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5.8 Summary (EMBHF)

  • Parabolic functions:

    Standard form: \(y = ax^2 + bx + c\)

    • \(y\)-intercept: \((0;c)\)
    • \(x\)-intercept: \(x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\)
    • Turning point: \(\left(-\frac{b}{2a}; -\frac{b^2}{4a}+c\right)\)
    • Axis of symmetry: \(x = -\frac{b}{2a}\)

    Completed square form: \(y = a(x+p)^2 + q\)

    • Turning point: \((-p;q)\)
    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
  • Average gradient:

    • Average gradient \(= \frac{y_2 - y_1}{x_2 - x_1}\)
  • Hyperbolic functions:

    Standard form: \(y = \frac{k}{x}\)

    • \(k > 0\): first and third quadrant
    • \(k < 0\): second and fourth quadrant

    Shifted form: \(y = \frac{k}{x+p} + q\)

    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
    • Asymptotes: \(x = -p\) and \(y = q\)
  • Exponential functions:

    Standard form: \(y = ab^x\)

    • \(a > 0\): above \(x\)-axis
    • \(a < 0\): below \(x\)-axis
    • \(b > 1\): increasing function if \(a > 0\); decreasing function if \(a < 0\)
    • \(0 < b < 1\): decreasing function if \(a > 0\); increasing function if \(a < 0\)

    Shifted form: \(y = ab^{(x +p)} + q\)

    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
    • Asymptotes: \(y = q\)
  • Sine functions:

    Shifted form: \(y = a \sin (k \theta + p) + q\)

    • Period \(= \frac{\text{360}\text{°}}{|k|}\)
    • \(k > 1\) or \(k < -1\): period decreases
    • \(0 <k <1\) or \(-1 <k <0\): period increases
    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
    • \(\sin (-\theta) = - \sin \theta\)
  • Cosine functions:

    Shifted form: \(y = a \cos (k \theta + p) + q\)

    • Period \(= \frac{\text{360}\text{°}}{|k|}\)
    • \(k > 1\) or \(k < -1\): period decreases
    • \(0 <k <1\) or \(-1 <k <0\): period increases
    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
    • \(\cos (-\theta) = \cos \theta\)
  • Tangent functions:

    Shifted form: \(y = a \tan (k \theta + p) + q\)

    • Period \(= \frac{\text{180}\text{°}}{|k|}\)
    • \(k > 1\) or \(k < -1\): period decreases
    • \(0 <k <1\) or \(-1 <k <0\): period increases
    • \(p > 0\): horizontal shift left
    • \(p < 0\): horizontal shift right
    • \(q > 0\): vertical shift up
    • \(q < 0\): vertical shift down
    • \(\tan (-\theta) = - \tan \theta\)
    • Asymptotes: \(\frac{\text{90}\text{°}-p}{k} \pm \frac{\text{180}\text{°} n}{k}\), \(n \in \mathbb{Z}\)