Chapter 9.2, Problem 28PPS

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 $g(x)=x^2-2$ and $f(x)=x^2+3$ .

Graph :

  

Interpretation :

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 $g(x)=x^2-2$ , the value of $k<0$ at $k=-2$ . 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 a downward translation by $2$ units.

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 $f(x)=x^2+3$ , the value of $k>0$ at $k=3$ . 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 a upward translation by $3$ units.

Therefore, as seen in the graph, the transformation of the function $g(x)=x^2-2$ to the graph of $f(x)=x^2+3$ is the vertical translation downwards by $5$ units.