Is there a name and formula for this curve? It isn't the "normal distribution"

Question

1. starts at -1,0 asymptotically to x axis
2. maxes out at 0,1
3. and back down again to mirror how it started.

Normal distributions keep tapering off well past the -1..1 endpoints, and I was looking for something simple that ended. And most importantly the formula for it!

Answer 1

Your conditions do not single out a particular function. For instance, $(x^2-1)^2, \frac{\cos(\pi x)+1)}2$ work. The first is a polynomial so that could be nice. The second is trigonometric.

If you want one that extends by 0 to a $C^\infty(\mathbb R)$ function, then the "bump functions" alluded to in the comment by D.B. would lead you to something proportional to $\exp(-1/(1-x^2))$.

Lets say $\mathcal F$ is the collection of functions $:[0,1]\to \mathbb R$ that satisfy the properties you laid out. Then I can see at least two things:

• $\mathcal F$ is convex: $f,g\in\mathcal F$ and $\lambda\in[0,1]$ implies $\lambda f + (1-\lambda) g \in \mathcal F$.
• if $f\in \mathcal F$, and $s \ge 1$, then $f^s\in\mathcal F$.

Here's some graphs on Desmos:

Answer 2

The curves you desire are a subset the Superparabola. These are specifically in domain $x\in[-1,1]$. It is also parameterized so that you can have smooth transition between the functions.