X 5 10 15 20.                                                       Y 88.574 52.302 30.884 18.236 . determine the function that best models the date

The polynomial that best fits the given data is:

\[ Y = 0.23624X^2 - 10.55464X + 135.137 \]

This quadratic function models the relationship between \( X \) and \( Y \) based on the provided data points.

If you have any questions or need further details, please let me know.

Here are some related questions to further your understanding:

1. How can we verify the accuracy of the fitted polynomial?
2. What other types of functions could be used to model this data?
3. How do we decide on the degree of the polynomial for fitting?
4. Can we use this polynomial to make predictions for \( X \) values outside the given range?
5. What is the method of least squares used in polynomial fitting?
6. How does the coefficient of determination \( R^2 \) relate to the fit of the model?
7. What are the potential issues with overfitting in polynomial regression?
8. How can we use software tools to perform polynomial regression on larger datasets?

**Tip:** Always visualize your data and the fitted curve to ensure the model accurately represents the underlying relationship.

Explore how to model data using quadratic functions with the polynomial Y = 0.23624X^2 - 10.55464X + 135.137. Understand the application of polynomial regression and the method of least squares in fitting data to mathematical models.

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