Chapter 2, Problem 2CA

To determine

To describe: The transformation from the graph of $f$ to the graph of g .

Answer to Problem 2CA

The combinations that describe the transformation from the graph of $f$ to the graph of g are,

A, C, D.

Explanation of Solution

Given Information: The function $g\left(x\right)=\frac{1}{2}\left|x-4\right|+4$ is a combination of transformations of $f\left(x\right)=\left|x\right|$ .

Calculation:

Consider the graph of $f\left(x\right)=\left|x\right|$ .

If the graph of $f$ is translated 4 units rights and vertical shrink by a factor of $\frac{1}{2}$ , followed by a translation 4 units up. The graph will be,

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

Which is the required graph of g.

If the graph of $f$ is translated 4 units rights and 4 units up followed by a vertical shrink by a factor of $\frac{1}{2}$ .The graph will be,

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

Which is not the required graph of g.

If the graph is vertically shrinked followed by translation 4 units rights and 4 units up. The graph will be,

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

Which is the required graph of g.

If the graph is translated 4 units rights and 8 units up, followed by a vertical shrink of $\frac{1}{2}$ The graph will be,

  $\begin{array}{c}f\left(x-4\right)=\left|x-4\right|\\ f\left(x-4\right)+8=\left|x-4\right|+8\\ \frac{1}{2}\left(f\left(x-4\right)+8\right)=\frac{1}{2}\left(\left|x-4\right|+8\right)\\ g\left(x\right)=\frac{1}{2}\left|x-4\right|+4\end{array}$

Which is the required graph of g.