Graph of Quadratic describes Parabola

Theorem
The locus of the equation defining a quadratic:
 * $y = a x^2 + b x + c$

describes a parabola.

Proof
Consider the focus-directrix property of a parabola $P$.

Let the focus of $P$ be the point $\left({0, f}\right)$ on a Cartesian plane.

Let the directrix of $P$ be the straight line $y = -d$.

Let $\left({x, y}\right)$ be an arbitrary point on $P$.

Then by the focus-directrix property:


 * $y + d = \sqrt {\left({x - k}\right)^2 + \left({y - f}\right)^2}$

where:
 * $y + d$ is the distance from $\left({x, y}\right)$ to the straight line $y = -d$
 * $\sqrt {\left({x - k}\right)^2 + \left({y - f}\right)^2}$ is the distance from $\left({x, y}\right)$ to the point $\left({k, f}\right)$ by the Distance Formula.

Hence:

This is in the form $y = a x^2 + b^2 + c$.

By setting $k$, $f$ and $d$ appropriately in terms of $a$, $b$ and $c$, the specific focus and directrix can be appropriately positioned.