Chapter 4.6, Problem 41E

To determine

To graph: the given function.

Explanation of Solution

Given information:

Given function is

  $y=-2\sec 4x$ .

Calculation:

The vertical asymptotes for $y=\tan x$ are at $x=-\left(\frac{\pi }{2}\right)$ and $x=\frac{\pi }{2}$ , and its period is $\pi$ .

So for the given equation, let us find out the values of the two consecutive asymptotes.

On comparing,

  $\begin{array}{c}4x=-\frac{\pi }{2}\\ x=-\frac{\pi }{8}\end{array}$

Also,

  $\begin{array}{c}4x=\frac{\pi }{2}\\ x=\frac{\pi }{8}\end{array}$

Therefore, the two consecutive asymptotes occur at $x=-\left(\frac{\pi }{8}\right)$ and $x=\left(\frac{\pi }{8}\right)$ .

Now, let us use the software to plot the graph including two full periods which means from $-\frac{\pi }{4}$ to $\frac{\pi }{4}$ , using the command

  $\text{plot}\left(-2\cdot \sec \left(4x\right),x=-\frac{\pi }{4}\mathrm{...}\frac{\pi }{4}\right)$

The above graph represents the equation $y=-2\sec 4x$ .