Sigmoid shaped function with fixed start of ascent, and tunable slope I am looking for the simplest sigmoid function that goes from 0 to 1 and has a fixed starting point and tunable slope. As I am not a mathematician, I am sure I already used a lot of improper terms, but I hope it will be clear:


*

*$$y=\frac{1}{1+e^{-x\alpha}}$$


The standard sigmoid (1.) is not such, as it's starting position depends on the slope (α).
I found that (2.) has the desired start:


*$$y=1-\frac{1}{\left( 1+\left( x \right)^{4} \right)}$$



Plot image
Now that is great, but I need a sigmoid, when below 0 (or actually below β in 3.) it is always zero and it is possibly differentiable.


*$$y=1-\frac{1}{\left( 1+\left( x-β \right)^{4} \right)}$$


My most sincere apologies if this question breaks the noob-o-meter. 
Please also correct me, if I used the wrong terms/ language.
 A: I recently developed a sigmoidal function that is based on the Superparabola as a differentiable model for the Heaviside step function. It has the following attributes:


*

*It is complete in a finite regime (i.e., the ends are absolute rather than asymptotic).

*It is fully differentiable over the entire regime.

*Parametrically, it can vary between a ramp function and a step function.


In addition, it can be moved and scaled as required. Without any further ado, the function is given by
$$f\left( x \right)=\frac{1}{2}\left[ 1+\text{sgn} (x)\,\frac{B\left( {1}/{2}\;,p+1,{{\left| x \right|}^{2}} \right)}{B\left( {1}/{2}\;,p+1 \right)} \right]$$
where the numerator and denominator $B$s are the incomplete and complete beta functions, respectively. When $p=0$ you get the ramp function and as $p\to\infty$ you get the step function. For all other $p$ you get a sigmoidal function with variable rise slope.
Also, note that the derivative of the incomplete beta function is given by
$$\frac{d}{dx}B(\nu,\mu,x)=x^{\nu-1}(1-x)^{\mu-1}$$
The figure below shows a typical sigmoidal function so-calculated (red) and the superparabola (blue) from which it was created by integration.

A: A simpler function that may be useful as a sigmoid function with tunable slope $m$ and range $[-1,1]$ for $x\in [-1,1]$ is
$$
f(x) = \frac{m x}{1+|x|(m-1)}
$$
This function is odd, i.e. $f(-x)=-f(x)$, and interpolates smoothly between $-1$ and $1$ for $x\in[-1,1]$, such that $f'(x)>0$, and $f'(0)=m$. When $m=1$, you get the ramp function (i.e., $f(x)=x$), and when $m\to\infty$ you get the Heaviside function. Note, however, that the slope at $x=-1$ and $x=1$ is not zero (impossible if you want to have the ramp function as one member of this family).
