an inflection point perspective
The sigmoid function of the form
\[f(x) = \frac{1}{1 + be^{-ax}}\]is plotted below with varying parameters a and b.
Notice how b shifts the point of inflection p along the x-axis and a changes the width of the sigmoid function.
Here is the plot of the function with varying a parameter:
Here is the plot of the function with varying b parameter:
Both of these together produce the effect of translating the point of inflection and controlling the width of the sigmoid function which produce a better classification. However the logistic function is of a sligtly different form and is given by:
\[f(x) = \frac{1}{1 + e^{-wx+c}}\]where we control b in the fuction above through C and a through W. Notice how they are the same