Final answer:
The rate of change for the gym membership cost, after accounting for the one-time registration fee, is the constant monthly fee of $8. This fee represents the steady monthly increase in total cost over time.
Step-by-step explanation:
The question is asking to determine the rate of change for a gym membership cost structure. In mathematical terms, the rate of change refers to how a certain quantity changes with respect to another quantity, which for this question, is the change in cost over time.
Given a one-time registration fee of $50 and a monthly fee of $8, it's important to recognize that the rate of change refers to the recurring cost, not the one-time fee. Therefore, the rate of change is the monthly fee, as that is what changes month-to-month.
Since the registration fee is a one-time payment, it does not affect the monthly rate of change. The monthly fee is constant and doesn't change over time, meaning the rate of change is $8 per month. In a graph of this scenario, where the x-axis represents time (in months) and the y-axis represents total cost, the slope of the line after the initial $50 fee would be $8 per month, illustrating a linear relationship between time and total cost, with the slope representing the rate of change.