**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.

**Return**