Your response is on the right track. To determine whether a function is linear or exponential, we can look at the changes in the values of \(y\) as \(x\) changes.
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Linear Function: A function is linear if it has a constant rate of change (slope). This means that if you calculate the difference in \(y\) values for equal intervals of \(x\), the change in \(y\) should remain constant.
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Exponential Function: A function is exponential if it multiplies by a constant factor for equal intervals of \(x\). That is, the ratio of consecutive \(y\) values should be approximately constant.
Let's analyze your data:
- Data Points:
- (1, 0.89)
- (3, 2.67)
- (5, 4.45)
- (9, 8.01)
- (12, 10.68)
1. Check for Linear Growth:
Calculate the differences in \(y\) values:
- From \(x = 1\) to \(3\): \(2.67 - 0.89 = 1.78\)
- From \(x = 3\) to \(5\): \(4.45 - 2.67 = 1.78\)
- From \(x = 5\) to \(9\): \(8.01 - 4.45 = 3.56\)
- From \(x = 9\) to \(12\): \(10.68 - 8.01 = 2.67\)
The differences in \(y\) values are not constant (1.78, 1.78, 3.56, 2.67), indicating a variable rate of change. Thus, the relationship is not linear.
2. Check for Exponential Growth:
Calculate the ratios of consecutive \(y\) values:
- Ratio from \(0.89\) to \(2.67\): \( \frac{2.67}{0.89} \approx 3.00 \)
- Ratio from \(2.67\) to \(4.45\): \( \frac{4.45}{2.67} \approx 1.67 \)
- Ratio from \(4.45\) to \(8.01\): \( \frac{8.01}{4.45} \approx 1.80 \)
- Ratio from \(8.01\) to \(10.68\): \( \frac{10.68}{8.01} \approx 1.33 \)
The ratios are also not constant.
Conclusion:
Given that neither the differences are constant nor the ratios, the relationship cannot be accurately modeled as linear or exponential with the provided data.
So, your answer to the question about a linear model is correct: NO, because the growth shows a variable rate of change. You may want to explore other types of functions or models to better fit the data.