Find the quadratic function that is the best fit for $f(x)$ defined by the table below.
\begin{tabular}{|c|c|c|c|c|}
\hline $\mathbf{x}$ & 0 & 2 & 4 & 6 \\
\hline $\mathbf{f}(\mathbf{x})$ & 2 & -3.6 & -8.4 & -12.4 \\
\hline
\end{tabular}
\[
f(x)=\square
\]
(Use integers or decimals for any numbers in the expression. Round to the nearest tenth as needed.)

Step 1 ：The quadratic function is generally in the form of \(f(x) = ax^2 + bx + c\). We can use the given points to form a system of equations to solve for a, b, and c.

Step 2 ：Given points are \((0,2)\), \((2,-3.6)\), \((4,-8.4)\), and \((6,-12.4)\).

Step 3 ：Substituting these points into the equation \(f(x) = ax^2 + bx + c\), we get:

Step 4 ：For \((0,2)\): \(2 = c\)

Step 5 ：For \((2,-3.6)\): \(-3.6 = 4a + 2b + 2\)

Step 6 ：For \((4,-8.4)\): \(-8.4 = 16a + 4b + 2\)

Step 7 ：For \((6,-12.4)\): \(-12.4 = 36a + 6b + 2\)

Step 8 ：We now have a system of equations:

Step 9 ：\[\begin{align*} 1) & c = 2 \\ 2) & 4a + 2b = -3.6 - 2 = -5.6 \\ 3) & 16a + 4b = -8.4 - 2 = -10.4 \\ 4) & 36a + 6b = -12.4 - 2 = -14.4 \end{align*}\]

Step 10 ：We can simplify equations 2, 3, and 4 by dividing by 2, 4, and 6 respectively:

Step 11 ：\[\begin{align*} 2) & 2a + b = -2.8 \\ 3) & 4a + b = -2.6 \\ 4) & 6a + b = -2.4 \end{align*}\]

Step 12 ：Subtracting equation 2 from equation 3, we get \(2a = 0.2\), so \(a = 0.1\).

Step 13 ：Substituting \(a = 0.1\) into equation 2, we get \(2*0.1 + b = -2.8\), so \(b = -2.8 - 0.2 = -3\).

Step 14 ：So, the quadratic function that best fits the given points is \(\boxed{f(x) = 0.1x^2 - 3x + 2}\).