## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 4 - Graphs of the Circular Functions - Section 4.3 Graphs of the Tangent and Cotangent Functions - 4.3 Exercises - Page 171: 26

#### Answer

Refer to the graph below. #### Work Step by Step

RECALL: (1) The function $y=\tan{[b(x-d)]}$ has: period = $\frac{\pi}{b}$ phase (horizontal) shift = $|d|$, (to the right when $d\gt 0$, to the left when $d\lt0$) (2) The consecutive vertical asymptotes of the function $y=\tan{x}$, whose period is $\pi$, are $x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$. Write the given funtion in $y=\tan{[b(x-d)]}$ form by factoring out $\frac{1}{2}$ within the tangent function to obtain: $y=\tan{[\frac{1}{2}(x+2\pi)]}$ Thus, with $b=\frac{1}{2}$ and $d=-2\pi$, the given function has: period = $\frac{\pi}{\frac{1}{2}}=2\pi$ phase (horizontal) shift = $2\pi$ to the left To find the consecutive vertical asymptotes, equate $(\frac{x}{2}+\pi)$ to $-\frac{\pi}{2}$ and to $\frac{\pi}{2}$, then solve each equation to obtain: \begin{array}{ccc} \frac{x}{2}+\pi&=-\frac{\pi}{2} &\text{ or } &\frac{x}{2}+\pi=\frac{\pi}{2} \\\frac{x}{2}&=-\frac{3\pi}{2} &\text{or} &\frac{x}{2}=-\frac{\pi}{2} \\x&=-3\pi &\text{or} &x=-\pi \end{array} This means that one period of the given function is in the interval $[-3\pi, -\pi]$ and the next period is $[-\pi, \pi]$. Dividing each interval into four equal parts give the key x-values: $-\frac{5\pi}{2}, -2\pi, -\frac{3\pi}{2}, -\frac{\pi}{2}, 0, \frac{\pi}{2}$. To graph the given function, perform the following steps: (1) Create a table of values for the given function using the key x-values listed above. (Refer to the attached image table below.) (2) Graph the consecutive vertical asymptotes. (3) Plot each point from the table then connect them using a smooth curve, making sure that the curves are asymptotic with the lines in Step (2) above. Refer to the graph in the answer part above. After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.