How to write equations to shape a polynomial function only in the upper right quadrant of a plot, starting from x=0 and y=0?

Question

I am searching for curve equations, but finding stuff like this gallery of curves, all of which aren't "growth" oriented sorts of curves, but more mathematically interesting possible curves.

I got to this point trying to think about how to "generate" random point distributions "along a growth curve", such as like this:

I like the shape of this curve, and am hoping to find equations to generate such a curve, and others similar to it, with slight fluctuations up and down and such. The goal is to plugin x and get a y, for each step through an equation.

Is there any place on the web that has a bunch of predefined equations to create such sorts of curves like in the red? Or if not, what are some basic equations beyond log(x) and $x^n$ which give you more varied and somewhat interesting (I would say "realistic looking") growth-like curves? By growth-like curves, I am looking for curves which would look like a line was best fit to some data point plot, but doesn't need to be that random, just a few ups and downs would be nice, slowly increasing to the right, or just like a few-pointed mountain would be interesting. I am no curve mathematician so I don't even know where to begin designing or finding equations to do such a thing, beyond like I said log(x) and $x^n$ type of simple curves.

Hoping there is a place someone has collected lots of various equations to show off some curves. But if not, just a good starting place in terms of equations would be all I need. Such as what types of things I should try (sin(x), etc.) and where I can plot them out for free online to see what they look like.

I guess something like this is a plotting tool I would use, but then I am not sure where to begin with finding the right equations/functions. Perhaps I should be able to find a list of "polynomial" type function plots?

Further notes, I would basically like to only go on the upper right quadrant of the graph, so it goes from 0 to n, incrementing by an integer on the x axis, but varying however on the y axis only in the positive direction.

Answer 1

For problems such as this I once developed something I called the gamma pulse. Briefly, in physical space

$$ \gamma(t;n,k)=k(kt)^ne^{-kt}u(t)\\ \int_0^\infty\gamma(t;n,k)=\Gamma(n+1) $$

Here, $k$ is the characteristic frequency, $n$ is the pulse order, $u$ is the Heaviside step function, and, of course, $\Gamma$ is the gamma function, hence the name of the function $\gamma$.

In similarity space, i.e., $\tau=t/k$, we have

$$ \gamma(\tau;n)=\tau^ne^{\tau}u(\tau)\\ \int_0^\infty\gamma(\tau;n)=\Gamma(n+1) $$

The following properties may help you customize this for your problem. The time at maximum pulse is given by

$$ \frac{\partial\gamma(\tau;n)}{\partial\tau}=0 \to\tau=n $$

The mean time is given by

$$ \bar{\tau}=\frac{\int_0^\infty\tau\gamma(\tau;n)}{\int_0^\infty\gamma(\tau;n)}=\frac{\Gamma(n+2)}{\Gamma(n+1)}=n+1 $$

The rms pulse width (left an exercise for the reader) is given by $\tau_{rms}^2=n+1$.

And finally, the $3dB$ width, i.e., the width at half-height is given by the following approximations,

$$ \text{Small} \ n(\le2):\quad \Delta\tau_{3dB}\approx\sqrt{6n}\\ \text{Large} \ n(\ge2):\quad \Delta\tau_{3dB}\approx2\sqrt{2\ln2\cdot n} $$

I have used this in a great number of problems similar to yours. There is a more complete write-up here and in the linked pdf file.