Chapter 9.2, Problem 3CYU

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: $h(x)=-x^2+8$ .

Graph :

Interpretation :

The graph of a quadratic equation ( $y=a{x}^2+bx+c$ ) is a parabola. The parabola is symmetric in nature and the point of symmetry is known as the vertex. This vertex lies on the line of symmetry and is symmetric about this line.

This vertex point shall be:

Highest point (if $a<0$ ) and be called the maximum. Here, the graph opens downwards.

Or, lowest point (if $a>0$ ) and can be called the minimum. Here the graph opens upwards.

In this case, ‘a’ is lesser than 0 hence the graph will have a maximum and will open downwards.

A parabola always points to infinity, either negative or positive.

In the function $h(x)=-x^2+8$ , the value of a is negative and thus it opens downwards. Therefore, the transformation of the function with respect to the graph of $f(x)=x^2$ , as seen in the graph shown above, is that the function has been inverted and now it open downwards.

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 $h(x)=-x^2+8$ , 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 inversion of the graph and vertical translation upward by 8 units.