**For this assignment we examine parametric curves.
A parametric curve in the plane represents a pair of continuous functions
that define ordered pairs (x,y). The equations are called the
parametric equations of the curve. The range of "t"
needs to be large enough so that the whole graph will be completed.**

**First let's see what the graph looks like
when a and b are equal to one.**

**We have a nice ellipse when a and b are
equal to one. Now let's see what happens when both a and b equal
two.**

**The general shape has remained the same,
except it is wider and longer. So let's vary a and b, not having
them equal. Set a = 2 and b = 3.**

**Let's do it again. This time b = 2 and a
= 3.**

**So far it seems that the only thing that
a and b change is the graph's width and/or length. I wonder what
will happen when a and b are negative numbers. Let a = -2 and
b = -3.**

**The graph looks the same. Since the positive
and negative values of a and b are used in the graphing, starting
with a negative value has no ill-effect on the graph.**

**Let's see what effect there is when a is
larger than b. Let a = 5 and b = 2.**

**It appears that if a is greater than b the
ellipse elongates along the x-axis. If b is greater than a, the
graph will elongate along the y-axis.**

**So for our original equation we see that
the variations in a and b, both positive and negative, cause the
graph to vary in its width and/or length.**

**Let's look at another equation that looks
very similar to our original one, but its results are quite different.
The equation is . First let a =
2 and b = 3 as we did in the first equation. Let's see what we
get.**

**This graph looks like our very first graph
when a and b both equaled one. So far there is no indication of
the two equations being different. Let's examine further before
we generalize that the equations are equivalent. Let's see if
there is a difference when a = 2 and b = 3 as we did in the other
equation.**

**WOW! Now it's different! Let's set a = 3
and b = 2 and see what happens.**

**It's definitely different! I like it. Let's
explore some more.**

**Why don't we let a = 4 and b = 5.**

**Let a = 6 and b = 2.**

**I like it! Now a is set at 4 and b is set
at 7.**

**Now let's switch the values around with
a = 7 and b = 4.**

**Another neat graph. I want to try one more
combination. I think that I see something happening. Let a = 2
and b = 7.**

**I noticed that whenever b was an odd number,
the graph was open at one end. Whenever b was an even number,
the graph was enclosed. Look at the graphs carefully. Do you see
any other similarities?**