Final answer:
The functions' initial values and rates of change for the costs of Fitness Center A and Fitness Center B are compared. Fitness Center A has an initial cost of $100 and a rate of change of $32 per month. Fitness Center B has an initial discount of $80 and a rate of change of $40 per month.
Step-by-step explanation:
To compare the functions' initial values and rates of change, we need to find the equations that represent the costs of the two fitness centers.
For Fitness Center A, the monthly cost can be represented by the equation y = 32x + 100, where x is the number of months and y is the total cost.
For Fitness Center B, the given table shows the values of the total cost for different numbers of months. We can use this information to find the equation. Since the relationship is linear, we can calculate the slope by finding the difference in y-values (change in cost) divided by the difference in x-values (change in months).
Using the first two points from the table, we have a slope of (120 - 80) / (3 - 2) = 40 / 1 = 40. We can then use one of the points, such as (2, 80), to find the y-intercept. Using the point-slope form, we have y - 80 = 40(x - 2), which simplifies to y = 40x - 80. Therefore, the equation that represents the cost for Fitness Center B is y = 40x - 80.
Comparing the initial values, the initial cost for Fitness Center A is $100 (start-up fee), while the initial cost for Fitness Center B is $-80 (negative because the center offers a discount or incentive of $80 at the start).
Comparing the rates of change, the rate of change for Fitness Center A is $32 per month, while the rate of change for Fitness Center B is $40 per month. This means that Fitness Center B's cost is increasing at a faster rate compared to Fitness Center A.