Introduction : Explorations of parametric curve in the plane which is a pair of functions x=f(t), y=g(t) ,where the two

continuous functions define ordered pairs (x,y).

**Explorations of x=cos(at), y=sin(bt).**

For various a and b, let's investigate x=cos(at), y=sin(bt) for 0 < t < 2 .

We can easily guess that if a and b are equal, a and b are related
with the period of the circle with radius 1.

After fixing a value, let's change b value.

From the graphs, we can guess the role of b value.

After fixing b value, let's change a value.

Similarly, we can guess the role of a value from the graphs.

**Exporations of x=(1-t^2)/(1+t^2), y= 2t/(1+t^2).**

Let's explore the graphs of

for
-< t <

If we would change the domain of t,

for -3< t <

for -< t < 3

As the domain increase, the ends of the circle approaches point ( -1,0
).

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