Chapter 9, Problem 13CR

To determine

To select the statement which best describes the function.

Answer to Problem 13CR

Option D. In standard form, the equation of the function is $f(x)=2x^2-8x+14$

Explanation of Solution

Given information :

Graph representing the function $h(x)$

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

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

Option C. In vertex form, the equation of the function is $f(x)=2{(x-2)}^2-6$

Option D. In standard form, the equation of the function is $f(x)=2x^2-8x+14$

:

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

It is difficult to conclude from the given graph if it is indeed vertically compressed.

Therefore, option A is rejected as it does not describe the graph of $h(x)$ in the best manner.

Option B. $h(x)$ is a reflection across 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 greater than 0 hence the graph will have a minimum and will open upwards.

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

Therefore, option B is rejected as it is incorrect.

Option C. In vertex form, the equation of the function is $f(x)=2{(x-2)}^2-6$

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.

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.

Therefore, as seen in the graph there is a vertical translation downwards by 6 units, a horizontal translation to the right by 2 units and a vertical stretch by a factor of 2.

Hence, option C is selected as the correct option describing the graph in the best way possible.

Option D. In standard form, the equation of the function is $f(x)=2x^2-8x+14$

Substituting $x=2$ in the expression above.

  $f(x)=2{(2)}^2-8(2)+14$

Simplifying.

  $=8-16+14$

Thus, $f(x)=6$ . According to the graph this value should be $f(x)=-6$

Therefore, option D. can be rejected.