Chapter 9.5, Problem 15E

a.

To determine

To calculate; to graph each function using the given graph.

Explanation of Solution

  $f(x)=3\sec 2x$

The function is of the form $g(x)=a\sec bx$ where $a=3$ and $b=2$ . So, the period is $\frac{2\pi }{\left|b\right|}=\frac{2\pi }{2}=\pi .$

Asymptotes: $x=\frac{\pi }{2\left|b\right|},-\frac{\pi }{2\left|b\right|}$

  $\begin{array}{l}x=\frac{\pi }{2(2)},-\frac{\pi }{2(2)}\\ x=\frac{\pi }{4},-\frac{\pi }{4}\end{array}$

  

The graph $g$ is a vertical stretch by a factor of $3$ and horizontal shrink by a factor of $\frac{1}{2}$ of the graph $f.$

b.

To determine

To calculate; to graph each function using the given graph.

Explanation of Solution

  $f(x)=4\csc 3x$

The function is of the form $g(x)=a\csc bx$ where $a=4$ and $b=3$ . So, the period is $\frac{2\pi }{\left|b\right|}=\frac{2\pi }{3}.$

Asymptotes: $x=\frac{\pi }{2\left|b\right|},-\frac{\pi }{2\left|b\right|}$

  $\begin{array}{l}x=\frac{\pi }{2(3)},-\frac{\pi }{2(3)}\\ x=\frac{\pi }{6},-\frac{\pi }{6}\end{array}$

  

The graph $g$ is a vertical stretch by a factor of $4$ and horizontal shrink by a factor of $\frac{1}{3}$ of the graph $f.$