How do I algebraically express transformations of a sigmoidal?

Question

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

I'm not really sure what you want exactly (what do you mean by 'express $a$ and $c$ alebraically' ? ), but you can play around with this: https://www.desmos.com/calculator/kbegf3a3j2 — Matti P., Mar 18 '19 at 11:12
@MattiP. I want to show how each parameter (a,b,c and d) changes the curve. I have found how is it done graphically, but want to be able to show these transformations algebraically as opposed to graphically. — Adsp, Mar 18 '19 at 11:27
I think anyway you will get a lot of insight from playing around with the graph. For example, the parameter $a$ bears a relationship between the difference in height of the two "legs" of the graph, namely the value at $-\infty$ and $+\infty$ ... — Matti P., Mar 18 '19 at 11:35

Jean Marie · Accepted Answer · 2019-03-19T10:34:54.483

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...

My function takes the form y=d+a/f(x−bc) Does the divide by a change any of your results or have you rearranged it? — Adsp, Mar 18 '19 at 20:30
I have considered that $f(x)=1/(1+e^{-x})$. If one considers $f(x)=1+e^{-x}$, we need indeed transformation $f(x) \to 1/f(x)$ which cannot be treated in a simple way (one could say that this transformation is a "violent" operation compared to others) — Jean Marie, Mar 18 '19 at 20:36

score 0 · Answer 2 · answered Mar 18 '19 at 11:26

As $d$ and $b$ are vertical/horizontal shifts respectively, then $a$ and $c$ can be interpreted as vertical/horizontal expansion/compression respectively ($a,c>1$ imply on expansion while $a,c<1$ imply compression). A typical shape for $d=b=0$ and $a=1$ is as follows:

How do I algebraically express transformations of a sigmoidal?

2 Answers2