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.
We will start the exploration by graphing
With 0 ≤ t ≤ 2π
This corresponds with the table of values for the equations:
t |
x |
y |
0 |
1 |
0 |
π/2 |
0 |
1 |
π |
-1 |
0 |
3π/2 |
0 |
-1 |
Let’s explore different graphs by plugging in numbers into the equations, starting in the form:
After using the same number for a in both equations, the resulting graph is the same:
So, instead, I will try when a < b and a > b, since we know we will get the unit circle when a = b.
Here is an example when a < b:
Also, there was a difference in graphs when b was odd and even; when a and b are even numbers, they did “overlap” each other:
The graph on the left was typical of a < b values and the right one for a > b
The graphs seem to be rotated 90 degrees.
And, there also seems to be a difference with odd and even values; and if a and b are divisible by each other:
Now let’s see what happens when we change the a and b values using this form of the equation:
Using a = b = 3, we get this:
Which corresponds to our table of values
t |
x |
y |
0 |
3 |
0 |
π/2 |
0 |
3 |
π |
-3 |
0 |
3π/2 |
0 |
-3 |
Next I tried a < b:
As I increased the values for b, the ellipse was stretched vertically.
And when a > b:
As the values for a increased, the ellipse was stretched horizontally.
Using some of the basic ideas for what the graph “does” for
and 
We can put these characteristics together and see how they as:
In this case a < c, which means a stretch in the vertical direction and b < d, which means the graph “flows” vertically.