Chapter 3.8, Problem 58HP

To determine

To determine the translation in the graph of the given function $g(x)=\left|x+8\right|$ .

Answer to Problem 58HP

  $8$ units

Explanation of Solution

Given:

Function: $g(x)=\left|x+8\right|$

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:

Consider the function $g(x)=\left|x+8\right|,$

Comparing with the general form $g(x)=a\left|x-h\right|+k.$

Here, the value of $a=1$ , so the graph opens downwards with a slope of $1$ and hence $f(x)=\left|x\right|$ is reflected across $x$ -axis.

The value of $h=-8$ and the value of $k=0$ , so the vertex of the graph is $\left(-8,0\right)$ .

Since $h=-8$ , the graph of the parent function $f(x)=\left|x\right|$ is translated $8$ units horizontally to the left of the graph.

Conclusion:

Therefore, the graph of the parent function $f(x)=\left|x\right|$ is translated $8$ units horizontally to the left of the graph.

Therefore, Darcy makes an error in determining the translation of the graph.