Chapter 3.3, Problem 20PPS

To determine

To find whether the given function is linear.

Answer to Problem 20PPS

The function is a linear function.

Explanation of Solution

Given:

A table is given as

Concept Used:

A function is linear if the rate of change is constant.

  $\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}$

Calculation:

Consider the given coordinates in the table.

$\left(4,-1\right),\left(2,1\right),\left(0,3\right),\left(-2,5\right)$ , and $\left(-4,7\right)$

In order to evaluate the rate of change

  $\begin{array}{c}\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{1-\left(-1\right)}{2-4}\\ =\frac{2}{-2}\\ =-1\\ \text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{3-1}{0-2}\\ =\frac{2}{-2}\\ =-1\end{array}$

And

  $\begin{array}{c}\text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{-2-0}{5-3}\\ =\frac{-2}{2}\\ =-1\\ \text{Rate of change}=\frac{\text{change in }y\text{coordinate}}{\text{change in }x\text{coordinate}}\\ =\frac{2-4}{7-5}\\ =\frac{-2}{2}\\ =-1\end{array}$

Thus, rate of change is constant and it is $-1$ .

Therefore, the function is a linear function.