**What is a parametric curve?**

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 and your work with parametric equations should pay close attention
to the range of **t**.

Let's consider the parametric equations:

x = cos t

y = sin t

for 0 £ t £ 2p

The above parametric equations graph like:

As you can see the graph is a circle; that is, the curve of
the parametric equations is a circle. Does the curve of a given
parametric equation always graphs to a circle? To verify the conjecture
you are going to explore** t **because it seems to affect the
change of the parametric curves.

The variable **t** is the angle of rotation. When you change
the rotation angle, the parametric curve might be changed or might
not be changed (when does the latter happen?). According to the
definition of the parametric curve, you can see the variables
__x__ and __y__ depending on **t**, which means that
if the range of **t** changes the parametric curves will also
be changed. So if 0 £ t £ p, the
parametric equations graphs as follows:

Now, let's try 4/p £
**t £ **4p
/3. Then we have

If t goes around from p to 2p, what will happen?

If you want to explore more curves according to different range
of t, **click here.**

**Trigonometric substitution**

You can also get a new equation from the parametric equations
as a trigonometric function of **t**,

is obtained. This equation is a standard equation of a circle centered at (0, 0) and with a radius 1.

From the properties of sine and cosine -1 £
cos**t** £ 1 and -1 £ sin **t** £
1 which implies that -1 £ x £ 1 and -1 £
y £ 1.

**Another parametric curve **

Consider the following equations as a variation of the above parametric equations.

Like the various curves we have already
investigated, the given equations have graphs depending on the
range of *t*. However, we have new variables *a* and
*b*. How do *a* and *b*, affect the graph of the
curve?

First, fix ** a = 1** and the
range of t to be 0 £

1. If **b = ****2, b = 3, b = 5,** then the curves
graph like this: (think about the reason why the case *b*
= 1 isn't considered):

2) If b is in between 0 and 1, how do the graphs change?

As you see with the above curves, when
the value of *b* increases to one, the shape of the graph
of the curve becomes more close to a circle, however never gets
to be a circle.

Second, fix now *b* = 2. Similarly,
you can observe the different graphs of the curves according to
*a* as follows:

Here you can notice the curve is a circle
when *a* is equal to 2 that is the very value of *b*
fixed. Then can you say that when *a* equals to *b*
the parametric curve is a circle? From the following you can see
how it is working.

**Trigonometric substitution**

In general, we can draw the standard ellipse equation from the parametric curve. That is,

In particular, when *a = b* we have

that represents a circle having a center
at (0, 0) and radius *a* (*a>0*).

Now, you can see the changes according
to the value of *a* and *b* in the equation

When** ****a
>
b,**
we get the major axis on the x-axis and the minor axis on the
y-axis. When **a = b**, we have a circle. When **a
< b, **we
get the major axis on the y-axis and the minor axis on the x-axis.