I am looking for alternative of sigmoid curves going through $(0,0)$, whose parameters can be sensed by eyeballing the function graph. As an example, consider this curve:

$$f(x) = {{a x ^ b} \over 1 + a x ^ b}$$

Where $ a, b $ are meaningless parameters without any straightforward interpretation. However, we can write the curve equation in such a way that both parameters will be meaningful. Imagine new parameters $c, d$ such that:

$$\begin{cases} f (c) = 0.1 \\ f (d) = 0.9 \end{cases}$$

Then expressing the curve with parameters a, b defined as follows, that is by parameters c, d, does the job.


$$\displaystyle{b}=\frac{{-{4}\cdot \log{{\left({3}\right)}}}}{ \log{{\left(\frac{c}{{d}}\right)}}}$$

So looking at the example below, we can easily guess the values of parameters c, d. The function is in 10% and 90% value in approximately $x=1$ (parameter c) and $x=3$ (parameter d). Job done.



Why I ask for alternatives? The curve equation lacks "symmetry." I do not ask for exact symmetry but what I mean is that the function above $f(d)=0.9$ approaches asymptote of 1 very slowly. While below $f(c)=0.1$ the function gets to zero quite quickly.


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