02.45tangraph

# Graphs of the Tangent Function

Recall, for all trig graphs,

- the
**median line**is a horizontal line through the centre of the graph - the
**amplitude**is the maximum height above the median line - the
**period**is the distance in the x-direction to complete one cycle

## Graph of y = tan(x)

For the graph $y = \tan(x)$

- Median line: $y = 0$

- Amplitude: (no amplitude)

- Period = $\pi$

- Asymptotes: $x=(2k+1)\dfrac{\pi}{2}, \; k\in Z$

- Domain: $x \in R\backslash \left\{ (2k+1)\dfrac{\pi}{2}, \; k\in Z \right\}$

- Extra point: $\left( \dfrac{\pi}{4} , \; 1 \right)$

- Range: $y \in R$

- x-intercepts: $x = k\pi, \; k \in Z$

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## Dilations: y = atan(nx)

$y = a\tan(nx)$ causes:

- a dilation by a factor of
**a**in the y-direction

- a dilation by a factor of $\frac{1}{n}$ in the x-direction

- Asymptotes: $x=(2k+1)\dfrac{\pi}{2n}, \; k\in Z$

- Domain: $x \in R\backslash \left\{ (2k+1)\dfrac{\pi}{2n}, \; k\in Z \right\}$

- extra point: $\left( \dfrac{\pi}{4n} , \; a \right)$

Recall that if **a** is negative, we get a reflection across the x-axis (in the y-direction)

And, if **n** is negative, we get a reflection across the y-axis (in the x-direction)

#### Example 1

Sketch the graph of $y = 2\tan \big(\frac{x}{2} \big)$

**Solution**

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- Median line: $y = 0$

- Period = $\dfrac{\pi}{1/2}=2\pi$

- Asymptotes: $x = (2k + 1)\pi, \; k \in Z$

- Domain: $x \in R\backslash \left\{ (2k + 1)\pi, \; k \in Z \right\}$

- Range: $y \in R$

- x-intercepts: $x = 2k\pi,\; k \in Z$ {found by solving $y = 0$}

## Translations: y = tan(x – b) + c

$y = tan(x - b) + c$ causes

- a translation by
**b**units to the right - a translation by
**c**units up

#### Example 2

Sketch $y = \tan \left( x + \dfrac{\pi}{6} \right) -1$

**Solution**

- Median line: $y = -1$

- Period = $\pi$

- Asymptotes: $x=k\pi+\dfrac{\pi}{3},\; k\in Z$

- Domain: $x \in R \backslash \left\{ k\pi+\dfrac{\pi}{3},\; k\in Z \right\}$

- Range: $y \in R$

The points on the median line can be found by solving y = –1

- Median points: $x=k\pi-\dfrac{\pi}{6},\; k \in Z$

x-intercepts can be found by solving y = 0

- x-intecepts: $x=k\pi+\dfrac{\pi}{12},\; k \in Z$

## Summary

$y = a \tan \big(n(x-b) \big)+c$

- Median line: $y = c$
- Dilated by a factor of
**a**from the x-axis (in the y-direction) - Dilated by a factor of
**1/n**from the y-axis (in the x-direction) - Period = $\frac{\pi}{n}$
- Translated
**b**units to the right - Translated
**c**units up - To find median points: solve $y = c$
- To find x-intercepts: solve $y = 0$

- Asymptotes: $x=(2k+1) \left(\dfrac{\pi}{2n} + b \right), \; k\in Z$

- Domain: $x \in R\backslash \left\{ (2k+1)\left( \dfrac{\pi}{2n} +b \right), \; k\in Z \right\}$

- extra point: $\left( \dfrac{\pi}{4n} + b , \; a + c \right)$

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