Questions: Find the quartic function that is the best fit for the data in the table below. Report the model with three significant digits in the coefficients. x: -2, -1, 0, 1, 2, 3, 4 y: 36, 3.75, 0, 3.75, 36, 159.75, 480 y= (Simplify your answer. Do not factor. Use integers or decimals for any numbers in the expression.)

Solution Steps

To find the quartic function that best fits the given data, we will use polynomial regression. This involves finding the coefficients of a polynomial of degree 4 that minimizes the difference between the predicted and actual values of \( y \). We will use the numpy library in Python to perform this regression and obtain the coefficients of the quartic polynomial.

We are given the following data points:

\[ \begin{array}{|c|c|} \hline \mathbf{x} & \mathbf{y} \\ \hline -2 & 36 \\ -1 & 3.75 \\ 0 & 0 \\ 1 & 3.75 \\ 2 & 36 \\ 3 & 159.75 \\ 4 & 480 \\ \hline \end{array} \]

We fit a quartic polynomial of the form:

\[ y = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 \]

The coefficients obtained from the fitting process are:

\[ \begin{align_} a_4 & = 1.75 \\ a_3 & = -1.77 \times 10^{-14} \\ a_2 & = 2 \\ a_1 & = 2.06 \times 10^{-14} \\ a_0 & = -6.95 \times 10^{-14} \\ \end{align_} \]

Substituting the coefficients into the polynomial expression, we have:

\[ y = -6.95 \times 10^{-14} x^0 + 2.06 \times 10^{-14} x^1 + 2 x^2 - 1.77 \times 10^{-14} x^3 + 1.75 x^4 \]

Final Answer