Constructing a Parabola Using GPS

A Parabola Construction

Recalling what we know about parabolas, they take the form algebraically as

and graphically looks like the following:

A parabola can be further defined as a set of points on the x, y Cartesian plane which are equal distance from a line, called the directrix, and a fixed point, called the focus.

Now, we will start our construction of the parabola using The Geometer's Sketchpad (GPS) by first defining the directrix, which can be any line line we choose. As an example, lets choose y = -2. Similarly, we can choose any point which is, in this case, equal distance from the x-axis, as y = 2.

From here lets identify a point on the directrix. When doing so, the point be made to move using the "Animate Object" under the "Display" option. Once the point is selected, attach a line segment from that point to the focus.

Using the line segment we just created, establish the mid-point of that segment and call it M.

Now, let create two perpendicular lines. One will be at the point we selected on the directrix and the other will be at the mid-point M and perpendicular to the red line.

With the two perpendiculars lines in place, select both lines and find their intersection. Call this intersection the locus.

For our final step, use the "Animate Object" under the "Display" option to move the point we selected on the directrix while "Tracing" the locus.