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 $\tuple {0, f}$ on a Cartesian plane.

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

Let $\tuple {x, y}$ be an arbitrary point on $P$.

Then by the focus-directrix property:


 * $y + d = \sqrt {\paren {x - k}^2 + \tuple {y - f}^2}$

where:
 * $y + d$ is the distance from $\tuple {x, y}$ to the straight line $y = -d$
 * $\sqrt {\paren {x - k}^2 + \paren {y - f}^2}$ is the distance from $\tuple {x, y}$ to the point $\tuple {k, f}$ 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.