**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__

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Seven (Parabolas)