Chapter 4.6, Problem 19E

To determine

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

Explanation of Solution

Given information:

Given function is

  $y=-2\tan 3x$ .

Calculation:

The vertical asymptotes for $y=-2\tan 3x$ are at $3x=-\left(\frac{\pi }{2}\right)$ and $3x=\frac{\pi }{2}$ , or $x=-\left(\frac{\pi }{6}\right)$ and $x=\left(\frac{\pi }{6}\right)$ . its period is $\frac{\pi }{3}$ .

So for the given equation, let us find out the values of the two consecutive asymptotes occur at $x=-\left(\frac{\pi }{6}\right)$ and $x=\left(\frac{\pi }{6}\right)$ .

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

For example,

Substitute $x=-\frac{\pi }{12}$ as shown:

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

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

$x$	$-\left(\frac{\pi }{6}\right)$	$-\left(\frac{\pi }{12}\right)$	0	$\left(\frac{\pi }{12}\right)$	$\left(\frac{\pi }{6}\right)$
$y$	Not defined	$-2$	$0$	$-2$	Not defined

Now using these values, plot the graph including two full periods which means from $-\frac{\pi }{3}$ to $\frac{\pi }{3}$ .

  

The above graph represents the equation $y=-2\tan 3x$ .