Find values for \(a\text{,}\) \(b\text{,}\) and \(c\) so that the points \((1, 3)\text{,}\) \((3, 5)\text{,}\) and \((4, 9)\) lie on the graph of \(y = ax^2 + bx + c\text{.}\)

## Solution.

We substitute the coordinates of each of the three points into the equation of the parabola to obtain three equations:

\begin{align*}
\alert{3} =\amp a(\alert{1})^2 + b(\alert{1}) + c \amp\\
\alert{5} =\amp a(\alert{3})^2 + b(\alert{3}) + c \amp\\
\alert{9} =\amp a(\alert{4})^2 + b(\alert{4}) + c \amp
\end{align*}

or, equivalently,

\begin{align*}
a + b + c \amp = 3 \amp\amp(1)\\
9a + 3b + c \amp = 5 \amp\amp(2)\\
16a + 4b + c \amp = 9 \amp\amp(3)
\end{align*}

This is a system of three equations in the three unknowns \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) To solve the system, we use Gaussian reduction. We first eliminate \(c\text{.}\) We subtract Equation (1) from Equation (2) to obtain

\begin{equation*}
8a + 2b = 2 \hphantom{blankblank}(4)
\end{equation*}

and then subtract Equation (1) from Equation (3) to get

\begin{equation*}
15a + 3b = 6 \hphantom{blankblank}(5)
\end{equation*}

We now have a system of two linear equations in two variables:

\begin{align*}
8a + 2b \amp = 2 \amp\amp(4)\\
15a +3b \amp = 6 \amp\amp(5)
\end{align*}

Next, we eliminate \(b\) from Equations (4) and (5): we add \(-3\) times Equation (4) to \(2\) times Equation (5) to get

\(-24a\) | \(-\) | \(6b\) | \(=\) | \(-6\) | \(\hphantom{blank}\) | \(-3\times(4)\) |

\(30a\) | \(+\) | \(6b\) | \(=\) | \(12\) | \(\) | \(2\times(5)\) |

\(6a\) | \(\) | \(\) | \(=\) | \(6\) | \(\) | \(\) |

or \(a = 1\text{.}\) We substitute \(1\) for \(a\) in Equation (4) to find

\begin{align*}
8(\alert{1}) + 2b \amp= 2 \amp\amp \blert{\text{Solve for }b.}\\
b \amp= -3
\end{align*}

Finally, we substitute \(-3\) for \(b\) and \(1\) for \(a\) in Equation (1) to find

\begin{align*}
\alert{1} + (\alert{-3}) + c\amp = 3\amp\amp \blert{\text{Solve for }c.}\\
c \amp = 5
\end{align*}

Thus, the equation of the parabola is

\begin{equation*}
y = x^2 - 3x + 5
\end{equation*}

The parabola and the three points are shown below.