The Parabola

The Northern Cross Radiotelescope

 The parabola is defined in analytic geometry as the set of all points P in a plane that are the same distance from a given line and a fixed point not on a line.  The fixed point is called the focus and the fixed line is called the directrix.

Activity

A fun classroom activity involving parabolas is paper folding with wax paper. The idea was obtained from an article by Scott Smith titled Paper Folding and Conic Sections.

Instructions

On the sheet of wax paper draw a fixed line and a fixed point.  Fold the fixed point onto any point on the fixed line and crease the paper.  Repeat this many times.  The creases form tangents to and envelope a parabola.  The following is a GSP simulation of what takes place with the paper folding.

This exercise leads to a nice presentation of the derivation of the standard equation of a parabola whose vertex is at the origin.

The figure below shows point P called the Focus as the point that is folded onto point P’ (located on The Directrix).  At P’ a perpendicular line to the Directrix was constructed.  This perpendicular line hits the crease line at point A.  The crease line is the perpendicular bisector of PP’, thus point A is equidistant from fixed point A and fixed the fixed line.  Point A is a point on the parabola whose focus is P and whose directrix is L. (this can be proved by showing that the crease line through A is tangent to the parabola).

By definition of a parabola the distance from point A to P is equal to the distance from A to P’.  Recalling the distance formula

we can write the following

the length of   = the length of

=

or by squaring both sides and multiplying out

or

the standard form of an up/down parabola whose vertex is at the origin

The following activity was obtained from Virginia Laird, Rockwell High School, Richardson, Texas.

Students should be provided with the below information through direct teaching and then complete the following problems.

• The standard form of a parabola with vertex  is  and.

• A parabola is the set of all points P in a plane that are the same distance from a given line and a fixed point not on the line.

• The fixed point is called the focus and the fixed line is called the directrix.

• The focus and the directrix are each located c units from the vertex, but lie on opposite sides of the parabola.

• The parabola cannot intersect the directrix.

• The width of the parabola at its focus is 4c units and is called the latus rectum.

• A parabola’s direction is dependent upon which term is squared in the standard form.

After the above information has been given to the students they should be grouped in pairs.  They should then explore the effects of (h,k) and c upon the parabola in

each of the forms.

Graph the following parabolas on the computer graphics program and draw conclusions about the direction of the curve, the effect of (h,k) and c.  Find the length of the latus rectum for each curve and verify from the graph.  Sketch the graph of each on your graph paper.  Also multiply each of the equations out and solve for x or y accordingly.

1.                2.                          3.                          4.

Based on the work that you have done above complete the following paragraph to enable you to make some conjectures about parabolas.

 A parabola in the form of (x-h)^2=4c(y-k) turns __________ or __________.  When 4c is a negative number, the parabola   turns __________, but if 4c is positive, the parabola turns __________.  (h,k) represents the __________ of the parabola.  A   parabola in the form of (y-k)^2=4c(x-h) turns __________ or __________.  If 4c is positive the parabola turns __________,   but if 4c is negative the parabola turns __________.  The focus is located__________ units from the vertex and the latus   rectum is __________ times this distance.

Describe in paragraph form the following parabolas.  Check your results by graphing each parabola on the computer.

1.                                                             2.

Put the following equations in standard form.  Use above conjectures to construct each parabola on graph paper and use Graphing Calculator to verify graph.

1.                                     2.