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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

Linear Function
Linear Function

Exponential Function
Exponential Function

Quadratic Function
Quadratic Function

cubic Function

1 answer

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

  • \( f(3) = 2771 \)
  • \( f(6) = 3491 \)
  • \( f(9) = 4398 \)
  • \( f(12) = 5540 \)
  • \( f(15) = 6979 \)

Let's calculate the first and second differences of the credit card balances to identify the pattern:

  1. First differences (subtract each successive value):

    • \( 3491 - 2771 = 720 \)
    • \( 4398 - 3491 = 907 \)
    • \( 5540 - 4398 = 1142 \)
    • \( 6979 - 5540 = 1439 \)

    First differences: \( 720, 907, 1142, 1439 \)

  2. Second differences (subtract each successive first difference):

    • \( 907 - 720 = 187 \)
    • \( 1142 - 907 = 235 \)
    • \( 1439 - 1142 = 297 \)

    Second differences: \( 187, 235, 297 \)

Since the first differences do not remain constant (indicating that it is not a linear function), and the second differences are also not constant (indicating it is not a quadratic function), we should consider other types.

Furthermore, the growth of the credit card balances appears to accelerate, which suggests an exponential relationship might be present.

Given this analysis, it’s likely that the best fitting model for Madison's credit card balances over time is:

Exponential Function.