Chapter 9.2, Problem 29PPS

To determine

Todescribe 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 $g(x)={(x-2)}^2+7$ and $f(x)=x^2-4$ .

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 $g(x)={(x-2)}^2+7$ , the value of $k>0$ at $k=7$ . 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 $7$ units.

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 $g(x)={(x-2)}^2+7$ , 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 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 $f(x)=x^2-4$ , the value of $k<0$ at $k=-4$ . 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 4 units.

Therefore, as seen in the graph, the transformation of the function $g(x)={(x-2)}^2+7$ to the graph of $f(x)=x^2-4$ is the vertical translation upwards by $11$ units and horizontal translation $2$ units to the right.

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