Chapter 3.8, Problem 55HP

To determine

To determine the function from the given graph.

Answer to Problem 55HP

  $f(x)=\frac{1}{2}\left|x-6\right|=\{\begin{array}{l}\frac{1}{2}\left(x-6\right)\text{ if }x\ge 6\\ -\frac{1}{2}\left(x-6\right)\text{ if }x<6\end{array}$

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 $\left(6,0\right)$ , so the graph is translated $6$ units horizontally and $0$ units vertically down.

So, $h=6$ and $k=0.$

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

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

  $\begin{array}{l}\Rightarrow 2=a\left|2-6\right|+0\\ \Rightarrow 2=a\left|-4\right|\\ \Rightarrow 2=4a\\ \Rightarrow a=\frac{1}{2}\end{array}$

Hence, the stretch factor $a=\frac{1}{2}$ .

The given function is $f(x)=\frac{1}{2}\left|x-6\right|=\{\begin{array}{l}\frac{1}{2}\left(x-6\right)\text{ if }x\ge 6\\ -\frac{1}{2}\left(x-6\right)\text{ if }x<6\end{array}.$

Conclusion:

Therefore, the given function is $f(x)=\frac{1}{2}\left|x-6\right|=\{\begin{array}{l}\frac{1}{2}\left(x-6\right)\text{ if }x\ge 6\\ -\frac{1}{2}\left(x-6\right)\text{ if }x<6\end{array}$ .