Chapter 4.3, Problem 68PPS

(a)

To determine

To Find: The graph for the function that is related to $a=1,p=2$ and $q=-3$ .

Answer to Problem 68PPS

The required graph is shown in Figure 1.

Explanation of Solution

Given:

The given general equation is $a\left(x-p\right)\left(x-q\right)=0$ .

The given diagram is shown in Figure 1

    Figure 1

Calculation:

To graph the given related function to show that the values of the quadratic function as,

$\begin{array}{l}f\left(x\right)=1\left(x-2\right)\left(x+3\right)\\ f\left(x\right)=x^2+x-6\end{array}$

The graph of the above function is shown in Figure 1

    Figure 1

(b)

To determine

To Find: The solution of the equation.

Answer to Problem 68PPS

The required solution of the given equation is $x=2,-3$ .

Explanation of Solution

Consider the equation is,

$\begin{array}{l}\left(x-2\right)\left(x+3\right)=0\\ x=2,-3\end{array}$

The required solution of the given equation is $x=2,-3$ .

(c)

To determine

To Find: The graph for the related function for $a=4$ , $-3$ and $\frac{1}{2}$ on the same graph.

Answer to Problem 68PPS

The graph for the related function is shown in Figure 1

Explanation of Solution

Consider the function is,

$\begin{array}{c}y=4\left(x^2+x-6\right)\\ =-3\left(x^2+x-6\right)\\ =\frac{1}{2}\left(x^2+x-6\right)\end{array}$

The required graph of the function is shown in Figure 2

    Figure 2

(d)

To determine

To Find: The similarities and the differences between the graphs.

Answer to Problem 68PPS

The similarities and the differences between the graphs is explained.

Explanation of Solution

The figure of the graph two shows that all the three curves have the same number of zeroes but have different slopes. The graph have a positive coefficient both have minimum but one that have the negative coefficient have maximum. The graph with $a=4$ have larger slope than one with the $a=\frac{1}{2}$ but its slope is close to the $a=-3$ .

(e)

To determine

To Find: The conclusion that can be made about the relationship between the factored form of the quadratic equation and the solutions.

Answer to Problem 68PPS

The conclusion made is described.

Explanation of Solution

The conclusion that can be made is that the relationship of the factored form of the quadratic equation and the solution is known already for the allowed factored form by the user of the zero product property. This occurs for the case when the factored form is equal to the 0 and can be isolated the variable to determine the solution.