Question

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

Answer 1

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=ax^2+bx+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 `(x_1,y_1)`, `(x_2,y_2)` and `(x_3,y_3)` that would produce three equations with three unknown variables `a`, `b` and `c`:
`y_1=ax_1^2+bx_1+c`
`y_2=ax_2^2+bx_2+c`
`y_3=ax_3^2+bx_2+c`

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