**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?**