5.8 Summary
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5.8 Summary (EMBHF)
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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
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Average gradient:
- Average gradient \(= \frac{y_2 - y_1}{x_2 - x_1}\)
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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 \(y = q\)
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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: \(y = q\)
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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
- \(\sin (-\theta) = - \sin \theta\)
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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
- \(\cos (-\theta) = \cos \theta\)
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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: \(\frac{\text{90}\text{°}-p}{k} \pm \frac{\text{180}\text{°} n}{k}\), \(n \in \mathbb{Z}\)
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