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 ($), f(x) 20, comma, 189, point, 9, 220,189.92 18, comma, 059, point, 2, 318,059.23 16, comma, 008, point, 4, 916,008.49 14, comma, 276, point, 1, 414,276.14
Answer
Attempt 1 out of 2
A linear or An exponential function would better model the data because as x increases, the y values change Additively or multiplicatively. The Common differences/slope or Common ratio/multiplier/base of this function is approximately ___
Can you fill all the sections out
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), f(x) 20, comma, 189, point, 9, 220,189.92 18, comma, 059, point, 2, 318,059.23 16, comma, 008, point, 4, 916,008.49 14, comma, 276, point, 1, 414,276.14
Answer
Attempt 1 out of 2
A linear or An exponential function would better model the data because as x increases, the y values change Additively or multiplicatively. The Common differences/slope or Common ratio/multiplier/base of this function is approximately ___
Can you fill all the sections out
Answers
GPT-4o mini
To determine whether a linear or exponential function better models the data, we need to look at how the value of the investment changes over time (as x increases).
### Data from the Table:
- For \(x = 11\): \(f(x) = 20,189.90\)
- For \(x = 22\): \(f(x) = 18,059.23\)
- For \(x = 33\): \(f(x) = 16,008.49\)
- For \(x = 44\): \(f(x) = 14,276.14\)
### Analysis:
1. **Calculate the differences between consecutive values** (to assess linearity):
- \(f(22) - f(11) = 18,059.23 - 20,189.90 = -2,130.67\)
- \(f(33) - f(22) = 16,008.49 - 18,059.23 = -2,050.74\)
- \(f(44) - f(33) = 14,276.14 - 16,008.49 = -1,732.35\)
These differences are not constant; their values are changing as x increases, indicating it is not a linear relationship.
2. **Calculate the ratio of consecutive values** (to assess exponential growth/decay):
- \(\frac{f(22)}{f(11)} = \frac{18,059.23}{20,189.90} \approx 0.893\)
- \(\frac{f(33)}{f(22)} = \frac{16,008.49}{18,059.23} \approx 0.886\)
- \(\frac{f(44)}{f(33)} = \frac{14,276.14}{16,008.49} \approx 0.891\)
These ratios are relatively consistent, suggesting that the values are changing multiplicatively.
### Conclusion:
An exponential function would better model the data because as \(x\) increases, the \(y\) values change multiplicatively.
### Fill in the Blank:
The common ratio/multiplier/base of this function is approximately **0.89** (an average of the ratios calculated above).
### Final Answer:
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 **0.89**.
### Data from the Table:
- For \(x = 11\): \(f(x) = 20,189.90\)
- For \(x = 22\): \(f(x) = 18,059.23\)
- For \(x = 33\): \(f(x) = 16,008.49\)
- For \(x = 44\): \(f(x) = 14,276.14\)
### Analysis:
1. **Calculate the differences between consecutive values** (to assess linearity):
- \(f(22) - f(11) = 18,059.23 - 20,189.90 = -2,130.67\)
- \(f(33) - f(22) = 16,008.49 - 18,059.23 = -2,050.74\)
- \(f(44) - f(33) = 14,276.14 - 16,008.49 = -1,732.35\)
These differences are not constant; their values are changing as x increases, indicating it is not a linear relationship.
2. **Calculate the ratio of consecutive values** (to assess exponential growth/decay):
- \(\frac{f(22)}{f(11)} = \frac{18,059.23}{20,189.90} \approx 0.893\)
- \(\frac{f(33)}{f(22)} = \frac{16,008.49}{18,059.23} \approx 0.886\)
- \(\frac{f(44)}{f(33)} = \frac{14,276.14}{16,008.49} \approx 0.891\)
These ratios are relatively consistent, suggesting that the values are changing multiplicatively.
### Conclusion:
An exponential function would better model the data because as \(x\) increases, the \(y\) values change multiplicatively.
### Fill in the Blank:
The common ratio/multiplier/base of this function is approximately **0.89** (an average of the ratios calculated above).
### Final Answer:
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 **0.89**.