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.

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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.

Eine Parabel

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.


The parabola chute at that doing Munich

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

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