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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

new3.JPG

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=0to 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.

 

new4.JPG

 

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

new5.JPG

 

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 a2  c2 = b2

The general equation is

 

 

new6.JPG

When a<b

 

new7.JPG

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 a2  c2 = b2

The general equation is

new8.JPG

 

 

 

 

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