06Etangraph

toc = 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) math \\ . \\ \text{ } \quad \centerdot \text{ Asymptotes: } \; x=(2k+1)\dfrac{\pi}{2}, \; k\in Z \\ \\ \text{ } \quad \centerdot \text{ Domain: } \; x \in R\backslash \left\{ (2k+1)\dfrac{\pi}{2}, \; k\in Z \right\} \\ \\ \text{ } \quad \centerdot \; \text{passes through the point } \left( \dfrac{\pi}{4}, \, 1 \right) math
 * Median line: y = 0
 * Amplitude = ¥ (no amplitude)
 * Period = p
 * Range: y Î R
 * x-intercepts: x = k p, k Î Z

Dilations: y = atan(nx)
y = atan(nx) causes: math \\ . \\ \text{ } \quad \centerdot \text{ Asymptotes: } \; x=(2k+1)\dfrac{\pi}{2n}, \; k\in Z \\ \\ \text{ } \quad \centerdot \text{ Domain: } \; x \in R\backslash \left\{ (2k+1)\dfrac{\pi}{2n}, \; k\in Z \right\} \\ \\ \text{ } \quad \centerdot \; \text{passes through the point } \left( \dfrac{\pi}{4n}, \, a \right) math
 * a dilation by a factor of **a** in the y-direction
 * a dilation by a factor of **1/n** in the x-direction

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


For the graph of y = 2tan(0.5x)
 * Median line: y = 0
 * Period = 2 p
 * Asymptotes: x = (2k + 1) p, k Î Z
 * Domain: x Î R\{ (2k + 1) p, k Î Z }
 * Range: y Î R

The x-intercepts can be found by solving y = 0
 * x-intercepts: x = 2k p, k Î Z

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 **


math \\ \text{For the graph } y = \tan \left( x + \dfrac{\pi}{6} \right) -1 \\ \\ \text{ } \quad \centerdot \text{ Median line: } y = -1 \\ \\ \text{ } \quad \centerdot \text{ Period } = \pi math math \\ . \\ \text{ } \quad \centerdot \text{ Asymptotes: } x=k\pi+\dfrac{\pi}{3},\; k\in J \\ \\ \text{ } \quad \centerdot \text{ Domain: } x \in R \backslash \left\{ k\pi+\dfrac{\pi}{3},\; k\in J \right\} \\ \\ \text{ } \quad \centerdot \text{ Range: } y \in R math

The points on the median line can be found by solving y = –1 math . \quad \centerdot \text{ Median points: } x=k\pi-\dfrac{\pi}{6},\; k \in Z math

x-intercepts can be found by solving y = 0 math . \quad \centerdot \text{ x-intecepts: } x=k\pi+\dfrac{\pi}{12},\; k \in Z math

Summary
math y = a \tan \big(n(x-b) \big)+c math math \\ . \\ \text{ } \quad \centerdot \text{ Asymptotes: } \; x=(2k+1) \left(\dfrac{\pi}{2n} + b \right), \; k\in Z \\ \\ \text{ } \quad \centerdot \text{ Domain: } \; x \in R\backslash \left\{ (2k+1)\left( \dfrac{\pi}{2n} +b \right), \; k\in Z \right\} \\ \\ \text{ } \quad \centerdot \; \text{passes through the point } \left( \dfrac{\pi}{4n} + b, \, a + c \right) math Go to top of page flat .
 * 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 = p /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