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:
\(\ds \paren {y + d}^2\) | \(=\) | \(\ds \paren {x - k}^2 + \paren {y - f}^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^2 + 2 y d + d^2\) | \(=\) | \(\ds x^2 - 2 k x + k^2 + y^2 - 2 f y + f^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 y \paren {f + d}\) | \(=\) | \(\ds x^2 - 2 k x + f^2 + k^2 - d^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \frac 1 {2 \paren {f + d} } x^2 - \frac k {\paren {f + d} } x + \frac {f - d} 2\) |
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.
$\blacksquare$