Chapter 4.5, Problem 28E

Solution

To find: The transformation required to obtain the graph of the function $y=2\tan \pi x-2$ from a basic trigonometric function.

The function $y=2\tan \pi x-2$ is obtained from $y=\tan x$ by the vertically stretched by 2 units horizontally shrink by factor of $\frac{1}{\pi }$ and vertical translation of 2 units down.

Given information:

The function $y=2\tan \pi x-2$ .

Explanation:

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

Its parent function is $y=\tan x$ .

Compare it with the function of the form as given below.

  $y=a\tan \left(b\left(x-h\right)\right)+k$  ...... (1)

The function $y=2\tan \pi x-2$ can be written as the following.

  $y=2\tan \left(\pi x-0\right)-2$  ...... (2)

Compare equation (1) and (2),

  $\begin{array}{c}a=2\\ b=\pi \\ h=0\\ k=-2\end{array}$

As $b=\pi$ , which is $\left|\pi \right|>1$ , the graph is obtained by horizontally shrinking by $\frac{1}{\pi }$ units.

As $a=2$ , the graph is stretched by 2 units vertically.

From the value of $k=-2$ , the graph is vertical translation of 2 units down.

Graph the functions $y=\tan x$ and $y=2\tan \pi x-2$ on the same window.

  

Thus, the function $y=2\tan \pi x-2$ is obtained from $y=\tan x$ by the vertically stretched by 2 units horizontally shrink by factor of $\frac{1}{\pi }$ and vertical translation of 2 units down.