Chapter 3.6, Problem 63S

To determine

To determine the transformation in the function $f(x)=\left|x-4\right|+3$ and graph it.

Answer to Problem 63S

The graph of $g(x)=\left|x\right|$ is translated $4$ units horizontally to the right and $3$ units vertically up to get the graph of $f(x)=\left|x-4\right|+3$ .

Explanation of Solution

Given:

Function: $f(x)=\left|x-4\right|+3$

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

Comparing with the parent function $g(x)=\left|x\right|$ ,

The graph of $g(x)=\left|x\right|$ is translated $4$ units horizontally to the right and $3$ units vertically up to get the graph of $f(x)=\left|x-4\right|+3$ .

Consider $f(x)=\left|x-4\right|+3$

Compare with the general form:

  $f(x)=a\left|x-h\right|+k$

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

The value of $h=4$ and the value of $k=3$ , so the vertex of the graph is $\left(4,3\right)$ .

Calculation for graph:

Consider $f(x)=\left|x-4\right|+3$

Values of x	Values of f (x)
0	$7$
1	$6$
-1	$8$
2	$5$
-2	$9$

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

Graph:

  

Conclusion:

Therefore, the transformation of $f(x)$ as the graph of $g(x)=\left|x\right|$ is that the graph of $g(x)=\left|x\right|$ is translated $4$ units horizontally to the right and $3$ units vertically up to get the graph of $f(x)=\left|x-4\right|+3$ .