To find the exponential model of best fit for the points \((-3, 5)\), \((1, 12)\), \((5, 72)\), and \((7, 137)\), we generally look for a function of the form:
\[ y = a \cdot b^x \]
where \(a\) and \(b\) are constants to be determined.
Steps to Determine the Model:
- Transform the Data: We can take the natural logarithm of \(y\). This transforms the exponential model into a linear model:
\[ \ln(y) = \ln(a) + x \cdot \ln(b) \]
Let \(Y = \ln(y)\), then the equation can be rewritten as:
\[ Y = \ln(a) + \ln(b) \cdot x \]
This suggests a linear relationship between \(x\) and \(Y\).
- Calculate \(Y\) Values: Calculate \(Y\) for each \(y\).
- For point \((-3, 5)\): \(Y = \ln(5) \approx 1.6094\)
- For point \((1, 12)\): \(Y = \ln(12) \approx 2.4849\)
- For point \((5, 72)\): \(Y = \ln(72) \approx 4.2767\)
- For point \((7, 137)\): \(Y = \ln(137) \approx 4.9220\)
Now we have the transformed points:
\[ (-3, 1.6094), (1, 2.4849), (5, 4.2767), (7, 4.9220) \]
- Use Linear Regression: We can use linear regression to find the best-fit line for these points in the transformed space. We need to find the slope (\(\ln(b)\)) and intercept (\(\ln(a)\)).
Let’s denote the points as \((x_1, Y_1), (x_2, Y_2), (x_3, Y_3), (x_4, Y_4)\). The slope (\(m\)) and intercept (\(c\)) for this linear model can be calculated as follows:
\[ m = \frac{N \sum (x_i Y_i) - \sum x_i \sum Y_i}{N \sum (x_i^2) - (\sum x_i)^2} \]
\[ c = \frac{\sum Y_i - m \sum x_i}{N} \]
Where \(N\) is the number of data points.
- Calculating the Necessary Sums:
Here’s what each sum equals for the given points:
- \(N = 4\)
- \(\sum x_i = -3 + 1 + 5 + 7 = 10\)
- \(\sum Y_i = 1.6094 + 2.4849 + 4.2767 + 4.9220 = 13.2930\)
- \(\sum (x_i^2) = (-3)^2 + 1^2 + 5^2 + 7^2 = 9 + 1 + 25 + 49 = 84\)
- \(\sum (x_i Y_i) = (-3)(1.6094) + (1)(2.4849) + (5)(4.2767) + (7)(4.9220) = -4.8282 + 2.4849 + 21.3835 + 34.4540 \approx 53.0942\)
- Calculate the Slope and Intercept:
Substituting these into the formulas gives:
\[ m = \frac{4(53.0942) - (10)(13.2930)}{4(84) - (10)^2} = \frac{212.3768 - 132.930}{336 - 100} = \frac{79.4468}{236} \approx 0.3364 \]
\[ c = \frac{13.2930 - 0.3364(10)}{4} = \frac{13.2930 - 3.364}{4} \approx 2.4823 \]
- Exponentiate to Find \(a\) and \(b\):
To find \(a\) and \(b\):
\[ \ln(a) = c \implies a = e^{c} \approx e^{2.4823} \approx 11.96 \] \[ \ln(b) = m \implies b = e^{m} \approx e^{0.3364} \approx 1.401 \]
Final Model:
Thus, the exponential model of best fit is:
\[ y \approx 11.96 \cdot 1.401^x \]
Conclusion:
By following the steps of logarithmic transformation, performing linear regression on the transformed data, and then converting back, we derived the exponential model. The rounded values for \(a\) and \(b\) to the nearest hundredth are:
- \(a \approx 11.96\)
- \(b \approx 1.40\)
So the final form of the exponential model is:
\[ y \approx 11.96 \cdot (1.40)^x \]