Chapter 9, Problem 19SGR

To determine

To graph and describe the relation between the given function to the graph of the function $f(x)=x^2$ .

Explanation of Solution

Given information :

Given functions is $f(x)=4x^2-18$ .

The graph of quadratic function $f(x)$ when multiplied by a positive constant a , the resulting graph $af(x)$ is a vertical dilation of $f(x)$ . The function is stretched or compressed vertically by a factor of $\left|a\right|$ .

If $\left|a\right|$$>1$ , the graph of $f(x)$ is stretched vertically, that is, all points on the graph $f(x)$ move farther away from the x -axis.

If $\left|a\right|<1$ , the graph of $f(x)$ is compressed vertically, that is, all points on the graph $f(x)$ move closer to the x -axis.

In the function $f(x)=4x^2-18$ , the value of $\left|a\right|>1$ at $a=4$ . Thus, the relation of $f(x)$ when multiplied by a , as seen in the graph, is that it is stretched vertically by a factor of $4$ .

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 $f(x)=4x^2-18$ , the value of $k<0$ at $k=-18$ . Therefore, the relation 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 $18$ units.