Chapter 3.8, Problem 62PFA

a.

To determine

To graph the function $g(x)=\left|\frac{1}{3}x\right|$ .

Explanation of Solution

Given:

Function: $g(x)=\left|\frac{1}{3}x\right|$

Concept used:

The general form of an absolute value function is $f(x)=a\left|x-h\right|+k$ .

The graph of $f(x)=a\left|x-h\right|+k$ is a “V” with vertex $\left(h,k\right)$ , slope $m=a$ on the right side of the vertex $\left(x>h\right)$ and slope $m=-a$ on the left side of the vertex $\left(x<h\right)$ .

The graph of $f(x)=-a\left|x-h\right|+k$ is an upside-down “V” with vertex $\left(h,k\right)$ , slope $m=-a$ for $\left(x>h\right)$ and slope $m=a$ for $\left(x<h\right)$ .

Calculation for graph:

Consider $g(x)=\left|\frac{1}{3}x\right|=\frac{1}{3}\left|x\right|$

Comparing it with the general form,

  $g(x)=a\left|x-h\right|+k$

Here, the value of $a=\frac{1}{3}$ , so the graph opens upwards with a slope of $\frac{1}{3}$ .

The value of $h=0$ and the value of $k=0$ , so the vertex of the graph is $\left(0,0\right)$ .

Values of x	Values of g (x)
0	$0$
1	$\frac{1}{3}$
-1	$\frac{1}{3}$
2	$\frac{2}{3}$
-2	$\frac{2}{3}$

By taking different values of x , the graph can be plotted.

Graph:

The graph of $g(x)=\left|\frac{1}{3}x\right|$ :

  

b.

To determine

To determine the relation of $g(x)=\left|\frac{1}{3}x\right|$ with the graph of the function $f(x)=\left|x\right|$ .

Answer to Problem 62PFA

The graph of $f(x)=\left|x\right|$ is compressed by $\frac{1}{3}$ units to get the graph of $g(x)=\left|\frac{1}{3}x\right|$ .

Explanation of Solution

Given:

Function: $g(x)=\left|\frac{1}{3}x\right|$

Concept used:

The general form of an absolute value function is $f(x)=a\left|x-h\right|+k$ .

The graph opens up if $a>0$ and opens down if $a<0$ . If $0<a<1$ , the graph is compressed, and if $a>1$ , it is stretched.

The graph of $f(x)=\left|x\right|$ is translated $h$ units horizontally and $k$ units vertically to get the graph of $f(x)=a\left|x-h\right|+k$ .

Consider $g(x)=\left|\frac{1}{3}x\right|=\frac{1}{3}\left|x\right|,$

The graph of $f(x)=\left|x\right|$ is compressed by $\frac{1}{3}$ , translated $0$ units horizontally and $0$ units vertically to get the graph of $g(x)=\left|\frac{1}{3}x\right|$ .

Conclusion:

Therefore, the transformation of $g(x)$ as the graph of $f(x)=\left|x\right|$ is that the graph of $f(x)=\left|x\right|$ is compressed by $\frac{1}{3}$ to get the graph of $g(x)=\left|\frac{1}{3}x\right|$ .