Question

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line.

x	$y=f\left(x\right)$
-2	-4
-1	-3.5
0	-3
1	-2.5
2	-2

Short answer

The function is linear and the equation of the line is$f\left(x\right)=0.5x-3$

Step-by-step solution

Step 1. Given Information  

Given data is

x	$y=f\left(x\right)$
-2	-4
-1	-3.5
0	-3
1	-2.5
2	-2

The data in the table represent values ofy with respect tox.

Compute the average rate of change of each function.

• If the average rate of change is constant, then the function is linear.
• If the average rate of change is not constant, then the function is nonlinear.

Step 2. Calculation   

Calculate average rate of change by the formula $\frac{∆y}{∆x}=\frac{y_2-y_1}{x_2-x_1}$

x	y	$\frac{∆y}{∆x}$
-2	-4	0.5
-1	-3.5	0.5
0	-3	0.5
1	-2.5	0.5
2	-2

The average rate of change is constant, so the function is linear.

Step 3. Find equation of line.  

The average rate of change is $m=0.5$and a point from given data $(-2,-4)$.

Use slope point form to find equation of the line.

$$\begin{gathered} y-y_1=m(x-x_1) \\ y-(-4)=0.5(x-(-2\left)\right) \\ y+4=0.5(x+2) \\ y+4=0.5x+1 \\ y=0.5x-3 \end{gathered}$$

Step 4. Conclusion  

As average rate of change is constant, so function is linear and the equation of the line is $f\left(x\right)=0.5x-3$