What do you call a curve that resembles a crooked S

Question

but it is tilted about theta=artan(x) wit the upper half of the '8' removed about the midpoint and the lower part removed about the of other half. It looks like a tilted 'S' but flipped, and it's monotonically increasing or decreasing.

Answer 1

Here's $\theta = \arctan{x}$,

Also note that the sigmoid function is a mathematical function having an "S" shape ("aka": a sigmoid curve). Often, the sigmoid function refers to the special case of the logistic function:

$y = \dfrac{1}{1 + e^{-x}}$

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If you are looking for the "reflection" or "inverse" of a "tilted $S$-shaped curve:

The inverse of $\;\theta = \arctan(x)\;$ is $\;x = \tan(\theta),\;$ and when plotted you get what you might be looking for if you restrict $\theta$ to $\theta \in (-\pi/2, \pi/2)$:

$x = \tan(\theta),\; \theta\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Somewhat similarly, what you seem to be describing (a reflected/flipped tilted "S") looks a bit like the simple cubic function: $y = x^3$, (the graph can be translated, or rotated to tilt it more.)

$y = x^3$

Answer 2

a sigmoid function http://en.wikipedia.org/wiki/Sigmoid_function. It has lots of applications in natural science, e.g., biological populations driven by evolution/natural selection.

Answer 3

It is also called the logistic curve.

Another curve that looks much like it is the $\int_{-\infty}^x e^{-t^2} dt$.