Chapter 1.7, Problem 57E

(a)

To determine

To find: The function for the given graph.

Answer to Problem 57E

The required function is $g\left(x\right)=-\frac{1}{2}\left|x\right|$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=\left|x\right|$ .

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The parent function is,

  $f\left(x\right)=\left|x\right|$

To obtain the equation $g\left(x\right)$ from $f\left(x\right)=\left|x\right|$ , reflect the equation across the x axis and vertical stretch it by a $k$ and at last vertical stretch the function by $k$ .

  $\begin{array}{c}g\left(x\right)=-\left|x\right|\\ =-\left|kx\right|\\ =-k\left|x\right|\end{array}$

The graph passes through the point $\left(4,-2\right)$ as,

  $\begin{array}{l}-k\left|4\right|=-2\\ 4k=2\\ k=\frac{1}{2}\end{array}$

Then, the equation of the graph is,

  $g\left(x\right)=-\frac{1}{2}\left|x\right|$

(b)

To determine

To find: The function for the given graph.

Answer to Problem 57E

The required function is $g\left(x\right)=3\left|x\right|-3$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=\left|x\right|$ .

The given graph is shown in Figure 2

  

Figure 2

Calculation:

The parent function is,

  $f\left(x\right)=\left|x\right|$

To obtain the equation $g\left(x\right)$ from $f\left(x\right)=\left|x\right|$ , vertical shifting the graph by 3 units downward and the horizontal stretching the graph by $k$ units where $k>0$ as,

  $\begin{array}{c}g\left(x\right)=\left|x\right|-3\\ =\left|kx\right|-3\end{array}$

The graph passes through the point $\left(-2,3\right)$ as,

  $\begin{array}{l}\left|k\left(-2\right)\right|-3=3\\ 2k=6\\ k=3\end{array}$

Then, the equation of the graph is,

  $g\left(x\right)=3\left|x\right|-3$