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In this section, you will:
  • Graph parabolas with vertices at the origin.
  • Write equations of parabolas in standard form.
  • Graph parabolas with vertices not at the origin.
  • Solve applied problems involving parabolas.
The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)

Did you know that the Olympic torch is lit several months before the start of the games? The ceremonial method for lighting the flame is the same as in ancient times. The ceremony takes place at the Temple of Hera in Olympia, Greece, and is rooted in Greek mythology, paying tribute to Prometheus, who stole fire from Zeus to give to all humans. One of eleven acting priestesses places the torch at the focus of a parabolic mirror (see [link] ), which focuses light rays from the sun to ignite the flame.

Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

Graphing parabolas with vertices at the origin

In The Ellipse , we saw that an ellipse    is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola    . See [link] .

Parabola

Like the ellipse and hyperbola    , the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points ( x , y ) in a plane that are the same distance from a fixed line, called the directrix    , and a fixed point (the focus ) not on the directrix.

In Quadratic Functions , we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See [link] . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.

The line segment that passes through the focus and is parallel to the directrix is called the latus rectum    . The endpoints of the latus rectum lie on the curve. By definition, the distance d from the focus to any point P on the parabola is equal to the distance from P to the directrix.

Key features of the parabola

To work with parabolas in the coordinate plane , we consider two cases: those with a vertex at the origin and those with a vertex    at a point other than the origin. We begin with the former.

Let ( x , y ) be a point on the parabola with vertex ( 0 , 0 ) , focus ( 0 , p ) , and directrix y = − p as shown in [link] . The distance d from point ( x , y ) to point ( x , p ) on the directrix is the difference of the y -values: d = y + p . The distance from the focus ( 0 , p ) to the point ( x , y ) is also equal to d and can be expressed using the distance formula    .

Questions & Answers

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
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Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
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archana Reply
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Kc Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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