Question

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

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 | |.

Explanation:

Step1: Check Linear Change (Difference)

Calculate the differences between consecutive \( f(x) \) values:
\( 28,207.11 - 23,729.61 = 4,477.5 \)
\( 32,684.61 - 28,207.11 = 4,477.5 \)
\( 37,198.11 - 32,684.61 = 4,513.5 \) (close to 4,477.5, slight variation likely due to rounding).

Step2: Check Exponential Change (Ratio)

Calculate the ratios between consecutive \( f(x) \) values:
\( \frac{28,207.11}{23,729.61} \approx 1.188 \)
\( \frac{32,684.61}{28,207.11} \approx 1.159 \) (not constant, so not exponential).

Since the differences (change in \( y \)) are approximately constant (linear change) as \( x \) increases, a linear function is better. The slope (rate of change) is the constant difference (or average difference). Using the first two points: slope \( m = \frac{28,207.11 - 23,729.61}{2 - 1} = 4,477.5 \) (approximate rate of change).

Answer:

Linear function would better model the data because as \( x \) increases, the \( y \) values change by a constant amount (or approximately constant). The rate of change (slope) of this function is approximately \( 4477.5 \).