Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works.

Specifically I'm trying to understand how to build proper polynomia parameters for the function to work properly.

Ive literally gone through dozens of web pages and downloadable opdf files loooking for good descriptions and examples of how to use this.

My main probloem so far has been that literally every single online resource Ive found always appears to leave out critically important descriptions of parts of the function, assuming that the reader will just know everything that is being talked about, so Im left with a large collection of incomplete descriptions - very frustrating!

Ive found some examples that appear to be better than others, so for example in one description I see an example of this sigmoid function ;

w = w.max / 1 + e ^-k (t - tm)

( from the online file ; "a flexible sigmoid function of determinate growth" )

here, I understand that the "-k" represents a value indicating how steep the slope of growth is and that t = time, and that tm represents the max ceiling of time, so Im guessing that here we'd subtract a current time from a max time??? That seems a bit confusing also.

Then I found a better example at this website ; http://www.cs.xu.edu/math/math120/01f/

The file here is called " logistic.pdf "

In this example they show the function of ;

y = C / 1 + Ae ^-Bx

A = # of times initial population must grow to reach (C) also A describes the relation betwen initial and limiting output values

B = Here in the example the definition is left ambiguous through out the document all it really says is that this value increases / decreases when the function is positive / negative

The document says (B) can be derived with the inflection point coordinates but it doesnt really say exactly why or how, it gives only a very vague generic description on page 6 where it says "ln 12.8 / 0.0266 = 95.8"

But on page 3 it says "The parameter B is much harder to interpret exactly. We will be content to simply mention that"

On page 4 it says "It turns out that A = 12.8, B = 0.0266, C = 11.5 are parameter values that yield a logistic function with a good fit to this data:"

This seems completely confusing, the value for B seems arbitrarily assigned and they simply say that this value "turns out" to be a good paramter fit.... I totally dont understand how or why this value of 0.0266 has been chosen and I dont understand what it means that this value "turns out " to be a good fit??!!

X = some input

C = A ceiling or limit but is representing long run behaviour of the function

So, if possible I would really honestly appreciate some basic answers to these questions ;

What IS (B) actually representing here?

And how, why, and where did they assign it the value of 0.0266 ???

And lastly, how can we determine what parameters to use for a sigmoid function?

I can understand that the first value passed to e, like ^- k can represent the growth slope angle;

w = w.max / 1 + e ^-k (t - tm)

But how do we determine what other parameters can be used and is there any online resource or simple method available that can describe how we choose the other parameters we want and then how do we determine the proper syntax of these parameters for the function?

Thanks for any meaningful feedback you can provide!

:)