Exercise 6 Page 174

A function is linear if its rate of change is constant. Looking at the table, we see that the number of months increases in increments of $3.$ If the change in total cost is constant as well, the function is linear.

We can see that the rate of change is constant. Thus, the function is linear.

Writing a Linear Model

Since the function is linear, we can write a linear model to represent the total cost as a function of the number of months. Let's use the point-slope form to write the equation. Here $m$ represents the slope of the line and $(x_1,y_1)$ is a point on the line. We have already found that the $y\text{-}$values increase by $126$ as the $x\text{-}$values increase by $3.$ We can use this to calculate the slope. We can use any point from the table for $(x_1,y_1).$ We will arbitrarily choose to use the point from first row, $(3,176).$ Let's substitute the point and the slope $42$ in the formula for the point-slope form.