability

This chapter provides good opportunity for experiments and activities in classroom where you can illustrate theoretical probability and number of possible arrangements in practice. Real-life examples have been used extensively in the exercise sections and you may choose to illustrate some of these concepts experimentally in class.

The terminology and usage of language in this section can be confusing, especially to second-language speakers. Discuss terminology regularly and emphasise the careful reading of questions.

Union and intersection symbols have been included, but “and” and “or” is the preferred notation in CAPS.

Make sure to outline the differences between 'and', 'or', 'only' and 'both'. For example, there may be no difference between tea and coffee drinkers and tea or coffee drinkers in common speech but in probability, the 'and' and 'or' have very specific meanings. Tea and coffee drinkers refers to the intersection of tea drinkers, i.e. those who drink both beverages, while tea or coffee drinkers refers to the union, i.e. those who drink only tea, those who drink only coffee and those who drink both.

Some learners may find factorial notation challenging. A lengthy problem-set has been provided to try and iron out some common misconceptions, such as or , but you may have to go over this more slowly with some learners.

Note that the formula for the arrangement of different items in different places i.e. is NOT included in CAPS and learners should therefore be able to solve these problems logically.

When applying the fundamental counting principle to probability problems, learners may struggle with knowing when to multiply and when to add probabilities. When a number of different outcomes fit a desired result, the probabilities of each outcome are added. When determining the probability of two or more events occurring, their individual probabilities are multiplied.

10.1 Revision

Terminology

Outcome: a single observation of an uncertain or random process (called an experiment). For example, when you accidentally drop a book, it might fall on its cover, on its back or on its side. Each of these options is a possible outcome.

Sample space of an experiment: the set of all possible outcomes of the experiment. For example, the sample space when you roll a single -sided die is the set . For a given experiment, there is exactly one sample space. The sample space is denoted by the letter .

Event: a set of outcomes of an experiment. For example, if you have a standard deck of 52 cards, an event may be picking a spade card or a king card.

Probability of an event: a real number between and inclusive of and that describes how likely it is that the event will occur. A probability of means the outcome of the experiment will never be in the event set. A probability of means the outcome of the experiment will always be in the event set. When all possible outcomes of an experiment have equal chance of occurring, the probability of an event is the number of outcomes in the event set as a fraction of the number of outcomes in the sample space. To calculate a probability, you divide the number of favourable outcomes by the total number of possible outcomes.

Relative frequency of an event: the number of times that the event occurs during experimental trials, divided by the total number of trials conducted. For example, if we flip a coin times and it landed on heads times, then the relative frequency of the heads event is .

Union of events: the set of all outcomes that occur in at least one of the events. For events called and , we write the union as “”. Another way of writing the union is using set notation: . For example, if is all the countries in Africa and is all the countries in Europe, or is all the countries in Africa and Europe.

Intersection of events: the set of all outcomes that occur in all of the events. For events called and , we write the intersection as “”. Another way of writing the intersection is using set notation: . For example, if is soccer players and is cricket players, and refers to those who play both soccer and cricket.

Mutually exclusive events: events with no outcomes in common, that is is an empty set. Mutually exclusive events can never occur simultaneously. For example the event that a number is even and the event that the same number is odd are mutually exclusive, since a number can never be both even and odd.

Complementary events: two mutually exclusive events that together contain all the outcomes in the sample space. For an event called , we write the complement as “”. Another way of writing the complement is as .

Dependent and independent events: two events, and , are independent if the outcome of the first event does not influence the outcome of the second event. For example, if you flip a coin and it lands on tails and flip it again and it lands on heads, neither outcome influences the other. Two events, C and D, are dependent if the outcome of one event influences the outcome of the other. For example, if your lunchbox contains sandwiches and apples, when you eat one of the items, this reduces the number of choices you have when deciding to eat a second item.

Video: 29FR

10.2 Identities

The addition rule (also called the sum rule) for any events, and is This rule relates the probabilities of events with the probabilities of their union and intersection.

The addition rule for mutually exclusive events is This rule is a special case of the previous rule. Because the events are mutually exclusive, .

The complementary rule is This rule is a special case of the previous rule. Since and are complementary, .

The product rule for independent events and is:

If two events and are dependent then:

Just because two events are mutually exclusive does not necessarily mean that they are independent. To test whether events are mutually exclusive, always check that . To test whether events are independent, always check that . See the exercises below for examples of events that are mutually exclusive and independent in different combinations.

Worked example 1: Dependent and independent events

Write down which of the following events are dependent and which are independent:

The student council chooses a head student and then a deputy head student.

A bag contains blue marbles and red marbles. You take a red marble out of the bag and then throw it back in again before you take another marble out of the bag.

Ask the question: Did the available choices change for the second event because of the first event?

Yes, because after selecting the head student there are fewer council members available to choose for the deputy head student position. Therefore the two events are dependent.

No, because when you throw the first marble back into the bag, there are the same number and colour composition of choices for the second marble. Therefore the two events are independent.

Worked example 2: Independent and dependent events

A bag contains yellow and black beads. We remove a random bead from the bag, record its colour and put it back into the bag. We then remove another random bead from the bag and record its colour.

What is the probability that the first bead is yellow?

What is the probability that the second bead is black?

What is the probability that the first bead is yellow and the second bead is black?

Are the first bead being yellow and the second bead being black independent events?

Probability of a yellow bead first

Since there is a total of beads, of which are yellow, the probability of getting a yellow bead is

Probability of a black bead second

The problem states that the first bead is placed back into the bag before we take the second bead. This means that when we draw the second bead, there are again a total of beads in the bag, of which are black. Therefore the probability of drawing a black bead is

Probability of yellow first and black second

When drawing two beads from the bag, there are possibilities. We can get

a yellow bead and then another yellow bead;

a yellow bead and then a black bead;

a black bead and then a yellow bead;

a black bead and then another black bead.

We want to know the probability of the second outcome, where we have to get a yellow bead first. Since there are yellow beads and beads in total, there are ways to get a yellow bead first. Now we put the first bead back, so there are again yellow beads and black beads in the bag. Therefore there are ways to get a black bead second if the first bead was yellow. This means that there are ways to get a yellow bead first and a black bead second. So, the probability of getting a yellow bead first and a black bead second is .

Dependent or independent?

According to the definition, events are independent if and only if

In this problem:

Since , the events are independent.

Video: 29FS

Worked example 3: Independent and dependent events

In the previous example, we picked a random bead and put it back into the bag before continuing. This is called sampling with replacement. In this worked example, we will follow the same process, except that we will not put the first bead back into the bag. This is called sampling without replacement.

So, from a bag with red and green beads, we remove a random bead and record its colour. Then, without putting back the first bead, we remove another random bead from the bag and record its colour.

What is the probability that the first bead is red?

What is the probability that the second bead is green?

What is the probability that the first bead is red and the second bead is green?

Are the first bead being red and the second bead being green independent events?

Count the number of outcomes

We will examine the number ways in which we can get the different possible outcomes when removing beads. The possible outcomes are

a red bead and then another red bead (RR);

a red bead and then a green bead (RG);

a green bead and then a red bead (GR);

a green bead and then another green bead (GG).

For the first outcome, we have to get a red bead first. Since there are red beads and beads in total, there are ways to get a red bead first. After we have taken out a red bead, there are now red beads and green beads left. Therefore there are ways to get a red bead second if the first bead was also red. This means that there are ways to get a red bead first and a red bead second. The probability of the first outcome is .

For the second outcome, we have to get a red bead first. As in the first outcome, there are ways to get a red bead first; and there are now red beads and green beads left. Therefore there are ways to get a green bead second if the first bead was red. This means that there are ways to get a red bead first and a green bead second. The probability of the second outcome is .

In the third outcome, the first bead is green and the second bead is red. There are ways to get a green bead first; and there are now green beads and red beads left. Therefore there are ways to get a red bead second if the first bead was green. This means that there are ways to get a red bead first and a green bead second. The probability of the third outcome is .

In the fourth outcome, the first and second beads are both green. Since there are green beads and beads in total, there are ways to get a green bead first. After we have removed a green bead, red beads and green beads remain in the bag. Therefore there are ways to get a green bead second if the first bead was also green. This means that there are ways to get a green bead first and a green bead second. Therefore the probability of the fourth outcome is .

To summarise, these are the possible outcomes and their probabilities:

first bead red and second bead red (RR): ;

first bead red and second bead green (RG): ;

first bead green and second bead red (GR): ;

first bead green and second bead green (GG): .

Probability of a red bead first

To determine the probability of getting a red bead on the first draw, we look at all of the outcomes that contain a red bead first. These are

a red bead and then another red bead (RR);

a red bead and then a green bead (RG).

The probability of the first outcome is and the probability of the second outcome is . By adding these two probabilities, we see that the probability of getting a red bead first is

This is the same as in the previous worked example, which should not be too surprising since the probability of the first bead being red is not affected by whether or not we put it back into the bag before drawing the second bead.

Probability of a green bead second

To determine the probability of getting a green bead on the second draw, we look at all of the outcomes that contain a green bead second. These are

a red bead and then a green bead (RG);

a green bead and then another green bead (GG).

The probability of the first outcome is and the probability of the second outcome is . By adding these two probabilities, we see that the probability of getting a green bead second is

Probability of red first and green second

We have already calculated the probability that the first bead is red and the second bead is green (RG). It is .

Dependent or independent?

According to the definition, events are independent if and only if

In this problem:

Since , the events are dependent.

Worked example 4: The addition rule for mutually exclusive events

A sample space, , consists of all natural numbers less than . is the event of drawing an even number at random. is the event of randomly drawing a prime number. Are and mutually exclusive events? Prove this using the addition rule.

Write down the sample space

The sample space contains all the natural numbers less than .

Write down the events

The even natural numbers less than are

The prime numbers less than are

We can already see from writing down our event sets that and share the event and are thus not mutually exclusive. However, the question asked us to prove this using the addition rule so let's go ahead and do that.

Compute the probabilities

The probability of an event is the number of outcomes in the event set divided by the number of outcomes in the sample space. There are outcomes in the sample space.

Since there are outcomes in the event set, .

Since there are outcomes in the event set, .

The event that is a prime number or an even number is the union of the above two event sets. There are elements in the union of the event sets, so .

Are the two events mutually exclusive?

To test whether two events are mutually exclusive, we can use the addition rule. For two mutually exclusive events, Therefore the intersection of and is nonzero. This means that the events and are not mutually exclusive.

Worked example 5: The addition rule

The probability that a person drinks tea is . The probability that a person drinks coffee is . The probability that a person drinks tea, coffee or both is . Determine the probability that a person drinks tea and coffee.

Determine if the events are mutually exclusive

Let the probability that a person drinks tea and the probability that a person drinks coffee .

From the information provided in the question, we know that:

Therefore the events are not mutually exclusive.

Compute the probability that a person drinks tea and coffee

Using the addition rule, we know that:

Therefore the probability that a person drinks tea and coffee is .

Video: 29FT

Worked example 6: The complementary rule

Joe wants to open a tuckshop at his school but is not sure which cool drinks to stock. Before opening, he interviewed a sample of learners to determine what types of cool drinks they like. From his research, he determined that the probability that a learner drinks cola is , the probability that a learner drinks lemonade is and the probability that a learner drinks neither is . Determine:

the probability that a learner drinks cola and lemonade.

the probability that a learner drinks only cola or only lemonade.

Determine the probability that a learner drinks cola or lemonade

Let the probability that a learner drinks cola and the probability that a learner drinks lemonade .

From the information provided in the question, we know that:

Using the complementary rule:

Calculate the probability that a learner drinks cola and lemonade

Using the addition rule:

The probability that a learner drinks both cola and lemonade is .

Determine the probability that a learner drinks only cola or only lemonade

This question requires us to calculate the probability that a learner likes lemonade or cola but not both of them. We can write this as: since a learner can like either cola or lemonade but not both.

We already know and , therefore the probability of a learner drinking only cola or only lemonade is .

Exercise 10.1

Determine whether the following events are dependent or independent and give a reason for your answer:

Joan has a box of yellow, green and orange sweets. She takes out a yellow sweet and eats it. Then, she chooses another sweet and eats it.

The two events are dependent because there are fewer sweets to choose from when she picks the second time.

Vuzi throws a die twice.

The two events are independent because the outcome of the first throw has no effect on the outcome of the second throw.

Celia chooses a card at random from a deck of 52 cards. She is unhappy with her choice, so she places the card back in the deck, shuffles it and chooses a second card.

The two events are independent because the set of cards in the deck is unchanged each time Celia chooses one randomly.

Thandi has a bag of beads. She randomly chooses a yellow bead, looks at it and then puts it back in the bag. Then she randomly chooses another bead and sees that it is red and puts it back in the bag.

The two events are independent because there are the same collection of beads each time Thandi chooses one.

Mark has a container with calculators. Some of them work and some are broken. He randomly chooses a calculator and sees that it does not work and throws it away. He then chooses another calculator, sees that it works and keeps it.

The two events are dependent because Mark has fewer calculators to choose from when he picks again.

Given that ; and ,

are events and mutually exclusive? Give a reason for your answer.

For the events to be mutually exclusive must be equal to . In this case = , so the events are not mutually exclusive.

are the events and independent? Give a reason for your answer.

For events to be independent: . . Therefore the events are independent.

In the following examples, are and dependent or independent?

Therefore the events are dependent.

Therefore the events are independent.

.

Are and mutually exclusive?

For and to be mutually exclusive: .

The events are therefore not mutually exclusive.

Are and independent?

For and to be independent,

Therefore the events are independent.

Simon rolls a die twice. What is the probability of getting:

two threes.

a prime number then an even number.

There are 3 possible prime numbers on a die, namely, 2, 3, and 5, and there are 3 possible even numbers, namely, 2, 4, and 6.

no threes.

If no threes are rolled then, for each of the events, 5 possibilities remain.

only one three.

In the sample space there are 36 possible outcomes. There are two ways of getting only one three: either getting a 3 on the first throw and a number other than three on the second throw or a 3 on the second throw and a number other than 3 on the first throw. The outcomes containing only one 3 are: ; ; ; ; ; ; ; ; ; .

at least one three.

In the sample space there are 36 possible outcomes. The outcomes containing at least one 3 are: ; ; ; ; ; ; ; ; ; ; .

The Mandalay Secondary soccer team has to win both of their next two matches in order to qualify for the finals. The probability that Mandalay Secondary will win their first soccer match against Ihlumelo High is and the probability of winning their second soccer match against Masiphumelele Secondary is . Assume each match is an independent event.

What is the probability they will progress to the finals?

What is the probability they will not win either match?

To calculate the probability of not winning a match use:

This solution makes use of the complementary rule which students should be familiar with. We will revise the rule in more detail later.

What is the probability they will win only one of their matches?

There are two possible outcomes: win-not win or not win-win. Let win .

You were asked to assume that the matches are independent events but this is unlikely in reality. What are some factors you think may result in the outcome of the matches being dependent?

This is an open-ended question designed to get learners to think critically about the dependent or independent nature of real life events. Example answers could include injuries to or suspensions of players during the first match, team morale if they win or lose the first match, etc.

A pencil bag contains 2 red pens and 4 green pens. A pen is drawn from the bag and then replaced before a second pen is drawn. Calculate:

The probability of drawing a red pen first if a green pen is drawn second.

The events are independent so:

The probability of drawing a green pen second if the first pen drawn was red.

The events are independent so:

The probability of drawing a red pen first and a green pen second.

A lunch box contains sandwiches and apples. Vuyele chooses a food item randomly and eats it. He then chooses another food item randomly and eats that. Determine the following:

The probability that the first item is a sandwich.

The probability that the first item is a sandwich and the second item is an apple.

First item sandwich and second item apple (SA):

The probability that the second item is an apple.

There are two possible outcomes of getting an apple second:

first item sandwich and second item apple (SA):

first item apple and second item apple (AA):

Are the events in a) and c) dependent? Confirm your answer with a calculation.

Therefore the events are dependent.

Given that ; and , determine by calculation whether events and are:

mutually exclusive

Therefore, and are not mutually exclusive.

independent

Therefore and are independent.

and are two events in a sample space where ; and . Determine the value of if:

and are mutually exclusive

For and to be mutually exclusive:

and are independent

For and to be independent:

and are two events in sample space where ; ; and . Determine:

whether events and independent. Justify your answer with a calculation.

For independent events

Therefore and are independent.

The probability that a Mathematics teacher is absent from school on a certain day is . The probability that the Science teacher will be absent that same day is .

Do you think these two events are independent? Give a reason for your answer.

Learner dependent. For example: No, there could a bug or illness spreading through the school, therefore the absence of both teachers may be dependent.

Assuming the events are independent, what is the probability that the Mathematics teacher or the Science teacher is absent?

Let the probability that the Mathematics teacher is absent and the probability that the Science teacher is absent .

What is the probability that neither the Mathematics teacher nor the Science teacher is absent?

Langa Cricket Club plays two cricket matches against different clubs. The probability of winning the first match is and the probability of winning the second match is . Assuming the results of the matches are independent, calculate the probability that Langa Cricket Club will:

win both matches.

Let the probability of winning the first match and the probability of winning the second match.

not win the first match.

win one or both of the two matches.

win neither match.

not win the first match and win the second match.

Two teams are working on the final problem at a Mathematics Olympiad. They have 10 minutes remaining to finish the problem. The probability that team A will finish the problem in time is % and the probability that team B will finish the problem in time is %. Calculate the probability that both teams will finish before they run out of time.

Let the probability that team A will finish and the probability team B will finish . The teams are working separately, therefore the two events are independent.

Thabo and Julia were arguing about whether people prefer tea or coffee. Thabo suggested that they do a survey to settle the dispute. In total, they surveyed 24 people and found that 8 of them preferred to drink coffee and 12 of them preferred to drink tea. The number of people who drink tea, coffee or both is 16. Determine:

the probability that a person drinks tea, coffee or both.

Let be the number of people who drink coffee and be the number of people who drink tea.

the probability that a person drinks neither tea nor coffee.

the probability that a person drinks coffee and tea.

the probability that a person does not drink coffee.

whether the event that a person drinks coffee and the event that a person drinks tea are independent.

Therefore the events are independent.

10.3 Tools and Techniques

Venn diagrams are used to show how events are related to one another. A Venn diagram can be very helpful when doing calculations with probabilities. In a Venn diagram each event is represented by a shape, often a circle or a rectangle. The region inside the shape represents the outcomes included in the event and the region outside the shape represents the outcomes that are not in the event.

Tree diagrams are useful for organising and visualising the different possible outcomes of a sequence of events. Each branch in the tree shows an outcome of an event, along with the probability of that outcome. For each possible outcome of the first event, we draw a line where we write down the probability of that outcome and the state of the world if that outcome happened. Then, for each possible outcome of the second event we do the same thing. The probability of a sequence of outcomes is calculated as the product of the probabilities along the branches of the sequence.

Two-way contingency tables are a tool for keeping a record of the counts or percentages in a probability problem. Two-way contingency tables are especially helpful for figuring out whether events are dependent or independent.

Worked example 7: Venn diagrams

There are 200 boys in Grade 12 at Marist Brothers High School. Their participation in sport can be broken down as follows:

107 play rugby

90 play soccer

63 play cricket

35 play rugby and soccer

23 play rugby and cricket

15 play rugby, soccer and cricket

190 boys play rugby, soccer or cricket

How many boys do not play any of these sports?

Draw a Venn diagram to illustrate the given information and use it to answer the following questions:

How many boys play soccer and cricket, but not rugby?

What is the probability that a randomly chosen Grade 12 boy at Marist Brothers High School will take part in at least two of the sports: rugby, soccer or cricket? Give your answer correct to 3 decimal places.

Calculate the number of boys playing none of the given sports

In order to calculate the number of boys playing none of the sports, we subtract the number of boys playing any of the three sports from the total number of boys in the sample space.

Therefore boys do not play rugby, cricket or soccer.

Draw the outline of the Venn diagram

Let the sample space; rugby; soccer and cricket. Put this information on a Venn diagram:

Calculate the counts for the different groupings

The following groupings exist:

Rugby, cricket and soccer:

Rugby and soccer but not cricket:

Rugby and cricket but not soccer:

Cricket and soccer but not rugby:

Only rugby:

Only soccer:

Only cricket:

Not rugby, cricket or soccer:

Fill in the counts on the Venn diagram

Calculate the unknown values

Since 190 of the boys play at least one of the sports, using the values on our Venn diagram, we can set up the following equation to solve for .

We know that:

Therefore there are 12 boys who play cricket and soccer but not rugby.

Calculate the probability that a randomly chosen Grade 12 boy plays at least two of the given sports

We know the number of boys who play two or more of rugby, cricket or soccer and we know the total number of boys. Therefore, we can calculate the probability using the following equation:

Therefore the probability that a randomly chosen Grade 12 boy plays at least 2 of either rugby, cricket or soccer or %

Video: 29GD

Worked example 8: Tree diagrams

The probability that the floor of a supermarket will be wet when it opens in the morning is % and there is a % probability of the floor being very wet. The probability that a person will slip and fall if the floor is dry is % and a person is three times as likely to fall if the floor is wet. If the floor is very wet, the probability that a person will fall is . Draw a tree diagram to represent the given information, showing the probabilities of each outcome, and use it to answer the following questions:

What is the probability that a person will fall on any given day?

What is the probability that a person will not fall on any given day?

Are the events of the floor being dry and a person falling independent? Justify your answer with a calculation.

Identify the events

There are three outcomes for the floor, namely, dry, wet and very wet, and two outcomes for a person, namely fall or not fall.

Draw the first level of the tree diagram

This tree diagram shows the possible outcomes and probabilities of the status of the floor.

Draw the second level of the tree diagram

This tree diagram shows the possible outcomes and probabilities based on whether the floor is very wet, wet or dry. Remember that the sum of the probabilities for any set of branches is 1. Use this as a logical check whenever you are constructing a tree diagram.

Compute the probabilities of the various outcomes

We can calculate the probability of each outcome by multiplying the probabilities along the path from the start of the tree to the end of the branch containing the desired outcome.

Compute the probability of falling or not falling

We can calculate the probability of falling or not falling by adding the probabilities of all the desired outcomes.

Therefore the probability of falling on a given day is % and the probability of not falling is %.

Determine whether the floor being dry and a person falling are independent events

Logically, it appears that these events are dependent but the question asked us to prove this using a calculation. We can do this using the rule for independent events:

Therefore we can conclude that the floor being dry and a person falling are dependent events.

Exercise 10.2

A survey was done on a group of learners to determine which type of TV shows they enjoy: action, comedy or drama. Let action, comedy and drama. The results of the survey are shown in the Venn diagram below.

Study the Venn diagram and determine the following:

the total number of learners surveyed

the number of learners who do not enjoy any of the mentioned types of TV shows

the probability of a learner enjoying at least two types of TV shows

Describe, in words, the meaning of each of the questions c) to h) in the context of this problem.

: the probability that learners do not enjoy action TV shows

: the probability that learners enjoy action or drama TV shows

: the probability that learners enjoy action, drama and comedy TV shows

: the probability that learners do not enjoy action and drama TV shows

: the probability that learners enjoy action TV shows or do not enjoy comedy TV shows

: the probability that learners do not enjoy action or comedy TV shows

At Thandokulu Secondary School, there are are learners in Grade 12, 270 of whom take one or more of Mathematics, History and Economics. The subject choice is such that everybody who takes Physical Sciences must also take Mathematics and nobody who takes Physical Sciences can take History or Economics. The following is known about the number of learners who take these subjects:

70 take History

50 take Economics

120 take Physical Sciences

200 take Mathematics

20 take Mathematics and History

10 take History and Economics

25 take Mathematics and Economics

learners take Mathematics and History and Economics

Represent the information above in a Venn diagram. Let Mathematics be , History be , Physical Sciences be and Economics be .

Determine the number of learners, , who take Mathematics, History and Economics.

Therefore learners take Mathematics, History and Economics.

Determine and state in words what your answer means.

This is the probability that a learner does not take Mathematics, History or Economics.

Determine the probability that a learner takes at least two of these subjects.

This question requires us to find the sum of the probabilities of all the learners who take at least two subjects. This includes the intersection of each of the subjects.

A group of people were asked about the kind of sports they watch on television. The information collected is given below:

watch rugby, cricket or soccer

watch rugby, cricket and soccer

watch rugby and cricket

watch rugby and soccer

watch rugby

watch cricket

watch soccer

watch cricket and soccer but not rugby

Represent all the above information in a Venn diagram. Let rugby watchers , cricket watchers and soccer watchers .

Find the value of .

Determine

Determine

Are watching cricket and watching rugby independent events? Confirm your answer using a calculation.

Therefore watching rugby and watching cricket are dependent events.

There are boys and girls in the English class. Each lesson, two learners are randomly chosen to do an oral.

Represent the composition of the English class in a tree diagram. Include all possible outcomes and probabilities.

Calculate the probability that a boy and a girl are chosen to do an oral in any particular lesson.

Calculate the probability that at least one of the learners chosen to do an oral in any particular lesson is male.

Note: This question can be answered by subtracting the outcome not containing a boy (girl; girl) from 1 (shown below) or by adding the three outcomes which include a boy. Either method is correct.

Are the events picking a boy first and picking a girl second independent or dependent? Justify your answer with a calculation.

Therefore picking a boy first and picking a girl second are dependent events.

During July in Cape Town, the probability that it will rain on a randomly chosen day is . Gladys either walks to school or gets a ride with her parents in their car. If it rains, the probability that Gladys’ parents will take her to school by car is . If it does not rain, the probability that Gladys' parents will take her to school by car is

Represent the above information in a tree diagram. On your diagram show all the possible outcomes and respective probabilities.

What is the probability that it is a rainy day and Gladys walks to school?

What is the probability that Gladys' parents take her to school by car?

There are two types of property burglaries: burglary of private residences and burglary of business premises. In Metropolis, burglary of a private residence is four times as likely as that of a business premises. The following statistics for each type of burglary were obtained from the Metropolis Police Department:

Burglary of private residences

Following a burglary:

% of criminals are arrested within .

% of criminals are arrested after .

% of criminals are never arrested for that particular burglary.

Burglary of business premises

Following a burglary:

% of criminals are arrested within .

% of criminals are arrested after .

% of criminals are never arrested for that particular burglary.

Represent the information above in a tree diagram, showing all outcomes and respective probabilities.

Calculate the probability that a private home is burgled and nobody is arrested.

Calculate the probability that burglars of private homes and business premises are arrested.

Use your answer in the previous question to construct a tree diagram to calculate the probability that a burglar is arrested after at most three burglaries.

This answer could also be reached by subtracting the probability of not being arrested after three burglaries from 1:

We will use this principle to answer the next question.

Determine after how many burglaries a burglar has at least a

% chance of being arrested.

% chance of being arrested.

Let the number of burglaries

After burglaries, there will be at least a % chance of being arrested.

After burglaries, there will be at least a % chance of being arrested.

Worked example 9: Two-way contingency tables

The table below shows the results of testing two different treatments on 240 fruit trees which have a disease causing the trees to die. Treatment involves the careful removal of infected branches and treatment involves removing infected branches as well as spraying the tree with antibiotic.

| Tree dies within 4 years | Tree lives 4 years | Total
Treatment A |  |  |
Treatment B |  |  |
Total |  |  |

Fill in the missing values on the table.

What is the probability the a tree received treatment B?

What is the probability that a tree will live beyond 4 years?

What is the probability that a tree is given treatment B and lives beyond 4 years?

Of the trees who were given treatment B, what is the probability that a tree lives beyond 4 years?

Are a tree given treatment B and living beyond 4 years independent events? Justify your answer with a calculation.

Complete the contingency table

Since each column has to sum to its total, we can work out the number of trees which fall into each category for treatments A and B. Then, we can add each row to get the totals on the right hand side of the table.

| Tree dies within 4 years | Tree lives 4 years | Total
Treatment A |  |  |
Treatment B |  |  |
Total |  |  |

Compute the required probabilities

For the second question, we need to determine the probability that a tree receives treatment B. This means that we do not include treatment A in this calculation. So, the probability that treatment B is given to a tree is the ratio between the number of trees that received treatment B and the total number of trees.

Similarly for the third question, the probability that a tree will live beyond years:

In the fourth question, we need to determine the probability that a tree receives treatment B and lives beyond years.

In the fifth question, there is a subtle change from the fourth question. Here, we need to determine the probability that of the trees which received treatment B, a tree lives beyond years. This means we are only concerned with those trees which received treatment B. We no longer need to care about the trees given treatment A, so our denominator needs to be adjusted accordingly.

Independence

We need to determine whether a tree given treatment B and living beyond years are dependent or independent events. According to the definition, two events are independent if and only if

From these probabilities we can see that and therefore the treatment of a tree with treatment B and living beyond years are dependent events.

Exercise 10.3

A number of drivers were asked about the number of motor vehicle accidents they were involved in over the last 10 years. Part of the data collected is shown in the table below.

| 2 accidents | 2 accidents | Total
Female |  |  |
Male |  |  |
Total |  |  |

What are the variables investigated here and what is the purpose of the research?

The variables are gender and number of accidents over a period of 10 years. The purpose of the research is to determine if gender is related to the number of accidents a driver is involved in.

Complete the table.

| 2 accidents | 2 accidents | Total
Female |  |  |
Male |  |  |
Total |  |  |

Determine whether gender and number of accidents are independent suing a calculation.

It can be seen that in all cases , therefore number of motor vehicle accidents is independent of the gender of the driver.

Researchers conducted a study to test how effective a certain inoculation is at preventing malaria. Part of their data is shown below:

| Malaria | No malaria | Total
Male |  |  |
Female |  |  |
Total |  |  |

Calculate the probability that a randomly selected study participant will be female.

Calculate the probability that a randomly selected study participant will have malaria.

If being female and having malaria are independent events, calculate the value .

Using the value of , fill in the missing values on the table.

| Malaria | No malaria | Total
Male | 27 | 189 |
Female | 81 | 567 |
Total |  |  |

The reaction time of drivers during an emergency stop was tested. Within the study cohort (the group of people being studied), the probability that a driver chosen at random was years old or younger is and the probability of a reaction time less than is .

Calculate the number of drivers who are years old or younger.

Calculate the number of drivers who have a reaction time of less than .

If age and reaction time are independent events, calculate the number of drivers years old and younger with a reaction time of less than .

Complete the table below.

| Reaction time | Reaction time | Total
years |  |  |
years |  |  |
Total |  |  | 400

| Reaction time /strong> | Reaction time | Total
years | 84 | 36 | 120
years | 196 | 84 | 280
Total | 280 | 120 | 400

A new treatment for influenza (the flu) was tested on a number of patients to determine if it was better than a placebo (a pill with no therapeutic value). The table below shows the results three days after treatment:

| Flu | No flu | Total
Placebo |  |  |
Treatment |  |  |
Total |  |  |

Complete the table.

| Flu | No flu | Total
Placebo |  |  |
Treatment |  |  |
Total |  |  |

Calculate the probability of a patient receiving the treatment.

Calculate the probability of a patient having no flu after three days.

Calculate the probability of a patient receiving the treatment and having no flu after three days.

Using a calculation, determine whether a patient receiving the treatment and having no flu after three days are dependent or independent events.

Therefore receiving treatment and having no flu after three days are dependent events.

Calculate the probability that a patient receiving treatment will have no flu after three days.

Calculate the probability that a patient receiving a placebo will have no flu after three days.

Comparing you answers in f) and g), would you recommend the use of the new treatment for patients suffering from influenza?

The probability of having no influenza after three days is much higher when on the new treatment so its use is recommended.

A hospital is trying to decide whether to purchase the new treatment. The new treatment is much more expensive than the old treatment. According to the hospital records, of the flu patients that have been treated with the old treatment, only still had the flu three days after treatment.

Construct a two-way contingency table comparing the old treatment data with the new treatment data.

Using the data from your table, advise the hospital whether to purchase the new treatment or not.

| Flu | No flu | Total
Old treatment |  |  |
New treatment |  |  |
Total |  |  |

The probability of not having flu after three days if given the new treatment is approximately the same if given the old treatment, therefore the hospital should not purchase the new, more expensive treatment.

Human immunodeficiency virus (HIV) affects % of the South African population.

If a test for HIV has a % accuracy rate (i.e. % of the time the test is correct, % of the time, the test returns a false result), draw a two-way contingency table showing the expected results if of the general population are tested.

If people are tested and the prevalence rate is %:

| Sick | Healthy | Total
Positive |  |  |
Negative |  |  |
Total |  |  |

If the test is % accurate:

| Sick | Healthy | Total
Positive |  |  |
Negative |  |  |
Total |  |  |

Calculate the probability that a person who tests positive for HIV does not have the disease, correct to two decimal places.

It is worth noting that this probability is bigger than the one suggested by the '% accuracy' of the test.

In practice, a person who tests positive for HIV is always tested a second time. Calculate the probability that an HIV-negative person will test positive after two tests, correct to four decimal places.

10.4 The fundamental counting principle

Mathematics began with counting. Initially, fingers, beans and buttons were used to help with counting, but these are only practical for small numbers. What happens when a large number of items must be counted?

This section focuses on how to use mathematical techniques to count different assortments of items.

Introduction

An important aspect of probability theory is the ability to determine the total number of possible outcomes when multiple events are considered.

For example, what is the total number of possible outcomes when a die is rolled and then a coin is tossed? The roll of a die has six possible outcomes ( or ) and the toss of a coin, outcomes (heads or tails). The sample space (total possible outcomes) can be represented as follows:

Therefore there are possible outcomes.

The use of lists, tables and tree diagrams is only feasible for events with a few outcomes. When the number of outcomes grows, it is not practical to list the different possibilities and the fundamental counting principle is used instead.

The fundamental counting principle

The fundamental counting principle states that if there are outcomes in event and outcomes in event , then there are outcomes in event and event combined.

If we apply this principle to our previous example, we can easily calculate the number of possible outcomes by multiplying the number of possible die rolls with the number of outcomes of tossing a coin: outcomes. This allows us to formulate the following:

If there possible outcomes for event and outcomes for event , then the total possible number of outcomes for both events is

This can be generalised to events, where is the number of events. The total number of outcomes for events is:

The order in which the experiments are done does not affect the total number of possible outcomes.

Worked example 10: Choices without repetition

A take-away has a 4-piece lunch special which consists of a sandwich, soup, dessert and drink for R . They offer the following choices for:

Sandwich: chicken mayonnaise, cheese and tomato, tuna mayonnaise, ham and lettuce

Soup: tomato, chicken noodle, vegetable

Dessert: ice-cream, piece of cake

Drink: tea, coffee, Coke, Fanta, Sprite

How many possible meals are there?

Determine how many parts to the meal there are

There are 4 parts: sandwich, soup, dessert and drink.

Identify how many choices there are for each part

Meal component

Sandwich

Soup

Dessert

Drink

Number of choices

Use the fundamental counting principle to determine how many different meals are possible

So there are possible meals.

In the previous example, there were a different number of options for each choice. But what happens when the number of choices is unchanged each time you choose?

For example, if a coin is flipped three times, what is the total number of different results? Each time a coin is flipped, there are two possible outcomes, namely heads or tails. The coin is flipped times. We can use a tree diagram to determine the total number of possible outcomes:

From the tree diagram, we can see that there is a total of 8 different possible outcomes.

Drawing a tree diagram is possible to draw for three different coin flips, but as soon as the number of events increases, the total number of possible outcomes increases to the point where drawing a tree diagram is impractical.

For example, think about what a tree diagram would look like if we were to flip a coin six times. In this case, using the fundamental counting principle is a far easier option. We know that each time a coin is flipped that there are two possible outcomes. So if we flip a coin six times, the total number of possible outcomes is equivalent to multiplying 2 by itself six times:

Another example is if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three-letter patterns if repetition is allowed, such as ABA, DCA, BBB etc. You will find that there are ways. This is because for the first letter of the pattern, you can choose any of the four available letters, for the second letter of the pattern, you can choose any of the four letters, and for the final letter of the pattern you can choose any of the four letters. Multiplying the number of available choices for each letter in the pattern gives the total available arrangements of letters:

This allows us to formulate the following:

When you have objects to choose from and you choose from them times, then the total number of possibilities is

Worked example 11: Choices with repetition

A school plays a series of soccer matches. For each match there are possibilities: a win, a draw or a loss. How many possible results are there for the series?

Determine how many outcomes you have to choose from for each event

There are outcomes for each match: win, draw or lose.

Determine the number of events

There are matches, therefore the number of events is .

Determine the total number of possible outcomes

There are possible outcomes for each of the events. Therefore, the total number of possible outcomes for the series of matches is

Exercise 10.4

Tarryn has five different skirts, four different tops and three pairs of shoes. Assuming that all the colours complement each other, how many different outfits can she put together?

In a multiple-choice question paper of questions the answers can be A, B, C or D. How many different ways are there of answering the question paper?

A debit card requires a five digit personal identification number (PIN) consisting of digits from 0 to 9. The digits may be repeated. How many possible PINs are there?

The province of Gauteng ran out of unique number plates in 2010. Prior to 2010, the number plates were formulated using the style LLLDDDGP, where L is any letter of the alphabet excluding vowels and Q, and D is a digit between 0 and 9. The new style the Gauteng government introduced is LLDDLLGP. How many more possible number plates are there using the new style when compared to the old style?

Therefore there are more possible number plates using the new style.

A gift basket is made up from one CD, one book, one box of sweets, one packet of nuts and one bottle of fruit juice. The person who makes up the gift basket can choose from five different CDs, eight different books, three different boxes of sweets, four kinds of nuts and six flavours of fruit juice. How many different gift baskets can be produced?

The code for a safe is of the form XXXXYYY where X is any number from 0 to 9 and Y represents the letters of the alphabet. How many codes are possible for each of the following cases:

the digits and letters of the alphabet can be repeated.

the digits and letters of the alphabet can be repeated, but the code may not contain a zero or any of the vowels in the alphabet.

We exclude the digit and the vowels (A; E; I; O; U), leaving other digits and letters to choose from.

the digits and letters of the alphabet can be repeated, but the digits may only be prime numbers and the letters X, Y and Z are excluded from the code.

The prime digits are , , and . This gives us 4 possible digits. If we exclude the letters X, Y and Z, we are left with letters to choose from.

A restaurant offers four choices of starter, eight choices for the main meal and six choices for dessert. A customer can choose to eat just one course, two different courses or all three courses. Assuming that all courses are available, how many different meal options does the restaurant offer?

A person who eats only a starter has 4 choices

A person who eats only a main meal has 8 choices

A person who eats only a dessert has 6 choices

A person who eats a starter and a main course has choices

A person who eats a starter and a dessert has choices

A person who eats a main meal and a dessert has choices

A person who eats all three courses has choices.

10.5 Factorial notation

Worked example 12: The arrangement of outcomes without repetition

Eight athletes take part in a race. In how many different ways can all places in the race be arranged?

Any of the athletes can come first in the race. Now there are only athletes left to be second, because an athlete cannot be both second and first in the race. After second place, there are only athletes left for the third place, athletes for the fourth place, athletes for the fifth place, athletes for the sixth place, athletes for the seventh place and athlete for the eighth place. Therefore the number of ways that the athletes can be ordered is as follows:

As in the example above, it is a common occurrence in counting problems that the outcome of the first event reduces the number of possible outcomes for the second event by exactly , and the outcome of the second event reduces the possible outcomes for the third event by more, etc.

As this sort of problem occurs so frequently, we have a special notation to represent the answer. For an integer, , the notation (read factorial) represents:

This allows us to formulate the following:

The total number of possible arrangements of different objects is

with the following definition: 0! = 1.

Worked example 13: Factorial notation

Determine

Show that

Show that

We know from the definition of a factorial that . However, it can be quite tedious to work this out by calculating each multiplication step on paper or typing each step into your calculator. Fortunately, there is a button on your calculator which makes this much easier. To use your calculator to work out the factorial of a number:

Input the number.

Press SHIFT on your CASIO or 2ndF on your SHARP calculator.

Then press on your CASIO or on your SHARP calculator.

Finally, press equals to calculate the answer.

If we follow these steps for , we get the answer .

Expand the factorial notation:

Expand the factorial notation:

If , we get . This is a special case. Both and , therefore so our identity still holds.

Exercise 10.5

Work out the following without using a calculator:

Calculate the following using a calculator:

Show that the following is true:

10.6 Application to counting problems

Worked example 14: Further arrangement of outcomes without repetition

Eight athletes take part in a race. In how many different ways can the first three places be arranged?

Eight different athletes can occupy the first places. For the first place, there are different choices. For the second place there are different choices and for the third place there are different choices. Therefore different athletes can occupy the first three places in:

Worked example 15: Arrangement of objects with constraints

In how many ways can seven boys of different ages be seated on a bench if:

the youngest boy sits next to the oldest boy?

the youngest and the oldest boys must not sit next to each other?

This question is a little different to the previous problems of arrangements without repetition. In this question, we have the constraint that the youngest boy and the oldest boy must sit together. The easiest way to think about this, is to see each set of objects which have to be together as a single object to arrange.

If we let boy and let the number subscript indicate order of age, we can view the objects to arrange as follows:

If the the youngest and oldest boys are treated as a single object, there are six different objects to arrange so there are different arrangements. However, the youngest and oldest boys can be arranged in different ways and still be together:

Therefore there are:

The arrangements where the youngest and oldest must not sit together is the total number of arrangements minus the number of arrangements where the oldest and youngest sit together. Therefore, there are:

Exercise 10.6

How many different possible outcomes are there for a swimming event with six competitors?

How many different possible outcomes are there for the gold (1st), silver (2nd) and bronze (3rd) medals in a swimming event with six competitors?

Susan wants to visit her friends in Pretoria, Johannesburg, Phalaborwa, East London and Port Elizabeth. In how many different ways can the visits be arranged?

A head boy, a deputy head boy, a head girl and a deputy head girl must be chosen out of a student council consisting of 18 girls and 18 boys. In how many ways can they be chosen?

Twenty different people enter a golf competition. Only the first six of them can win prizes. In how many different ways can the prizes be won?

Three letters of the word 'EMPTY' are arranged in a row. How many different arrangements are possible?

Pool balls are numbered from to . You have only one set of pool balls. In how many different ways can you arrange:

all balls. Write your answer in scientific notation, rounding off to two decimal places.

four of the balls.

The captains of all the sports teams in a school have to stand next to each other for a photograph. The school sports programme offers rugby, cricket, hockey, soccer, netball and tennis.

In how many different orders can they stand in the photograph?

In how many different orders can they stand in the photograph if the rugby captain stands on the extreme left and the cricket captain stands on the extreme right?

Since we have no choice about where to put the rugby and cricket captains, there are only people left to arrange.

In how many different orders can they stand if the rugby captain, netball captain and cricket captain must stand next to each other?

The rugby captain, netball captain and cricket captain are treated as a single object, as they must stand together. So there are four different objects to arrange therefore there are different arrangements. The rugby captain, netball captain and cricket captain can also swop positions between themselves in different ways, therefore there are:

How many three-digit numbers can be made from the digits 1 to 6 if:

repetition is not allowed?

repetition is allowed?

There are two different red books and three different blue books on a shelf.

In how many different ways can these books be arranged?

If you want the red books to be together, in how many different ways can the books be arranged?

If the red books are treated as a single object, there are four different objects to arrange therefore there are different arrangements. The red books can also be rearranged between themselves in different ways, therefore there are:

If you want all the red books to be together and all the blue books to be together, in how many different ways can the books be arranged?

There are two groups of books, red and blue, which can be arranged ways. Then there are two red books which can be arranged in ways and there are three blue books which can be arranged in ways. Therefore, there are

There are two different Mathematics books, three different Natural Sciences books, two different Life Sciences books and four different Accounting books on a shelf. In how many different ways can they be arranged if:

the order does not matter?

all the books of the same subject stand together?

There are four groups of books, which can be arranged in different ways. Of the books, two are Mathematics books, three are Natural Sciences books, two are Life Sciences books and four are Accounting books. Therefore, there are:

the two Mathematics books stand first?

The Mathematics books can be arranged in ways while the remaining books can be arranged in ways. Therefore, there are:

the Accounting books stand next to each other?

The Accounting books can be arranged in ways and, if treated as a single object, can be arranged with the remaining books in ways. Therefore, there are:

Worked example 16: Arrangement of letters

If you take the word, 'OMO', how many letter arrangements can we make if:

we consider the two O’s as different letters?

we consider the two O’s as identical letters?

Since we consider the two O’s as different letters, write the first O as and the second . The different letter arrangements are as follows:

We can see from writing out all the arrangements that there are different ways for the letters to be arranged. This is not practical if there are a large number of letters. Instead, we can work this out more easily using the fundamental counting principle.

Using the fundamental counting principle, as there are letters in the word OMO if we treat each O as a separate letter, there are different arrangements.

If we consider the two O’s as identical letters, only arrangements are possible:

We can also work this out using a modified version of the previous identity. We know that if we treat each letter as different, the number of arrangements is . However, when we have duplicate letters, we have to remove the identical arrangements of these letter from our final answer. So we divide by the factorial of the number of times a letter is repeated.

In this example, O appears twice so we divide by :

Worked example 17: The number of letter arrangements for a longer word

If you take the word 'BASSOON', how many letter arrangements can you make if:

repeated letters are treated as different?

repeated letters are treated as identical?

the word starts with an O and repeated letters are treated as identical?

the word starts and ends with the same letter and repeated letters are treated as identical?

There are letters in the word 'BASSOON'. If we treat each letter as a different letter, there are arrangements.

If repeated letters are treated as identical characters, there are two S's and two O's. This is similar to the previous worked example except now we have more than one letter repeated. When more than one letter is repeated, we have to divide the total number of possible arrangements by the product of the factorials of the number of times each letter is repeated.

If the word starts with an 'O', there are still 6 letters left of which two are S's.

If the word starts and ends with the same letter, there are two possibilities:

S - - - - - S with the letters in between consisting of 'B','A','O','O' and 'N'. Therefore:

O - - - - - O with the letters in between consisting of 'B','A','S','S' and 'N'. Therefore:

Therefore, the total number of arrangements .

This allows us to formulate the following:

For a set of objects, of which are the same, are the same are the same, the number of arrangements

Exercise 10.7

You have the word 'EXCELLENT'.

If the repeated letters are regarded as different letters, how many letter arrangements are possible?

If the repeated letters are regarded as identical, how many letter arrangements are possible?

There are 3 E's and 2 L's so we have to divide by and .

If the first and last letters are identical, how many letter arrangements are there?

The word could start and end in E or L. With an E, there are letter arrangements (divide by 2 L's) and with an L there are letter arrangements (divide by 3 E's). Therefore, there are:

How many letter arrangements can be made if the arrangement starts with an L?

This is equivalent to removing one L from the letters available for arrangement. Therefore, there are:

How many letter arrangements are possible if the word ends in a T?

This is equivalent to removing the T from the letters available for arrangement. Therefore, there are:

You have the word 'ASSESSMENT'.

If the repeated letters are regarded as different letters, how many letter arrangements are possible?

If the repeated letters are regarded as identical, how many letter arrangements are possible?

If the first and last letters are identical, how many letter arrangements are there?

The word could start and end in S or E. With an S, there are letter arrangements (divide by 2 E's and 2 remaining S's) and with an E there are letter arrangements (divide by 4 S's). Therefore, there are:

How many letter arrangements can be made if the arrangement starts with a vowel?

The word could start with A or E. With an A, there are letter arrangements (divide by 2 E's and 4 S's) and with an E there are letter arrangements (divide by 4 S's). Therefore, there are:

How many letter arrangements are possible if all the S's are at the beginning of the word?

This is equivalent to removing all the S's from the letters available for arrangement. Therefore, there are:

On a piano the white keys represent the following notes: C, D, E, F, G, A, B. How many tunes, seven notes in length, can be composed with these notes if:

a note can be played only once?

the notes can be repeated?

the notes can be repeated and the tune begins and ends with a D?

The tune starting and ending with a D leaves five possible positions in which to arrange the seven notes. Therefore, there are:

the tune consists of 3 D’s, 2 B’s and 2 A’s.

There are three black beads and four white beads in a row. In how many ways can the beads be arranged if:

same-coloured beads are treated as different beads?

same-coloured beads are treated as identical beads?

There are eight balls on a table. Some are white and some are red. The white balls are all identical and the red balls are all identical. The balls are removed one at a time. In how many different orders can the balls be removed if:

seven of the balls are red?

three of the balls are red?

there are four of each colour?

How many four-digit numbers can be formed with the digits 3, 4, 6 and 7 if:

there can be repetition?

each digit can only be used once?

if the number is odd and repetition is allowed?

For the number to be odd, it must end in 3 or 7. For the numbers ending in 3, there are different arrangements of the first three digits and similarly, for 7, there are different arrangements. Therefore, there are

10.7 Application to probability problems

When needing to determine the probability that an event occurs, and the total number of arrangements of the sample space, , and the total number arrangements for the event, , are very large, the techniques used earlier in this chapter may no longer be practical. In this case, the probability may be determined using the fundamental counting principle. The probability of the event, , is the total number of arrangements of the event divided by the total number of arrangements of the sample space or .

Worked example 18: Personal Identification Numbers (PINs)

Every client of a certain bank has a personal identification number (PIN) which consists of four randomly chosen digits from 0 to 9.

How many PINs can be made if digits can be repeated?

How many PINs can be made if digits cannot be repeated?

If a PIN is made by selecting four digits at random, and digits can be repeated, what is the probability that the PIN contains at least one eight?

If a PIN is made by selecting four digits at random, and digits cannot be repeated, what is the probability that the PIN contains at least one eight?

If digits can be repeated: you have digits to choose from and you have to choose four times, therefore the number of possible PINs .

If digits cannot be repeated: you have 10 digits for your first choice, nine for your second, eight for your third and seven for your fourth. Therefore, .

Let be the event that at least one eight is chosen. Therefore the complement of is the event that no eights are chosen.

If no eights are chosen, there are only nine digits to choose from. Therefore, .

The total number of arrangements in the set, as calculated in Question 1, is . Therefore:

Let be the event that at least one eight is chosen. Therefore the complement of , is the event that no eights are chosen.

If no eights are chosen, there are only 9 then 8 then 7 then 6 digits to choose from as we cannot repeat a digit once it is chosen. Therefore, .

The total number of arrangements in the set, as calculated in Question 1, is . Therefore:

Worked example 19: Number plates

The number plate on a car consists of any letters of the alphabet (excluding the vowels and 'Q'), followed by any digits ( to ). For a car chosen at random, what is the probability that the number plate starts with a 'Y' and ends with an odd digit?

Identify what events are counted

The number plate starts with a 'Y', so there is only option for the first letter, and ends with an odd digit, so there are options for the last digit ().

Find the number of events

Use the counting principle. For the second and third letters, there are possibilities ( letters in the alphabet, minus vowels and 'Q'). There are possibilities for the first and second digits.

Find the total number of possible number plates

Use the counting principle. This time, the first letter and last digit can be anything.

Calculate the probability

The probability is the number of outcomes in the event, divided by the total number of outcomes in the sample space.

Worked example 20: Probability of word arrangements

Refer to worked example 16 for context. If you take the word, 'BASSOON' and you randomly rearrange the letters, what is the probability that the word starts and ends with the same letter if repeated letters are treated as identical?

If the word starts and ends with the same letter, there are a total number of 120 possible arrangements (from worked example 16). Let this event .

The total number of possible arrangements if repeated letters are treated as identical (from worked example 16).

Therefore, the probability of an arrangement beginning and ending with the same letter

Exercise 10.8

A music group plans a concert tour in South Africa. They will perform in Cape Town, Port Elizabeth, Pretoria, Johannesburg, Bloemfontein, Durban and East London.

In how many different orders can they plan their tour if there are no restrictions?

In how many different orders can they plan their tour if their tour begins in Cape Town and ends in Durban?

This reduces the available objects (cities) by two, hence

If the tour cities are chosen at random, what is the probability that their performances in Cape Town, Port Elizabeth, Durban and East London happen consecutively? Give your answer correct to 3 decimal places.

If these cities are grouped together, they can be treated as a single object in the arrangement, hence there are different ways to order the objects. Within the grouped cities, there are different ways to order them. Therefore, there are

Therefore, the probability of any of the orders having Cape Town, Port Elizabeth, Durban and East London happen consecutively is:

A certain restaurant has the following course options available for a three-course set menu:

STARTERS | MAINS | DESSERTS
Calamari salad | Fried chicken | Ice cream and chocolate sauce
Oysters | Crumbed lamb chops | Strawberries and cream
Fish in garlic sauce | Mutton Bobotie | Malva pudding with custard
| Chicken schnitzel | Pears in brandy sauce
| Vegetable lasagne |
| Chicken nuggets |

How many different set menus are possible?

What is the probability that a set menu includes a chicken course?

Eight different pairs of jeans and 5 different shirts hang on a rail.

In how many different ways can the clothes be arranged on the rail?

In how many ways can the clothing be arranged if all the jeans hang together and all the shirts hang together?

The shirts and jeans form two groups, which can be arranged ways. The five shirts can be arranged ways and the eight pairs of jeans can be arranged ways.

What is the probability, correct to three decimal places, of the clothing being arranged on the rail with a shirt at one end and a pair of jeans at the other?

The five different choices of shirt and eight different choices of pairs of jeans can form different arrangements at the ends of the rail.

There are different ways to arrange a shirt at one end and a pair of jeans on the other: S - - - - - - - - - - - J and J - - - - - - - - - - - S

If a shirt is at one end and a pair of jeans at the other, there remains different arrangements of the remaining clothing items.

Therefore, there are:

The probability of a clothing arrangement with a shirt at one end and a pair of jeans at the other

A photographer places eight chairs in a row in his studio in order to take a photograph of the debating team. The team consists of three boys and five girls.

In how many ways can the debating team be seated?

What is the probability that a particular boy and a particular girl sit next to each other?

Regard the particular boy and girl as one group. The group can be seated ways. The number of ways that this group and the remaining 6 people can sit . Therefore the total number of ways this particular boy and girl can sit together in the photograph . The probability of a particular boy and girl sitting together

If the letters of the word 'COMMITTEE' are randomly arranged, what is the probability that the letter arrangements start and end with the same letter?

There are 2 M’s, 2 T’s and 2 E’s and a total of 9 letters.

Total number of letter arrangements

Possibilities of the first and last letter being the same:

M(COITTEE)M \\ Total of 7 letters of which there are 2E’s and 2T’s \\ Number of letter arrangements

T(MCOIEEM)T\\ Total of 7 letters of which there are 2E’s and 2M’s \\ Number of letter arrangements

E(TTMCOIM)E \\ Total of 7 letters of which there are 2T’s and 2M’s \\ Number of letter arrangements

Total number of letter arrangements if the letter arrangement starts and ends with the same letter .

Four different Mathematics books, three different Economics books and two different Geography books are arranged on a shelf. What is the probability that all the books of the same subject are arranged next to each other?

Total number of different ways the books can be arranged .There are 3 subjects of books which can be arranged ways.

Therefore, the total number of arrangements if the subjects are arranged together

A number plate is made up of three letters of the alphabet (excluding F and S) followed by three digits from 0 to 9. The numbers and letters can be repeated. Calculate the probability that a randomly chosen number plate:

starts with the letter D and ends with the digit 3.

Total number of arrangements

Total number of arrangements beginning with D and ending with

has precisely one D.

The 'D' could be at the first, second or third positions. The other two letters cannot include a 'D', leaving 23 other letters. Therefore there are

contains at least one 5.

In the 13-digit identification (ID) numbers of South African citizens:

The first six numbers are the birth date of the person in YYMMDD format.

The next four digits indicate gender, with and above being male and to being female.

The next number is the country ID; 0 is South Africa and 1 is not.

The second last number used to be a racial identifier but it is now 8 for everybody.

The last number is a control digit, which verifies the rest of the number.

Assume that the control digit is a randomly generated digit from 0 to 9 and ignore the fact that leap years have an extra day.

Calculate the total number of possible ID numbers.

For all available arrangements, if an ID number is structured ABCDEFGHIJKLM:

A and B are any digits between 00 and 99 (year)

C and D are any digits from 01 to 12 (month)

E and F are any digits from 01 to 28, 30 or 31 dependent on month (day)

G, H, I and J are any digits from 0001 to 9999 (gender)

K is either a 0 or 1

L is an 8

M is any digit between 0 and 9

Therefore, the total number of possible ID numbers for 30-day months is: Therefore, the total number of possible ID numbers for 31-day months is: Therefore, the total number of possible ID numbers for February is: Therefore, the total number of possible ID numbers is:

Calculate the probability that a randomly generated ID number is of a South African male born during the 1980s. Write your answer correct to two decimal places.

There is a lot of information in the problem and we can simplify it by identifying the relevant information. We want to calculate the probability that an ID number is for a South African male born during the 1980s. This means that we have to look at the digits for country, gender and year of birth.

If an ID number is structured ABCDEFGHIJKLM:

For a South African, K is 0.

For a male, GHIJ are from to .

For someone born during the 1980s, AB are between 80 and 89.

This gives combinations for ABGHIJK

Without any restrictions, the total combinations for ABGHIJK is .

10.8 Summary

The addition rule (also called the sum rule) for any events, and is This rule relates the probabilities of events with the probabilities of their union and intersection.

The addition rule for mutually exclusive events is This rule is a special case of the previous rule. Because the events are mutually exclusive, .

The complementary rule is This rule is a special case of the previous rule. Since and are complementary, .

The product rule for independent events and is:

If two events and are dependent then:

Venn diagrams are used to show how events are related to one another. A Venn diagram can be very helpful when doing calculations with probabilities. In a Venn diagram each event is represented by a shape, often a circle or a rectangle. The region inside the shape represents the outcomes included in the event and the region outside the shape represents the outcomes that are not in the event.

Tree diagrams are useful for organising and visualising the different possible outcomes of a sequence of events. Each branch in the tree shows an outcome of an event, along with the probability of that outcome. For each possible outcome of the first event, we draw a line where we write down the probability of that outcome and the state of the world if that outcome happened. Then, for each possible outcome of the second event we do the same thing. The probability of a sequence of outcomes is calculated as the product of the probabilities along the branches of the sequence.

Two-way contingency tables are a tool for keeping a record of the counts or percentages in a probability problem. Two-way contingency tables are especially helpful for figuring out whether events are dependent or independent.

The fundamental counting principle states that if there are outcomes for event and outcomes for event , then there are different possible outcomes for both events.

When you have objects to choose from and you choose from them times, if the number of choices remains the same after each choice, then the total number of possibilities is

The number of arrangements of different objects is

For a set of objects, of which there are k subsets with repeated objects i.e. are the same, are the same, are the same, the number of arrangements are

Exercise 10.9

An ATM card has a four-digit PIN. The four digits can be repeated and each of them can be chosen from the digits 0 to 9.

What is the total number of possible PINs?

What is the probability of guessing the first digit correctly?

What is the probability of guessing the second digit correctly?

If your ATM card is stolen, what is the probability, correct to four decimal places, of a thief guessing all four digits correctly on his first guess?

After three incorrect PIN attempts, an ATM card is blocked from being used. If your ATM card is stolen, what is the probability, correct to four decimal places, of a thief blocking the card? Assume the thief enters a different PIN each time.

The LOTTO rules state the following:

Six numbers are drawn from the numbers 1 to 49 - this is called a 'draw'.

Numbers are not replaced once drawn, so you cannot have the same number more than once.

The order of the drawn numbers does not matter.

You decide to buy one LOTTO ticket consisting of numbers.

How many different possible LOTTO draws are there? Write your answer in scientific notation, rounding to two digits after the decimal point.

Complete the tree diagram below after the first two LOTTO numbers have been drawn showing the possible outcomes and probabilities of the numbers on your ticket.

What is the probability of getting the first number drawn correctly?

What is the probability of getting the second number drawn correctly if you get the first number correct?

What is the probability of getting the second number drawn correct if you do not get the first number correctly?

What is the probability of getting the second number drawn correct?

This is the sum of the probabilities of the outcomes where the second number drawn is correct:

Notice that the answer is the same as the probability of getting the first number correct. If you are unaware of the outcome of prior events, the probability of the outcome of a certain event, is equal to the probability of that outcome of the first event. Learners are not required to learn this concept but it is interesting to note.

What is the probability of getting all LOTTO numbers correct? Write your answer in scientific notation, rounding to two digits after the decimal point.

The population statistics of South Africa show that % of all babies born are female. Calculate the probability that a couple planning to have children will have a boy followed by a girl and then a boy. Assume that each birth is an independent event. Write your answer as a percentage, correct to two decimal places.

Fezile and Vuzi write a Mathematics test. The probability that Fezile will pass the test is . The probability that Vuzi will pass the test is . What is the probability that only one of them will pass the test?

Landline telephone numbers are 10 digits long. Numbers begin with a zero followed by digits chosen from the digits 0 to 9. Repetitions are allowed.

How many different phone numbers are possible?

The first three digits of a number form an area code. The area code for Cape Town is . How many different phone numbers are available in the Cape Town area?

What is the probability of the second digit being an even number?

There are even numbers between 0 and 9, therefore:

Ignoring the first digit, what is the probability of a phone number consisting of only odd digits? Write your answer correct to three decimal places.

Take the word 'POSSIBILITY'.

In how many way can the letters be arranged if repeated letters are considered identical?

There are two S's and three I's, therefore there are:

What is the probability that a randomly generated arrangement of the letters will begin with three I's? Write your answer as a fraction.

If the arrangement begins with three I's, that leaves POSSBLTY to be arranged.

The code to a safe consists of digits chosen from the digits to . None of the digits are repeated. Determine the probability of a code where the first digit is odd and none of the first three digits may be a zero. Write your answer as a percentage, correct to two decimal places.

There are odd numbers from to , so there are five arrangements for the first digit.

For the second digit there are digits left to arrange, as the first digit and zero are removed.

For the third digit there are digits left to arrange as zero and the first two digits are removed.

For the fourth digit, there are digits ( is now included) left which can be combined in ! ways for the remaining digits.

Four different red books and three different blue books are to be arranged on a shelf. What is the probability that all the red books and all the blue books stand together on the shelf?

Total number of arrangements

If the red and blue books stand together, there are two groups of books which can be arranged ways.

Then there are four red books which can be arranged in ways and the three blue books can be arranged in ways.

The probability that Thandiswa will go dancing on a Saturday night (event ) is and the probability that she will go watch a movie is (event ). Determine the probability that she will:

go dancing and watch a movie if and are independent.

go dancing or watch a movie if and are mutually exclusive.

go dancing and watch a movie if .

not go dancing or go to a movie if .

Three boys and four girls sit in a row.

In how many ways can they sit in the row?

What is the probability that they sit in alternating gender positions?

There is only one way they can sit alternately: GBGBGBG

The number of ways they can sit alternately

The number plate on a car consists of any 3 letters of the alphabet (excluding the vowels, J and Q), followed by any 3 digits from 0 to 9. For a car chosen at random, what is the probability that the number plate starts with a Y and ends with an odd digit? Write your answer as a fraction.

The number plate starts with a Y, so there is only 1 choice for the first letter.

The number plate ends with an odd digit, so there are 5 choices (1, 3, 5, 7, 9)

There are 19 letters available because the 5 vowels (A, E, I, O, U), J and Q are excluded.

There are four black balls and yellow balls in a bag. Thandi takes out a ball, notes its colour and then puts it back in the bag. She then takes out another ball and also notes its colour. If the probability that both balls have the same colour is , determine the value of .

Using a tree diagram, the different outcomes and probabilities can be illustrated as follows:

Solving for :

A rare kidney disease affects only in people and the test for this disease has a % accuracy rate.

Draw a two-way contingency table showing the results if of the general population are tested.

If people are tested and the prevalence rate is %:

| Sick | Healthy | Total
Positive |  |  |
Negative |  |  |
Total |  |  |

If the test is % accurate:

| Sick | Healthy | Total
Positive |  |  |
Negative |  |  |
Total |  |  |

Calculate the probability that a person who tests positive for this rare kidney disease is sick with the disease, correct to two decimal places.

Notice that this means that a positive result is wrong % of the time! This is an important concept in medical science. For very rare diseases, tests have to be highly accurate otherwise the result is meaningless.