Answer
$f(x) = -x^{2} + x + 1$
Work Step by Step
We must first solve for the functions in terms of a, b, and c:
f(1) = a + b + c = 1
f(2) = 4a + 2b + c = -1
f(3) = 9a + 3b + c = -5
We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix:
$\begin{bmatrix}
1 & 1 & 1 & |1\\
4 & 2 & 1 & |-1\\
9 & 3 & 1 & |-5\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -2 & -3 & |-5\\
0 & -6 & -8 & |-14\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & 2 & 3 & |5\\
0 & -3 & -4 & |-7\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & 2 & 3 & |5\\
0 & 0 & 1 & |1\\
\end{bmatrix}$
C:
c = 1
B:
2b + 3(1) = 5
2b + 3 = 5
2b = 2
b = 1
A:
a + 1 + 1 = 1
a + 2 = 1
a = -1
Using the solution above, the quadratic function is:
$f(x) = -x^{2} + x + 1$
