A quadratic function is just one possible polynomial. They're also not really related to big-O complexity, where you mostly just care about what the highest exponent is: O(n^2) vs O(n^3).

For most short programs it's fairly easy to determine the complexity. Just count how many nested loops you have. If there's no loops, it's probably O(1) unless you're calling other functions that hide the complexity.

If there's one loop that runs N times, it's O(n), and if you have a nested loop, it's likely O(n^2).

You throw out any constant-time portion, so your function's actual runtime might be the polynomial: 5n^3 + 2n^2 + 6n + 20. But the big-O notation would simply be O(n^3) in that case.

I'm simplifying a little, but that's the overview. I think a lot of people just memorize that certain algorithms have a certain complexity, like binary search being O(log n) for example.