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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