Question

Madison's credit card balances, f(x), over the span of x months are shown in the table below.

x

(number of Months)

f(x)

(credit card Balance

3

$2,771
6 $3,491
9 $4,398
12 $5,540
15 $6,979


Which type of function is the best model for this data?





(1 point)
Responses

Quadratic Function
Quadratic Function

Linear Function
Linear Function

Exponential Function
Exponential Function

cubic Function

Answer 1

To determine which type of function best models Madison's credit card balances over time, we can first analyze the data provided:

- As the number of months (x) increases, the credit card balance (f(x)) increases consistently.
- We can calculate the differences in the balances for equal intervals (every 3 months):

1. Between 3 and 6 months:
\( f(6) - f(3) = 3,491 - 2,771 = 720 \)

2. Between 6 and 9 months:
\( f(9) - f(6) = 4,398 - 3,491 = 907 \)

3. Between 9 and 12 months:
\( f(12) - f(9) = 5,540 - 4,398 = 1,142 \)

4. Between 12 and 15 months:
\( f(15) - f(12) = 6,979 - 5,540 = 1,439 \)

The differences (720, 907, 1,142, and 1,439) are not constant, indicating it is not a linear function.

Let's find the second differences:

1. Between the first differences (720 and 907):
\( 907 - 720 = 187 \)

2. Between the next differences (907 and 1,142):
\( 1,142 - 907 = 235 \)

3. Between the last differences (1,142 and 1,439):
\( 1,439 - 1,142 = 297 \)

The second differences (187, 235, and 297) are not constant either, indicating it isn’t a quadratic function either.

Since the growth appears to be accelerating and the balance increases more rapidly over time, this suggests exponential growth.

Based on the analysis, the best model for Madison's credit card balances over time fits an **exponential function**.

So, the answer to your question is:

**Exponential Function**.