This is the write-up for assignment #10.

This write-up explores a parametric curve in the plane. A parametric curve in the plane is a pair of functions, x=f(t) and y=g(t), where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent to which the curve will vary will depend on the range of the value of t.

Let's first consider a very basic set of parametric equations:

The graph of these two parametric equations
for 0 __<__ **t **__< __6.28 is shown below.

Now, let's consider the graph of:

Let's take a look at several graphs where we
let **a** and **b** vary in the parametric equations above.

First we will let the value of **b** remain
1 and we will let the value of **a **= -7(purple), -5(red),
-3(blue), 2(green), 4(lt.blue), 6(yellow).

We can see that the value of **a** will
stretch the curve toward the right and left (while **b** remains
a constant of 1).

Click here
for a dynamic presentation where the value of **a **changes
from -10 to 10.

Now, let's keep the value of **a** equal
to 1 and vary the **b** value. We will let **b** =-7(purple),
-5(red), -3(blue), 2(green), 4(lt.blue), 6(yellow).

We can see that the value of **b** will
stretch the curve up and down (while **a** remains a constant
of 1).Click here
for a dynamic presentation where the value of **b **changes
from -10 to 10.

Now, let's take a look at several graphs where
we let **a** and **b** both vary at the same time.

This is the graph of x=2cos(t) and y=6sin(t):

This is the graph of x=5cos(t) and y=2sin(t):

This is the graph of x=**-**7cos(t) and y=3sin(t):

This is the graph of x=**-**2cos(t) and y=**-**sin(t):

After exploring the parametric curves of x=acos(t)
and y=bsin(t), it looks as though the **a **and **b** values
will determine the intercepts on the x and y axis. The **a**
value will determine the x-intercept and the **b** value will
determine the y-intercept. Also, when a<b, the graph is elliptical
with the major axis on the y-axis. When a>b, the graph is elliptical
with the major axis on the x-axis. And when a=b, the graph is
a circle with center at the origin.

Now, let's explore various values of **a**
and **b** in the parametric equation:

We will first let **b** value be 1 and we
will vary the **a** value.

Here is the graph of x=cos (**-**5t) and y=sin(t):

This is also the graph of x=cos(5t) and y=sin(t).

Here is the graph of x=cos (**-**3t) and y=sin(t):

Also, this is the graph of x=cos(3t) and y= sin(t).

Here is the graph of x=cos(**-**1t) and y=sin(t):

Once again, this is also the graph when a=1 in x=cos(at) and y=sin(t).

Here is the graph of x=cos(**-.**5t) and y=sin(t):

This is also the graph for x=cos (**.**5t) and y=sin(t).

Here are the graphs of x=cos(**a**t) and
y=sin(t) for **a**=**-.**4 (purple), **-.**3 (red), **-.**2 (blue), **-.**1 (green), 0 (lt.blue).

If we let the values of a = **.**4, **.**3, **.**2,
**.**1, we will
get the same graphs as above.

** Click here
for a dynamic presentation where the value of **a **changes
from -5 to 5.

Now, let the **a** value be 1 and we will
vary the **b** value.

Here is the graph of x=cos (t) and y=sin(**-**5t):

This is also the graph of x=cos(t) and y=sin(5t). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos (t) and y=sin(**-**3t):

Also, this is the graph of x=cos(t) and y= sin(3t). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos(t) and y=sin(**-**1t):

Once again, this is also the graph when b=1 in x=cos(t) and y=sin(bt). (This is the reflection of the above graph over the x-axis.)

Here is the graph of x=cos(t) and y=sin(**-.**5t):

If we looked at the graph for x=cos (t) and
y=sin(**.**5t),
we would reflect it over the x-axis like:

Here are the graphs of x=cos(t) and y=sin(**b**t)
for **b**=**-.**4
(purple), **-.**3
(red), **-.**2
(blue), **-.**1
(green), 0 (lt.blue).

If we let the values of b = **.**4, **.**3, **.**2,
**.**1, we will
get this graph:

Which is once again a reflection over the x-axis.

** Click here
for a dynamic presentation where the value of **b **changes
from -5 to 5.

If we look at varying the **a** and **b**
values at the same time, we could generate an infinite number
of graphs. One thing that I observed is that when the **a**
and **b** values are both the same in x=cos(at) and y=sin(bt)
(a>1) then a graph of a circle is always formed. When 0<a<1
, then the graph is a fraction of the circle. For example, if
x=cos(**.**25t)
and y=sin(**.**25t),
then 1/4 of the total circle with radius 1 and center at the origin
is graphed.