I am graphing a sigmoidal of the form $$ y=d+\frac{a}{1+e^{-(x-b)/c}} $$
I am investigating how the shape of the graph changes when each of the parameters a, b, c, and d are altered. I understand that d will shift the graph vertically, b will shift the graph horizontally, c will dilate the graph and a will change the size of the graph. However, I am unsure how to express these parameters change the graph in algebraic form.
So far I have: d=constant (k) and has no bearing on x, so it simply shifts the graph up or down by a value of d x-b=0, so x=b. The horizontal shift of x is equal to the value of b. Am I on the right track? Is there a more fluid way of expressing these values algebraically? How can I express a and c as well?
Thankyou
You are in a particular case of the following general setting :
How to describe in a geometrical way the transformation of the graphical representation of $y=f(x)$ into the graphical representation of
$$y=d+a.f\left(\frac{x-b}{c}\right) \ \ ?\tag{1}$$
Here are the successive actions, in this order :
1) $x$-axis translation $b$ units rightwards (this must be considered algebraically : if $b<0$, the translation is $|b|$ units on the left).
2) $x$-axis directional enlargment if $c<1$, shrinking if $c>1$ by a factor $c$.
3) $y$-axis directional enlargment if $a>1$, shrinking if $a<1$ by a factor $a$.
4) $y$-axis translation $d$ units upwards (considered algebraically as for 1)).
Important remark : there is an equivalent way to write down (1):
$$\underbrace{\frac{y-d}{a}}_Y=f\left(\underbrace{\frac{x-b}{c}}_X\right) \tag{2}$$
which is symmetrical in $x$ and $y$.
(2) can be written as well under the form :
$$Y=f(X) \ \ \text{with} \ \ \begin{cases}x&=&cX+b\\y&=&aY+d\end{cases} \ \ \ \ (3)$$
(old coordinates expressed as - affine - functions of the new ones, as usual).
(3) provides a "dual view" : the new curve can be interpreted "statically" as the ancient curve "seen" with respect to a change of origin and scaling on both axes...