Chapter 2.2, Problem 63E

To determine

To identify: The x- intercepts of the function and describe where the graph is increasing and decreasing and verify using a graphical calculator.

Answer to Problem 63E

The x-intercepts are $2\text{ and 4}$ .

The function decreases on the right hand side of the axis of symmetry $x=3$ and increases for the left hand side of the values of the axis of symmetry $x=3$ .

Explanation of Solution

Given Information:The function $f\left(x\right)=-4\left(x-4\right)\left(x-2\right)$ is given.

Calculation:

The function $f\left(x\right)=-4\left(x-4\right)\left(x-2\right)$ is in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ .

On comparing, the x- intercepts and the value of a is,

  $\begin{array}{c}p=4\\ q=2\\ a=-4\end{array}$

Using x- intercepts, the axis of symmetry is,

  $\begin{array}{c}\text{S}=\frac{p+q}{2}\\ =\frac{4+2}{2}\\ =\frac{6}{2}\\ =3\end{array}$

• Here, S is the axis of symmetry.

Therefore, the axis of symmetry is $x=3$ .

Now, since the value of $a=-4<0$ , the parabola is inverted.

The function increases in the interval $\left(-\infty ,3\right)$ and decreases in the interval $\left(3,\infty \right)$ .

The graph of the function is,

  

Clearly, the function decreases on the right hand side of the axis of symmetry $x=3$ and increases for the left hand side of the values of the axis of symmetry $x=3$ .