Chapter 2.2, Problem 64E

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 64E

The x-intercepts are $-5\text{ and }-1$ .

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 $f\left(x\right)=-5\left(x+5\right)\left(x+1\right)$ is given.

Calculation:

The function $f\left(x\right)=-5\left(x+5\right)\left(x+1\right)$ in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ is,

  $\begin{array}{c}f\left(x\right)=-5\left(x+5\right)\left(x+1\right)\\ =-5\left(x-\left(-5\right)\right)\left(x-\left(-1\right)\right)\end{array}$

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

  $\begin{array}{c}p=-5\\ q=-1\\ a=-5\end{array}$

Using x- intercepts, the axis of symmetry is,

  $\begin{array}{c}\text{S}=\frac{p+q}{2}\\ =\frac{-5-1}{2}\\ =\frac{-6}{2}\\ =-3\end{array}$

• Here, S is the axis of symmetry.

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

Now, since the value of $a=-5<0$ , 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 $x=-3$ .