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
1
1
2
2
3
3
4
4
Value of Investment ($), f(x)
17, comma, 113, point, 3, 8
17,113.38
14, comma, 022, point, 6, 8
14,022.68
10, comma, 922, point, 6, 8
10,922.68
7, comma, 807, point, 1, 8
7,807.18
A linear or An exponential function would best fit the data because as x increases, the y values change Additively or multiplicatively. The differences/slope or Common ratio/multiplier/base of this function is approximately____
Number of Years Since Investment Made, x
1
1
2
2
3
3
4
4
Value of Investment ($), f(x)
17, comma, 113, point, 3, 8
17,113.38
14, comma, 022, point, 6, 8
14,022.68
10, comma, 922, point, 6, 8
10,922.68
7, comma, 807, point, 1, 8
7,807.18
A linear or An exponential function would best fit the data because as x increases, the y values change Additively or multiplicatively. The differences/slope or Common ratio/multiplier/base of this function is approximately____
Answers
GPT-4o mini
To determine whether the investment is best described by a linear or exponential function, let's analyze the data you've provided.
### Data:
- Number of Years Since Investment Made (x): 1, 2, 3, 4
- Value of Investment ($) f(x):
- f(1) = 17,113.38
- f(2) = 14,022.68
- f(3) = 10,922.68
- f(4) = 7,807.18
### Analyzing the Change in Values:
1. **Calculating the Differences** (for linearity):
- f(2) - f(1) = 14,022.68 - 17,113.38 = -3,090.70
- f(3) - f(2) = 10,922.68 - 14,022.68 = -3,100.00
- f(4) - f(3) = 7,807.18 - 10,922.68 = -3,115.50
The differences between consecutive values are not constant, indicating that the relationship is not linear.
2. **Calculating the Ratios** (for exponential nature):
- f(2) / f(1) = 14,022.68 / 17,113.38 ≈ 0.819
- f(3) / f(2) = 10,922.68 / 14,022.68 ≈ 0.779
- f(4) / f(3) = 7,807.18 / 10,922.68 ≈ 0.715
While these ratios are not constant either, they are relatively close together, suggesting a multiplicative relationship.
### Conclusion:
The investment value decreases in a way that suggests it is better modeled by an exponential function since the values change multiplicatively rather than additively. Hence, an exponential function would best fit the data.
### Final Statements:
An exponential function would best fit the data because as x increases, the y values change **multiplicatively**. The approximate **common ratio/multiplier** can be taken as an average of the ratios calculated.
Average multiplier = (0.819 + 0.779 + 0.715) / 3 ≈ **0.771**.
Therefore, the common ratio/multiplier/base of this function is approximately **0.771**.
### Data:
- Number of Years Since Investment Made (x): 1, 2, 3, 4
- Value of Investment ($) f(x):
- f(1) = 17,113.38
- f(2) = 14,022.68
- f(3) = 10,922.68
- f(4) = 7,807.18
### Analyzing the Change in Values:
1. **Calculating the Differences** (for linearity):
- f(2) - f(1) = 14,022.68 - 17,113.38 = -3,090.70
- f(3) - f(2) = 10,922.68 - 14,022.68 = -3,100.00
- f(4) - f(3) = 7,807.18 - 10,922.68 = -3,115.50
The differences between consecutive values are not constant, indicating that the relationship is not linear.
2. **Calculating the Ratios** (for exponential nature):
- f(2) / f(1) = 14,022.68 / 17,113.38 ≈ 0.819
- f(3) / f(2) = 10,922.68 / 14,022.68 ≈ 0.779
- f(4) / f(3) = 7,807.18 / 10,922.68 ≈ 0.715
While these ratios are not constant either, they are relatively close together, suggesting a multiplicative relationship.
### Conclusion:
The investment value decreases in a way that suggests it is better modeled by an exponential function since the values change multiplicatively rather than additively. Hence, an exponential function would best fit the data.
### Final Statements:
An exponential function would best fit the data because as x increases, the y values change **multiplicatively**. The approximate **common ratio/multiplier** can be taken as an average of the ratios calculated.
Average multiplier = (0.819 + 0.779 + 0.715) / 3 ≈ **0.771**.
Therefore, the common ratio/multiplier/base of this function is approximately **0.771**.