Chapter 5.6, Problem 1E

Solution

To sketch: The graph of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ and to check the answer with the help of a graphing calculator.

Given:

The function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ .

Graph:

Identify the zeroes of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ :

Observe that the zeroes of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ are those pairs $\left(x,0\right)$ for which $h\left(x\right)=0$ .

Substitute $h\left(x\right)=0$ in $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ :

  $0=\left(x+6\right)\left(x-7\right)$ .

Solve the equation $0=\left(x+6\right)\left(x-7\right)$ for $x$ :

  $\begin{array}{c}0=\left(x+6\right)\text{or }0=\left(x-7\right)\\ -6=x\text{ or }7=x\end{array}$ .

Thus, the zeroes of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ are $\left(-6,0\right)$ and $\left(7,0\right)$ .

Plot the zeroes $\left(-6,0\right)$ and $\left(7,0\right)$ of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$

on a coordinate grid:

  

Determine whether $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ is positive or negative on each interval determined by the zeroes $\left(-6,0\right)$ and $\left(7,0\right)$ :

Observe that the intervals determined by the zeroes $\left(-6,0\right)$ and $\left(7,0\right)$ are $x<-6$ , $-6<x<7$ and $x>7$ .

To determine whether $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ is positive or negative over the interval $x<-6$ , evaluate $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ for some value of $x$ that lies in $x<-6$ , say $x=-7$ , for its sign.

That is, substitute $x=-7$ in $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ and evaluate $h\left(-7\right)$ :

  $\begin{array}{l}h\left(-7\right)=\left(-7+6\right)\left(-7-7\right)\\ h\left(-7\right)=14\end{array}$ .

Observe that $h\left(-7\right)=14$ is positive.

Thus, $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ is positive over $x<-6$ .

Similarly, it may be observed that $h\left(0\right)=-42$ and $h\left(8\right)=14$ .

That is, $h\left(x\right)=\left(x+6\right)\left(x-7\right)$ is negative over $-6<x<7$ and is positive over $x>7$ .

Sketch the graph of the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$

incorporating all the information gained during the analysis:

  

Verify the sketch with the help of a graphing calculator:

Graph the function $h\left(x\right)=\left(x+6\right)\left(x-7\right)$

with the help of a graphing calculator on a similar grid:

  

Observe that the graph so obtainedlooks exactly like the one sketched previously.

Thus, it is verified that the sketch is correct.