Looking for a math expression to fit these series of inputs/graphs

Question

I require a math formula that lets me move smoothly between these graphs:

It must always intersect x=0 and x=1 as seen, and its peak must always be at 1. It can be a parametric equation. How might I generate something like this.

For example that middle graph might be equal to $y=-\left(2x-1\right)^{2}+1$

But that equation needs to somehow be modified to produce those other graphs.

Its almost like its interpolating between these 3 graphs:

https://www.desmos.com/calculator/2cofwiga9k

An attempt might be $$ \sin(\pi x^{p}) $$
I like the results for 0<p<1 — Jaume Oliver Lafont, Jun 24 '20 at 03:38
ok..its not a bad approximation..but its not really perfect. Here you can slide it here, you can see the issues: https://www.desmos.com/calculator/dkkxhft1dn — user1436508, Jun 24 '20 at 03:41
$$x^{p}(1-x)^{1-p}$$ has the right symmetry but fails the scaling... Dividing it by the maximum value should work. — Jaume Oliver Lafont, Jun 24 '20 at 03:54
Jaume Oliver..almost there! Perhaps the dividing parameter/scaling value can be a function of p somehow — user1436508, Jun 24 '20 at 03:57

score 1 · Accepted Answer · answered Jun 24 '20 at 04:45

1

A fit is given by $$y = \frac{x^{p}(1-x)^{1-p}}{p^p(1-p)^{1-p}}$$

with $$0<p<1$$

It can be checked at https://www.desmos.com/calculator/es1znohisl

answered Jun 24 '20 at 04:45

Jaume Oliver Lafont

5,212

excellent! hopefully this was a fun problem to work on for you – user1436508 Jun 24 '20 at 14:42
May I ask about the context? What are you using it for? – Jaume Oliver Lafont Jun 24 '20 at 16:45
a controls related task – user1436508 Jun 24 '20 at 17:10

Looking for a math expression to fit these series of inputs/graphs

1 Answers1