Chapter 9, Problem 13MCQ

To determine

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

Explanation of Solution

Given information :

Given functions is $g(x)=-x^2+1$ .

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 greater than 0 hence the graph will have a minimum and will open upwards.

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

In the function $g(x)=-x^2+1$ , the value of a is negative and thus it opens downwards. Thus, it has a maximum value.

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 $g(x)=-x^2+1$ , the value of $k>0$ at $k=1$ . 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 1 unit.