*Parametric Equations*
*By Joshua Singer*
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)**.
These two equations are usually called *parametric equations*.
Usually we think of these equations as **x** and **y** "vary
with time **t**" or the angle of rotation that some line
makes from an initial location. The extent of the curve will depend
on the range of **t** and close attention should be given to
the range of **t**.

In this exploration, we are going to investigate
the parametric equations *x = a cos (t)* and *y
= b sin (t)* for 0 __<__ **t** __<__ 2p and different values of **a** and **b**. We
will observe and describe the characteristics of the curves when
**a** < **b**, **a** = **b**, and **a** >
**b**. Then we will investigate the parametric equations *x
= a cos (t) + h sin(t)* and *y = b sin(t) + h cos
(t)* for 0 __<__ **t** __<__ 2p where **h**
is any real number.

First, let's look at the graph of the parametric
equations *x = a cos (t)* and *y = b sin (t)*
for 0 __<__ **t** __<__ 2p where **a** and **b** equal 1.

We notice that the graph of the parametric
equations *x = a cos (t)* and *y = b sin (t)*
for 0 __<__ **t** __<__ 2p
is the unit circle. From what we know of
the cosine and sine functions, the graphs are periodic (repeat
over intervals) and have a range [-1.1]. The period of the functions
cosine and sine is 2p, so once **t** = 2p, the graph will just repeat. So, in the graph above,
**x** and **y** will range from -1 to 1. To see why the
graph is a circle, let's look at the parametric equations:

So, what happens to the graphs when we change
**a** and **b** with **a** < **b**? Let's look
at the graphs where **a** = 1 and **b** = 2, **a** =
2 and **b** = 3, and **a** = 3 and **b** = 5.

We notice when **a** < **b**, the
graph of the parametric equations *x = a cos (t)*
and *y = b sin (t)* for 0 __<__ **t** __<__
2p is
an ellipse with the major axis on the y-axis and the minor axis
on the x-axis. The vertices of the major axis are at **b**
and **-b** while the vertices of the minor axis are at **a**
and **-a**. Click HERE to change **a** and **b** to negative numbers
(with **a** < **b**) and observe the graphs. Do they
change?

Let's take a look at the graphs of the parametric
equations *x = a cos (t)* and *y = b sin (t)*
for 0 __<__ **t** __<__ 2p where **a** = **b**.

When **a** = **b**, the graph of the
parametric equations *x = a cos (t)* and *y =
b sin (t)* for 0 __<__ **t** __<__ 2p is a circle with radius of **a** (or **b** since
**a** = **b**). Click HERE to change **a** and **b** to negative numbers
(with **a** = **b**) and observe the graphs. Do they change?

Now, what happens to the graph when **a**
> **b**? Let's look at the graphs when **a** = 2 and
**b** = 1, **a** = 3 and **b** = 2, and **a** = 5
and **b** = 3.

We notice that when **a** > **b**,
the graph of the parametric equations *x = a cos (t)*
and *y = b sin (t)* for 0 __<__ **t** __<__
2p
is an ellipse with the major axis on the x-axis and the minor
axis on the y-axis. The vertices of the major axis are **a**
and **-a** while the vertices on the minor axis are **b**
and **-b**. Click HERE to change **a** and **b** to negative numbers
(with **a** > **b**) and observe the graphs. Do they
change?

Now let's investigate what is changed if the
parametric equations are *x = a cos (t) + h sin (t)*
and *y = b sin (t) + h cos(t)* for 0 __<__ **t**
__<__ 2p where **h** is any real number. First, we will
look at seven graphs where **a** < **b** and **h **=
-3, -2, -1, 0, 1, 2, 3.

Notice that we have an ellipse since **a**
< **b** and that when **h** = 0, the axes of symmetry
are the y-axis and x-axis, the major axis is the y-axis, and the
minor axis is the x-axis. As **h** changes, the axes of symmetry
change. Also the length of the major axis increases as the absolute
value of **h** increases. Notice that the graphs of *x
= cos (t) - 2 sin (t)* and *y = 4 sin (t) - 2 cos(t)*
for 0 __<__ **t** __<__ 2p and *x = cos (t) + 2 sin (t)* and *y
= 4 sin (t) + 2 cos(t)* for 0 __<__ **t** __<__
2p
are line segments. This is because *y* is a multiple
of *x* in the form *y = mx* where *m*
is the slope of the line (in this case, 2 and -2). Click HERE to change **a**, **b**, and **h** (with **a**
< **b**) and observe the graphs.

Now, we will look at seven graphs where **a**
= **b** and **h **= -3, -2, -1, 0, 1, 2, 3.

We notice that when **a** = **b** and
**h** = 0, we have a circle with radius **a** (or **b**)
as before. As **h** changes, the graph becomes an ellipse with
axes of symmetry changing and the length of the minor and major
axes changing. Only when **a** = **b** = **h** will we
get a line segment (since **y = x** or **y = -x**). Click
HERE to change **a**, **b**, and **h** (with **a**
= **b**) and observe the graphs.

Finally, we will look at seven graphs where
**a** > **b** and **h** = -3, -2, -1, 0, 1, 2, and
3.

Notice that we have an ellipse since **a**
> **b** and that when **h** = 0, the axes of symmetry
are the x-axis and y-axis, the major axis is the x-axis, and the
minor axis is the y-axis. As **h** changes, the axes of symmetry
change. Also the length of the major axis increases as the absolute
value of **h** increases. Notice that the graphs of *x
= 4 cos (t) - 2 sin (t)* and *y = sin (t) - 2 cos(t)*
for 0 __<__ **t** __<__ 2p and *x = 4 cos (t) + 2 sin (t)* and *y
= sin (t) + 2 cos(t)* for 0 __<__ **t** __<__ 2p are line segments. This is because *y*
is a multiple of *x* in the form *y = mx*
where *m* is the slope of the line (in this case,
-2 and 2). Click HERE to change **a**, **b**, and **h** (with **a**
> **b**) and observe the graphs.

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