Chapter 3.1, Problem 27AYU

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 27AYU

Solution:

The given function is a linear function and the equation of the line is $y = 8$.

Explanation of Solution

Given:

The given function is

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

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

$Avg = \frac{8 - 8}{-1 - (-2)} = 0$

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

$Avg = \frac{8 - 8}{0 - (-1)} = 0$

Similarly, the rate of change between the other rows also can be found to be 0.

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 $m = 0$ and let $x_1 = -2$ and $y_1 = 8$.

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

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

$\Rightarrow y-8=0(x-(-2))$

$\Rightarrow y-8=0$

$\Rightarrow y=8$

Therefore, the equation of the line is $y = 8$.