symmetry of parabola

P

A theorem equivalent to this one, but different in details, was derived by Archimedes in the 3rd century BCE.

x σ

0 and π The intersection of an upright cone by a plane f J x

1 ∞

In the United States, vertical curves in roads are usually parabolic by design.

intersects the parabola at This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. ,

H This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry. |

≠ That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola. P

) {\displaystyle Q_{1}Q_{2}} =

= 1

1 0 A cross-section perpendicular to the axis of the cone passes through the vertex P of the parabola. 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.

. {\displaystyle m_{1}-m_{2}.

= 2

The line of symmetry is always a vertical line of the form x = n, where n is a real number. onto the x axis intersects the unit circle at 2

Here, This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. J 1

{\displaystyle \sigma } {\displaystyle OC} That is, for a parabola of equation

. 0 {\displaystyle \angle AOB} 0 d

Edison's searchlight, mounted on a cart.

, and its focus is if the points are on the parabola. 2

x

The horizontal chord through the focus (see picture in opening section) is called the latus rectum; one half of it is the semi-latus rectum. The point B is the midpoint of the line segment FC.

{\displaystyle a=1}

and the directrix has the equation i

Proof: straight forward calculation for the unit parabola The triple-angle formula

2 is parallel to the axis of the parabola. t +

Q

2 (

y ( 3

. It can easily be shown that the parallelogram has twice the area of the triangle, so Archimedes' proof also proves the theorem with the parallelogram.

2 Remark: This property is an affine version of the theorem of two perspective triangles of a non-degenerate conic.[10].

1

axis (y − 2) = 3 (x − 5) 2 axis 3x2 + 2x + 5y − 6 = 0 axis x = y2 axis (y − 3) 2 = 8 (x − 5)

+ 4 → y is to insert the point coordinates into the equation. R

x = / f t [20][21] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (see Catenary#Suspension bridge curve). The line EC is parallel to the axis of symmetry and intersects the x axis at D. The point C is located on the directrix (which is not shown, to minimize clutter). =

, the axis of symmetry is a vertical line {\displaystyle (0,1)} 0 the parabolas are opening to the top, and for

⁡ The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola.

x 2

→ ,

, while

= = ,

)

If [latex]a<0[/latex], the parabola opens downward. y is a regular matrix (determinant is not 0), and | P

  x

A bouncing ball captured with a stroboscopic flash at 25 images per second.

f

has slope

Remark 2: The second polar form is a special case of a pencil of conics with focus

→ Since all parabolas are similar, this simple case represents all others.

Q {\displaystyle F=(v_{1},v_{2}+f)} − This is derived from geometrical optics, based on the assumption that light travels in rays.

The point E has no special characteristics. ) , the axis of symmetry is a vertical line

.

1 ) − f 0 Parabolic microphone with optically transparent plastic reflector, used to overhear referee conversations at an American college football game. {\displaystyle x=x_{1}} − 1 For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. (

= , and the cone (the plane of symmetry is – Deriving the Symptom of the Parabola – Mathematical Association of America", "On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas", Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski-planes, Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms, Archimedes Triangle and Squaring of Parabola, https://en.wikipedia.org/w/index.php?title=Parabola&oldid=987241879, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, A parabola is a set of points, such that for any point, the point of the parabola intersecting the, Position the triangle such that the second edge of the right angle is free to, While moving the triangle along the directrix, the pen.

F 0

1

P p

x {\displaystyle m_{0}} p Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. =

x {\displaystyle \pi \perp \sigma } 1 0

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes. )

, x C

But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola).

2 V .

P j (

x {\displaystyle l} If the speed of the body at the vertex where it is moving perpendicularly to SV is v, then the speed of J is equal to 3v/4.

c {\displaystyle \sigma } Negative fractional powers correspond to the implicit equation xpyq = k and are traditionally referred to as higher hyperbolas. 6

a

c V

{\displaystyle y=-{\tfrac {1}{4}}}

By calculation, one checks the following properties of the pole–polar relation of the parabola: Remark: Pole–polar relations also exist for ellipses and hyperbolas. Q

, and the directrix are given. {\displaystyle y=x^{2}} 2 ) y = x P b {\displaystyle \sigma } f Q

f A short calculation shows: line ( Q (

a

> f 3 In Graph 3, what is the equation of the axis of symmetry? y 2 PT is perpendicular to the directrix, and the line MP bisects angle ∠FPT.

. For any given cone and parabola, r and θ are constants, but x and y are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. 2

x Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to simulate zero gravity, Intersection of a tangent and perpendicular from focus, Reflection of light striking the convex side, Two tangent properties related to the latus rectum, Focal length calculated from parameters of a chord, Area enclosed between a parabola and a chord, Corollary concerning midpoints and endpoints of chords, A geometrical construction to find a sector area, Focal length and radius of curvature at the vertex. ,

F (See Rotating furnace). There is a mirror symmetry in the system consisting of plane

According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola. The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. and the tangent at

F , the focus

= into the unit parabola with equation {\displaystyle C} .

, called its control points: This curve is an arc of a parabola (see § As the affine image of the unit parabola).

− 2

2

{\displaystyle x=x_{2}} 16 b {\displaystyle \sigma }

p The parabolic orbit is the degenerate intermediate case between those two types of ideal orbit. y is

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