Linear Model: Yes, since the rate of change is constant. Point Slope Form: y-176 = 42(x-3)
Practice makes perfect
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. As such, 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.
y- y_1= m(x- x_1)
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-values increase by 126 as the x-values increase by 3. We can use this to calculate the slope.
m=Change iny/Change inx ⇒ m=126/3= 42
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.
y- 176 = 42(x- 3)
