Chapter 3.6, Problem 19E

To determine

The transformation of $r\left(x\right)=f\left(\frac{1}{2}x\right)$ from $f\left(x\right)=-2x-4$ .

Answer to Problem 19E

The graph of function $r\left(x\right)$ is horizontal stretch of $f\left(x\right)$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=-2x-4$ .

Calculation:

Graph is transform from the parent function $f\left(x\right)$ by multiplication of any constant $\left(a>1\right)$ with the x -coordinate (inputs); the transformation of graph is a horizontal shrink as the graph shrinks toward the y -axes. If $\left(0<a<1\right)$ , the transformation of graph is a horizontal stretch as the graph stretches away from the y-axis.

Here, the multiplication constant is $\frac{1}{2}$ which is in between 0 and 1. Thus, the graph will transform as horizontal stretch.

Function $r\left(x\right)=f\left(\frac{1}{2}x\right)$ is calculated as follows:

  $\begin{array}{c}r\left(x\right)=f\left(\frac{1}{2}x\right)=\frac{1}{2}\left(-2x-4\right)\\ =-x-2\end{array}$

The graph of $r\left(x\right)$ and $f\left(x\right)$ is shown below:

  

Thus, from above graph, the graph of function $r\left(x\right)$ is horizontal stretch of $f\left(x\right)$ .