Chapter 4.6, Problem 30E

To determine

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

Explanation of Solution

Given information:

Given function is

  $y=\tan 4x$ .

Calculation:

The vertical asymptotes for $y=\tan 4x$ are at $4x=-\left(\frac{\pi }{2}\right)$ and $4x=\frac{\pi }{2}$ , or $x=-\left(\frac{\pi }{8}\right)$ and $x=\left(\frac{\pi }{8}\right)$ . its period is $\frac{\pi }{4}$ .

So for the given equation, let us find out the values of the two consecutive asymptotes occur at $x=-\left(\frac{\pi }{8}\right)$ and $x=\left(\frac{\pi }{8}\right)$ .

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

For example,

Substitute $x=-\frac{\pi }{16}$ as shown:

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

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

$x$	$-\left(\frac{\pi }{8}\right)$	$-\left(\frac{\pi }{16}\right)$	0	$\left(\frac{\pi }{16}\right)$	$\left(\frac{\pi }{8}\right)$
$y$	Not defined	$-1$	$0$	$1$	Not defined

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

  

The above graph represents the equation $y=\tan 4x$ .