Chapter 9.2, Problem 35PPS

To determine

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

Explanation of Solution

Given information :

Given functions are $y=\frac{1}{2}{x}^2$ and $y=3x^2$ .

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 $y=\frac{1}{2}{x}^2$ , the value of $\left|a\right|<1$ at $a=\frac{1}{2}$ . Thus, the transformation of $f(x)$ when multiplied by a , as seen in the graph, is that it is compressed vertically by a factor of $\frac{1}{2}$ .

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 $y=3x^2$ , 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 $3$ .

Therefore, as seen in the graph, the contrast of the function $y=\frac{1}{2}{x}^2$ to the graph of $y=3x^2$ is that the latter is vertically stretched by a factor of $6$ when compared to the former.