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!
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I found a function $(Sin[(x - 0.5)\pi] + 1)0.5$ looks like close. But the intersection point is a little bit right side. It is best if I could control the location of the intersection point. – Tommy Zenth Jun 26 '21 at 08:36
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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.
Sayan Dutta
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Thanks for your advice! But the derivative of this curve at (0,0) and (1,1) probably are not correct. – Tommy Zenth Jun 26 '21 at 08:25
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If you have a specific choice of derivative, then you can use $y=kx(x-\sqrt{2})\left(x-\frac{\sqrt{2}}{2}\right)$ instead and fit $k$ according to the required derivative. – Sayan Dutta Jun 26 '21 at 08:39