Assignment #10
Nicole Mosteller
EMAT 6680

Explorations of Parametric Equations
My understanding of parametric equations is that the set of equations defines both the x component
and the y component of a relation that occurs on some defined interval of t.

Take for example the parametric equations defined below
.
The x component is the cosine function on the interval of 0 < t < 2P, and 
the y component is the sine function on the interval of 0 < t < 2P.

The graph of this parametric equations shows the relation in Figure 1.

Figure 1:  The unit circle.
Its from this example that we arrive at a most common and useful trig identity:
The equation of this circle is 
.
Using the information from the parametric equation we see that
.
This trig identity will be used in the following investigation.

Question 3:  For various a and b, investigate 

for 0 < t < 2P.

To begin this investigation, let's first look at the case when a = b.


Figure 2:  The case when a = b, and a and b have the following values.
      
Notice that each of the circles that are produced from the parametric equations are
still centered at the origin.  Also each circle has a radius the value of a and b.

Proof:  For these parametric equations when a = b (let this value be r), the resulting graph is 
a circle with radius equal to r.

Looking at this parametric equation as the combination of the x component and y component, we see
.
To simplify each term, square them.
.
Using the distributive property, we have
.
Substituting the trig identity that we established earlier,
.
We have the equation for a circle with radius r.

To continue the investigation, let's first look at the case when a and b are not equal.
The easiest way to see the impact that a and b have on the parametric equations
is to change one value while maintaining the other as a constant.

Figure 3 shows with a variation in the value of a while keeping b constantly 1.

Figure 3: The value of a varies with b = 1.
       

By varying the value of a, we see that the original circle has been stretched along the x-axis.
We can anticipate a similar stretch only along the y-axis by varying the value of b and allowing a to remain constant.
See Figure 4.

Figure 4:  The value of b varies while a remains constant.

Interested to see why the parametric equations in the form of 

(when a and b are not equal) always give the graph of an ellipse?
Click Here! for the proof.

To get a good idea what happens as a and b varies, 
Click Here! for a Graphing Calculator animation.

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