# Alternatives for sigmoid curve starting from 0 with interpretable parameters

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.

$$\displaystyle{a}={9}\cdot{d}^{{-{b}}}$$

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

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.

https://www.desmos.com/calculator/fmalvakguo

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.