11.5 Parametric Equations
In this section we will discuss parametric equations. A curve in the Cartesian plane, sometimes called
a plane curve, is said to be parameterized when the set of all ordered pairs  x, y  on the curve are
represented as functions of another variable. You have already seen an example of a set of parametric
equations earlier in the text. Recall that for any real number t, if  x, y  is a point on the unit circle
corresponding to t, then the unit circle definitions of sine and cosine were defined as
x  cos t , y  sin t
.
The figure below illustrates that every point that lies on the graph of the unit circle can be described by
the ordered pair  x, y    cos t ,sin t  where t is the corresponding central angle or arc length from the
point 1,0  to  x, y  . The variable t is called a parameter. If 0  t  2 , then the parametric equations
x  cos t , y  sin t represent another way to describe the exact same set of points as the rectangular
equation x 2  y 2  1 .
x2  y 2  1
The unit circle can be described
using the rectangular equation x 2  y 2  1
or by using the parametric equations
x  cos t , y  sin t for 0  t  2.
When a plane curve, such as the graph of the unit circle, does not pass the vertical line test, then it is
impossible to describe the curve with a single function of the form y  f ( x) . However, it is often be
possible to describe the plane curve using parametric equations of the form x  f (t ) and y  g (t ) . If you
study calculus or physics you will see that we need parametric equations to model many situations
where one single function will not adequately suffice.
Definition
Parametric Equations of a Plane Curve
A plane curve is the set of all ordered pairs  x, y  such that x  f (t ) and y  g (t ) where
t is in an interval I. The variable t is called a parameter, and the equations x  f (t ) and
y  g (t ) are called parametric equations of the curve.
OBJECTIVE 1 Plotting Points to Sketch the Graph of a Plane Curve Defined Parametrically
One way to sketch a plane curve that is defined parametrically is to plot ordered pairs using various
values of the parameter. We then connect those ordered pairs with a smooth curve. If a parametric
equation is defined on some predetermined interval such as a  t  b , then it is important to
determine the initial point and the terminal point.
Definition
Initial Point and Terminal Point of a Plane Curve Defined Parametrically
If a plane curve is defined parametrically as x  f (t ) and y  g (t ) on a closed interval
a  t  b ,then the ordered pair  f (a), g (a)  is called the initial point and the ordered pair
 f (b), g (b) 
is called the terminal point.
Determining the initial point and terminal point will help to understand the proper direction, or
orientation, in which the plane curve should be sketched. Not all plane curves described by parametric
equations have an initial point and terminal point. However, if a plane curve does have these two points,
then the graph always “starts” at the initial point and “ends” at the terminal point.
OBJECTIVE 2 Determining a Rectangular Representation for a Curve Defined Parametrically
Using Substitution
It is sometimes possible to convert a given set of two parametric equations into an equivalent rectangular
equation. We do this by eliminating the parameter in attempt to write one equation in terms of x and y.
One way to eliminate the parameter is to try to solve one of the parametric equations for the variable
representing the parameter, then substitute that expression into the second equation. When using this
method, carefully consider which of the two parametric equations is the easiest to solve for the
parameter. The five-step method for determining a rectangular representation for a curve defined
parametrically using substitution is stated below.
Determining a Rectangular Representation for a Curve Defined Parametrically Using Substitution
Step 1.
Carefully observe the given parametric equations in an attempt to identify a relationship
between the variables such that a substitution could be made to eliminate the parameter t.
Step 2.
Choose a parametric equation and solve for t or rewrite the parametric equation based on
the observation made in Step 1.
Step 3.
Substitute the expression from Step 2 into the other parametric equation in place of the
parameter to obtain the rectangular equation.
Step 4.
Write the rectangular equation in a familiar form whenever possible.
Step 5.
If necessary, restrict the domain and range of the rectangular equation to be consistent
with any restrictions of the parameter or any restrictions of the parametric equations.
Sometimes we can chose one parametric equation and solved for the parameter, t. This allows us to
substitute the expression that represented t into the other equation, thus eliminating the parameter.
Although this is often a convenient way to eliminate the parameter to determine the rectangular
representation of parametric equations, it is not the only way. Sometimes the two given parametric
equations are related to each other in some way. If we can recognize the relationship, then we might be
able rewrite one of the parametric equations in terms of the other. This technique is especially useful
when one of the parametric equations is a power of the other as in the following example.
OBJECTIVE 3 Determining a Rectangular Representation for a Curve Defined Parametrically
Using a Trigonometric Identity
Sometimes we can eliminate the parameter by substituting an appropriate form of one of the equations
into the other. When parametric equations are defined using trigonometric functions, this approach is
not always possible. An alternate approach is to substitute both parametric equations into a known
trigonometric identity to eliminate the parameter. It may be a good idea to review some of the
fundamental trigonometric identities before proceeding. We now outline a five-step process for
eliminating the parameter using trigonometric identities.
Determining a Rectangular Representation for a Curve Defined Parametrically Using a
Trigonometric Identity
Step 1.
Isolate the trigonometric expression in each parametric equation.
Step 2.
Select a trigonometric identity (by observation) that relates the two parametric equations.
Step 3.
Substitute the rectangular expressions for the trigonometric expressions into the chosen
trigonometric identity to eliminate the parameter.
Step 4.
Write the rectangular equation in a familiar form whenever possible.
Step 5.
If necessary, restrict the domain and range of the rectangular equation to be consistent
with any restrictions of the parameter or parametric equations.