Chapter 3.1, Problem 21AYU

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 21AYU

Solution:

The given function is a linear function and the equation of the line is $y=-3x-2$.

Explanation of Solution

Given:

The given function is

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

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

$Avg=\frac{1-4}{-1-(-2)}=-3$

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

$Avg=\frac{-2-1}{0-(-1)}=-3$

Similarly, the rate of change between the other rows also can be found to be $-3$.

Therefore, the average rate of change is a constant.

Thus, the given function is linear.

Now, we need to determine the equation of the line.

Here, we have the slope $m=-3$ and let $x_1=-2$ and $y_1=4$.

Now, by using the point-slope formula, we can write the equation of the line as

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

$\Rightarrow y-4=-3(x-(-2))$

$\Rightarrow y-4=-3x-6$

$\Rightarrow y=-3x-6+4$

$\Rightarrow y=-3x-2$

Therefore, the equation of the line is $y=-3x-2$.