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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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  Let’s look at the following parametric equations and vary a and b:

 

 

for  0 < t < 2pi

 

 

 

  What happens when we set a and b equal?

 

  Let’s look at  a = 2  and  b = 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.

 

 

 

 

 

 

  Let’s see what happens when a and b aren’t equal.

  We’ll 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 let’s see what happens when we hold b constant and vary a.

  Let’s 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 let’s see an example for large values of a and b.

 

  Here a = 19 and b = 29

 

 

 

 

 

 

 

 

 

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