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	1
0	-2
1	-5
2	-8

Short answer

The function is linear and the equation of the line is $f\left(x\right)=-3x-2$.

Step-by-step solution

Step 1. Given Information.

x	$y=f\left(x\right)$
-2	4
-1	1
0	-2
1	-5
2	-8

The data in the table represent values of y with respect to x.

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	-3
-1	1	-3
0	-2	-3
1	-5	-3
2	-8

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

Step 3. Find the equation of line.

The average rate of change is $m=-3$and a point from given data $\left(-2,4\right)$.

Use slope point form to find the equation of the line.

$\left(y-y_1\right)=m\left(x-x_1\right)$

localid="1646218447844" $\begin{array}{rcl}y-4& =& -3\left(x-\left(-2\right)\right)\\ y-4& =& -3\left(x+2\right)\\ y& =& -3x-6+4\\ y& =& -3x-2\end{array}$

Step 4. Conclusion

As average rate of change is constant, so function is linear and the equation of the line is

$f\left(x\right)=-3x-2$