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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