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The value of a certain investment over time is given in the table below. Answer the questions below to explain what kind of function would better model the data, linear or exponential.
Number of Years Since Investment Made, x
1
2
3
4
Value of Investment ($), f(x)
20,189.92
18,059.23
16,008.49
14,276.14
Answer
Attempt 1 out of 2

function would better model the data because as x increases, the y values change
. The
of this function is approximately
.

Answer 1

[AI-generated answer — Bot GPT 3.5]

In order to determine whether a linear or exponential function would better model the data, we need to analyze the relationship between the number of years since the investment was made (x) and the value of the investment (f(x)).

If the value of the investment increases or decreases at a constant rate as the number of years increases, then a linear function would better model the data. This can be determined by calculating the differences between consecutive values of f(x).

For example, the difference between the first and second values of f(x) is 20,189.92 - 18,059.23 = 2,130.69. Similarly, the difference between the second and third values of f(x) is 18,059.23 - 16,008.49 = 2,050.74.

However, if the differences between consecutive values of f(x) are not constant, but instead increase or decrease at an increasing or decreasing rate, respectively, then an exponential function would better model the data.

To determine if the differences between consecutive values of f(x) are decreasing or increasing, we can calculate the differences between consecutive differences.

For example, the difference between the second and third differences is (2,050.74 - 2,130.69) = -79.95.

Since the differences between consecutive differences are decreasing, this suggests that an exponential function would not be the best model for the data.

Based on this analysis, it would be better to model the data with a linear function.