What is a parabola?

A parabola is a figure in the plane consisting of all points equidistant from the following:
  • a line, called the directrix of the parabola,
  • and a point not on that line, called the focus of of the parabola.

Parabolas are incredibly useful and important. For example, the shapes themselves are used as parabolic reflectors and parallel ray collectors. So, we want to study all we can about these mathematical objects.

If the directrix is parallel to the x-axis in the coordinate plane, the parabola can be described by a function. We'll call these parabolas standard parabolas.

Functions describing standard parabolas are called quadratic functions. Every quadratic function can be expressed in the following standard form:
f(x)=ax2 + bx + c


The functions apply to a number of situations. The height of a falling object, the time it takes to naively sort objects, and the area of a circle as a function of diameter are just a few examples we'll investigate.














Construction Activity
Directions: Perform the steps carefully; turn in written responses to Questions 1-6.

  • On a blank piece of paper, use a marker to draw a line across one edge. We'll call this the directrix.
  • Near the middle of the page and near directrix, draw a point. We'll call this the focus.
  • Match the focus with a point on the directrix and fold the paper; make a distinct crease. Repeat this at least ten times, then use a marker to trace the creases.

    1. What happens? Compare with a partner.
    2. A parabola is the set of all points equidistant from a given point and line. Is this a parabola? Justify your answer. What are those creases you made?

  • With a partner, choose one paper with which to proceed. Recall Snell's Law, which says that when an object hits a wall, the angle of incidence equals the angle of reflection.
  • Imagine a ray protruding from the focus. When it hits a crease, it bounces off adhering to Snell's Law. Draw ten such rays.

    3. What do you notice? Can you verify a pattern?
    4. How could a shape like this be used?
    5. Why do we call the point we created in the first step a focus?

  • On the other partner's paper, construct a set parallel rays that are not perpendicular to the directrix. Draw at least five rays and their behavior bouncing off the creases.

    6. What do you observe? Describe any patterns that occur.







The following applet demonstrates the activity.
  • Points A and B can be moved. The tools at top can be used to zoom or move the window.
  • By right-clicking, you can restore the axes. By clicking the top-right corner, you can reset the applet.
  • The fold tool works by first clicking point B, then clicking on a point on the line.
  • The focus-reflect tool works by clicking point A, point B, then a point on the parabola.
  • The checkboxes only work when the Move tool, the arrow, is selected.
    Enjoy!


    Oops! You're not seeing a graph! Check your message bar at the top of the screen; make sure ActiveX controls are enabled.

    Else, sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

    Do we need a webstart?














    Now, let's derive the quadratic equation for a parabola in standard form:



Go home
or go home.

Tim Cieplowski, July 29, 2011, Created with GeoGebra