Chapter 4.5, Problem 26E

Solution

The transformation required to obtain the graph of the function $y=-2\sec \frac{1}{2}x$ 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 $y=-2\sec \frac{1}{2}x$ .

Explanation:

Consider the function $y=-2\sec \frac{1}{2}x$ ,

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

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

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

The function $y=-2\sec \frac{1}{2}x$ can be written as the following.

  $y=-2\sec \left(\left(\frac{1}{2}\right)x-0\right)+0$  ...... (2)

Compare equation (1) and (2),

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

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

As $a=-2$ , the graph is stretched by 2 units vertically and reflects about the $x$ -axis.

Graph the functions $y=\sec x$ and $y=-2\sec \frac{1}{2}x$ 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.