Chapter 3.7, Problem 40E

To determine

ToDescribe:Thetransformations from the graph of $f\left(x\right)=\left|x\right|$ to graph of $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$ . Then graph the function $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$

Answer to Problem 40E

The graph of $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$ is

  

Explanation of Solution

Given:

The function $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$

Calculation:

From the graph of $f\left(x\right)=\left|x\right|$ . There are certain transformations which will get the graph of $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$

Those transformations are as follows:

First transformation

A vertical shrink of the graph of $n\left(x\right)$ by a factor of $\frac{1}{3}$ and a reflection in the $x\text{-axis}$ .

Second transformation

A horizontal translation $1$ unit to the left of the resultant graph after first transformation.

Third transformation

After that a vertical translation $2$ units up of the resultant graph after second transformation.

Transforming the graph of $f\left(x\right)=\left|x\right|$ by using the transformations described above, the graph of $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$ is

  

Conclusion:

The graph of $n\left(x\right)=\left|-\frac{1}{3}x+1\right|+2$ is