## Write up #10 Parametric Curves

## by Kent Wiginton

#1. Graph

We can easily view this parametric equation using Graphing
Calculator 3.0.

Anyone can observe that this graph is a circle. Now we want
to try to explore different variations of this parametric equation.

We are going to replace x= cos (t) with x= cos(at). What do
you think will happen. We will be varing the x coordinate of the
parametric curve.

Notice when a is an integer (whole number).

Let a = 0

Let a = 1

What about when a = 2?

a = 3?

Click **here** to see
a movie of the curve as a varies from -3 to 3. Just click on the
play button next to the a at the bottom.

What do you notice? Observe that the open end moves back and
forth along the x-axis from x= -1 to 1. Try to imagine that someone
is twisting the y-axis and the curve is wrapping around it. Look
at the movie again and see if you can see what I am describing?

Now let's focus on the other part of the parametric equation
**sin (t).**

We know that when x= cos (t) and y= sin(t) we have a circle.
We have already looked at varying t in the cos (t) by multiples
of a. Now let's consider what would happen when we vary by a in
the sin (t). My hypothesis is that just as we can see that the
sin and cos functions are just shifted by one-half pi; the graph
will be shifted by pi/2.

**Let's take a look.**

Just as before, we have the some of the same types of curves
only this time they are rotated by 90 degrees. Let's look at the
differences. When a= 2 the graphs are very different.

The next link has both graphs on the same axis. It is easier
to see the similarities when x is an integer. Try to see how the
graphs are similar. The red graph is the variation of the cosine part of
the parametric equation plus a shift over 2 units so that we can
compare the two at the same time. The purple is the variation of the sine part of the parametric
equation.

**Both Graphs**

With further observation one will notice that the graphs are
not the same. It is true that they are rotated by 90 degrees,
however, something else is happening. If we could look step by
step we would notice that the red graph is where we altered the
cosine part and left the sine alone. Thus at the beginning we
can see the familar sine graph. The purple graph is where we altered
the sine part and left the cosine alone. Just as before we can
see the cosine graph at the beginning of the cycle (when a is
between -.5 and .5).

**Return**