Answer
$f(x) = -2x^{2} - 2x + 1$
Work Step by Step
We must first solve for the functions in terms of a, b, and c:
f(-2) = 4a - 2b + c = -3
f(1) = a + b + c = -3
f(2) = 4a + 2b + c = -11
We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix:
$\begin{bmatrix}
4 & -2 & 1 & |-3\\
1 & 1 & 1 & |-3\\
4 & 2 & 1 & |-11\\
\end{bmatrix}$ ~ $\begin{bmatrix}
4 & -2 & 1 & |-3\\
0 & -6 & -3 & |9\\
0 & -4 & 0 & |8\\
\end{bmatrix}$ ~ $\begin{bmatrix}
4 & -2 & 1 & |-3\\
0 & 2 & 1 & |-3\\
0 & -2 & 0 & |4\\
\end{bmatrix}$ ~ $\begin{bmatrix}
4 & -2 & 1 & |-3\\
0 & 2 & 1 & |-3\\
0 & 0 & 1 & |1\\
\end{bmatrix}$
C:
c = 1
B:
2b + 1 = -3
2b = -4
b = -2
A:
4a - 2(-2) + 1 = -3
4a + 4 + 1 = -3
4a + 5 = -3
4a = -8
a = -2
Using the solution above, the quadratic function is:
$f(x) = -2x^{2} - 2x + 1$
