Chapter 9, Problem 9SGR

To determine

To state if true or false.

Answer to Problem 9SGR

True.

Explanation of Solution

Given information :

The solutions of a quadratic equation are called roots.

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.

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 $g(x)={(x-h)}^2$ 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.

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 graph of quadratic function $f(x)$ when x is multiplied by a positive constant a before the quadratic function $f(x)$ is evaluated, the resultant function $f(ax)$ is horizontally stretched or compressed vertically by a factor of $\frac{1}{\left|a\right|}$ .

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

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

If the quadratic function $y=a{x}^2+bx+c$   has real solutions can be found out by using the quadratic formula:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The formula $b^2-4ac$ , also known as the discriminant can be used to determine the number of roots of solutions in a quadratic equation. There are three cases:

• $b^2-4ac<0$: The equation has 0 real solutions. The graph does not cross the  x -axis.
• $b^2-4ac=0$: The equation has 1 real solution. The graph touches the  x -axis at one point.
• $b^2-4ac>0$: The equation has 2 real solutions. The graph crosses through the  x -axis at two points.

A real number  x  will be called a solution or a root iff it satisfies the equation, meaning   $a{x}^2+bx+c=0$.