How would one find a equation or a function to describe a custom curve in a coordinate system?
Like for example, when someone presents you with a graph of a repeating curve, how would one find its equation to describe it?
Here is my sketch of the example:
What would be the equation to describe the curve that looks something like the repeating curve on my sketch, which follows through a given line in a coordinate system?
I guess we would need to have the parameters in our equation to determine the width and height of the loops, and the length of the curved lines connecting the loops.
All in all, I wouldn't know where to start, so any hints you can provide are helpful.
There's no general process for finding a parametrisation of a given curve. It is a question of having some experience and then trying things until you find something that looks good.
If we look at one copy of your curve, we can se that $y$ increases, then decreases about the double of the increase and then increases to the starting level. That sounds like how $\sin t$ behaves.
If we then look at $x$ we can see that it changes direction four times, so if we want to use $\sin$/$\cos$ we'll need a factor $2$ on the period, and as it has to end in a different position than where it started we'll have to add something to $\sin$/$\cos$, a good starting point is just to take the parameter.
That leads to the guess that $(t+\sin(2t), \sin(t))$ might look like the curve. If you try to draw that (wolfram alpha is good for that), you get a promising result - that can be improved, I would start by trying a lower factor to the $t$ in the equation for $x$, but I'll leave the that to you, then you can some experience in what changes have what effect.
Edit: Regarding the added question in a comment: One way of making the curve shrink (or grow) would be to multiply the equation for $y$ with something. For high values of $x$, $\frac{1}{x}$ has a nice slow decrease, so a factor like $\frac{1}{t+N}$ sounds like it make have the desired effect. Again I recommend you try for yourself.