Chapter 4.6, Problem 43E

To determine

To graph: the given function.

Explanation of Solution

Given information:

Given function is

  $y=\tan \left(x-\frac{\pi }{4}\right)$ .

Calculation:

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

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

On comparing,

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

Also,

  $\begin{array}{c}x-\frac{\pi }{4}=\frac{\pi }{2}\\ x=\frac{\pi }{2}+\frac{\pi }{4}\\ x=\frac{3\pi }{4}\end{array}$

Therefore, the two consecutive asymptotes occur at $x=-\left(\frac{\pi }{4}\right)$ and $x=\frac{3\pi }{4}$ .

Now, let us use the software to plot the graph including two full periods which means from $-2\pi$ to $2\pi$ , using the command:

  $\text{plot}\left(\tan \left(x-\frac{\pi }{4}\right),x=-2\pi \mathrm{...2}\pi \right)$

  

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