**Exploring Parametric
Curves**

By: Diana Brown

A parametric curve in the plane is
a pair of functions

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.**

**EXPLORATIONS:**

Graph

x = cos (t)

y = sin (t)

for 0≤ t ≤ 30

How would you change the equations
to explore other graphs?

For example: for various **a **and
**b **of the graphs:

x = cos (at)

y = sin (bt)

for 0 ≤ t ≤ 30

First lets investigate different
values of a and b, but keeping a = b.

a=2=b a=3=b

a=4=b a=10=b

All circles.

Now lets investigate when a = 1
and b varies

b = ½ b = 2

b = 3 b = 4

Notice that when a = 1, b
determines the number of loops

Now lets investigate when a = ½
and b varies

b = 1 b = 2

This pair creates two loops This pair
creates 4 loops

b = 3 b = 4

This pair creates 6 loops This pair
creates 8 loops

Notice that when a = ½ then the
number of loops is determined by 2b.

Now lets investigate when a = 2
and b varies

b = ½ b = 1

b = 3 b = 4

It now seems as though the number
of loops is determined by ½b, let’s try two more to see what happens:

b = 5 b = 6

It seems to be so.

Now lets hold b to be constant and
vary a. For the following graphs b =1.

a = ½ a = 2

a = 3 a = 4

It seems to do the same as above
except about the y axis and when a is even it is open curves. Let’s try two more to see if this is true.

a = 5 a = 6

It looks as though we will get
something similar about the y axis when b is held constant and a is changing,
just as the curves were changing above when a was held constant and b was
changing.

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