Analytic formula for parameterizing the below family of curves

Question

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.