Parametric Equations 

A curve in the plane is said to be parameterized if the set of
coordinates on the curve, (x,y), are represented
as functions of a variable t. Namely,

Conversely, given a pair of parametric equations with parameter t, the set of points (f(t), g(t)) form a curve in the plane.
As an example, the graph of any function can be parameterized. For, if y = f(x) then let t = x so that
Example 
The parametric equations

In the last example there were no restrictions on the parameter. The curve defined parametrically by the equations was identical to the graph of the function. Note that by restricting the values of the parameter, we can parameterize part of the graph of the function.
Example 
The parametric equations

There are several techniques we use to sketch a curve generated by a pair of parametric equations. The simplest is to evaluate f(t) and g(t) for several values of t. We then plot the points (f(t), g(t)) in the plane and through them draw a smooth curve (assuming this is valid!!!). This idea of plotting points is identical to the elementary graphing techniqus of graphing functions and is illustrated in the following two diagrams.
The orientation of a parameterized curve is
the direction determined by increasing values of the parameter.
Sometimes arrows are drawn on the curve to denote the orientation.
The diagram shows the same parametric curve we have just studied
where we have included some arrows to illustrate the orientation.
In this case, the direction of t increasing is from left to right.
Plotting a parametric curve 

A second technique to identifying the curve of the parametric equations
is to try to eliminate the parameter from the equations. This will
result in an equation involving only x and y which
we may recognize. For example, let's look again at the previous example.
Since x = 2t then solving for t gives
t = x/2. Substitute for t into the equation for
y to get y = (x/2)^{2} = x^{2}/4. This we recognize as the graph of a parabola and we can sketch its
graph usinfromg function graphing techniques.
Be careful, however, to take into account any restrictions on the value of
the parameter. If, in this example we add the condition t > 0, then
the curve defined by the parametric equations would be the graph of
y=x^{2}/4 on the positive x axis.
Eliminating the parameter 

A parametric representation of a curve is not unique. That is, a curve C may be represented by two (or more) different pairs of parametric equations.
Example 
We saw earlier that the parametric equations
However, the equations 
Example 
Consider Now, consider This parametric curve is also the unit circle and we have found two different parameterizations of the unit circle. 
A thing to note in this previous example was how we obtained an equation in terms of x and y. We eliminated the parameter but not in a direct way by solving one of the equations. This technique is useful in many parametric equations involving sine and cosine. That is, solve the equations for sin(t) and cos(t) then square and add the equations.
Position of a moving object 

One nice interpretation of parametric equations is to think of the parameter as
time (measured in seconds, say) and the functions f and g as functions that describe the x and
y position of an object moving in a plane.
We give four examples of parametric equations that describe the motion of
an object around the unit circle.
Example 
The parametric equations

Example 
The parametric equations

Example 
The parametric equations

Example 
The parametric equations

We illustrate these four examples with the applets below. Click the start on each one to see the motion we just described.
Four examples of motion  

Projectiles 

A projectile is an object moving only under the force of gravity such as when we throw a basketball, or fire a missile or water the garden. In modeling the motion of a projectile we assume no air resistance. In this case, the projectile follows a parabolic path.
In order to obtain parametric equations that describe the motion of a projectile it is useful to set up a rectangular coordinate system with the positive xaxis along the (horizontal) ground and in the direction we will send the projectile and the yaxis perpendicular to the ground. With that done, if an object is projected at an angle to the ground with an initial speed V, then the path of the projectile is given by the parametric equations
From these equations we can find how long the object is in the air
(time of flight),
how far horizontally the object will travel (range of the projectile)
and the maximum height reached by the object.
Time of flight 
The projectile reaches the ground at y=0. Solving
2Vsin()/g. 
Range 
The range is the value of x when the projectile reaches the
ground. Since we have just found the time of flight, we substitute
this time of flight into the equation connecting x and t
to get

Maximum height 
By the symmetry of the path of the object, you can intuitively reason
that the projectile reaches its greatest height at onehalf of the
time of flight. That is when t=Vsin()/g.
If we substitute this value of t into the equation for y
we obtain

We summarize these formulas for the general case and for the typical system of units.
Time of flight  Range  Max Height  
General  2Vsin()/g.  V^{2}sin(2)/g.  V^{2}sin^{2}()/2g 
Metric  Vsin()/4.9  V^{2}sin(2)/9.8  V^{2}sin^{2}()/19.6 
Feet  Vsin()/16  V^{2}sin(2)/32  V^{2}sin^{2}()/64 
Example 
A projectile is fired at an angle of /4
to the horizontal with an initial speed of 100 m/s.
The range is The time of flight is 100sin(/4)/4.9 which is approximately 14.4 seconds. The maximum height is 10,000sin^{2} (/4)/19.6 which is approximately 255 meters. 
Projectiles 
