Chapter 9, Problem 9PT

To determine

To select correct option that best represents the graph shown.

Answer to Problem 9PT

Option D. $y=-3x^2+2$

Explanation of Solution

Given information :

Option A. $y=-3x^2$

Option B. $y=3x^2+1$

Option C. $y=x^2+2$

Option D. $y=-3x^2+2$

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.

The graph of quadratic function $f(x)$ when multiplied by a positive constant a , the resulting graph $af(x)$ is a vertical dilation of $f(x)$ . The function is stretched or compressed vertically by a factor of $\left|a\right|$ .

If $\left|a\right|$ $>1$ , the graph of $f(x)$ is stretched vertically, that is, all points on the graph $f(x)$ move farther away from the x -axis.

If $\left|a\right|<1$ , the graph of $f(x)$ is compressed vertically, that is, all points on the graph $f(x)$ move closer to the x -axis.

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.

Option A. $y=-3x^2$

This option is incorrect as there is no upward translation of the function as shown in the graph.

Option B. $y=3x^2+1$

This option is incorrect as there is no inversion in the graph of the function as has been shown.

Option C. $y=x^2+2$

This option is incorrect as there is no inversion and vertical dilation in the graph of the function as has been shown.

Option D. $y=-3x^2+2$

This option is correct as it is inverted, stretched and translated when compared to the parent function, as shown in the graph