Chapter 9.2, Problem 30PPS

To determine

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

Explanation of Solution

Given information :

Given functions are $g(x)=-3x^2$ and $f(x)=-6x^2$ .

Graph :

  

Interpretation :

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 $g(x)=-3x^2$ , the value of $\left|a\right|>1$ at $a=-3$ . Thus, the transformation of $f(x)$ when multiplied by a , as seen in the graph, is that it is stretched vertically by a factor of $3$ .

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 $f(x)=-6x^2$ , the value of $\left|a\right|>1$ at $a=-6$ . Thus, the transformation of $f(x)$ when multiplied by a , as seen in the graph, is that it is stretched vertically by a factor of $6$ .

Therefore, as seen in the graph, the transformation of the function $g(x)=-3x^2$ to the graph of $f(x)=-6x^2$ is the vertical stretch by the factor of 2 .