Chapter 4.5, Problem 5E

Solution

To identify: The graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function $y=\tan 2x$ .

The vertical asymptotes of the graph $y=\tan 2x$ lies at odd multiple of $\frac{\pi }{4}$ and the periods are $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ .

Given information:

The function $y=\tan 2x$ .

Explanation:

Consider the function $y=\tan 2x$ ,

Graph the function

  

Compare the function $y=\tan 2x$ with standard function $y=\tan bx$ .

Then,

  $\begin{array}{c}\left|b\right|=\left|2\right|\\ =2\end{array}$

As $\left|b\right|>1$ , the graph is shrunk horizontally by the factor of $\frac{1}{2}$ .

The shrink factor causes the vertical asymptote of $y=\tan 2x$ to be odd multiple of $\frac{\pi }{4}$ . The two periods as can be seen from the graph are $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ .

Thus, the vertical asymptotes of the graph $y=\tan 2x$ lies at odd multiple of $\frac{\pi }{4}$ and the periods are $-\frac{\pi }{2}$ and $\frac{\pi }{2}$ .