# How do I find the standard equation of a parabola given 3 points?

Jan 21, 2015

A parabola is given by the equation $y = a {x}^{2} + b x + c$ which means that if the three coefficients $a$, $b$ and $c$ are known, the parabola is uniquely identified. Each point gives you a condition, and so, given three points you'll end up with three conditions for three variables, and thus there will be one solution, or no solutions at all.

To solve the problem, simply start from the generic formula written above, and substitute the points given. In fact, if a point $\left({x}_{0} , {y}_{0}\right)$ lies on the parabola, it must verify that ${y}_{0} = a {x}_{0}^{2} + b {x}_{0} + c$. So, with three given points $\left({x}_{1} , {y}_{1}\right)$ $\left({x}_{2} , {y}_{2}\right)$ and $\left({x}_{3} , {y}_{3}\right)$ , you'll have the following system:
${y}_{1} - a {x}_{1}^{2} - b {x}_{1} - c = 0$
${y}_{2} - a {x}_{2}^{2} - b {x}_{2} - c = 0$
${y}_{3} - a {x}_{3}^{2} - b {x}_{3} - c = 0$

Remember that the variables are $a$, $b$ and $c$. If you can solve this sistem, you can find the parabola.

Let me give you a simple example: pick the points $\left(0 , 0\right) , \left(1 , 1\right)$ and $\left(- 1 , 1\right)$, belonging to the parabola $y = {x}^{2}$.
If $\left(0 , 0\right)$ must lie on the parabola, then it must be
$0 = a \left({0}^{2}\right) + b \left(0\right) + c$, and thus $c = 0$.
If $\left(1 , 1\right)$ must lie on the parabola, then it must be
$1 = a \left({1}^{2}\right) + b \left(1\right)$, and thus $a + b = 1$
If $\left(- 1 , 1\right)$ must lie on the parabola, then it must be
$1 = a \left(- {1}^{2}\right) + b \left(- 1\right)$, and thus $a - b = 1$

So far, we know that $c = 0$ and that $a + b = 1$, and $a - b = 1$, and one can easily check that this means $a = 1$ and $b = 0$.

We have thus solved the problem, because the solution $\left(a , b , c\right) = \left(1 , 0 , 0\right)$ identifies the parabola $y = {x}^{2}$