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 11 22 33 44
Value of Investment ($), f(x) 16, comma, 890, point, 3, 216,890.32 14, comma, 259, point, 0, 914,259.09 12, comma, 001, point, 5, 712,001.57 10, comma, 051, point, 1, 610,051.16
Answer
Attempt 1 out of 2
An exponential
function would better model the data because as xx increases, the yy values change
multiplicatively
. The
common ratio/multiplier/base
of this function is approximately
.
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), f(x) 16, comma, 890, point, 3, 216,890.32 14, comma, 259, point, 0, 914,259.09 12, comma, 001, point, 5, 712,001.57 10, comma, 051, point, 1, 610,051.16
Answer
Attempt 1 out of 2
An exponential
function would better model the data because as xx increases, the yy values change
multiplicatively
. The
common ratio/multiplier/base
of this function is approximately
.
Answers
GPT-4o mini
To determine whether an exponential function or a linear function better models the investment data, we need to analyze how the values change as the number of years since the investment increases.
Given the values:
- At 11 years: $16,890.32
- At 22 years: $14,259.09
- At 33 years: $12,001.57
- At 44 years: $10,051.16
Let's calculate the percentage change between each year interval to see if the values decline multiplicatively (as seen in exponential functions) or if they decrease by a fixed absolute amount (as seen in linear functions).
1. **From 11 to 22 years**:
\[
\text{Percentage Change} = \frac{14,259.09 - 16,890.32}{16,890.32} \times 100 \approx -15.4\%
\]
2. **From 22 to 33 years**:
\[
\text{Percentage Change} = \frac{12,001.57 - 14,259.09}{14,259.09} \times 100 \approx -15.9\%
\]
3. **From 33 to 44 years**:
\[
\text{Percentage Change} = \frac{10,051.16 - 12,001.57}{12,001.57} \times 100 \approx -16.3\%
\]
The percentage changes appear to be consistently around -15% to -16%, indicating a multiplicative decrease as time goes on.
### Conclusion:
An exponential function would better model the data because, as \( x \) increases, the \( y \) values change multiplicatively.
The common ratio/multiplier of this function can be approximated from the percentage change. If we represent the average percentage reduction as a decimal, we can interpret a multiplier of approximately \( 0.84 \) (which corresponds to a decrease of about 16%).
So to complete your answer:
- The common ratio/multiplier/base of this function is approximately \( 0.84 \).
Thus, the full response would be:
**An exponential function would better model the data because as \( x \) increases, the \( y \) values change multiplicatively. The common ratio/multiplier/base of this function is approximately \( 0.84 \).**
Given the values:
- At 11 years: $16,890.32
- At 22 years: $14,259.09
- At 33 years: $12,001.57
- At 44 years: $10,051.16
Let's calculate the percentage change between each year interval to see if the values decline multiplicatively (as seen in exponential functions) or if they decrease by a fixed absolute amount (as seen in linear functions).
1. **From 11 to 22 years**:
\[
\text{Percentage Change} = \frac{14,259.09 - 16,890.32}{16,890.32} \times 100 \approx -15.4\%
\]
2. **From 22 to 33 years**:
\[
\text{Percentage Change} = \frac{12,001.57 - 14,259.09}{14,259.09} \times 100 \approx -15.9\%
\]
3. **From 33 to 44 years**:
\[
\text{Percentage Change} = \frac{10,051.16 - 12,001.57}{12,001.57} \times 100 \approx -16.3\%
\]
The percentage changes appear to be consistently around -15% to -16%, indicating a multiplicative decrease as time goes on.
### Conclusion:
An exponential function would better model the data because, as \( x \) increases, the \( y \) values change multiplicatively.
The common ratio/multiplier of this function can be approximated from the percentage change. If we represent the average percentage reduction as a decimal, we can interpret a multiplier of approximately \( 0.84 \) (which corresponds to a decrease of about 16%).
So to complete your answer:
- The common ratio/multiplier/base of this function is approximately \( 0.84 \).
Thus, the full response would be:
**An exponential function would better model the data because as \( x \) increases, the \( y \) values change multiplicatively. The common ratio/multiplier/base of this function is approximately \( 0.84 \).**