# Parametric Curves

#### Various graphing technology, such as the TI-81, TI-82, TI-83, TI-85, TI-86, TI-89, TI-92. Ohio State Grapher, xFunction, Theorist, Graphing Calculator 3.2, and Derive, can be readily used with parametric equations. Try Graphing Calculator 3.2 or xFunction for what is probably the friendliest software.

1. Graph

As you can see, the solution appears to be a circle with center at the origin and a radius of 1.

In order to further investigate, I will observe the parametric equations

for various a's and b's.

Let's observe some examples:

1) a=b

It is obvious that when a=b, then the solution will always be a circle about the origin with a radius of 1.

2)a=2

As you can see, when a=2, then the solution is a series of curves that look like loops.

The number of loops depends on b. You can see that the number of loops = 1/2(b).

3) b=2

These graphs look very different, so I will look at the ones where a is odd first. When a is odd, you can see that there are always 2 maxs and mins for y, and there are a maxs and mins for x.

When a is even, there appears to only be one max and min for y and 1/2(a) maxs and mins for x. When a=4, this notion is not true. This makes me wonder if this is because a=2b. So, now I will look at some of these examples.

4)a=2b

So, it is true that when a=2b the graph is always in this shape.

5) Some other fun examples

As you can see from these examples, the number of maxs and mins for x is always a, and the number of maxs and mins for y is always b. Another observation is that the graph is always in the boundaries -1<x<1 and -1<y<1.

Another way to help me further my investigation is to look at the parametric equations

for various a's and b's.

Let's observe some examples.

1)a=b

It is obvious that when a=b, the solution is a circle with center at the origin and radius equal to a and b (r=a=b).

2)Some other fun examples

As you can see, the solution is always an ellipse with the center at the origin and the x max=a, x min=-a, y max=b, and y min= -b.