# Parabola (mathematics)

in mathematics is **a parabola** (v. griech.: παραβολή *parabole* = *the besides-going thing; the comparison*, v. altgriech.: *paraballein* = put side by side) a conic section, which develops, if one cuts the cone with one level, which is parallel to a production of the cone.(If the level one tangential level of the cone is, keeps one a parabola degenerated, those a straight line is simple.)

in addition the function graphs of square functions represent parabolas.

## To table of contents |

## there

are representational forms apart from the definition as conic section to specify still further possibilities a parabola:

A parabola is the quantity of all points *X*, whose distance to a firm point ( the focus *F*) and a straight line (the Leitgeraden *l*) is alike.

- <math> \ operator name {par} = \ left \ {X |\ overline {XF} = \ overline {Xl} \ right \}< /math>

That point, which lies exactly in the center between focus and Leitgerade, is called **vertex** *A* of the parabola. The connecting straight line of focus and vertex *is called* axle of the parabola. It is also the only symmetry axis.

The coordinate system becomes inThe following so fixed that <math> A= (0,0)< /math> and <math> F= (0, f)< /math>. For each point <math> P= (x, y)< /math> on the parabola then math <\> overline {PF applies} = \ overline {PQ}< for /math> and thus

- < math> \ {(y-f) ^2+x^2 sqrt} =y+f< /math>.

From this directly the functional connection between math <x> /math< follows> and <math> y< /math> for all points <math> P< /math>:

- <math> y=x^2 \ frac {1} {4f}< /math>

Each square function of the form <math> y=ax^2< /math> is thus oneParabola with the focus <math> f= \ frac {1} {4a}< /math>.

## characteristics

the parabola only on a parameter is dependent there (the distance from Leitgerade and focus <math> 2f< /math> and/or. *the parameter* <math> A< /math> in the equation), all parabolas are to each other similar. The differences in the curvature develop only throughthe enlargement ratio. In particular is the numeric eccentricity ε = 1.

Parabolas can be regarded as border line of an ellipse or a hyperbola, if a focus is fixed, and which is removed others arbitrarily far in or other direction.

Becomes parallel a jet, thatto the axle breaks in, at the parabola reflected, then the resulting jet goes in reverse through the focus, and. This characteristic has also a Rotationsparaboloid, thus the surface, which develops, if one turns a parabola around its axle; it is used frequently in the technology (see Parabolic reflector).

- Proof: The upward gradient of the tangent to the parabola in the point <math> P< /math> results from the derivative of <math> ax^2< /math> and is <math> 2ax< /math>. The zero of this tangent is to 2 <}> /math with math \ frac {x}< {> and thus the point math <G=> (\ frac {x} {2}, 0) forms< for /math>. This lies thus exactly in the center between <math> F< /math>and <math> Q= (x, - f)< /math>. Thus the gleichschenkliche triangle becomes <math> \ delta FPQ< /math> into 2 congruent triangles divides. Reflection at the parabola corresponds to reflection at the tangent.
- The angle of incidence <math> \ fishes GPQ< /math> FPG /math is equal to <>the loss angle math \< fishes>. With it all jets meet on <math> F< /math>.

Each particle, itselfin a homogeneous gravitational field without effect of other forces moved (for example a baseball, if one ignores air resistance), follows a parabelförmigen course (*trajectory parabola*). In radialsymmetrical gravitational fields, how it ideal-proves around a heavenly body prevails, the parabola is one of the solutions of a Keplerbahn.

## sometimes parabolas

function graphs are called all graphs of polynomial functions as parabolas. For example the graph of a polynomial of degrees of 4 is a parabola 4. Order. With the definition of the parabola as conic section only parabolas of second order tune, thus *f* (*x*) = *ax* ²* +* bx* +* C*.*

## special parabolas

in the building of the faculty for mathematics and computer science at the technical University of Munich was installed into the Magistrale a parabola chute. This chute consists of twoParts and the form of a parabola shows.

## see also

## Web on the left of

- http://mathworld.wolfram.com/Parabola.html Mathworld - parabola (English)
- http://home.telebel.de/~kuweber/Parabel/Parabel.html animated parabola (Java applet produces with Geogebra)