## Assignment 6

Construction of a Parabola## Nicolina Scarpelli

First let's begin by recalling what a parabola is and what properties it attains. A parabola is a member of the conic sections, and algebraically takes the form of the quadratic equation y = ax

^{2}+ bx + c. Geometrically, a parabola is the set of all points in the plane equidistant from a given line L, called the directrix, and a given point F not on the line called the focus.In this exploration I will focus on the geometric aspect of the parabola and use Geometer's Sketchpad to construct a parabola using the directrix and the focus.Steps to constructing a parabola in Geometer's Sketchpad (GSP):

(1) Begin first by constructing an arbitrary line and define this line as the directrix. For my example, I chose the directrix to be the line y = -3. Next, construct a point for the focus, assuming the focus is not on the line. Ultimately, you want to find the locus of points that are equidistant from the focus and the directrix. For my example, I chose the point (0,3) as my focus. Just as a reminder, you can choose any point you want for the focus and any point for the point on the directrix. To label points or objects in GSP, use the text tool on the toolbox labeled "A". Once you click on that a little hand should show up, drag the hand to what you want to label, double click on what you want to label, and a little box will show up allowing you to give the object a label. Once you are done, click OK.

(2) Choose an arbitrary point on the directrix. Do not plot the point under the graph menu because you want the point to be able to move. Thus, use the point button in the toolbox and place a point on the directrix. Then, once you have created the point, click on the point then click on the point labeled focus and go up to the construct menu and click segment. Now you should have a segment from your focus to the point on the directrix.

(3) Click on the segment from the focus to the selected point on the directrix, go to the construct menu and click midpoint to construct the midpoint of the segment. Label this point "midpoint".

(4) Next, construct a perpendicular line through the point on the directrix. Do this by selecting the point on the directrix and selecting the directrix line, go up to construct, and click perpendicular line.

(5) Now, construct a perpendicular line through the midpoint which is perpendicular to the segment from the focus to the point on the directrix. To do this, click on the midpoint and click on the segment, go to the construct menu and click perpendicular line.

(6) Construct the intersection of the two perpendicular lines by dragging your arrow pointer near the intersection until the pink text pops up in the bottom right hand corner that says "click to construct intersection". When you click, a point will appear and this point will be the intersection point of the two perpendicular lines. This intersection is the locus.

(7) Now in order to construct a parabola, we can drag the point on the directrix to the left and right and it will show the movement of the locus point. However, if you trace the locus and animate the point on the directrix, it is clear that the locus of points is a parabola. In order to do this, click on the locus point, go to the Display menu and click on trace intersection. Then click on the point on the directrix, and go to the Display menu, and click on Animate Point. When this point is animated, it should trace out the graph of a parabola.

Click here to see the full animation of the construction of a parabola. Once open, click the "Animate Point" button to watch the full movement and creation of a parabola.