3. 3
Evaluate sets of parametric equations for given
values of the parameter.
Sketch curves that are represented by sets of
parametric equations.
Rewrite sets of parametric equations as single
rectangular equations by eliminating the
parameter.
Find sets of parametric equations for graphs.
Objectives
5. 5
Plane Curves
Up to this point you have been representing a graph by a
single equation involving two variables such as x and y.
In this section, you will study situations in which it is useful
to introduce a third variable to represent a curve in the
plane.
To see the usefulness of this procedure, consider the path
of an object that is propelled into the air at an angle of 45.
6. 6
Plane Curves
When the initial velocity of the object is 48 feet per second,
it can be shown that the object follows the parabolic path
Rectangular equation
7. 7
Plane Curves
However, this equation does not tell the whole story.
Although it does tell you where the object has been, it does
not tell you when the object was at a given point (x, y) on
the path.
To determine this time, you can introduce a third variable t,
called a parameter. It is possible to write both x and y as
functions of t to obtain the parametric equations
x = 24
y = –16t2
+ 24 .
Parametric equation for x
Parametric equation for y
8. 8
Plane Curves
From this set of equations you 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 , and so
on, as shown below.
Curvilinear Motion: Two Variables for Position, One Variable for Time
9. 9
Plane Curves
For this particular motion problem, x and y are continuous
functions of t, and the resulting path is a plane curve.
(We know that a continuous function is one whose graph
has no breaks, holes, or gaps.)
11. 11
Sketching a Plane Curve
When sketching a curve represented by a pair of
parametric equations, you still plot points in the xy-plane.
Each set of coordinates (x, y) is determined from a value
chosen for the parameter t.
Plotting the resulting points in the order of increasing
values of t traces the curve in a specific direction. This is
called the orientation of the curve.
12. 12
Example 1 – Sketching a Curve
Sketch the curve given by the parametric equations
x = t2
– 4 and y = , –2 t 3.
Solution:
Using values of t in the specified
interval, the parametric equations
yield the points (x, y) shown in the
table.
13. 13
Example 1 – Solution
By plotting these points in the order of increasing t, you
obtain the curve shown in Figure 10.40.
The arrows on the curve
indicate its orientation as t increases
from –2 to 3.
So, when a particle moves on
this curve, it starts at (0, –1)
and then moves along the curve
to the point .
Figure 10.40
cont’d
14. 14
Sketching a Plane Curve
Note that the graph shown in Figure 10.40 does not define
y as a function of x. This points out one benefit of
parametric equations—they can be used to represent
graphs that are more general than graphs of functions.
Two different sets of parametric equations can have the
same graph. For example, the set of parametric equations
x = 4t2
– 4 and y = t, –1 t
has the same graph as the set of parametric equations
given in Example 1.
15. 15
Sketching a Plane Curve
However, by comparing the values of t in Figures 10.40 and
10.41, you can see that the second graph is traced out
more rapidly (considering t as time) than the first graph.
Figure 10.41
Figure 10.40
16. 16
Sketching a Plane Curve
So, in applications, different parametric representations can
be used to represent various speeds at which objects travel
along a given path.
18. 18
Eliminating the Parameter
Example 1 uses simple point plotting to sketch the curve.
This tedious process can sometimes be simplified by
finding a rectangular equation (in x and y) that has the
same graph. This process is called eliminating the
parameter.
x = t2
– 4 t = 2y x = (2y)2
– 4 x = 4y2
– 4
y =
19. 19
Eliminating the Parameter
Now you can recognize that the equation x = 4y2
– 4
represents a parabola with a horizontal axis and vertex at
(–4, 0).
When converting equations from parametric to rectangular
form, you may need to alter the domain of the rectangular
equation so that its graph matches the graph of the
parametric equations. Such a situation is demonstrated in
Example 2.
20. 20
Example 2 – Eliminating the Parameter
Sketch the curve represented by the equations
and
by eliminating the parameter and adjusting the domain of
the resulting rectangular equation.
21. 21
Example 2 – Solution
Solving for t in the equation for x produces
which implies that
22. 22
Example 2 – Solution
Now, substituting in the equation for y, you obtain the
rectangular equation
From this rectangular equation, you can recognize that the
curve is a parabola that opens downward and has its vertex
at (0, 1).
cont’d
23. 23
Example 2 – Solution
Also, this rectangular equation is defined for all values of x.
The parametric equation for x, however, is defined only
when t –1.
This implies that you should
restrict the domain of x to positive
values, as shown in Figure 10.42.
Figure 10.42
cont’d
24. 24
Eliminating the Parameter
In Example 2, it is important to realize that eliminating the
parameter is primarily an aid to curve sketching.
When the parametric equations represent the path of a
moving object, the graph alone is not sufficient to describe
the object’s motion.
You still need the parametric equations to tell you the
position, direction, and speed at a given time.
26. 26
Finding Parametric Equations for a Graph
You have been studying techniques for sketching the graph
represented by a set of parametric equations.
Now consider the reverse problem—that is, how can you
find a set of parametric equations for a given graph or a
given physical description? You know that such a
representation is not unique.
27. 27
Finding Parametric Equations for a Graph
That is, the equations
x = 4t2
– 4 and y = t, –1 t
produced the same graph as the equations
x = t2
– 4 and y = , –2 t 3.
This is further demonstrated in Example 4.
28. 28
Example 4 – Finding Parametric Equations for a Graph
Find a set of parametric equations to represent the graph of
y = 1 – x2
, using the following parameters.
a. t = x b. t = 1 – x
Solution:
a. Letting t = x, you obtain the
parametric equations
x = t and y = 1 – x2
The curve represented by the
parametric equations is shown
in Figure 10.43.
= 1 – t2
.
Figure 10.43
29. 29
Example 4 – Solution
b. Letting t = 1 – x, you obtain the parametric equations
x = 1 – t and y = 1 – x2
The curve represented by the
parametric equations is shown
in Figure 10.44.
Note that the graphs in
Figures 10.43 and 10.44 have
opposite orientations.
= 1 – (1 – t)2 = 2t – t2
.
cont’d
Figure 10.44
30. 30
Finding Parametric Equations for a Graph
A cycloid is a curve traced out by a point P on a circle as
the circle rolls along a straight line in a plane.
31. 31
Example 5 – Parametric Equations for a Cycloid
Write parametric equations for a cycloid traced out by a
point P on a circle of radius a as the circle rolls along the
x-axis given that P is at a minimum when x = 0.
Solution:
As the parameter, let be the measure of the circle’s
rotation, and let the point P(x, y) begin at the origin.
When = 0, P is at the origin; when = , P is at a
maximum point (a, 2a); and when = 2, P is back on the
x-axis at (2a, 0).
32. 32
Example 5 – Solution
From the figure below, you can see that
cont’d
33. 33
Example 5 – Solution
So, you have
which implies that BD = a sin and AP = –a cos .
Because the circle rolls along the x-axis, you know that
Furthermore, because BA = DC = a, you have
and
cont’d
34. 34
Example 5 – Solution
So, the parametric equations are
cont’d
