Chapter 3, Problem 2CT

Determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line.

To determine

Whether the given function is linear or non linear. If linear, then find the equation of the line.

Answer to Problem 2CT

The equation of the given function is

$\Rightarrow y=-5x+2$

Explanation of Solution

Given:

The given function is

Formula used:

A function is said to be linear if it has a constant slope.

That is, we have to check if the average rate of change of the function is constant.

The average rate of change is $\frac{\Delta y}{\Delta x} = \frac{y_2 - y_2}{x_2 - x_1}$

The point-slope forms of the line through the point $(x_1, y_1)$ with slope $m$ is

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

Calculation:

The average rate of change of the given function is

Considering the first and the second points, we get $\frac{\Delta y}{\Delta x}=\frac{y_2-y_2}{x_2-x_1}=\frac{7-12}{-1-(-2)}=-5$

Considering the second and the third points, we get $\frac{\Delta y}{\Delta x}=\frac{y_2-y_2}{x_2-x_1}=\frac{2-7}{0-(-1)}=-5$

Considering the third and the forth points, we get $\frac{\Delta y}{\Delta x}=\frac{y_2-y_2}{x_2-x_1}=\frac{-3-2}{1-(0)}=-5$ and so on.

Thus, we can see that the average rate of change of the given function is a constant.

Thus, the slope of the given function is $m=-5$

Now, we have to find the equation of the line.

Thus, let us use the point slope form of the line.

Let $(x_1, y_1) = (-2, 12)$ and the slope is $m=-5$

Thus, the equation of the line joining the above 2 points is

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

$\Rightarrow y-12=-5(x-(-2))$

$\Rightarrow y-12=-5(x+2)$

$\Rightarrow y-12=-5x-10$

$\Rightarrow y=-5x-10+12$

$\Rightarrow y=-5x+2$

Thus, the equation of the given function is

$\Rightarrow y=-5x+2$