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
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
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