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 ($), f(x) 19, comma, 908, point, 1, 819,908.18 22, comma, 275, point, 5, 722,275.57 24, comma, 578, point, 1, 224,578.12 27, comma, 254, point, 3, 827,254.38
{linear or exponential} function would best fit the data because as x increases, the y values change {additively or multiplicatively} The {common difference/slope or common ration/mulitpier/base} of this function is approximately {??? round to the nearest whole number}
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), 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 {??? round to the nearest whole number}
Answers
GPT-4o mini
To determine if the investment value data represents a linear or exponential function, we can analyze how the values change as the years increase.
### Given Data:
- Year 11: $19,908.18
- Year 22: $22,275.57
- Year 33: $24,578.12
- Year 44: $27,254.38
### Check for Linear Growth:
1. **Calculate the differences between consecutive years:**
- From year 11 to 22:
$$ 22,275.57 - 19,908.18 = 2,367.39 $$
- From year 22 to 33:
$$ 24,578.12 - 22,275.57 = 2,302.55 $$
- From year 33 to 44:
$$ 27,254.38 - 24,578.12 = 2,676.26 $$
The differences are not constant but vary (2,367.39, 2,302.55, 2,676.26), suggesting the data does not fit a linear function.
### Check for Exponential Growth:
2. **Calculate the ratios between consecutive values:**
- From year 11 to 22:
$$ \frac{22,275.57}{19,908.18} \approx 1.118 $$
- From year 22 to 33:
$$ \frac{24,578.12}{22,275.57} \approx 1.104 $$
- From year 33 to 44:
$$ \frac{27,254.38}{24,578.12} \approx 1.109 $$
The ratios are close to each other but not exact, yet they reflect a multiplicative pattern, suggesting that the data might fit an exponential function.
### Conclusion:
The **exponential** function would best fit the data because as \( x \) increases, the \( f(x) \) values change **multiplicatively**.
The **common ratio/base** of this function is approximately:
- The average of the calculated ratios is:
$$ \text{average} \approx \frac{1.118 + 1.104 + 1.109}{3} \approx 1.110 $$, which rounds to the nearest whole number, yielding approximately 1.
Thus, the final answer would be:
The **exponential** function would best fit the data because as \( x \) increases, the \( y \) values change **multiplicatively**. The **common ratio/base** of this function is approximately **1** when rounded to the nearest whole number.
### Given Data:
- Year 11: $19,908.18
- Year 22: $22,275.57
- Year 33: $24,578.12
- Year 44: $27,254.38
### Check for Linear Growth:
1. **Calculate the differences between consecutive years:**
- From year 11 to 22:
$$ 22,275.57 - 19,908.18 = 2,367.39 $$
- From year 22 to 33:
$$ 24,578.12 - 22,275.57 = 2,302.55 $$
- From year 33 to 44:
$$ 27,254.38 - 24,578.12 = 2,676.26 $$
The differences are not constant but vary (2,367.39, 2,302.55, 2,676.26), suggesting the data does not fit a linear function.
### Check for Exponential Growth:
2. **Calculate the ratios between consecutive values:**
- From year 11 to 22:
$$ \frac{22,275.57}{19,908.18} \approx 1.118 $$
- From year 22 to 33:
$$ \frac{24,578.12}{22,275.57} \approx 1.104 $$
- From year 33 to 44:
$$ \frac{27,254.38}{24,578.12} \approx 1.109 $$
The ratios are close to each other but not exact, yet they reflect a multiplicative pattern, suggesting that the data might fit an exponential function.
### Conclusion:
The **exponential** function would best fit the data because as \( x \) increases, the \( f(x) \) values change **multiplicatively**.
The **common ratio/base** of this function is approximately:
- The average of the calculated ratios is:
$$ \text{average} \approx \frac{1.118 + 1.104 + 1.109}{3} \approx 1.110 $$, which rounds to the nearest whole number, yielding approximately 1.
Thus, the final answer would be:
The **exponential** function would best fit the data because as \( x \) increases, the \( y \) values change **multiplicatively**. The **common ratio/base** of this function is approximately **1** when rounded to the nearest whole number.
thats not right, please relook the problem