Chapter 3.1, Problem 22AYU

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

To determine

To calculate: Whether the given function is linear or non-linear. If it linear, then we have to determine the equation of the line.

Answer to Problem 22AYU

Solution:

The given function is non-linear.

Explanation of Solution

Given:

The given function is

$x$	$y=f(x)$
$-2$	$\mathrm{¼}$
$-1$	$\mathrm{½}$
0	1
1	2
2	4

Formula used:

A function is said to be linear, if it has a constant average rate of change otherwise the function is non-linear.

The formula for finding the average rate of change is

$Avg=\frac{f(x_2)-f(x_1)}{x_2-x_1}$

We know that the average rate of change of a linear function is the slope of that function.

Therefore, we can find the equation of the line using the point-slope formula.

Thus, we get

$y-y_1=m(x-x_1)$

Calculation:

Now, we have to find the average rate of change.

The rate of change between the first and the second row is

$\text{Avg}=\frac{\frac{1}{2}-\frac{1}{4}}{-1-(-2)}$

$=\frac{\frac{2-1}{4}}{-1-(-2)}$

$=\frac{\frac{1}{4}}{-1-(-2)}=\frac{1}{4}$

The rate of change between the second and the third row is

$Avg=\frac{1-\frac{1}{2}}{0-(-1)}$

$=\frac{\frac{1}{2}}{0-(-1)}=\frac{1}{2}$

Here, we can see that the rate of change of the given function differs.

Therefore, the average rate of change is not a constant.

Thus, the given function is non-linear.