Which comes first, quadratic polynomials or parabolas?

Question

Ok, this may well go into philosophy but here's my dilemma.

Take a quadratic polynomial, for instance $x^2-5x+6$. This polynomial can be factored in $\mathbb{R}[x]$ and so it's considered fundamentally different from, say $x^2+1$.

But if we consider these two polynomials as parabolas, they are essentially the same (in a certain sense there exists only one parabola).

In the polynomial setting the zeroes play a fundamental role; in the geometric setting they essentially mean nothing, but there is a relation between the two entities. It seems to me that a quadratic polynomial and a parabola share similarities but I cannot understand if one is more fundamental than the other or if there is a fundamental notion at the base of both.

My question is this: is there a mathematical sense in which one of the two notions (quadratic polynomial, parabola) is more essential than the other?

(Of course the question is generalizable to any degree I guess)

Answer 1

There are two issues here that are worth separating; one of them is about quadratic polynomials vs. parabolas but the other one is about symmetries (when you say "in a certain sense there exists only one parabola").

Regarding the first issue, in general which concepts a mathematician regards as prior to or more fundamental than other concepts is to a large extent a matter of personal taste. Some people will prefer to make geometry fundamental, others algebra. Historically geometry was fundamental but after Descartes and so forth we built geometry out of algebra and that's been a very useful move. Nowadays we can make sense of quadratic polynomials over an arbitrary field or even commutative ring, whereas parabolas are sort of stuck in $\mathbb{R}$.

Regarding the second issue, the sense in which there exists only one parabola is that any two parabolas $y = a(x - b)^2 + c$ can be related by a combination of

1. vertical translation $y \mapsto y + v$,
2. horizontal translation $x \mapsto x + h$, and
3. vertical scaling $y \mapsto sy$.

(We can also throw horizontal scaling in there.) However, in any particular situation where we might want to discuss quadratic polynomials, these may or may not be relevant symmetries. For example if we're trying to understand the roots of a polynomial, meaning we want to set $y = 0$, then vertical translation is not a relevant symmetry. And in complex dynamics people do things like try to understand the behavior of quadratic polynomials under iteration, which leads to things like Julia sets. In this case neither vertical nor horizontal translation is a relevant symmetry because $x$ and $y$ are the same variable and so need to be transformed together.