Assignment #10

By Nikki Masson

Parametric Curves

A parametric curve in the plane is a pair of functions

x=f(t)

y=g(t)

where the two continuous functions define ordered pairs (x,y).

Part I: Get familiar with graphing parametric curves in Graphing Calculator.

Using graphing calculator, draw x=cos(t) and y=sin(t) from 0<t<pi

Using graphing calculator, draw x=cos(t) and y=sin(t) from 0<t<2pi

Part II: Investigate various values for a and b, when x=cox(at) and y=sin(bt) for 0<t>2pi

A. a>b

1. a=2 and b=1, x=cos(2t) and y=sin(t) for 0<t>2pi

2. a=3 and b=1, x=cos(3t) and y=sin(t) for 0<t>2pi

3. a=4 and b=1, x=cos(4t) and y=sin(t) for 0<t>2pi

4. a=5 and b=1, x=cos(5t) and y=sin(t) for 0<t>2pi

Observation, when a is even the grah is not closed and when a is odd, the graph is closed.

B. a=b: This is the unit circle from Part I, no matter what the values of a and b are as long as they are equal.

a=5 and b=5, x=cos(5t) and y=sin(5t) for 0<t>2pi

C. a<b

1. a=1 and b=2, x=cos(t) and y=sin(2t) for 0<t>2pi

2. a=1 and b=3, x=cos(t) and y=sin(3t) for 0<t>2pi

3. a=1 and b=4, x=cos(t) and y=sin(4t) for 0<t>2pi

You can do further investigations into these parametric curves by changing values for a and b.

Part II: Investigate various values for a and b, when x=acox(t) and y=bsin(t) for 0<t>2pi.

A. a>b

1. a=2 and b=1, x=2cos(t) and y=sin(t) for 0<t>2pi

2. a=3 and b=1, x=3cos(t) and y=sin(t) for 0<t>2pi

We can see that as we increase a, the ellipse will go from -a to a.

3. a=3 and b=2, x=3cos(t) and y=2sin(t) for 0<t>2pi

We can see from the patterns that the coefficent in front of the cos(t) and sin(t) tell us the vertices of the ellipse or circle.

B. a=b: x=acox(t) and y=bsin(t) for 0<t>2pi

From our patterns we have seen in Part III: when a=b, we will have a circle with the vertices at x=y=a.

1. a=2 and b=2, x=2cos(t) and y=2sin(t) for 0<t>2pi

C. a<b

1. a=1 and b=2, x=cos(t) and y=2sin(t) for 0<t>2pi

1. a=2 and b=3, x=2cos(t) and y=3sin(t) for 0<t>2pi

Just as we suspected, we have ellipse with the vertices at y=3 and x=2.