Chapter 4.6, Problem 23E

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function is

  $y=\frac{1}{2}\sec \pi x$ .

Calculation:

The vertical asymptotes for $y=\frac{1}{2}\sec \pi x$ are at $x=-\frac{\pi }{2}$ and $x=\frac{\pi }{2}$ and its period is $2p$ .

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

On comparing we get,

  $\begin{array}{c}\pi x=-\frac{\pi }{2}\\ x=\left(-\frac{\pi }{2}\right)\left(\frac{1}{\pi }\right)\\ x=-\frac{1}{2}\end{array}$

Also,

  $\begin{array}{c}\pi x=\frac{\pi }{2}\\ x=\frac{\pi }{2\pi }\\ x=\frac{1}{2}\end{array}$

Therefore, the two consecutive asymptotes occur at $x=-\frac{1}{2}$ and $x=\frac{1}{2}$ .

Next, we calculate the values of $y$ foe a few values of $x$ between these two asymptotes.

For example,

Substituting $x=-\frac{1}{4}$

We get,

  $\begin{array}{c}y=\frac{1}{2}\sec \left(\pi \left(-\frac{1}{4}\right)\right)\\ y=\frac{1}{2}\sec \left(-\frac{\pi }{4}\right)\\ y=\frac{1}{2}\left(\sqrt{2}\right)\\ y=\frac{1}{\sqrt{2}}\end{array}$

Similarly, we calculate other values of $y$ and write them down in tabular form.

$x$	$-\left(\frac{1}{2}\right)$	$-\left(\frac{1}{4}\right)$	$0$	$\frac{1}{4}$	$\frac{1}{2}$
$y$	Not defined	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	Not defined

Now using these values, plot the graph including two full periods which means from $-2$ to $2$ .

  

The above graph represents the equation $y=\frac{1}{2}\sec \left(\pi x\right)$ .