Chapter 3.8, Problem 29PPS

To determine

To graph the function $g(x)=\left|x+2\right|-3$ and determine its vertex and its relation with the graph of the function $f(x)=\left|x\right|$ .

Answer to Problem 29PPS

Vertex: $\left(-2,-3\right)$

Translation: $2$ units left and $3$ units down

Explanation of Solution

Given:

Function: $g(x)=\left|x+2\right|-3$

Concept used:

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

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 for graph:

Consider $g(x)=\left|x+2\right|-3$

Compare with the general form:

  $g(x)=a\left|x-h\right|+k$

Here, the value of $a=1$ , so the graph opens upwards with a slope of $1$ .

The value of $h=-2$ and the value of $k=-3$ , so the vertex of the graph is $\left(-2,-3\right)$ .

Values of x	Values of g (x)
0	$-1$
1	$0$
-1	$-2$
2	$1$
-2	$-2$

By taking different values of x , the graph can be plotted.

Graph:

The graph of $g(x)=\left|x+2\right|-3$ :

  

Interpretation:

From the graph, it can be observed that the vertex of the graph is $\left(-2,-3\right)$ .

Comparison:

Now, comparing the graph of $g(x)$ with the graph of $f(x)=\left|x\right|$ .

The graph of $f(x)=\left|x\right|$ is translated $2$ units horizontally to the left and $3$ units vertically down to get the graph of $g(x)=\left|x+2\right|-3$ .

Conclusion:

Therefore, the vertex of $g(x)=\left|x+2\right|-3$ is $\left(-2,-3\right)$ .

The graph of $f(x)=\left|x\right|$ is translated $2$ units horizontally to the left and $3$ units vertically down to get the graph of $g(x)=\left|x+2\right|-3$ .