Chapter 9.2, Problem 12PPS

To determine

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

Explanation of Solution

Given information :

The function that is provided: $g(x)=2{(x-3)}^2+8$

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 $g(x)=2{(x-3)}^2+8$ , 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 stretched vertically by a factor of $2$ .

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 $g(x)=2{(x-3)}^2+8$ , the value of $h>0$ at $h=3$ . Therefore, the translation of the function with respect to the graph of $f(x)=x^2$ , as seen in the graph shown above, is $3$ units to the right.

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 $g(x)=2{(x-3)}^2+8$ , the value of $k>0$ at $k=8$ . Therefore, the translation of the function with respect to the graph of $f(x)=x^2$ , as seen in the graph shown above, can be described as an upward translation by $8$ units.

Therefore, as seen in the graph, the transformation of the function with respect to the graph of $f(x)=x^2$ is the horizontal translation $3$ units to the right, vertically stretched by a factor of $2$ and a vertical translation upwards by $8$ units.