A parametric curve in the plane is a pair of functions

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 (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.

Here is the graph of

What if we change our parameter to ?

Let's examine why this happens. Consider a particular value for t: zero. Then we have , or , and thus the point (1, 0) is graphed. Now increase our particular value of t to . Then we have, and thus the point (0, 1) is graphed. Use your calculator to find some other points on this graph.

Get two points in each of the four quadrants and graph them on graph paper to turn in.

Email questions to me.

For various **a** and **b**, investigate

x = a cos (t)

y = b sin (t)

for . Try a=1 and b=1, so we have the parametric equation with a=1, b=1.

It's a circle, radius 1, center (0,0).

Now try a=2, b=2 in :

It's a circle, radius 2, center (0,0).

Now try a=1, b=2 in :

Now we have an ellipse, height 4, width 2, center (0,0).

Now switch them so a=2, b=1 in :

Now we have an ellipse, height 2, width 4, center (0,0).

Can you form a conjecture? What is your hypothesis? Use your hypothesis to predict what the graph of will look like with a=3, b=5. Email your prediction.

What about noninteger values of t, say a=2.5, b=4.5? Will noninteger values fit your hypothesis? Here's the graph:

Does your hypothesis still hold?

What if one of the values is zero? Let a=0, b=4. What do you think the graph will look like? Use your hypothesis to predict.

Here's the graph:

It's a line 0 wide and 4 tall. Does this fit your hypothesis? How about a=4, b=0?

It's a line 4 wide and 0 tall.

Finally, what would the graph look like with a=0, b=0 and why? Email your answer.

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