# Math Help - Help me define this curve!!!!!

1. ## Help me define this curve!!!!!

What is the equation of a curve like this (constant then an inflection then straight again)? Thanks in advance.

2. If you want the function to be smooth then one which approximates the shape you drew is a sigmoid:

$
f(x) = \frac{1}{1+e^{-x}}
$

This function has a range of (0,1) but is not constant anywhere (though you can tighten up the function to make the derivative very small as x goes away from 0)

You can also define such a function peicewise if you want the function to be constant on some intervals.

3. ## Thanks!

Thats great. How can I move this so it is all above 0 in the x plane?

4. Shifting a function up (or down) by a fixed amount can be done by adding (or subtracting) a constant from the function. The sigmoid function is already above the x=0 line in the plane since
$
x \rightarrow \infty,
f(x) \rightarrow 1
$

while

$
x \rightarrow -\infty,
f(x) \rightarrow 0
$

To shift a function left (or right) change the input by a constant.
exp:
f(x) shifted to the right by 5 is f(x-5)
f(x) shifted to the left by 5 is f(x+5).

$f(x) = \frac{\pi}{2} + \arctan(8x)$