Chapter 9.2, Problem 41PPS

To determine

To describe and graph the contrast in the pair of functions.

Explanation of Solution

Given information :

Given functions are $y=x^2-4$ and $y={(x-4)}^2$ .

Graph :

  

Interpretation :

The graph represented by $g(x)=x^2+k$ is equivalent to the graph of $f(x)=x^2$ that has been translated vertically. The direction of translation or movement of the graph is dependent on the value of k .

If $k>0$ , the graph of $f(x)=x^2$ is translated $\left|k\right|$ units upwards.

If $k<0$ , the graph of $f(x)=x^2$ is translated $\left|k\right|$ units downwards.

In the function $y=x^2-4$ , the value of $k<0$ at $k=-4$ . Therefore, the translation of the function with respect to the graph of $f(x)=x^2$ , as seen in the graph shown above, can be described as a downward translation by $4$ unit.

The graph represented by $g(x)={(x-h)}^2$ is equivalent to the graph of $f(x)=x^2$ that has been translated horizontally. The direction of translation or movement of the graph is dependent on the value of $h$ .

If $h>0$ , the graph of $f(x)=x^2$ is translated $\left|h\right|$ units to the right.

If $h<0$ , the graph of $f(x)=x^2$ is translated $\left|h\right|$ units to the left.

In the function $y={(x-4)}^2$ , the value of $h>0$ at $h=4$ . Therefore, the translation of the function with respect to the graph of $y=x^2$ , as seen in the graph shown above, is $4$ units to the right.

Therefore, as seen in the graph, the contrast of the function $y=x^2-4$ to the graph of $y={(x-4)}^2$ is the vertical translation downwards by $4$ units and horizontal translation to the left by $4$ units.