Let’s look at several graphs where x and y are
expressed in terms of parameter t.
We will be evaluating the changes that occur in the graphs when we vary the coefficients of a and b in the following parametric equations:
x = a cos (t)
y = b sin (t)
Let 0< t < for all the graphs investigated.
Let b = 1 and vary a.
When a =1 and b=1 we have our unit circle. ()
When a and b are different we have an ellipse. ()
t 
Cos (t) 
Sin (t) 
0 
1 
0 










0 
1 

0 
1 
Since the cosine of zero is one, our values for x will be a
factor of a.
When a = 3 the graph looks the same as when a = 3. The orientation or direction of the graphs is different.
This time we will let a=1 and vary b for all t such that 0 < t < .
The purple graph is still our unit circle. However, this time our ellipse is elongated vertically instead of horizontally.
What happens when we vary both a and b?
Since a=b, the graph is of a circle with radius equal to a and b. If a or b had been negative, the graph would still be a circle of radius equal to the absolute value of a or b.
When a and b are different values our graph is different than the circle. It is now the ellipse.
When a=2, 2 the ellipse crosses the x axis at (2,0) (2,0). When
a=3, 3, the ellipse crosses the x axis at (3,0) (3,0).
When b =2, 2 the graph crosses the y axis at (0,2) and (0,2). When
b = 5, 5, the graph crosses the y axis at (0,5) and (0,5).
Click HERE to see an example of changing graph for a parametric equation where t has a changing coefficient.