Chapter 9.2, Problem 5AGP

To determine

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

Explanation of Solution

Given information :

The function that is provided: $j(x)=2(-x^2)$ .

Graph :

  

Interpretation :

In the graph of quadratic function $f(x)$ when x is multiplied by a positive constant a before the quadratic function $f(x)$ is evaluated, the resultant function $f(ax)$ is horizontally stretched or compressed vertically by a factor of $\frac{1}{\left|a\right|}$ .

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

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

In the function $j(x)=2(-x^2)$ , the value of $\left|a\right|=1$ at $a=-1$ . Therefore, this has no transformational effect of horizontal dilation on $f(x)=x^2$ , as shown in the graph above.

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