What would be the simplest formula to describe a sigmoid graph with asymptotes at 0 and 100 and at the same time ensuring that two (X,Y) values are satisfied?

For example, a sigmoid graph with minimum value approaching 0 and maximum value approaching 100, however that passes through (3000,50) and (4,000, 75).

Here is a picture of what this might look like.


Consider your standard logistic function

$$y = \frac{L}{1+e^{-k(x-x_0)}}$$

Filling in the data about minimum, maximum, and midpoint, we get

$$y = \frac{100}{1 + e^{-k(x-3000)}}$$

So it remains to find the correct value for "steepness" which will pass through the remaining point.

Plugging in our other point, we get

$$75 = \frac{100}{1 + e^{-k(4500-3000)}}$$

$$\frac{3}{4} = \frac{1}{1 + e^{-1500k}}$$ $$\frac{4}{3} = 1 + e^{-1500k}$$ $$e^{-1500k} = \frac{1}{3}$$ $$e^{1500k} = 3$$ $$k = \frac{\ln{3}}{1500} \approx 0.00073240819$$

Which we can confirm by plugging back in to the earlier equation.


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