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Parametric
Curves
(Assignment 10)
by
Robin Kirkham, Cara Haskins , and Matt Tumlin
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Parametric curves in the
plane x = f(t) and y = g(t)
are pairs of functions such
that there are two continuous functions defined by an ordered pair (x,y). These equations are usually
called the parametric equations of a curve. The extent of the curve depends on the range of t and the
work with parametric equations while paying close attention to the range of
t. In many applications, think of
x and y as they vary with respect to time t or the angle of rotation that some line makes from an
initial location.
There are various
technology that can be used to demonstrate these curves such as: TI-81, TI-82,
TI-83, TI-85, TI-86, TI-89, Ohio state Grapher, xFunction, theorist, Graphing
Calculator 3.2, and Derive. This investigation
is performed using the Graphing Calculator 3.2.
1. Graph
y = sin ( t ) for 0 £ t £ 2p
As you observe the solution appears to be a circle
with center at the orogin and a radius of 1.
Further, let us observe the parametric equations
x = cos ( at )
y =
sin ( bt ) for 0 £ t £ 2p
for various aÕs and bÕs.
Let us observe some examples:
1)
a = b
The observation is that although the values of a and b are changed
the circle remains about the origin with a radius of 1.
2)
Let us observe what happens when we let a = 2 and we vary b in
each graph such that b = 3, then b= 4, then b=5, and finally b= 6.
As we observe when a=2 ten the solution is a series of curves that
look like loops.
The number of loops depends on what the value of b is set at. The
number of loops = 1/2(b).
3)
This asks the question: what happens then when b= 2 and we vary a as we have
already done for b?
These graphs look very
different; it appears that some of the oddity is when a is an odd number versa
a is an even number.
When a is an odd
number, there is always 2 local maximums and minimums for y and there is
maximum and minimum for x.
When a is an even number, there
appears to be only one maximum and minimum for y and 1/2 of a maximums and
minimum for x.
When a=4, however, this is not true.
Could there be a relationship change since at that point a=2b.
4)
Let us observe what happens then when a=2b.
It seems true that when a=2b the graph is always in
the above shape.
Conclusion:
This investigation shows a small sampling of what
can be determined with the use of parametric curves. There are many interesting
investigations that we could continue with yet this surely provides enough such
that every new observation opens the doors to many other variations.
Simple using the basic curves and varying the a and b
parameters to observe how the curves react provides us with much more that can
be used and expanded on in the classroom.