I had this question and saw the other thread, and the proof of why all quadratics were parabolas started with the axiom that $y = x^2$ is a parabola.
I don't really understand why $y = x^2$ is a parabola
I get that if you plot it you get a parabola, but why is that the case?
I didn't know about the multiple definitions of a parabola, I haven't yet learned about that I am in highschool.
I don't really know what to say, I am just gonna go with the definition from the other thread: the parabola is the curve that is equidistant from a point (focus) and a line that is perpendicular to the axis of the parabola.
link to other thread: Why is the graph of a quadratic function a parabola?