MATH 2024
Parametric Curves
Adams
Plane Curves and Parametric Equations
Until now, we have represented graphs using a single equation involving two variables. Today we
begin examining situations in which three variables are used to represent a curve in the plane. Let’s
begin by considering the path followed by an object that is propelled into the air at an angle of
45
◦
. If the initial velocity of the object is 48 feet per second, the object travels the parabolic path
given by the rectangular equation
y
=

1
72
x
2
+
x
as shown in the figure below:
This equation, however, does not tell the whole story. Although it tells you where in the plane the
object travels, it does not tell you when the object was at any given point (
x, y
). To determine
this time, we will introduce a third variable
t
, called a
parameter
. By writing both
x
and
y
as
functions of the parameter
t
, we obtain the set of
parametric equations
x
(
t
) = 24
√
2
t
and
y
(
t
) =

16
t
2
+ 24
√
2
t
From this set of equations, we can determine that at time
t
= 0, the object is at the point (0
,
0).
Similarly, at time
t
= 1, the object is at the point
(
24
√
2
,
24
√
2

16
)
, and so on. For this particular
motion problem,
x
and
y
are continuous functions of
t
, and the resulting path is called a
plane
curve
. If
f
and
g
are continuous functions of
t
on an interval
I
, then the equations
x
=
f
(
t
)
and
y
=
g
(
t
)
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MATH 2024
Parametric Curves
Adams
are called
parametric equations
and
t
is called the
parameter
. The set of points (
x, y
) obtained
as
t
varies over the interval
I
is called the
graph
of the parametric equations. Taken together, the
parametric equations and the graph are called a
plane curve
, denoted by
C
.
Sketching a Curve
When sketching a curve represented by a set of parametric equations, you can plot points in the
xy

plane. Each set of coordinates (
x, y
) is determined from a value chosen for the parameter
t
. By
plotting the resulting points in order of increasing values of
t
, the curve is traced out in a specific
direction. This is called the
orientation
of the curve.