Chapter 9.2, Problem 50PFA

To determine

To select from options describing the relationship between $g(x)$ and $f(x)=x^2$ which is not true.

Answer to Problem 50PFA

Option C. In vertex form, the equation of the function is $g(x)=-2{(x-1)}^2-1$

Explanation of Solution

Given information :

Option A. $g(x)$ is a vertical stretch of the graph of $f(x)=x^2$

Option B. $g(x)$ is a reflection across the x -axis of the graph of $f(x)=x^2$

Option C. In vertex form, the equation of the function is $g(x)=-2{(x-1)}^2-1$

Option D. In standard form, the equation of the function is $g(x)=-2x^2-4x-3$

Option A. $g(x)$ is a vertical stretch of the graph of $f(x)=x^2$

It cannot be concluded from the graph if $g(x)$ is not a vertical stretch of the graph of $f(x)=x^2$ .

Hence, cannot select option A. as the incorrect one.

Option B. $g(x)$ is a reflection across the x -axis of the graph of $f(x)=x^2$

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 $f(x)=(-a)x^2$ , the value of a is negative and thus it opens downwards. Therefore, the transformation of the function with respect to the graph of $f(x)=x^2$ is that the function has been inverted and now it open downwards.

This is true for the graph shown. Hence, option B is eliminated.

Option C. In vertex form, the equation of the function is $g(x)=-2{(x-1)}^2-1$

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 $g(x)=-2{(x-1)}^2-1$ , the value of $h>0$ at $h=1$ . Therefore, the translation of the function with respect to the graph of $y=x^2$ , as seen in the graph shown above, is $1$ unit to the right.

This is not true as the horizontal translation in graph shown is to the left.

Therefore, option C is selected as the correct answer.