Chapter 2.2, Problem 62E

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 62E

The x-intercepts are $-1\text{ and 3}$ .

The function decreases on the left hand side of the axis of symmetry $x=1$ and increases for the right hand side of the values of the axis of symmetry $x=1$ .

Explanation of Solution

Given Information: The function $f\left(x\right)=\frac{3}{4}\left(x+1\right)\left(x-3\right)$ is given.

Calculation:

The function $f\left(x\right)=\frac{3}{4}\left(x+1\right)\left(x-3\right)$ in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ is,

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

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

  $\begin{array}{c}p=-1\\ q=3\\ a=\frac{3}{4}\end{array}$

Using x- intercepts, the axis of symmetry is,

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

• Here, S is the axis of symmetry.

Therefore, the axis of symmetry is $x=1$ .

Now, since the value of $a=\frac{3}{4}>0$ , the function increases in the interval $\left(1,\infty \right)$ and decreases in the interval $\left(-\infty ,1\right)$ .

The graph of the function is,

  

Clearly, the function decreases on the left hand side of the axis of symmetry $x=1$ and increases for the right hand side of the values of the axis of symmetry $x=1$ .