Assignment #10 – Parametric Curves

Jackie Gammaro

A parametric curve in a
plane is a pair of functions *x* = *f(t)*, *y *= *g(t)* where the two
continuous functions define ordered pairs (*x*,*y*). (Wilson 2010):-).

One basic parametric curve
defines* x *= cos(*t*) and *y* = sin(*t*) for value of *t* from 0 to 2p. One
should recognize the unit circle.

To explore the graph in Graphing Calculator 3.5,
click on the link below.

How
can we change the equation to explore other graphs?

First, lets investigate *x* = cos (*at*) and *y* = sin (*bt*) for 0 £ *t *£ 2p.
If *a* = *b* = 1, then the graph
is the same as above . What if *a* = *b*
= 2? Somewhat surprising to me at
first, we get the same parametric curve. If youÕre wondering why, recall sin^{2}*x* + cos^{2}*x* = 1, and let *x* = *at*, *a*
= *b*.

I think it will be fun to
explore what happens when *x* = cos (*at*) and *y* = sin (*bt*). Lets vary
the values of *a* and *b*, such that *a* *b*.

Let a = 1 and let b vary.

To view in Graphing Calculator 3.5, click on the
link.

Let a = 1, b = 3.

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link.

Conjecture:

It seems as if, when we fix *a* = 1, and *b* = *x*, where *x* is an integer, then we have *x* loops. So the number of ÒloopsÓ is equal to *b*. Let b = 5.

To view in Graphing Calculator 3.5, click on this
link.

It seems as if my conjecture is holding
true, one more time.

Let
a = 20.

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link.

II. Let *b*
= 1 and *a *=* x*, where *x* is an
integer.

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link.

Let a = 3, b = 1.

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link.

This looks like a 90¡ rotation of the graph when a = 1 and b =3. It seems as if perhaps the number of
loops will be dependent upon a, but this time the loops are vertically atop
each other, rather than horizontally next to each other.

Lets test the conjecture for
a = 4.

To view in Graphing Calculator 3.5, click on this
link.

Ok, conjecture does not hold
true. Now it seems as if when *a* is odd, it is a 90 degree rotation of
the graph when the values of *a* and *b* are switched. When *a* is even, the figure in the graph does not appear to be a closed
figure with loops. It seems as if
the number of *y*-intercepts is equal
to the value of *a.*

Lets test this conjecture
for *a* =10, *b* = 1.

To view in Graphing Calculator 3.5, click on this
link.

Let *a* = 11.

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link.

It
seems as if my conjecture is true.

Now, itÕs your turn. Change values of *a* and *b* and see what the
curve looks like.

So, Ill get you started,
just one more! Lets see what happens when *a*
= 4 and *b* = 5.

Let
*a* = 4 and *b* = 5.

To view in Graphing Calculator 3.5, click on this
link.

Pretty neat is look like a
pretzel. Now fix the value of *a* to be 4, and let *b* vary, or fix *b* = 5, and
let *a* vary and make some conjectures,
better yet, try to prove your conjecture!