# Thread: Equation of a Parabola

1. ## Equation of a Parabola

Find the equation of the parabola y = ax² + bx + c that contains the points (1, -2), (2, 0), and (4, 22).

2. Originally Posted by bearej50
Find the equation of the parabola y = ax² + bx + c that contains the points (1, -2), (2, 0), and (4, 22).

Plug in the points and you'll get three equations.

$\displaystyle a+b+c = -2$

$\displaystyle 4a+2b+c = 0$

$\displaystyle 16a+4b + c = 22$

You can solve this system of equations in a number of ways. I recommend setting up a matrix.

$\displaystyle \bigg(\begin{array}{ccc} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 16 & 4 & 1 \end{array}\bigg) \bigg( \begin{array}{ccc} a \\ b \\ c \end{array}\bigg) = \bigg(\begin{array}{ccc} -2 \\ 0 \\ 22 \end{array} \bigg)$

So let's write that as $\displaystyle AX = B$. We want to find x, hence: $\displaystyle X = BA^{-1}$

And you know how to calculate the inverse of a 3x3 matrix hopefully! $\displaystyle A^{-1} = \frac{adj(A)}{det(A)}$

3. I was taught a different way. Subtract the equations from one another.
I got a = 13/5, b = -23/5, and c = 0. Is this right?

4. Originally Posted by bearej50
I was taught a different way. Subtract the equations from one another.
I got a = 13/5, b = -23/5, and c = 0. Is this right?
Check yourself if it is right. If it satisfies the original equations it's right.

Unfortunately yours are not. I get a = 3 ,b = -7, c = 2, and that is correct.