# How do you find the vertex of a parabola with two points?

Apr 24, 2015

There are infinite number of parabolas passing through two given points. Therefore, this problem has no solution.

The logic behind this is simple. Generally speaking, parabola is determined by a quadratic polynomial
$y = a {x}^{2} + b x + c$.
There are three unknown coefficients here: $a$, $b$ and $c$.

To determine three unknown variables we need three equations. That's why we need three given points $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{2} , {y}_{2}\right)$ and $\left({x}_{3} , {y}_{3}\right)$ that would produce three equations with three unknown variables $a$, $b$ and $c$:
${y}_{1} = a {x}_{1}^{2} + b {x}_{1} + c$
${y}_{2} = a {x}_{2}^{2} + b {x}_{2} + c$
${y}_{3} = a {x}_{3}^{2} + b {x}_{2} + c$

Solving this system of three linear equations with three unknown variable produces coefficients $a$, $b$ and $c$, thereby defining a parabola.