Question

the value of a certain investment over time is given in the table below. answer the questions below to determine what kind of function would best fit the data, linear or exponential.

number of years since investment made, $x$: 1, 2, 3, 4
value of investment ($), $f(x)$: 28,900.70, 33,424.24, 38,772.12, 44,930.54

answer
the __ function would best fit the data because as $x$ increases, the $y$ values change of this function is approximately __.

Explanation:

Step1: Check linear differences

Calculate the differences between consecutive $f(x)$ values:
$33,424.24 - 28,900.70 = 4,523.54$
$38,772.12 - 33,424.24 = 5,347.88$
$44,930.54 - 38,772.12 = 6,158.42$

Step2: Check exponential ratios

Calculate the ratios of consecutive $f(x)$ values:
$\frac{33,424.24}{28,900.70} \approx 1.1565$
$\frac{38,772.12}{33,424.24} \approx 1.1600$
$\frac{44,930.54}{38,772.12} \approx 1.1588$

Step3: Identify best fit type

The ratios are nearly constant, while linear differences vary significantly.

Step4: Calculate average growth factor

Find the average of the ratios:
$\frac{1.1565 + 1.1600 + 1.1588}{3} \approx 1.1584$

Answer:

The exponential function would best fit the data because as $x$ increases, the $y$ values change by a nearly constant multiplicative factor.
The growth factor of this function is approximately $\boldsymbol{1.16}$ (or 15.8% as a percentage, rounded to two decimal places).