Chapter 3, Problem 4RE

In Problems 4 and 5, determine whether the function is linear or nonlinear. If the function is linear, find the equation of the line.

4.

To determine

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

Answer to Problem 4RE

Solution:

The given function is linear and its equation is $y=5x+3$.

Explanation of Solution

Given:

The given function is

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

$Avg=\frac{-2-3}{-1-(0)}=5$

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

$Avg=\frac{3-8}{0-(1)}=5$

Similarly, the rate of change between the other rows also can be found to be 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 $m=5$ and let $x_1=-1$ and $y_1=-2$.

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

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

$\Rightarrow y-(-2)=5(x-(-1))$

$\Rightarrow y+2=5x+5$

$\Rightarrow y=5x+5-2$

$\Rightarrow y=5x+3$

Therefore, the equation of the line is

$y=5x+3$