# Analytic formula for parameterizing the below family of curves

I'm trying to find an analytic formula for a curve that can look like any of the curves below depending on one or more parameters. My initial thought was to use exponentials, something that might include functions like $1-e^{-x}$ or $e^x-1$, but I don't think this quite works to capture most or all of the curves below. I'd preferably like something of the form y=f(x), being a single variable function rather than using a parametric equation.

• $x^n$, for any positive $n$. – Chappers Sep 30 '15 at 4:11
• @Chappers Can you post as an answer. – MHH Oct 17 '16 at 12:05

The functions $y=x^{\alpha}$, where $\alpha$ is a positive real number, will do this: $\alpha = 1$ is the diagonal, as $\alpha \to 0$, the line approaches the left and top sides of the square, and reflecting in $y=x$ gives $y=x^{1/\alpha}$, which also corresponds to the cases with $\alpha<1$.