Find a quadratic function that best fits the data. Give answers to the nearest hundredth.
\begin{tabular}{r|r|r|r}
$x$ & -4 & 0 & 7 \\
\hline$y$ & -8 & 12 & -17
\end{tabular}
A. $y=-x^{2}+12$
B. $y=0.29-6.14 x+12$
C. $y=-0.83 x^{2}+1.68 x+12$
D. $y=0.83 x^{2}+1.68 x+12$

Step 1 ：The problem is asking to find the best fit quadratic function for the given data points. The general form of a quadratic function is \(y = ax^2 + bx + c\). We can use the method of least squares to find the coefficients a, b, and c that minimize the sum of the squared residuals (the differences between the observed and predicted values of y). We can solve this problem by setting up and solving a system of linear equations.

Step 2 ：The given data points are \(x = [-4, 0, 7]\) and \(y = [-8, 12, -17]\).

Step 3 ：We set up the system of equations as follows: \[A = \begin{bmatrix} 16 & -4 & 1 \ 0 & 0 & 1 \ 49 & 7 & 1 \end{bmatrix}\]

Step 4 ：After solving the system of equations, we get the coefficients \(a = -0.83\), \(b = 1.68\), and \(c = 12.0\).

Step 5 ：Finally, we form the quadratic function using these coefficients. The quadratic function that best fits the data is \(\boxed{y = -0.83x^{2} + 1.68x + 12}\).