Notice in this definition that
x and
y are used in two ways. The first is as functions of the independent variable
t. As
t varies over the interval
I , the functions
and
generate a set of ordered pairs
This set of ordered pairs generates the graph of the parametric equations. In this second usage, to designate the ordered pairs,
x and
y are variables. It is important to distinguish the variables
x and
y from the functions
and
Graphing a parametrically defined curve
Sketch the curves described by the following parametric equations:
To create a graph of this curve, first set up a table of values. Since the independent variable in both
and
is
t , let
t appear in the first column. Then
and
will appear in the second and third columns of the table.
t
−3
−4
−2
−2
−3
0
−1
−2
2
0
−1
4
1
0
6
2
1
8
The second and third columns in this table provide a set of points to be plotted. The graph of these points appears in
[link] . The arrows on the graph indicate the
orientation of the graph, that is, the direction that a point moves on the graph as
t varies from −3 to 2.
To create a graph of this curve, again set up a table of values.
t
−2
1
−3
−1
−2
−1
0
−3
1
1
−2
3
2
1
5
3
6
7
The second and third columns in this table give a set of points to be plotted (
[link] ). The first point on the graph (corresponding to
has coordinates
and the last point (corresponding to
has coordinates
As
t progresses from −2 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.
In this case, use multiples of
for
t and create another table of values:
t
t
0
4
0
2
−2
0
−4
0
4
2
−2
2
2
4
0
−4
0
The graph of this plane curve appears in the following graph.
This is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates
To better understand the graph of a curve represented parametrically, it is useful to rewrite the two equations as a single equation relating the variables
x and
y. Then we can apply any previous knowledge of equations of curves in the plane to identify the curve. For example, the equations describing the plane curve in
[link] b. are
Solving the second equation for
t gives
This can be substituted into the first equation:
This equation describes
x as a function of
y. These steps give an example of
eliminating the parameter . The graph of this function is a parabola opening to the right. Recall that the plane curve started at
and ended at
These terminations were due to the restriction on the parameter
t.
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Mohamed
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