Chapter 2.2, Problem 61E

To determine

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

Answer to Problem 61E

The x-intercepts are $-\text{6 and 2}$ .

The function decreases on the left hand side of the axis of symmetry $x=-2$ and increases for the right hand side of the values of the axis of symmetry $x=-2$ .

Explanation of Solution

Given Information:The function $f\left(x\right)=\frac{1}{2}\left(x-2\right)\left(x+6\right)$ is given.

Calculation:

The function $f\left(x\right)=\frac{1}{2}\left(x-2\right)\left(x+6\right)$ in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ is,

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

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

  $\begin{array}{c}p=2\\ q=-6\\ a=\frac{1}{2}\end{array}$

Using x- intercepts, the axis of symmetry is,

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

• Here, S is the axis of symmetry.

Therefore, the axis of symmetry is $x=-2$ .

Now, since the value of $a=\frac{1}{2}>0$ , the function increases in the interval $\left(-2,\infty \right)$ and decreases in the interval $\left(-\infty ,-2\right)$ .

The graph of the function is,

  

Clearly, the function decreases on the left hand side of the axis of symmetry $x=-2$ and increases for the right hand side of the values of the axis of symmetry $x=-2$ .