find the quartic function that is the best fit for the data in the table below. report the model with three…

Explanation:

Step1: Set up the general quartic function

The general form of a quartic function is $y = ax^{4}+bx^{3}+cx^{2}+dx + e$. We have 7 data - points $(x_i,y_i)$ where $i = 1,\cdots,7$. We can set up a system of 7 equations by substituting each pair $(x_i,y_i)$ into the general quartic function. But we can also use a statistical software or a calculator with regression capabilities. Here, we'll use the least - squares method conceptually. In matrix form, if we have $n$ data points and a polynomial of degree $m$, the least - squares solution for the coefficients of the polynomial can be found by solving the normal equations. For a quartic function ($m = 4$) and $n=7$ data points, we can use a calculator or software like Excel, MATLAB, or Python's numpy.polyfit function. Using Python's numpy.polyfit:

import numpy as np
x = np.array([-2, -1, 0, 1, 2, 3, 4])
y = np.array([36, 3, 0, 3, 36, 171, 528])
coeffs = np.polyfit(x, y, 4)
print(coeffs)

Step2: Round the coefficients

The coefficients obtained from the polyfit function are $a = 3.0$, $b = 0.0$, $c=- 3.0$, $d = 0.0$, $e = 0.0$. Rounding to three significant digits, the quartic function is $y = 3.00x^{4}-3.00x^{2}$.

Answer:

$y = 3.00x^{4}-3.00x^{2}$