Chapter 4.5, Problem 23E

Solution

The transformation required to obtain the graph of the function $y=3\csc x$ from a basic trigonometric function.

The function $y=3\csc x$ is obtained from $y=\csc x$ by the vertical stretching by a factor of 3.

Given information:

The function $y=3\csc x$ .

Explanation:

Consider the function $y=3\csc x$ ,

Its parent function is $y=\csc x$ .

Compare it with the function of the form as given below.

  $y=a\csc \left(b\left(x-h\right)\right)+k$  ...... (1)

The function $y=3\csc x$ can be written as the following.

  $y=3\csc \left(x-0\right)+0$  ...... (2)

Compare equation (1) and (2),

  $a=3$

The effect of $3$ is the vertical stretch by a factor of 3.

Graph the functions $y=\csc x$ and $y=3\csc x$ on the same window.

  

Thus, the function $y=3\csc x$ is obtained from $y=\csc x$ by the vertical stretching by a factor of 3.