x	y
1	3
2	4.2426
3	5.1962
4	6
5	6.7082
6	7.3485
Find the power function that is the best fit for the given data. (Round your numerical values to two decimal places.)
 

To find the power function that best fits the given data, we can express the function in the form:

\[
y = a \cdot x^b
\]

where \(a\) and \(b\) are constants to be determined.

### Steps:
1. **Linearize the power function** by taking the natural logarithm (ln) of both sides:

\[
\ln(y) = \ln(a) + b \cdot \ln(x)
\]

This equation is now in the form of a linear equation \(y = mx + c\), where \(\ln(y)\) corresponds to the dependent variable, \(\ln(x)\) is the independent variable, \(b\) is the slope, and \(\ln(a)\) is the intercept.

2. **Apply linear regression** on the transformed data, where \( \ln(y) \) is regressed on \( \ln(x) \).

I will now calculate the best-fit values of \(a\) and \(b\) for the power function. Let me compute that for you.The power function that best fits the given data is:

\[
y = 3.00 \cdot x^{0.50}
\]

Would you like further details or clarification on how this was calculated?

Here are five related questions to explore further:

1. How does the natural logarithm help linearize non-linear equations like power functions?
2. Can the power function method be used for all types of data trends?
3. What is the significance of the coefficient \(a\) in the power function?
4. How can we assess the goodness of fit for a power function?
5. What happens to the power function if \(b\) is negative?

**Tip**: Always check the residuals of your regression to ensure the model fits well.

Find the power function that is the best fit for the given data x = [1, 2, 3, 4, 5, 6], y = [3, 4.2426, 5.1962, 6, 6.7082, 7.3485]. Round your numerical values to two decimal places.

Learn how to find the best-fit power function for data points using regression analysis. This solution demonstrates the process of linearizing a power function, applying linear regression to the transformed data, and computing the constants a and b for the power function y = 3.00 * x^0.50.

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