Parabola

Pronunciation: /pəˈr æ bə lə/ Explain

Right conic intersected with a plane parallel to the generator forming a parabola.
Figure 1: Parabola from a conic section.

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Manipulative 1: Parabola Manipulative created with GeoGebra.

A parabola is the shape made when graphing a quadratic equation.[1] A parabola can also be described as a conic section formed by intersecting a right circular conic surface with a plane parallel to the generator of the cone.

A parabola can also be defined as all points in a plane equidistant from a line, called the directrix, and a point, called the focus, not on the line. Every parabola also has a vertex and a axis of symmetry.

Discovery

In manipulative 1, click on the purple focus, the red point on the directrix, and the blue point on the parabola and drag them change the figure.

  1. How is the parabola different if the focus and directrix are close together as opposed to far apart?
  2. How does the parabola change if the focus is moved to the left or to the right?
  3. Click on the focus (purple point) and drag it until the focus is on the directrix. What happens to the parabola? When this happens, mathematicians say that the line is a degenerate parabola.
  4. Click on the blue point on the parabola and drag it. Notice that the length of the two black lines remains equal. Why do these values remain equal?

Parts of a Parabola

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Manipulative 2: Parts of a parabola Manipulative created with GeoGebra.

The vertex of a parabola is at the inflection point of the parabola. The inflection point is the point where the parabolic curve changes direction. In manipulative 2, the vertex is blue. The vertex can not be dragged in this manipulative, because the vertex is dependent upon the focus and directrix. Click on the check boxes in manipulative 2 to see the parts of a parabola.

The distance from the focus to any point on the parabola is the same as the distance from that point to the directrix. Click on the blue point on the parabola and drag it to demonstrate this property.

Conic Section Equation for Parabolas

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Manipulative 3: Parabolic Equation in Conic Form. Manipulative created with GeoGebra.

The conic section form of a parabolic equation is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where B2 = 4AC and either A ≠ 0 or C ≠ 0. This form allows one to draw parabolas where the directrix is not parallel to the x-axis or y-axis. The sliders in manipulative 6 allow you to change the figure. Note that for some values of A through F, the parabola will be out of the display area of the graph. Also, both A and C must be either negative or positive or the equation is undefined.

Parabolas in the Physical World

A bouncing basketball captured with a stroboscopic flash at 25 images per second. The trajectory of the basketball approximates a parabola
Figure 2: A bouncing ball captured with a stroboscopic flash at 25 images per second.

The best known physical example of a parabola is a ball in free flight. Once the baseball leaves contact with the bat, the trajectory of a baseball hit by a batter closely follows a parabolic path. Another example, shown in figure 2, is a basketball bouncing on a hard floor. Each bounce approximates a parabola. The physical factors that keep it from being a perfect parabola are the deformation of the ball during the bounce and air resistance.

A rotating liquid approximates a parabola.
Figure 3: A rotating liquid approximates a parabola.

A lesser known example of a parabola is a rotating liquid. Figure 3 shows a rotating liquid with the interior of the parabola colored orange.

Parabolic Reflector

A flashlight shining a beam of light on a wooden table.
Figure 4: A flashlight throwing a focused beam.

Another common use of a parabola is a parabolic reflector, used in automobile headlights, flashlights and spotlights, also called searchlights. Figure 3 shows the focused beam a flashlight produces using a parabolic reflector.

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Manipulative 4: Parabolic reflector.

When light reflects from a surface, the angle of reflection is twice the angle between the light ray and the normal to the surface. Manipulative 4 shows a blue line that is normal to the surface of the parabola, and how the light beam, shown in green, would reflect off of the surface. Click on the red point on the directrix and the purple focus to change the image.

Discovery

  1. If you wanted to make two searchlights, one with a narrow beam and one with a broad beam, would the focus be closer to the directrix for a narrow beam or a broad beam. Why?

Cite this article as:

McAdams, David E. Parabola. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/parabola.html.

Image Credits

Revision History

8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
1/7/2010: Added "References". (McAdams, David E.)
2/10/2009: Simplified primary definition. (McAdams, David E.)
10/13/2008: Initial version. (McAdams, David E.)

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