Parametric equations are quite different from conventional equations like
y = cos (x). You can think of parametric points as representing positions
of an object, and of t as representing time in seconds. Evaluating the parametric
equations for a value of t gives us the coordinates of the position of the
object after t seconds have passed.
Now, let's start our investigation by graphing the parametric curve that
is formed by the equations : x = cos (t) and y = sin (t).
The curve as you probably guessed is a circle. Let's look at the table below
to define some important points.
t |
x |
y |
0 |
1 |
0 |
p/4 |
Ö 2/2 |
Ö 2/2 |
p/2 |
0 |
1 |
3p/4 |
-Ö 2/2 |
Ö 2/2 |
p |
-1 |
0 |
5p/4 |
-Ö 2/2 |
-Ö 2/2 |
6p/4 |
0 |
-1 |
7p/4 |
Ö 2/2 |
-Ö 2/2 |
2p |
1 |
0 |
t |
|
y |
0 |
1 |
0 |
p/4 |
Ö 2/2 |
Ö 2 |
p/2 |
0 |
2 |
3p/4 |
-Ö 2/2 |
Ö 2 |
p |
-1 |
0 |
5p/4 |
-Ö 2/2 |
-Ö 2 |
6p/4 |
0 |
-2 |
7p/4 |
Ö 2/2 |
-Ö 2 |
2p |
1 |
0 |
The graph of the curve changes to an oblong shape. Look on the table. The
y coordinates are all multiplied by 2, but the x coordinates all remain
the same. Below, is a graph of the curve above and the new equation
|
x |
y |
0 |
2 |
0 |
p/4 |
Ö 2 |
Ö 2 |
p/2 |
0 |
2 |
3p/4 |
-Ö 2 |
Ö 2 |
p |
-2 |
0 |
5p/4 |
-Ö 2 |
-Ö 2 |
6p/4 |
0 |
-2 |
7p/4 |
Ö 2 |
-Ö 2 |
2p |
2 |
0 |
|
x |
y |
0 |
-2 |
0 |
p/4 |
-Ö 2 |
Ö 2/2 |
p/2 |
0 |
1 |
3p/4 |
Ö 2 |
Ö 2/2 |
p |
2 |
0 |
5p/4 |
Ö 2 |
-Ö 2/2 |
6p/4 |
0 |
-1 |
7p/4 |
-Ö 2 |
-Ö 2/2 |
2p |
-2 |
0 |
From the curve, it looks identical to x = 2 cos (t) and y = sin (t), but
if you look at the tables you would see the coordinates do not match. For
example at t = 0, the positive 2 curve's x coordinate is 2, while the negative
2 curve's x coordinate is -2. Since we are doing a full cycle of t values
(from 0 to 2
|
x |
y |
0 |
-2 |
0 |
p/4 |
-Ö 2 |
-Ö 2 |
p/2 |
0 |
-2 |
3p/4 |
Ö 2 |
-Ö 2 |
p |
2 |
0 |
5p/4 |
Ö 2 |
Ö 2 |
6p/4 |
0 |
2 |
7p/4 |
-Ö 2 |
Ö 2 |
2p |
-2 |
0 |
Again, we form a circle, but the manner in which the circle is formed is
different from the positive 2 curve. The positive curve starts at point
(2, 0) and goes counterclockwise to form the circle, but the negative curve
starts at (-2,0) and goes counterclockwise. So the positive curve forms
the top half of the circle and then forms the bottoms half, but the negative
curve does the opposite.
For the next exploration, I would like to graph the parametric curve x =
1/2 cos (t) and y = 1/2 sin (t).
|
x |
y |
0 |
.5 |
0 |
p/4 |
Ö 2/4 |
Ö 2/4 |
p/2 |
0 |
-.5 |
3p/4 |
-Ö 2/4 |
Ö 2/4 |
p |
.5 |
0 |
5p/4 |
-Ö 2/4 |
-Ö 2/4 |
6p/4 |
0 |
.5 |
7p/4 |
Ö 2/4 |
-Ö 2/4 |
2p |
.5 |
0 |