A little bit about this: for the economy aspect of a plugin that I'm writing, I need to determine the price of something based on 1) the initial price and 2) a maximum price.
With this, if we create a parabola (and cut it off where x >= 0) with the axis of symmetry as 0 (x = 0), we can get this value. Since the parabola is dynamic based on the item needing to be charged, we have to create something to return the y of a point when given the x.
Our two points for testing were (0, 100) and (2500, 6000). Where 100 is base, 6000 is max, 2500 is maxLevel and
point is the point we want to get. HOWEVER with the idea that this parabola starts at 0, a good idea is to treat this as
something that is ZERO INDEXED by setting the max level point to maxLevel - 1 and by getting the max level point as maxLevel - 1
/**
* @param base the base value
* @param max the maximum value when x >= 0, extrema
* @param point, the x value to get from
*
* @return y value from point
*/
private static double fromCurve(double base, double max, int maxLevel, int point) {
// calculate quadratic regression
// coords: {(0, base),(-maxLevel, -max), (maxLevel, max)}
// we learn that (-maxLevel, max) creates a slightly better curve!
// corellation
double[] x = {0, (-1 * maxLevel), maxLevel};
double[] y = {base, (-1 * max), max};
// funcs and val storage
double[] fVals = vals(x[0], x[1], y[0], y[1]);
double[] sVals = vals(x[1], x[2], y[1], y[2]);
double bMul = (-1 * (sVals[1]/fVals[1]));
double a3 = (bMul * fVals[0]) + sVals[0];
double d3 = (bMul * fVals[2]) + sVals[2];
final double a = d3/a3;
final double b = (fVals[2] - fVals[0] * a) / fVals[1];
final double c = y[0] - (a * Math.pow(x[0], 2)) + (b * x[0]);
return (a * Math.pow(point, 2)) + (b * point) + c;
}// stored: a, b, d
private static double[] vals(double x1, double x2, double y1, double y2) {
double[] arr = new double[3];
arr[0] = (-1 * Math.pow(x1, 2)) + Math.pow(x2, 2);
arr[1] = (-1 * x1) + x2;
arr[2] = (-1 * y1) + y2;
return arr;
}