Chapter 3.8, Problem 61PFA

To determine

To determine the translation in the graph of the given function $g(x)=-\left|x-1\right|$ .

Answer to Problem 61PFA

Reflection across $x$ -axis (A), translation 1 unit right (C)

Explanation of Solution

Given:

Function: $g(x)=-\left|x-1\right|$

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 $g(x)=-\left|x-1\right|,$

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 $h=1$ and the value of $k=0$ , so the vertex of the graph is $\left(1,0\right)$ .

Since $h=1$ , the graph of the parent function $f(x)=\left|x\right|$ is translated 1unit right.

Conclusion:

Therefore, the graph of the parent function $f(x)=\left|x\right|$ is translated 1 unit right and reflected across $x$ -axis.

Options (A) and (C) are correct.