Parametric Curves

by

Sharon Sewell

Fall 2001

This is my last write up for EMAT 6680 and I have come to some conclusions. One I enjoyed working on the computers, but found it equally as frustrating. They are definitely a great tool for the classroom. So many students are visual learners and the programs we explored in this class were good examples of how to enable the visual learner better "see" the material being taught. I had never really worked with parametric curves before so the graphing calculator was very helpful.

I changed every kind of variable possible in the equations
x = cos t and y = sin t. Then I graphed each change to see how each
picture changed. This was my version of a beginners exploration of
parametric curves. Let's start with the most basic graph, where x
= cos t and y = sin t. The graph is a circle as long as the variable
t goes from 0 to 2¹. Click here to change the range of t. When the variable
range is reduced parts of the circle disappear but the diameter does not
change.

Now if the value is multiplied to the front of the equation
while the values multiplied to t stays at one, the circle is amplified.
If the value is multiplied to the equations when the t values are different
then bow tie shape is amplified.

There are probably very involve proofs and explanations
of why all of these things happen, but this would be a good starting point
for a lesson plan on parametric curves. It would catch the student's
attention and get them asking questions on why the changes take place.
As the lesson progresses these pictures could be referred back to for visual
explanations.

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