I need to design an analytical function that looks like this (See figure bellow). The idea is to control the angles "a" at the beginning and at the end. If the function depends on x (any kind of parameter, angle, value in [-1,1], etc.), we should have f(x) symmetric to f(-x) towards the line y=-x+1 (orange line). I mean, I should be able to build in some ways with the symmetric function. For instance, we have x^2 symmetric to x^(1/2).
I can build this function with a Bezier curve, but I need an analytical form that would give something close to the Bezier solution.