Chapter 4.6, Problem 31E

To determine

To sketch: the graph of the given function.

Explanation of Solution

Given information:

Given function is

  $y=2\csc \left(x-\pi \right)$ .

Calculation:

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

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

On comparing,

  $\begin{array}{l}x-\pi =0\\ x=\pi \end{array}$

Also,

  $\begin{array}{l}x-\pi =\pi \\ x=\pi +\pi \\ x=2\pi \end{array}$

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

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

For example,

Substituting $x=\frac{5\pi }{4}$

  $\begin{array}{c}y=2\csc \left(\frac{5\pi }{4}-\pi \right)\\ y=2\csc \left(\frac{\pi }{4}\right)\\ y=2\left(\sqrt{2}\right)\\ y=2\sqrt{2}\end{array}$

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

$x$	$\pi$	$\frac{5\pi }{4}$	$\frac{6\pi }{4}$	$\frac{7\pi }{4}$	$2\pi$
$y$	Not defined	$2\sqrt{2}$	2	$2\sqrt{2}$	Not defined

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

  

The above graph represents the equation $y=2\csc \left(x-\pi \right)$ .