In the last section, 7.5U Waxed-Paper Models, you saw a parabola traced out by crease lines in a sheet of waxed paper. Since the parabola wasn't actually traced, but enclosed by a series of lines, we say that the parabola is the envelope of the series of traced lines. In the first example below, C is the point that you would draw on the waxed paper, and AB is the edge of the paper. When the waxed paper is folded so that C is on the edge, D is the point on the line that corresponds to C. You can move the point D to get new locations of the fold line. The motion of D can be animated by using the animate button. By tracing the fold line you get the pattern that would emerge on the waxed paper. After tracing has started, the red X at the bottom right can be clicked to erase any traced lines.
In the applet below you can experiment with the locus definition of a parabola: the locus of points, P, such that the distance from a fixed point, called the focus F, is equal to the distance to a fixed line, called the directrix.
Drag D below and observe the path that P follows. Trace the path to get the graph of the locus. Using the animate button will move D on AB for you. After points have been traced, clicking the red X in the bottom right corner will erase the traced points. Finally, selecting to show the locus keeps the path visible as you experiment.
Make sure the trace is turned off. Turn the locus on.
1. As you move the point D, notice the measurements of PF and PD. Why are the measurements as they appear?
2. Move point A to move the directrix. How is the shape affected as the directrix is moved? Move F and try to describe how the relative position of the focus and the directrix determine the shape of the parabola.
3. Try to locate the vertex of the parabola. Write the coordinates down and compare them to the coordinates of the focus and the equation of the directrix. Move the focus and the directrix. How is the position of the vertex related to the focus and directrix? Why must this be so?
This was created with a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. Portions of this work were funded by the National Science Foundation (awards DMI 9561674 & 9623018).