Chapter 9.2, Problem 2CYU

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: $h(x)=\frac{1}{2}{(x-2)}^2$ .

Graph :

Interpretation :

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 $h(x)=\frac{1}{2}{(x-2)}^2$ , the value of $h>0$ at $h=2$ . Therefore, the translation of the function with respect to the graph of $f(x)=x^2$ , as seen in the graph shown above, is 2 units to the right.

The graph of quadratic function $f(x)$ when multiplied by a positive constant a , the resulting graph 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 $h(x)=\frac{1}{2}{(x-2)}^2$ , the value of $\left|a\right|<1$ at $a=\frac{1}{2}$ . Thus, the transformation of $f(x)$ when multiplied by a is vertical compression by a factor of $\frac{1}{2}$ .

Therefore, as seen in the graph, the transformation of the function with respect to the graph of $f(x)=x^2$ is horizontal translation to the right by 2 units and vertical compression by a factor of $\frac{1}{2}$ .