Chapter 3.1, Problem 26AYU

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 26AYU

Solution:

The given function is a linear function and the equation of the line is $y=0.5x-3$.

Explanation of Solution

Given:

The given function is

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

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

$Avg=\frac{-3.5-(-4)}{-1-(-2)}=0.5$

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

$Avg=\frac{-3-(-3.5)}{0-(-1)}=0.5$

Similarly, the rate of change between the other rows also can be found to be $0.5$.

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 $m=0.5$ 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

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

$\Rightarrow y-(-4)=0.5(x-(-2))$

$\Rightarrow y+4=0.5x+1$

$\Rightarrow y=0.5x+1-4$

$\Rightarrow y=0.5x-3$

Therefore, the equation of the line is $y=0.5x-3$.