Chapter 4.5, Problem 27E

Solution

The transformation required to obtain the graph of the function $y=-\tan \frac{\pi }{2}x+2$ from a basic trigonometric function.

The unction $y=-2\sec \frac{1}{2}x$ is obtained from $y=\sec x$ by the horizontally stretching by 2 units and stretched by 2 units vertically and reflects about the $x$ -axis.

Given information:

The function $y=-\tan \frac{\pi }{2}x+2$ .

Explanation:

Consider the function $y=-\tan \frac{\pi }{2}x+2$ ,

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 $y=-\tan \frac{\pi }{2}x+2$ can be written as the following.

  $y=-\tan \left(\left(\frac{\pi }{2}\right)x-0\right)+2$  ...... (2)

Compare equation (1) and (2),

  $\begin{array}{c}a=-1\\ b=\frac{\pi }{2}\\ h=0\\ k=2\end{array}$

As $b=\frac{\pi }{2}$ , which is $\left|\frac{\pi }{2}\right|>1$ , the graph is obtained by horizontally shrinking by $\frac{2}{\pi }$ units.

As $a=-1$ , the graph is stretched by 1 unit vertically and reflects about the $x$ -axis.

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

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

  

Thus, the function $y=-2\sec \frac{1}{2}x$ is obtained from $y=\sec x$ by the horizontally stretching by 2 units and stretched by 2 units vertically and reflects about the $x$ -axis.