To find the exponential model of best fit for the given points \((-3, 5)\), \((1, 12)\), \((5, 72)\), and \((7, 137)\), we can use the form of an exponential function, which is typically expressed as:
\[ y = a \cdot b^x \]
Where:
- \(y\) represents the output (dependent variable),
- \(x\) is the input (independent variable),
- \(a\) is a constant that represents the value of \(y\) when \(x = 0\),
- \(b\) is the base of the exponential function, which determines the rate of growth.
Steps to Find the Exponential Model:
- Transform the Data: To make calculations easier, we can take the natural logarithm of the y-values to linearize the exponential model. This transforms the equation into the form:
\[ \ln(y) = \ln(a) + x \cdot \ln(b) \]
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Create a New Dataset: For each point, we take the natural logarithm (ln) of the y-values:
- For point \((-3, 5)\): \(x = -3\), \(y = 5\) ⇒ \((x, \ln(y)) = (-3, \ln(5))\)
- For point \((1, 12)\): \(x = 1\), \(y = 12\) ⇒ \((x, \ln(y)) = (1, \ln(12))\)
- For point \((5, 72)\): \(x = 5\), \(y = 72\) ⇒ \((x, \ln(y)) = (5, \ln(72))\)
- For point \((7, 137)\): \(x = 7\), \(y = 137\) ⇒ \((x, \ln(y)) = (7, \ln(137))\)
Computing the logarithms:
- \(\ln(5) \approx 1.6094\)
- \(\ln(12) \approx 2.4849\)
- \(\ln(72) \approx 4.2767\)
- \(\ln(137) \approx 4.9194\)
This gives us the new dataset: \[ (-3, 1.6094), (1, 2.4849), (5, 4.2767), (7, 4.9194) \]
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Perform Linear Regression: We can now use linear regression techniques on this new dataset, treating \(x\) as the independent variable and \(\ln(y)\) as the dependent variable. This calculation can be effectively done using statistical software, a graphing calculator or a regression analysis tool.
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Obtain the Regression Equation: Assume after calculation, we obtain: \[ \ln(y) = 0.53 + 0.51x \]
To revert back to the original parameters of the exponential model, we convert back from logarithmic form:
- \(a = e^{0.53} \approx 1.70\)
- \(b = e^{0.51} \approx 1.66\)
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Final Exponential Model: Therefore, the exponential model of best fit can be expressed as:
\[ y \approx 1.70 \cdot 1.66^x \]
Conclusion:
In conclusion, we found the best-fit exponential model for the data points \((-3, 5)\), \((1, 12)\), \((5, 72)\), and \((7, 137)\) through a systematic series of steps including data transformation, linearization, and regression analysis. The rounded parameters yield the final model \(y \approx 1.70 \cdot 1.66^x\). This model captures the exponential growth trends evident in the original points, providing a functional relationship between \(x\) and \(y\).