Question

Kirk’s parents will pay for him to take swimming lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear.Lessons1234Group ($)5585115145Private ($)75125175225Find the rate of change and the initial value for the group lessons.Find the rate of change and the initial value for the private lessons.Compare and contrast the rates of change and the initial values.

Answer 1

The rate of change of a relationship is the change in output divided by the change in input.

In this case, the rate of change m is:

$m=\frac{\text{ change in cost}}{\text{change in number of lessons}}$

And, calling x the number of lessons and y the cost, we can write that linear relation as:

$y=mx+b$

where b is the initial value.

Thus, for the group lessons, we have:

$\begin{gathered} m=\frac{85-55}{2-1}=30 \\ \\ b=y-mx=55-30(1)=25 \end{gathered}$

And for private lessons, we have:

$\begin{gathered} m=\frac{125-75}{2-1}=50 \\ \\ b=y-mx=75-50(1)=25 \end{gathered}$

Therefore, the rate of change for the group lessons is 30, which is smaller than the rate of change for the private lessons, 50.

And the initial values for the group and private lessons are both the same: 25.