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 ($), f(x) 20, comma, 189, point, 9, 220,189.92 18, comma, 059, point, 2, 318,059.23 16, comma, 008, point, 4, 916,008.49 14, comma, 276, point, 1, 414,276.14
Answer
Attempt 1 out of 2
A linear or An exponential function would better model the data because as x increases, the y values change Additively or multiplicatively. The Common differences/slope or Common ratio/multiplier/base of this function is approximately ___
.
Number of Years Since Investment Made, x 11 22 33 44
Value of Investment ($), 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 ___
.
Answers
GPT-4o mini
To determine whether a linear or exponential function would better model the investment value data provided, we need to analyze the changes in the value of the investment as the number of years increases.
### Data Summary:
- Years (x): 11, 22, 33, 44
- Value (f(x)): 20,189.92; 18,059.23; 16,008.49; 14,276.14
### Evaluating Linear vs. Exponential:
1. **Linear Function**: A linear function increases or decreases by a constant amount (additively). If the differences between subsequent investment values are constant or near constant, then a linear model may be suitable.
2. **Exponential Function**: An exponential function increases or decreases by a constant percentage (multiplicatively). If the investment values change by a consistent ratio or multiplier as time progresses, an exponential model may be more appropriate.
### Calculating Differences:
Let's calculate the differences in investment values and see if they suggest a linear trend:
- f(22) - f(11) = 18,059.23 - 20,189.92 = -2,130.69
- 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
### Notable Observations:
- The differences between values are not constant and are decreasing but do not closely align with a linear model since they indicate a greater reduction in the earlier years and a smaller reduction in the later years.
### Calculating Ratios:
Next, we can analyze the ratios between successive values:
- f(22) / f(11) = 18,059.23 / 20,189.92 ≈ 0.895
- f(33) / f(22) = 16,008.49 / 18,059.23 ≈ 0.886
- f(44) / f(33) = 14,276.14 / 16,008.49 ≈ 0.891
### Conclusion:
The ratios between consecutive values are somewhat consistent (hovering around 0.89), suggesting that the investment value decreases by a consistent percentage rather than a consistent additive amount.
Thus, an exponential function would better model the data because as \( x \) increases, the values of \( f(x) \) change multiplicatively.
### Filling the 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 Summary:
- Years (x): 11, 22, 33, 44
- Value (f(x)): 20,189.92; 18,059.23; 16,008.49; 14,276.14
### Evaluating Linear vs. Exponential:
1. **Linear Function**: A linear function increases or decreases by a constant amount (additively). If the differences between subsequent investment values are constant or near constant, then a linear model may be suitable.
2. **Exponential Function**: An exponential function increases or decreases by a constant percentage (multiplicatively). If the investment values change by a consistent ratio or multiplier as time progresses, an exponential model may be more appropriate.
### Calculating Differences:
Let's calculate the differences in investment values and see if they suggest a linear trend:
- f(22) - f(11) = 18,059.23 - 20,189.92 = -2,130.69
- 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
### Notable Observations:
- The differences between values are not constant and are decreasing but do not closely align with a linear model since they indicate a greater reduction in the earlier years and a smaller reduction in the later years.
### Calculating Ratios:
Next, we can analyze the ratios between successive values:
- f(22) / f(11) = 18,059.23 / 20,189.92 ≈ 0.895
- f(33) / f(22) = 16,008.49 / 18,059.23 ≈ 0.886
- f(44) / f(33) = 14,276.14 / 16,008.49 ≈ 0.891
### Conclusion:
The ratios between consecutive values are somewhat consistent (hovering around 0.89), suggesting that the investment value decreases by a consistent percentage rather than a consistent additive amount.
Thus, an exponential function would better model the data because as \( x \) increases, the values of \( f(x) \) change multiplicatively.
### Filling the 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**.