Chapter 9.2, Problem 13PPS

To determine

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

Explanation of Solution

Given information :

The function that is provided: $h(x)=6+\frac{2}{3}{x}^2$ .

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 $h(x)=6+\frac{2}{3}{x}^2$ , the value of $\left|a\right|<1$ at $a=\frac{2}{3}$ . 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 $\frac{2}{3}$ .

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 $h(x)=6+\frac{2}{3}{x}^2$ , the value of $k>0$ at $k=6$ . 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 $6$ units.

Therefore, as seen in the graph, the transformation of the function with respect to the graph of $f(x)=x^2$ is the vertical translation upwards by 6 units and vertical compression by a factor of $\frac{2}{3}$ .