u/AQuietLilyInTheWind

I've just recently finished Calculus II and was dabbling around with parametric equations in desmos graphing because they seemed like they'd be useful for creating curved paths based on some time value.

Upon messing around with the variables a lot, I figured out that I could make a curved path connecting (0,0) and (a,b) using the equation (t(t+a-1), t(t+b-1)), with 0<= t <= 1.

A couple of questions about this:

1. Is there a reason why this works?
2. Is there a reason changing the interval of t from [0,1] to something like [0,3] screws up the whole equation? (obviously making the interval cover a wider range of values than [0,1] still hits the point (a,b), but can the interval be changed in ways to make the curve have more unique properties that could be used?)
3. Are there ways to chain more points that you want the curve to hit?
4. Are there any resources that you know of that cover this / similar topics like this?