 The Conics

By: Diana Brown

Day Six:

Parabola Introduction

How to Construct a Parabola Using Wax Paper:

Start out with a piece of uncrumbled wax paper about the size of half a sheet of standard notebook paper (8.5 x 11).  Draw a line in the top of the wax paper without writing off the paper.  Then draw a point anywhere an inch or two below the line.  The next step is to fold up the line so it touches the point inside.  When this portion of the line is aligned with the point, crease the paper and fold it accordingly.  Choose another part of the line and align this with the point, creasing the wax paper.  Repeat this step several times until the majority of the line has touched the point inside or until a parabola is visible.

How to Construct a Parabola Using Geometer’s Sketchpad:

Draw a line and construct a point (Point A) on the line.  Put another point (Point B) anywhere a little above the line Draw a segment from point A to point B and construct the perpendicular bisector of segment AB.  After constructing the perpendicular line select Display and Trace Perpendicular Line. Highlight Point A, go to Display and Animate Point.  Watch the parabola appear! The standard form equation of a general quadratic (polynomial functions of degree 2) function is
f(x) = ax2 + bx + c where a ≠ 0.

If b = 0, the quadratic function has the form f(x) = ax² + c.
Since f(-x) = a(-x) ² + c = ax² +c = f(x),

Such quadratic functions are even functions, which mean that the y-axis is a line of symmetry of the graph of f.

The graph of a quadratic function is a parabola, a line-symmetric curve whose shape is like the graph of y = x² shown  in figure. The point of intersection of the parabola and its line of symmetry is the vertex of the parabola and is the lowest or highest point of the graph. The graph of a parabola either opens upward like y=x² or opens downward like the graph of y = -x² .

In the figure to the left, the vertex of the graph of y=x²  is (0,0) and the line of symmetry is x = 0.

Labeled Diagram of a Parabola 