Chapter 4.5, Problem 10E

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 $y=2\tan \left(\frac{x}{2}\right)$ .

The vertical asymptote of $y=2\tan \left(\frac{x}{2}\right)$ to be odd multiple of $\pi$ and the two periods are $-2\pi$ and $2\pi$ .

Given information:

The function $y=2\tan \left(\frac{x}{2}\right)$ .

Explanation:

Consider the function $y=2\tan \left(\frac{x}{2}\right)$ ,

Graph the function

  

Compare the function $y=2\tan \left(\frac{x}{2}\right)$ with standard function $y=a\tan bx$ .

Then,

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

  $\begin{array}{c}\left|b\right|=\left|\frac{1}{2}\right|\\ =\frac{1}{2}\end{array}$

As $\left|a\right|>1$ and $\left|b\right|<1$ , the graph of $y=\tan x$ is stretched horizontally by the factor of $2$ and vertically stretched by the factor of 2.

The stretched factor causes the vertical asymptote of $y=2\tan \left(\frac{x}{2}\right)$ to be odd multiple of $\pi$ and the two periods are $-2\pi$ and $2\pi$ .

Thus, the vertical asymptote of $y=2\tan \left(\frac{x}{2}\right)$ to be odd multiple of $\pi$ and the two periods are $-2\pi$ and $2\pi$ .