**Assignment 6: Problem 9**

**Jason P. Pickhardt**

__Problem:__

A parabola is the set of points equidistant from a line, called the directrix, and a fixed point, called the focus. Assume the focus is not on the line. Construct a parabola given a fixed point for the focus and a line (segment) for the directrix.

a. Use an Action Button to generate the parabola from an animation and trace of a constructed point.

b. Repeat *a* with a
trace of the tangent line at the constructed point.

c. Use the locus command to generate the parabola from a constructed point or the tangent line at that point.

__Exploration:__

This exploration is best suited for the use of Geometer’s Sketchpad. Thus all of the work was done in GSP here.

__Conclusion:__

It is interesting to see in this exploration the connectedness of geometry and algebra. In most classrooms students are taught to graph parabolas in the form, and never see any other ways to develop the set of points described by this algebraic expression. It was completely fair for someone with some basic knowledge about properties of triangles and GSP to make the constructions called for in this exploration and find the set of points equidistant from the directrix and focus.