Question

1. anytime fitness charges its members $32 per month with a start - up fee of $100.

compare the functions’ rates of change.
compare the functions’ initial value.
the table shows fees for orange theory fitness

number of months, x	total cost ($), y
3	120
4	160

Explanation:

Part 1: Compare the functions’ rates of change

We need to find the rate of change (slope) for both Anytime Fitness and Orange Theory Fitness.

Step 1: Rate of change for Anytime Fitness

The cost function for Anytime Fitness can be written as \( y = 32x + 100 \), where \( x \) is the number of months and \( y \) is the total cost. The rate of change (slope) of a linear function \( y = mx + b \) is \( m \). So, the rate of change for Anytime Fitness is \( 32 \) dollars per month.

Step 2: Rate of change for Orange Theory Fitness

To find the rate of change for Orange Theory Fitness, we use the formula for slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), which is \( \frac{y_2 - y_1}{x_2 - x_1} \). Let's take the points \( (2, 80) \) and \( (3, 120) \).
\[
\text{Rate of change} = \frac{120 - 80}{3 - 2} = \frac{40}{1} = 40
\]
We can check with another pair, say \( (3, 120) \) and \( (4, 160) \):
\[
\text{Rate of change} = \frac{160 - 120}{4 - 3} = \frac{40}{1} = 40
\]
So, the rate of change for Orange Theory Fitness is \( 40 \) dollars per month.

Step 3: Compare the rates of change

The rate of change for Anytime Fitness is \( 32 \) dollars per month, and for Orange Theory Fitness, it is \( 40 \) dollars per month. So, the rate of change of Anytime Fitness is less than that of Orange Theory Fitness (\( 32 < 40 \)).

Part 2: Compare the functions’ initial value

The initial value of a linear function \( y = mx + b \) is the value of \( y \) when \( x = 0 \) (i.e., \( b \)).

Step 1: Initial value for Anytime Fitness

The cost function for Anytime Fitness is \( y = 32x + 100 \). When \( x = 0 \) (0 months), \( y = 100 \). So, the initial value (start - up fee) is \( 100 \) dollars.

Step 2: Initial value for Orange Theory Fitness

We can find the equation of the line for Orange Theory Fitness. We know the slope \( m = 40 \) (from part 1). Using the point - slope form \( y - y_1 = m(x - x_1) \), let's use the point \( (2, 80) \).
\[
y - 80 = 40(x - 2)
\]
\[
y - 80 = 40x - 80
\]
\[
y = 40x
\]
When \( x = 0 \), \( y = 0 \). So, the initial value (cost when \( x = 0 \) months) is \( 0 \) dollars.

Step 3: Compare the initial values

The initial value of Anytime Fitness is \( 100 \) dollars, and the initial value of Orange Theory Fitness is \( 0 \) dollars. So, the initial value of Anytime Fitness is greater than that of Orange Theory Fitness (\( 100>0 \)).

Answer:

s:

• Rates of change comparison: The rate of change of Anytime Fitness (\( \$32 \) per month) is less than the rate of change of Orange Theory Fitness (\( \$40 \) per month).
• Initial values comparison: The initial value of Anytime Fitness (\( \$100 \)) is greater than the initial value of Orange Theory Fitness (\( \$0 \)).