Question

Using the table below, create a function that represents the cost of a monthly museum membership.
months cost
1 63
2 87
3 111
4 135
5 159


This question is worth 3 points. Be sure to show your work.

Describe the initial value, the rate of change, and provide the function in the form y = mx + b.

Answer 1

To create a function that represents the cost of a monthly museum membership based on the provided data, let's identify the key components needed for the function: the initial value (y-intercept), the rate of change (slope), and the linear equation of the form \( y = mx + b \).

### Step 1: Identify the Initial Value (b)

The initial value (y-intercept) is the cost of the membership when \( x = 0 \). Although your table only provides data starting from 1 month, we can extrapolate that the membership cost \( C(0) \) for 0 months would be \( C(1) - \Delta C \), where \( \Delta C \) is the increase in cost from 1 month.

Given that:
- The cost for 1 month (x = 1) is 63.
- The cost for 2 months (x = 2) is 87.

The difference in costs from one month to the next is:
\[
\Delta C = C(2) - C(1) = 87 - 63 = 24
\]

If we assume the linear relationship continues backward (from \( C(1) \) to \( C(0) \)), we can find the initial value:
\[
C(0) = C(1) - \Delta C = 63 - 24 = 39
\]
So, the initial value \( b = 39 \) when \( x = 0 \).

### Step 2: Identify the Rate of Change (m)

The rate of change, or slope \( m \), can be found by calculating the change in cost per month. We can use the same differences in costs:

Using the costs for the first and second months:
- \( C(1) = 63 \)
- \( C(2) = 87 \)

The slope \( m \) is:
\[
m = \frac{C(2) - C(1)}{2 - 1} = \frac{87 - 63}{1} = 24
\]

We can double-check this by calculating the slope between additional points, for example, between months 2 and 3:
\[
m = \frac{C(3) - C(2)}{3 - 2} = \frac{111 - 87}{1} = 24
\]
The rate of change is consistent at \( 24 \).

### Step 3: Write the Function

Now we can compile the information into the linear function format:
- Slope \( m = 24 \)
- Initial value \( b = 39 \)

Thus, the function \( C(x) \) representing the cost of a monthly museum membership is:
\[
C(x) = 24x + 39
\]

### Summary
- **Initial value (b):** 39
- **Rate of change (m):** 24
- **Function in the form \( y = mx + b \):** \( C(x) = 24x + 39 \)

This function provides a way to determine the cost of the membership for any number of months \( x \) starting from \( x = 0 \).