Chapter 3.8, Problem 64PFA

To determine

To determine the function from the given graph.

Answer to Problem 64PFA

  $f(x)=2\left|x\right|-3$ , (B)

Explanation of Solution

Given:

  

Concept used:

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

The vertex of the function is $\left(h,k\right)$ .

The variables $h$ and $k$ tell how far the graph shifts horizontally and vertically.

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 $f(x)=a\left|x-h\right|+k$ .

Calculation:

Let the function be $f(x)=a\left|x-h\right|+k$ .

From the graph, it can be observed that the vertex of the function is $\left(0,-3\right)$ , so the graph is translated $0$ units horizontally and $3$ units vertically down.

So, $h=0$ and $k=-3.$

To determine the stretch factor, input the known pair of value. From the graph, $\left(2,1\right)$ satisfies the given function.

So, substituting the values in the function $f(x)=a\left|x-h\right|+k.$

  $\begin{array}{l}\Rightarrow 1=a\left|2-0\right|-3\\ \Rightarrow 1+3=a\left|2\right|\\ \Rightarrow 4=2a\\ \Rightarrow a=2\end{array}$

Hence, the stretch factor $a=2$ .

Conclusion:

Therefore, the given function is $f(x)=2\left|x\right|-3$ .

Option (B) is correct.