1. non-x^2 subset

The graphs of quadratic function is subset of the set of parabolas.
What the description (or equation) of the other subsets of the set of parabolas (that are not quadratic function)? [The complementary sets...)?

2. Re: non-x^2 subset

A general parabola can be written as $ap^2x^2+2apqxy\ +aq^2y^2+\left(bp-q\right)x+\left(bq+p\right)y+c=0$

$p$ and $q$ determine the orientation (and scaling) of the parabola. $a$, $b$ and $c$ serve the same function (with respect to the orientation) as in $y = ax^2+bx+c$.

See this.

I think that restricting $p$ and $q$ to values such that $p^2+q^2=1$ removes the scaling effects of $p$ and $q$.