Mathematics » Functions II » The Tangent Function

# Summary and Main Ideas

## Summary

• Parabolic functions:

Standard form: $$y = ax^2 + bx + c$$

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

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 $$= \cfrac{y_2 – y_1}{x_2 – x_1}$$
• Hyperbolic functions:

Standard form: $$y = \cfrac{k}{x}$$

• $$k > 0$$: first and third quadrant
• $$k < 0$$: second and fourth quadrant

Shifted form: $$y = \cfrac{k}{x+p} + q$$

• $$p > 0$$: 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 $$= \cfrac{\text{360}\text{°}}{|k|}$$
• $$k > 1$$ 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 $$= \cfrac{\text{360}\text{°}}{|k|}$$
• $$k > 1$$ 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 $$= \cfrac{\text{180}\text{°}}{|k|}$$
• $$k > 1$$ 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: $$\cfrac{\text{90}\text{°}-p}{k} \pm \cfrac{\text{180}\text{°} n}{k}$$, $$n \in \mathbb{Z}$$