Chapter 4.6, Problem 21E

To determine

To sketch: the graph of two full period of the given function.

Explanation of Solution

Given information:

Given function is

  $y=\csc \pi x$ .

Calculation:

The vertical asymptotes for $y=\csc \pi x$ are at $x=0$ and $x=\pi$ and its period is $2$ .

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

On comparing we get,

  $\begin{array}{c}\pi x=0\\ x=\frac{0}{\pi }\\ x=0\end{array}$

Also,

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

Therefore, the two consecutive asymptotes occur at $x=0$ and $x=1$ .

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,

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

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

$x$	0	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{6}$	1
$y$	Not defined	$\sqrt{2}$	1	2	Not defined

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

  

The above graph represents the equation $y=\csc \left(\pi x\right)$ .