Chapter 4.6, Problem 29E

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function is

  $y=\tan \frac{\pi x}{4}$ .

Calculation:

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

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

On comparing,

  $\begin{array}{c}\frac{\pi x}{4}=-\frac{\pi }{2}\\ x=\left(-\frac{\pi }{2}\right)\left(\frac{4}{\pi }\right)\\ x=-2\end{array}$

Also,

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

Therefore, the two consecutive asymptotes occur at $x=-2$ and $x=2$ .

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

For example,

Substituting $x=-1$

We get,

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

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

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	Not defined	$-1$	$0$	1	Not defined

Now using these values, plot the graph including two full periods which means from $-4$ to $4$ .

  

The above graph represents the equation $y=\tan \left(\frac{\pi x}{4}\right)$ ..