Answer
$f(x) = -x^{2} + 2x + 8$
Work Step by Step
We must first solve for the functions in terms of a, b, and c:
f(1) = a + b + c = 9
f(2) = 4a + 2b + c = 8
f(3) = 9a + 3b + c = 5
We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix:
$\begin{bmatrix}
1 & 1 & 1 & |9\\
4 & 2 & 1 & |8\\
9 & 3 & 1 & |5\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |9\\
0 & -2 & -3 & |-28\\
0 & -6 & -8 & |-76\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |9\\
0 & 2 & 3 & |28\\
0 & -6 & -8 & |-76\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |9\\
0 & 2 & 3 & |28\\
0 & 0 & 1 & |8\\
\end{bmatrix}$
C:
c = 8
B:
2b + 3(8) = 28
2b + 24 = 28
2b = 4
b = 2
A:
a + 2 + 8 = 9
a + 10 = 9
a = -1
Using the solution above, the quadratic function is:
$f(x) = -x^{2} + 2x + 8$
