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Exploring Geometric Constructions of Parabolas

 Activity Description Activity Guide

In 350 B.C. Menaechmus, a pupil of Plato and Eudoxus, studied the curves formed when a plane intersects the surface of a cone. In how many different ways can you slice a cone to result in a uniquely shaped cross-section? Compare your results with the diagram below.

Picture can be found at Eric Weisstein's World of Mathematics. (http://mathworld.wolfram.com/ConicSection.html)

We give credit to Apollonius for naming the conic sections (parabola, hyperbola, and ellipse) around 220 B.C. Only a few short years later in 212 B.C., Archimedes studied the properties of the conic sections. In this activity, we are going to explore the characteristics of one specific conic section, the parabola.

We will investigate the locus of points in the plane that are equidistant from a point and a line by using a paper-folding technique and by using The Geometer's Sketchpad. Below you can find the definitions for three geometric terms that we'll be using frequently in this activity.

 directrix - a fixed line that is the same distance as the focus from each point on the parabola.   focus - a fixed point that is the same distance as the directrix from each point on the parabola.   locus - the set of all points that satisfy a given condition or a set of given conditions.

Part 1: Parabola - paper folding activity

Each student will need a 6-inch square piece of wax paper (or patty paper) and a straight edge.

• With a straight edge, construct a line on your piece of wax paper. This line will serve as the directrix.

• Construct a point anywhere on the paper, except on the line, and label it F. This point will serve as a focus. Construct another point anywhere on the line, and label it G.

• Using your straightedge, construct the line segment FG. Fold your paper so that the two points are concurrent, and deliberately crease it so that you can easily see the fold mark when the paper has been flattened out. Conjecture any relationships you might see between the line segment FG and the folded line.

• Fold your wax paper so that point F falls anywhere along the directrix. Again, deliberately crease your wax paper so that you can easily see the fold. Continue this process approximately ten more times so that each time point F falls on a different location of the directrix.

We are interested in the pattern of the creases that are formed when point F is folded along the directrix. Flatten out your paper and look for any geometric patterns. Describe the boundary of the shape of the area containing the focus that is bounded by all of the fold lines.

• Make a conjecture about the relationship of the distance from the focus to the boundary, and the distance from the boundary to the directrix. How could you test your conjecture?

Part 2: Parabola as loci of lines

To see the pattern described in Part 1 distinctly, we would have to fold the paper dozens of times. Instead, we are going to simulate the activity using The Geometer's Sketchpad.

• With your neighbor, plan a geometric construction we could use to simulate the folding process as described in Part 1. Write down the key steps of the construction and share your ideas with the class. Carry out your construction using The Geometer's Sketchpad.

There are several ways to simulate the activity in Part 1, using different features of The Geometer's Sketchpad. Specific steps have been recorded below for three different methods. (Instructor Note: A Sketchpad file illustrating the simulation using the animate feature has been saved as parabola-animate.gsp.)

 All three constructions start in a similar manner. Construct a line (d) that will serve as the directrix.  Construct two points, one as the focus (F) and one on the directrix (G).  Construct a segment from the focus to the point on the directrix (segment FG).  Construct the perpendicular bisector segment FG.

 Simulation using the Trace feature: Select the perpendicular bisector, and choose the Trace Perpendicular Line command under the Display menu.  Drag the point G along the directrix. To erase the traces, choose the Erase Traces command under the Display menu.

 Simulation using the Animate feature: Select the perpendicular bisector, and choose the Trace Perpendicular Line command under the Display menu.  Select point G, and choose Animate Point from the Display menu. To cease animation, you may click on the solid square button ( ■ ) on the Motion Controller window, or choose Stop Animation under the Display menu.

 Simulation using the Locus feature: Select point G (the driver point) and the perpendicular bisector. Select Locus from the Construct menu.

Note: There is an advantage to the construction using the Locus feature over the others. Once you have constructed the sketch, you can easily manipulate the position of the focus and directrix and investigate the connection between them. (Instructor Note: A Sketchpad file illustrating the simulation using the locus of lines has been saved as parabola-locus.gsp.)

• What is the general shape formed by the loci of the perpendicular bisector? Drag the focus point to manipulate the loci. What do you observe about the shape formed?

• Conjecture what happens when the focus is below the directrix. Test your conjecture.

• What is the relationship between the location of the focus and directrix and the general shape formed by the loci of lines?

• What is the relationship of each of the perpendicular bisectors to the parabola?

Why does the construction tracing perpendicular bisectors of segment from the focus to the directrix produce a parabola? To be able to answer this question we will need to slightly modify our construction. In Part 2, we constructed a parabola by loci of lines. Since each of the lines were tangent to the parabola, we could not locate any specific points on the parabola. We would like to be able to construct only those points on the parabola.

• Re-simulate the activity in Part 2 to construct a parabola as a locus of points. (Instructor Note: A Sketchpad file illustrating the simulation using the locus of points has been saved as parabola-locuspts.gsp.)

 To construct a parabola as a locus of points: Construct a dashed line through point G that is perpendicular to the directrix. Construct the point of intersection of the dashed line and the perpendicular bisector of segment FG. Call this point of intersection P. Construct the locus of point P as G moves along the directrix.
• What is the general shape formed by the loci of point P? Drag the focus point to manipulate the loci. What do you observe about the shape formed?

• Confirm that point P satisfies the definition of a parabola, that is, measure the distance from P to G and from P to F. Formally prove that these distances are equal.

• Algebraically, derive the standard form of a quadratic function. (Hints: Use the labeled sketch below to equate the distances from P to G and from P to F. Let the distance (FV) = p.)

 Algebraic derivation of the standard form of a quadratic function: distance (P to G) = distance (P to F)  = (x-x)2 + (k-p-y) 2  = (x-h)2 + (y-k-p) 2 k2 -kp-ky-kp+p2 +py-ky+py+y2 = (x-h)2 + y2-ky- py-ky+k2+kp-py+pk+ p2 k2 +p2 + y2 –2kp-2ky +2py = (x-h)2 + k2 +p2 + y2 +2kp –2ky-2py-2kp +2py = (x-h)2+2kp -2py 4py = (x-h)2+4kp  y = 1/4p(x-h)2+k

• Manipulate the locations of the directrix and the focus to classify the movement of the graph.

• Another common form of a quadratic function is y = ax2 + bx + c. Further explore how the constant a, b, and c affect the graph of the parabola. Compare and contrast these two algebraic forms of quadratic functions.

Extensions:

• In Part 2 of this activity, instead of using the perpendicular bisector of FG, conjecture what would happen when an arbitrary perpendicular line to FG is used.

• Construct a line perpendicular to FG through any point other than the midpoint. Trace this perpendicular line. Is the shape formed a parabola? Does it meet the definition of a parabola? If so, where are its focus and directrix? If not, how would you classify its shape?

• Further manipulate the sketch to find when point F is the focus? When is line d the directrix? What is special about the relationship of the perpendicular bisector of segment FG to the parabola?

• Refer back to Part 3 of this activity. Since segment GP = segment FP, a circle centered at P will pass through both F and G and be tangent to the directrix at G. On the basis of these relationships, make up an alternative definition of a parabola.

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