Assignment Six

Ma 668, Fall 1997
For: Dr. J. Wilson
From: Sandra McAdams

PARABOLA CONSTRUCTION

In the typical high school algebra I and II courses students generally study quadratic functions of the form . They find the axis of symmetry, vertex and maximum or minimum value of the function. They rarely discuss the construction of the parabola as a conic section and generally do not understand its geometric construction. This write up will discuss the ways a student might construct the parabola given a line and a point. A parabola is a 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. One might begin by having students do a paper folding activity. Using any type of paper draw a line j on your paper. This will be the directrix. Chose an arbitrary point F on the paper to represent the focus. Instruct the students to fold the paper in such a way that F falls on line j. After making several folds they should begin to see the form of a parabola taking place. This should include a discussion of the perpendicular bisector being formed. Follow up with a Geometer's Sketch Pad lab in which they construct the parabola with similar reasoning. See a GSP diagram by clicking HERE


Ask student to use GSP to answer the following questions:



1. Using the GSP sketch move point F further away from the line j and animate, then move it closer to the line j. What do you notice about the parabola?





Observations should include the fact that as F moves closer to the directrix the parabola becomes more narrow. The further from j the focus is moved the more relaxed the curve becomes.

 

 

2. Find the line of symmetry for several sketches. Is there a characteristic point included in each sketch? What relationship is there between the line and symmetry and the line j, the directrix? A possible solution is found below.



The line of symmetry ( the pink line) goes through the focus F and is perpendicular to the directrix j.

 

3. Select point A and mark it as center. Rotate line j for different angle measures. What effect does rotating the directrix have on the parabolic shape?



We see that by changing the directrix we change the opening of the parabola. The direction of the opening is controlled by the directrix.

 


The Parabola and Its Equation

 

To examine the relationship between the parabola and its equation we could use the coordinate plane and the distance formula to develop an equation.

Example: Determine an equation of the parabola that is equidistant from y = 0 and the point (0,2).

Solution: Let P be a general point that is equidistant from y = 0 and ( 0,2) . See diagram below.

Distance 1 (d1) = y - 0

Distance 2 ( d2) =

Since d1 = d2, then





After the equation is developed then a natural discussion could take place about the effects of the coefficient of the squared term and the constant term in the equation. The discussion might extend on into other conics and their constructions.

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