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

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 Commented Jun 24, 2020 at 3:38
• sorry your link seems corrupted Commented Jun 24, 2020 at 3:39
• ok..its not a bad approximation..but its not really perfect. Here you can slide it here, you can see the issues: desmos.com/calculator/dkkxhft1dn Commented Jun 24, 2020 at 3:41
• @Jaume Oliver Lafont Nice find, well done. Commented Jun 24, 2020 at 3:44
• $$x^{p}(1-x)^{1-p}$$ has the right symmetry but fails the scaling... Dividing it by the maximum value should work. Commented Jun 24, 2020 at 3:54

A fit is given by $$y = \frac{x^{p}(1-x)^{1-p}}{p^p(1-p)^{1-p}}$$
with $$0