Editing Parabola (section)

== As a graph of a function ==
[[File:Parabeln-var-s.svg|thumb|Parabolas <math>y = ax^2</math>]]
The previous section shows that any parabola with the origin as vertex and the ''y'' axis as axis of symmetry can be considered as the graph of a function
<math display="block">f(x) = a x^2 \text{ with } a \ne 0.</math>

For <math>a > 0</math> the parabolas are opening to the top, and for <math>a < 0</math> are opening to the bottom (see picture). From the section above one obtains:
* The ''focus '' is <math>\left(0, \frac{1}{4a}\right)</math>,
* the ''focal length'' <math>\frac{1}{4a}</math>, the ''semi-latus rectum'' is <math>p = \frac{1}{2a}</math>,
* the ''vertex'' is <math>(0, 0)</math>,
* the ''directrix'' has the equation <math>y = -\frac{1}{4a}</math>,
* the ''[[tangent]]'' at point <math>(x_0, ax^2_0)</math> has the equation <math>y = 2a x_0 x - a x^2_0</math>.

For <math>a = 1</math> the parabola is the '''unit parabola''' with equation <math>y = x^2</math>.
Its focus is <math>\left(0, \tfrac{1}{4}\right)</math>, the semi-latus rectum <math>p = \tfrac{1}{2}</math>, and the directrix has the equation <math>y = -\tfrac{1}{4}</math>.

The general function of degree 2 is
<math display="block">f(x) = ax^2 + bx + c ~~\text{ with }~~ a, b, c \in \R,\ a \ne 0.</math>
[[Completing the square]] yields
<math display="block">f(x) = a \left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a},</math>
which is the equation of a parabola with
* the axis <math>x = -\frac{b}{2a} </math> (parallel to the ''y'' axis),
* the ''focal length'' <math>\frac{1}{4a}</math>, the ''semi-latus rectum'' <math>p = \frac{1}{2a}</math>,
* the ''vertex'' <math>V = \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right)</math>,
* the ''focus'' <math>F = \left(-\frac{b}{2a}, \frac{4ac - b^2 + 1}{4a}\right)</math>,
* the ''directrix'' <math>y = \frac{4ac - b^2 - 1}{4a}</math>,
* the point of the parabola intersecting the ''y'' axis has coordinates <math>(0, c)</math>,
* the ''tangent'' at a point on the ''y'' axis has the equation <math>y = bx + c</math>.