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PARAMETRIC EQUATIONS

 

By:Lauren Lee

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†††† A parametric curve in the plane is a pair of functions

(x = f(t) and y = g(t)), where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t.In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.

 

 

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Letís look at the following parametric equations and vary a and b:

 

 

for 0 < t < 2pi

 

 

 

What happens when we set a and b equal?

 

Letís look ata = 2 andb = 2

 

 

 

 

 

 

 

 

 

 

You will notice that we get a circle with radius one.

 

 

Further investigations revealed to me that this will always be the case††††† when a = b.

 

 

 

 

 

 

Letís see what happens when a and b arenít equal.

Weíll let a = 2 and vary the values of b.

 

 

What happens when b = 4?

 

 

 

 

 

 

 

 

 

 

 

What happens when b = 6?

 

 

 

 

 

 

 

 

 

 

 

And when b = 10 :

 

 

 

 

 

 

 

 

 

Notice in these examples that the number of circular shapes created is equal to the value of b divided by the value of a.

 

 

 

 

 

Now letís see what happens when we hold b constant and vary a.

Letís make b = 2.

 

 

 

Let a = 4

 

 

 

 

 

 

 

 

 

 

 

 

Let a = 6

 

 

 

 

 

 

 

 

 

 

 

 

Let a = 10

 

 

 

 

 

 

 

 

 

 

 

 

These values of b seem to produce the same curves for a, except that they are rotated 90 degrees.

 

 

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Just for fun

 

Now letís see an example for large values of a and b.

 

Here a = 19 and b = 29

 

 

 

 

 

 

 

 

 

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