Chapter 8.5, Problem 1E

To determine

To fill: The blank in the statement.

Answer to Problem 1E

The complete statement is “The values $p\text{ and }q$ are $x-\text{intercepts}$ of the graph of the function $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ ”.

Explanation of Solution

The given equation is the $x$ -intercept form of the parabola.

The x -intercepts are the points on the graph of the parabola that intersects the x -axis.

The $x$ -intercepts represents zeros, or roots, of the quadratic function where the value of $y=0$ .

Equate the equation $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ with 0 to obtain the zeroes of the equation.

  $\begin{array}{c}a\left(x-p\right)\left(x-q\right)=0\\ x-p=0\text{ or }x-q=0\left(\text{zero product property}\right)\\ x=p\text{ or }x=q\end{array}$

It can be seen $x=p\text{ and }x=q$ are the $x$ intercepts of the quadratic equation $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ .

Therefore, The complete statement is “The values $p\text{ and }q$ are $x-\text{intercepts}$ of the graph of the function $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ ”.