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 11 22 33 44
Value of Investment ($), f(x) 19, comma, 908, point, 1, 819,908.18 22, comma, 275, point, 5, 722,275.57 24, comma, 578, point, 1, 224,578.12 27, comma, 254, point, 3, 827,254.38
{linear or exponential} function would best fit the data because as x increases, the y values change {additively or multiplicatively} The {common difference/slope or common ration/mulitpier/base} of this function is approximately {???}
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), f(x) 19, comma, 908, point, 1, 819,908.18 22, comma, 275, point, 5, 722,275.57 24, comma, 578, point, 1, 224,578.12 27, comma, 254, point, 3, 827,254.38
{linear or exponential} function would best fit the data because as x increases, the y values change {additively or multiplicatively} The {common difference/slope or common ration/mulitpier/base} of this function is approximately {???}
Answers
GPT-4o mini
To determine which kind of function best fits the provided data, we can analyze how the value of the investment changes as the number of years increases.
### Given Values:
- For \( x = 11 \), \( f(x) = 19,908.18 \)
- For \( x = 22 \), \( f(x) = 22,275.57 \)
- For \( x = 33 \), \( f(x) = 24,578.12 \)
- For \( x = 44 \), \( f(x) = 27,254.38 \)
### Analyzing Differences:
1. **Calculate the differences between consecutive values:**
- From 11 to 22: \( 22,275.57 - 19,908.18 = 2,367.39 \)
- From 22 to 33: \( 24,578.12 - 22,275.57 = 2,302.55 \)
- From 33 to 44: \( 27,254.38 - 24,578.12 = 2,676.26 \)
The differences are not constant, suggesting that it is not a linear function.
### Analyzing Ratios:
2. **Calculate the ratios of consecutive values:**
- Ratio from \( f(11) \) to \( f(22) \): \( \frac{22,275.57}{19,908.18} \approx 1.118 \)
- Ratio from \( f(22) \) to \( f(33) \): \( \frac{24,578.12}{22,275.57} \approx 1.103 \)
- Ratio from \( f(33) \) to \( f(44) \): \( \frac{27,254.38}{24,578.12} \approx 1.110 \)
The ratios are relatively consistent and suggest a multiplicative change, indicative of an exponential function.
### Conclusion:
Thus, a **{exponential} function** would best fit the data because as \( x \) increases, the \( f(x) \) values change **{multiplicatively}**.
The **common ratio** (base of the exponential function) can be estimated as approximately **{1.110}** (average of the ratios calculated).
Final answer format:
- The **exponential** function would best fit the data because as \( x \) increases, the \( y \) values change **multiplicatively**. The **common ratio** of this function is approximately **1.110**.
### Given Values:
- For \( x = 11 \), \( f(x) = 19,908.18 \)
- For \( x = 22 \), \( f(x) = 22,275.57 \)
- For \( x = 33 \), \( f(x) = 24,578.12 \)
- For \( x = 44 \), \( f(x) = 27,254.38 \)
### Analyzing Differences:
1. **Calculate the differences between consecutive values:**
- From 11 to 22: \( 22,275.57 - 19,908.18 = 2,367.39 \)
- From 22 to 33: \( 24,578.12 - 22,275.57 = 2,302.55 \)
- From 33 to 44: \( 27,254.38 - 24,578.12 = 2,676.26 \)
The differences are not constant, suggesting that it is not a linear function.
### Analyzing Ratios:
2. **Calculate the ratios of consecutive values:**
- Ratio from \( f(11) \) to \( f(22) \): \( \frac{22,275.57}{19,908.18} \approx 1.118 \)
- Ratio from \( f(22) \) to \( f(33) \): \( \frac{24,578.12}{22,275.57} \approx 1.103 \)
- Ratio from \( f(33) \) to \( f(44) \): \( \frac{27,254.38}{24,578.12} \approx 1.110 \)
The ratios are relatively consistent and suggest a multiplicative change, indicative of an exponential function.
### Conclusion:
Thus, a **{exponential} function** would best fit the data because as \( x \) increases, the \( f(x) \) values change **{multiplicatively}**.
The **common ratio** (base of the exponential function) can be estimated as approximately **{1.110}** (average of the ratios calculated).
Final answer format:
- The **exponential** function would best fit the data because as \( x \) increases, the \( y \) values change **multiplicatively**. The **common ratio** of this function is approximately **1.110**.