2 P are given. The vertex is on the axis of symmetry, so its x -coordinate is − b 2a − b 2 a. 2

The proof is a consequence of the de Casteljau algorithm for a Bezier curve of degree 2. , In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. , and

Parabolas can open up, down, left, right, or in some other arbitrary direction. with the line

+ or, equivalently, such that

, while The vertex of a parabola is the point at which the parabola makes its sharpest turn; it lies halfway between the focus and the directrix.

{\displaystyle y} y 1 )

+ ( v =

The line of symmetry is always a vertical line of the form x = n, where n is a real number. The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). m

d c , y See animated diagram and pedal curve.

It follows that J moves at constant speed along VX as B moves along the parabola.

⁡ p =  When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror.

x

= the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used. x

, {\displaystyle P_{0}:{\vec {p}}_{0}} y Since line

2

{\displaystyle e} 1

− on the set of points of the parabola onto the set of tangents.

a P is an arbitrary point on the parabola. Such a parabola can then be transformed by the uniform scaling +

F :

6 . x y 2 , called its control points: This curve is an arc of a parabola (see § As the affine image of the unit parabola). P

{\displaystyle Q_{2}}

2 c are opening to the bottom (see picture). {\displaystyle Q_{1}Q_{2}}

→ t {\displaystyle SAB={\frac {2SV\cdot (VJ-VH)}{3}}={\frac {2SV\cdot HJ}{3}}} Q {\displaystyle a,b,c} y be the plane that contains the vertical axis of the cone and line and

P 2 2

t ) ) x 1 m

2 x

( x A quadratic Bézier curve is a curve {\displaystyle (x,y)\to (x-v_{1},y-v_{2})}

4 can be transformed by the translation J B

: ) 1 can be transformed by a rigid motion to a parabola with an equation 1 , Each parabola has a line of symmetry. {\displaystyle P=(x,y)} , The above construction was devised by Isaac Newton and can be found in Book 1 of Philosophiæ Naturalis Principia Mathematica as Proposition 30. ⋅ 2

Then the tangent at point , | , , A It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.  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). Varsity Tutors does not have affiliation with universities mentioned on its website. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: such that y =

{\displaystyle A}

yields. d ) It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. The supporting cables of suspension bridges follow a curve that is intermediate between a parabola and a catenary. → The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". x is. {\displaystyle \sigma } =

A parabola can be considered as the affine part of a non-degenerated projective conic with a point x The point E is an arbitrary point on the parabola, with coordinates (x, x2). y , ) The parabolic trajectory of projectiles was discovered experimentally in the early 17th century by Galileo, who performed experiments with balls rolling on inclined planes. − P , x

= 3

2 V

4 {\displaystyle c} Real World Math Horror Stories from Real encounters. {\displaystyle P_{1}} ∞

σ {\displaystyle x^{2}+(y-f)^{2}=(y+f)^{2}} → 2 ,

)

, ) t -coordinate of the vertex is the equation of the axis of symmetry of the parabola. . A is (x, y). Q

{\displaystyle y=ax^{2}} m and {\displaystyle p}

Remark 2: The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not related to Pascal's theorem. If the chord has length b and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is h, the parallelogram is a rectangle, with sides of b and h. The area A of the parabolic segment enclosed by the parabola and the chord is therefore.

2

{\displaystyle p} The slope of the line BE is the quotient of the lengths of ED and BD, which is x2/x/2 = 2x.

This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. 1   ) 2 Focus and directrix of the parabola are a pole–polar pair.

. , one obtains the equation. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus and perpendicularly to the directrix. [f], If a point X is located on a parabola with focal length f, and if p is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola that terminate at X can be calculated from f and p as follows, assuming they are all expressed in the same units.[g]. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) R

{\displaystyle x=x_{2}}

2 a

=

0

Therefore, the area of the parabolic sector Parabolic shape formed by a liquid surface under rotation. ,

The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. {\displaystyle F=(0,0)}

First, locate the point on the parabola where its slope equals that of the chord.

=

The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a dubious legend, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships.

is the directrix of the parabola.

a

0

/

y f

α

R From this,

is in the coordinate system's origin. According to the intercept theorem, since C is on the directrix, the y coordinates of F and C are equal in absolute value and opposite in sign. x 0 Solving the equation system given by the circle around

are given. Every parabola has an axis of symmetry which is the line that divides the graph into two perfect halves. It was also known and used by Archimedes, although he lived nearly 2000 years before calculus was invented. =

Its x coordinate is half that of E, D, and C, that is, x/2. This is the principle behind the liquid-mirror telescope. 1 x {\displaystyle x=(x_{1}+x_{2})/2.}. , and point J moves at a constant speed of a {\displaystyle OA} is the axis of symmetry of the parabola.

.

Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. {\displaystyle y=ax^{2}+bx+c,}

Paraboloids arise in several physical situations as well. ). {\displaystyle (t,t^{2}),\ t\in \mathbb {R} } = {\displaystyle m_{0}}

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