Outline
I.
What a parabola is
A.
The
history of a parabola
B.
Definition of a parabola
C. Parts and types of parabolas
II. The Parabolic Formula
A. The Formula
B. How you get the formula
C. Different Formulas
III. Parabolas in Real Life
A. Basketball
IV. How our exhibits utilize parabolas
A. Fluid Parabola (Twirl-a-Slope)
B. The Monkey and the Hunter
V. Conclusion
What a Parabola is
The man, Menaechmus,
received credit for first discovering conics, or, more commonly known as, conic
sections. They are named this because
they are the intersections of circular cones with planes. If the plane passes through the vertex of the
right circular cone exactly parallel to the side of the cone, 
it
is known as degenerate conics.
Furthermore, there are four different types of non-degenerate conics,
known as the ellipse, circles, hyperbola, and as displayed in this exhibit, the
parabola.
The term parabola comes from the Greek word para, meaning “alongside” and the word bola, meaning “to cast or throw.” Because of this, many people interpret the meaning of the word parabola as “thrown parallel.” A numerous amount of people believe the figure was given its name by Apollonius of Perga. However, some question the accuracy of this because Archimedes was already using the name “Parabola.”
A parabola can also be defined as a
locus of points (a collection of points that share the same property). This is because the distances from the points
that make up a parabola to the focus and to the directrix are equal. An example of this is shown in the diagram to
left. The height of the focus from zero
equals “f” and the height of the directrix from zero equals “-f”, which are the
same distance apart from one another.
The lowest point of the parabola is called the vertex. The axis of the parabola is the line through
“f” and perpendicular to the directrix.
Assuming that the parabola is concave upward, the parabola has a single low point, known as the minimum. A parabola will always demonstrate vertical symmetry, no matter what the width or location of the parabola on a coordinate grid. The slope of the parabola continuously gets steeper as you move away from the minimum value; yet, the curve will never become vertical. These characteristics remain true with the three other types of parabolas. There are downward opening parabolas, which have the same equation as upward opening parabolas. Similarly, there are right and left opening parabolas. These are inverse relations of upward and downward parabolas.
The Parabolic Formula
There are several steps which must be taken in order to generate an equation for upward and downward opening parabolas with the vertex at the origin. First, the distance formula must be used. (All of the variables are taken from the diagram shown up above.)
(x-x)
2 (y-- p) 2
(x-0) 2 + (y-p) 2 = (x-x) 2 + (y+p)
Square both sides:
(x-0)
2 + (y-p) 2 = (y+p) 2
The (x-x) 2 cancels out:
Factor both sides of the equation:
Move y2 -2py + p2 from the left side of the equation to the right:
The standard form of a parabolic equation is found, as shown above. If p>0 the parabola opens upward. If p<0 the parabola opens downward. In the equation, the value of P is the focal length of the parabola, which is the directed distance from the vertex to the focus of the parabola. The absolute value of 4P is the focal width, which is the length of the chord through the focus and perpendicular to the axis.
For a more familiar form, the equation can be written as:
Or y=1
x2
4p
More common forms of parabolic equations are:
y=ax2 + bx + c
and
y= a(b(x-h)) 2 + k
y2=4px
If p>0 the parabola opens to the right. If p<0 it opens to the left.
Parabolas in Real Life
Studying parabolas is important
because they appear quite often in the world today. In fact, by just watching a basketball game,
a person can witness parabolas numerous times.
Throughout the duration of a game, parabolas are extremely apparent when
a person shoots. The path of the ball
through the air as the ball heads towards the hoop is in the shape of a
parabola. In fact, this is one of the
most common types of parabolas in our world today. As the ball is suspended in the air,
projectile motion is occurring. Projectile
motion is commonly defined as any body thrown in
space, moving under the force of a gravitational field and without the
influence of air resistance. This motion
traces out a parabola. Ultimately, once the ball is shot the vertical velocity
takes the ball to it’s highest point and then gravity brings the ball down
while it is still moving horizontally through the air, thus, creating a
parabola.
How the Exhibits Utilize Parabolas
The exhibit, Fluid Parabola
(Twirl-a-slope), demonstrates the natural occurrence of parabolas. The exhibit consists of a clear rectangular
box mounted on a lazy susan. A red liquid is then poured into the clear
box. Spinning the lazy susan then
creates the desired affect. The liquid
rises at the two ends of the rectangle, thus creating a parabolic curve. As the container spins, each element of the
water is subject to the force of gravity acting downward. The water is also subject to centrifugal
force, which pushes the liquid outward.
The resulting effect is a parabola.
This experiment demonstrates the relationship between the physical
forces of nature and the mathematical concept of a parabola.
The other exhibit, The Monkey and the Hunter, also displays the concept of parabolas. This exhibit simulates the use of parabolas in a hunter’s journey. The hunter’s gun is replaced by a spring-and-plunger type ball launcher. When the plunger is pulled back and locked into place, a battery powers a small electromagnet that holds the monkey, or as used in this exhibit, a ball, to the tree, which is a standard laboratory stand. When the plunger is released, a ball is shot from the gun and the electromagnet is simultaneously turned off, causing the monkey to fall. As the ball moves throughout the air, a parabolic shape is formed because of projectile motion. This happens because the bullet, which was shot off at angle, has an initial vertical velocity that takes the bullet to its highest point. Once the highest point is reached gravity pulls the bullet down in a parabolic motion. Because of gravity, the acceleration of the monkey and the bullet are the same. Since gravity is accelerating both the dart and the monkey downward at the same rate, the dart still hits the monkey.
Overall,
Menaechmus’ discovery of parabolas has had an important impact on the
world. It has been discovered that a
parabola is the set of points in a plane that are equidistant from a fixed line
and a fixed point in the same plane or in a parallel plane. Parabolas are found
throughout the world, from manmade technology to everyday occurrences, such as
satellite dishes and a game of basketball. Ultimately, parabolas have improved
the lives of people and have advanced technology.
Works Cited
“4 the Parabola.” IntMath.com.
http://www.intmath.com/PlnAnalGeom/4_Para.php.
Calvert J.B. “Parabola.”
Mathematical Physics and Mathematics.
http://www.du.edu/~jcalvert/math/parabola.htm.
“Parabola.” History.mcs. Jan. 1997. The MacTutor
History of Mathematics.
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Parabola.html.
“Water Balloons: The physics of projectile motion.” Thinkquest.com.
1996. Oracle Education Foundation.
http://library.thinkquest.org/2779/Even_more.html.
Monkey and the Hunter Main Page
Monkey and the Hunter Study Guide
Monkey and the Hunter/Water Spinner Construction Directions