**Mathematics
Education**

**EMAT 6680,
Professor Wilson**

**Exploration 10,
Parametric Curves by Ursula Kirk**

A
parametric curve in the plane is a pair of functions. Where x=f (t) and y=g (t)
and the two continuous functions defined ordered pairs (x, y). The two
equations are called the parametric equation of a curve. The extend of the
curve will depend on the range of *t.*

My
investigation will be based on exploring how the various values for *a*
and *b* affect the parametric curves for the equation below.

When a=b

**Observations:**

The purple parametric equation yields the purple circle in the picture below. Since a=b=1, the ratio of our circle is also 1. In order to see the whole circle, we must graph our parametric equation from t=0 to t=2pi.

Also, we can convert our original parametric equation by squaring both sides of
the equation when a=b.

By adding both equations we obtain which reduces to our unit circle.

Therefore, our purple circle in the
picture below of radius 1 produced by our first parametric equation is our unit
circle.

Similarly, our red equation where a=b=2,
yields a circle of radius 2, our blue equation where a=b=3, yields a circle of radius
3 and finally our green parametric equation where a=b=4, yields a circle of
radius 4.

When a>b

**Observations:**

In
the next case, the value of **a** is greater than the value of **b**. As
we can see in the image below, these parametric equations yield ellipses. The
ellipses are all horizontal with a major vertex equal to **a** and a minor
vertex equal to **b. **

The center of the ellipse is at (0, 0) and the foci
of the ellipse can be found by solving for **c**, using the equation *a*^{2} – *c*^{2} = *b*^{2}

The general equation is

When a<b

**Observations**

In
the next case, the value of **b** is greater than the value of **a**. As
we can see in the image below, these parametric equations yield ellipses. The
ellipses are all vertical with a major vertex equal to **b** and a minor
vertex equal to **a. **The center of the ellipse is at (0, 0) and the foci
of the ellipse can be found by solving for **c**, using the equation *a*^{2} – *c*^{2} = *b*^{2}

The
general equation is