### Assignment 10

Parametric Curves

by: Kelli Nipper

A parametric curve is a pair of functions x=f(t), y=g(t) that give x
and y as contiuous functions of the real number t (the parameter). Each
value determines a point ( f(t), g(t) ). The graph is the set of all such
points in the interval assigned.

### x= cos t

### y= sin t

### [0, 2 pi]

The point (cos t, sin t) begins at (1, 0). As t moves counterclockwise from
0 to 2 pi, it gives points that satisfy: x = cos t, y = sin
t

All points lie on the unit circle, thus the graph is the unit circle.

A given figure in the plane may have different parameterizations.
For example:

This parametric curve also lies on the unit circle because

To explore other graphs I changed the parameters:

For various **a** and** b**, x = cos (**a**t) ; y = sin (**b**t)

Changes the period.

For various **a** and **b**, x = **a** cos (t) ; y = **b**
sin (t)

Changes the amplitude. In general, when **a** and **b** increase,
the amplitude increases. When **a** and **b** decreases, the period
decreases.

In closing, to describe arbitrary amplituted and periods,
for x = **a** sin **b**t and y = **a** cos **b**t
a is the amplitude and 2pi/**b** is the period.

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