Helping find the S-Curve function from the graph

Question

I would like to model a function for the S-curve in the S-Curve Plot. The line in the graph is $y=x$ and the range of the x-axis and y-axis are both (0,1). It seems to be an incomplete beta function or something else. A simpler form would be better. Any help would be appreciated!

Answer 1

Take the cubic $y=x(x-\sqrt{2})\left(x-\frac{\sqrt{2}}{2}\right)$ and rotate the $X$-axis to $y=x$ (i.e., rotate the axes by $-45°$)

That would do I guess.