Graph of Quadratic describes Parabola

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Theorem

The locus of the equation defining a quadratic:

$y = a x^2 + b x + c$

describes a parabola.


Corollary 1

The locus of the equation of the square function:

$y = x^2$

describes a parabola.


Corollary 2

The locus of the equation of the square root function on the non-negative reals:

$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$

describes half of 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:

\(\displaystyle \paren {y + d}^2\) \(=\) \(\displaystyle \paren {x - k}^2 + \paren {y - f}^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle y^2 + 2 y d + d^2\) \(=\) \(\displaystyle x^2 - 2 k x + k^2 + y^2 - 2 f y + f^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 y \paren {f + d}\) \(=\) \(\displaystyle x^2 - 2 k x + f^2 + k^2 - d^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle \frac 1 {2 \paren {f + d} } x^2 - \frac k {\paren {f + d} } x + \frac {f - d} 2\)

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.

$\blacksquare$