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Parabola by Definition

You’ve learned that the algebraic graph of a quadratic equation is a parabola. The geometric definition of a parabola, however, is the set of all points whose distance from a fixed point, the focus, is equal to its distance from a fixed line, the directrix.

On this web page, you will explore the geometric definition of a parabola and see how the focus and directrix effect the parabola’s shape. You’ll also see how parabolas can model different real-world situations, including water fountains, projectiles, and the cables supporting the Verrazano-Narrows Bridge. This exploration extends the project on page 559 of Discovering Algebra: An Investigative Approach, but it can enrich your study of quadratic models at any point within Chapter 10.

Sketch

This sketch shows a directrix and a focus, and the parabola that they determine. You can drag the directrix and the focus to change their locations. You can also drag point D to see how the distances to the focus and directrix from any point on the parabola (the blue segments) change.

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Investigate

  1. Drag point D and observe how the distances from the focus and the directrix change. Explain how the distances support the geometric definition of a parabola.
  2. Drag the focus and the directrix. Watch how each changes the parabola.
  3. How can you make the parabola open upward instead of downward? What is the relationship between how a parabola opens and where the focus is in relation to the directrix?
  4. How can you make the parabola narrower? Wider? What is the relationship between the width of a parabola and where the focus is in relation to the directrix?
  5. How can you move the vertex of the parabola left or right?
  6. What happens when the focus is on the directrix?

Sketch

Each of the next four sketches shows a real-world photo that might be modeled by a parabola. Drag the focus and the directrix and see if you can make the parabola match the photo.

Investigate

  1. This photo shows a girl throwing a ball. (You might recognize the photo from page 532 of Discovering Algebra.) Try to match the parabola to the trajectory of the ball. Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.
  2. This photo shows several streams of water from a fountain. Try to match the parabola to any one path of water. Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.
    Photo: © Painet.
  3. This photo shows the Verrazano-Narrows Bridge in New York City. Try to match the parabola to the cable that supports the middle of the bridge. Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.
    Photo: © Corbis.
  4. This photo shows the Gateway to the Dargah of Khwaja Mu’inuddin Chrishti in Ajmer, India. Try to match the parabola to the arch. Do you think the arch was built in the shape of a parabola? Why or why not? Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled.
    Photo: © Corbis.

Summarize

  1. What is the shape of the path of a ball thrown into the air? What is the shape of a stream of water in a fountain? Why do you think these have the same shape?
  2. Give an example of a man-made structure that was built in the shape of a parabola. What other examples can you think of?
  3. Why do you think engineers and architects use parabolas? What other geometric shapes are frequently used by engineers and architects? Why do you think these shapes are used?
  4. If you saw a new curved shape, how would you determine whether it could be modeled by a parabola?