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 ($), f(x) 19, comma, 908, point, 1, 819,908.18 22, comma, 275, point, 5, 722,275.57 24, comma, 578, point, 1, 224,578.12 27, comma, 254, point, 3, 827,254.38
Answer
Attempt 1 out of 2
{blank} function would best fit the data because as x increases, the y values change {blank}
. The {blank}
of this function is approximately?
.
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), 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
Answer
Attempt 1 out of 2
{blank} function would best fit the data because as x increases, the y values change {blank}
. The {blank}
of this function is approximately?
.
Answers
GPT-4o mini
To determine the best-fitting function for the given data, we can analyze how the value of the investment increases over time.
Let's look at the values in the table:
- At year 11: $19,908.18
- At year 22: $22,275.57
- At year 33: $24,578.12
- At year 44: $27,254.38
### Step 1: Analyze Growth
To see if the growth is linear, we can look at the differences between consecutive years:
1. From year 11 to year 22:
- $22,275.57 - $19,908.18 = $2,367.39
2. From year 22 to year 33:
- $24,578.12 - $22,275.57 = $2,302.55
3. From year 33 to year 44:
- $27,254.38 - $24,578.12 = $2,676.26
The differences are not constant, indicating that this is **not linear** growth.
### Step 2: Check for Exponential Growth
To check for exponential growth, we can look at the ratios of consecutive years:
1. From year 11 to year 22:
- \( \frac{22,275.57}{19,908.18} \approx 1.119 \)
2. From year 22 to year 33:
- \( \frac{24,578.12}{22,275.57} \approx 1.103 \)
3. From year 33 to year 44:
- \( \frac{27,254.38}{24,578.12} \approx 1.110 \)
The ratios are fairly close to each other, suggesting that the investment value is indeed increasing at an exponential rate.
### Conclusion
Based on this analysis:
- An **exponential** function would best fit the data because as \( x \) increases, the values of \( f(x) \) change **multiplicatively** rather than additively.
- The **growth factor** of this function is approximately **1.1** (or about 10% growth per interval).
Now filling in the blanks:
- **Exponential** function would best fit the data because as x increases, the y values change **multiplicatively**. The **growth factor** of this function is approximately **1.1**.
Let's look at the values in the table:
- At year 11: $19,908.18
- At year 22: $22,275.57
- At year 33: $24,578.12
- At year 44: $27,254.38
### Step 1: Analyze Growth
To see if the growth is linear, we can look at the differences between consecutive years:
1. From year 11 to year 22:
- $22,275.57 - $19,908.18 = $2,367.39
2. From year 22 to year 33:
- $24,578.12 - $22,275.57 = $2,302.55
3. From year 33 to year 44:
- $27,254.38 - $24,578.12 = $2,676.26
The differences are not constant, indicating that this is **not linear** growth.
### Step 2: Check for Exponential Growth
To check for exponential growth, we can look at the ratios of consecutive years:
1. From year 11 to year 22:
- \( \frac{22,275.57}{19,908.18} \approx 1.119 \)
2. From year 22 to year 33:
- \( \frac{24,578.12}{22,275.57} \approx 1.103 \)
3. From year 33 to year 44:
- \( \frac{27,254.38}{24,578.12} \approx 1.110 \)
The ratios are fairly close to each other, suggesting that the investment value is indeed increasing at an exponential rate.
### Conclusion
Based on this analysis:
- An **exponential** function would best fit the data because as \( x \) increases, the values of \( f(x) \) change **multiplicatively** rather than additively.
- The **growth factor** of this function is approximately **1.1** (or about 10% growth per interval).
Now filling in the blanks:
- **Exponential** function would best fit the data because as x increases, the y values change **multiplicatively**. The **growth factor** of this function is approximately **1.1**.
instead of "growth factor" please use either common difference/slope or common ration/multiplyer/base. also round last number to the nearest whole number