The parametric equations above graphed from t= 0 to
6.28 yield a circle.The intercepts are all equal to 1.
Now let us change the equations to explore. Using 2
t or 4 t instead of t appeared to give the same graph.
Now let us change the equations in a different manner.Using
2 cos (t) and 2 sin (t) give a graph with intercepts = 2. Hence
the coefficients determine the intercepts.
If we use
we see that it is an ellipse with major axis on the
x axis. The intercepts still match the coefficients of cosine
and sine. Notice we still have an ellipse with the coefficient
of t = 1.
Given: x=a cos (t) and y = b sin (t) and b>a then
the major axis will be on the y axis. We will also change the
coefficient of t. This produces the graph below.
Zero is an intercept now along with x = 2 since that
is the number with cosine above. The Maximum and minimum values
of the graph are at 4 the value with sine above. Let's change
those values to:
Using 2 cos ( 4 t ) gives a partial spiral, if the
4 is changed to an odd number, a whole spiral is formed a seen
If the coefficient of the "t" in sine is
changed and the coefficient of t in cosine is "1", then
the spiral goes around the x axis . Above, it spirals around the
y axis since the "1" is with the sine function.
The coefficient of sine determine the max and min for
the graph and the coeffiecient of cosine determine the x intercepts.
Below we observe that the max, min, and intercepts
and the number of spirals changed when we changed our values above.
There are endless combinations one could experiment
with to see the changes the graphs can make. A very interesting