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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

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The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
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Kingsley Reply
The numbers cannot be combined with the x
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humberto
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Yvonne
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what is the value of x in 4x-2+3
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Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
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Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
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LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
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( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
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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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