Finding quadratic function from two points and known max value

[grsbanks]

Short answer: nope.

Just like a linear function can be defined with 2 points, a quadratic function needs 3 points to be defined.

The best you can get is a family of functions that pass through the two points you know and have the known maximum.

In the more general case, a polynomial of order n (ax^n+bx^(n-1)+cx^(n-2)...) requires n+1 points and you end up solving a system of n+1 simultaneous equations to get your coefficients.