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 ($), f(x) 16, comma, 890, point, 3, 216,890.32 14, comma, 259, point, 0, 914,259.09 12, comma, 001, point, 5, 712,001.57 10, comma, 051, point, 1, 610,051.16
Answer
Attempt 1 out of 2
function would better model the data because as xx increases, the yy values change
. The
of this function is approximately
.
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), 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
function would better model the data because as xx increases, the yy values change
. The
of this function is approximately
.
Answers
GPT-4o mini
To determine whether a linear or exponential function better models the given investment data, we need to analyze the pattern of the values over time.
Here are the data points:
- For x = 11 years, f(x) = $16,890.32
- For x = 22 years, f(x) = $14,259.09
- For x = 33 years, f(x) = $12,001.57
- For x = 44 years, f(x) = $10,051.16
### Analysis:
1. **Change in Values:**
- From year 11 to 22, the investment value decreases from $16,890.32 to $14,259.09 (a decrease of approximately $2,631.23).
- From year 22 to 33, it decreases from $14,259.09 to $12,001.57 (a decrease of approximately $2,257.52).
- From year 33 to 44, it decreases from $12,001.57 to $10,051.16 (a decrease of approximately $1,950.41).
The amount of decrease in the investment value is not constant; it shows decreasing values. This suggests that the investment’s value is not decreasing linearly.
2. **Percent Decrease:**
- The percentage decrease from year to year can be calculated, indicating that the investment may be following a multiplicative behavior rather than an additive one common in linear functions.
Based on these observations, we conclude that the investment is not experiencing a steady decline but rather a decreasing trend that could potentially fit an exponential decay model (which is common in investments decreasing in value). However, based on the specific values, it could also suggest a non-linear pattern.
### Conclusion:
**Final Answer**:
An **exponential** function would better model the data because as \(x\) increases, the \(y\) values change more rapidly (decreasing at a decreasing rate). The **trend** of this function is approximately **exponential decay**.
Here are the data points:
- For x = 11 years, f(x) = $16,890.32
- For x = 22 years, f(x) = $14,259.09
- For x = 33 years, f(x) = $12,001.57
- For x = 44 years, f(x) = $10,051.16
### Analysis:
1. **Change in Values:**
- From year 11 to 22, the investment value decreases from $16,890.32 to $14,259.09 (a decrease of approximately $2,631.23).
- From year 22 to 33, it decreases from $14,259.09 to $12,001.57 (a decrease of approximately $2,257.52).
- From year 33 to 44, it decreases from $12,001.57 to $10,051.16 (a decrease of approximately $1,950.41).
The amount of decrease in the investment value is not constant; it shows decreasing values. This suggests that the investment’s value is not decreasing linearly.
2. **Percent Decrease:**
- The percentage decrease from year to year can be calculated, indicating that the investment may be following a multiplicative behavior rather than an additive one common in linear functions.
Based on these observations, we conclude that the investment is not experiencing a steady decline but rather a decreasing trend that could potentially fit an exponential decay model (which is common in investments decreasing in value). However, based on the specific values, it could also suggest a non-linear pattern.
### Conclusion:
**Final Answer**:
An **exponential** function would better model the data because as \(x\) increases, the \(y\) values change more rapidly (decreasing at a decreasing rate). The **trend** of this function is approximately **exponential decay**.