1.4

Parametric Equations

RelationsCirclesEllipsesLines and Other Curves

What you’ll learn about…

…and why

Parametric equations can be used to obtain graphs of relations and functions.

Relations

A relation is a set of ordered pairs (x, y) of real numbers.

The graph of a relation is the set of points in a plane

that correspond to the ordered pairs of the relation.

If x and y are functions of a third variable t, called a parameter, then we can use the parametric mode of a grapher to obtain a graph of the relation.

Parametric Curve, Parametric Equations

( ) ( )

( ) ( ) ( )( )

If and are given as functions

,

over an interval of -values, then the set of points , ,

defined by these equations is a . The equations are

of th

x y

x f t y g t

t x y f t g t

= =

=

parametric curve

parametric equations e curve.

Relations

( ) ( )( ) ( ) ( )( )

The variable is a parameter for the curve and its domain is the

parameter interval. If is a closed interval, , the point

, is the initial point of the curve and the point ,

is the te

t I

I a t b

f a g a f b g b

£ £

rminal point of the curve. When we give parametric equations

and a parameter interval for a curve, we say that we have parametrized

the curve. A grapher can draw a parametrized curve only over a closed

interval, so the portion it draws has endpoints even when the curve being

graphed does not.

Example Relations

2Describe the graph of the relation determined by , 1 .x t y t= = -

21 1Set , 1 , and use the parametric mode

of the grapher to draw the graph.

x t y t= = -

Circles

In applications, t often denotes time, an angle or the distance a particle has traveled along its path from a starting point.

Parametric graphing can be used to simulate the motion of a particle.

Example CirclesDescribe the graph of the relation determined by

3cos , 3sin , 0 2 .

Find the initial points, if any, and indicate the direction in which the curve

is traced.

Find a Cartesian equation for a curve

x t y t t p= = £ £

that contains the parametrized curve.

( ) ( )2 2 2 2 2 2

2 2

9cos 9sin 9 cos sin 9 1 9

Thus, 9

This represents the equation of a circle with radius 3 and center at the origin.

As increases from 0 to 2 the curve is traced in a counter-clock

x y t t t t

x y

t p

+ = + = + = =

+ =

wise direction

beginning at the point (3,0).

2 2 9x y+ =

Ellipses

2 2

2 2

2 2

2 2

Parametrizations of ellipses are similar to parametrizations of circles.

Recall that the standard form of an ellipse centered at (0, 0) is

1.

For cos and sin , we have 1

which

x y

a b

x yx a t y a t

a b

+ =

= = + =

is the equation of an ellipse with center at the origin.

Lines and Other Curves

Lines, line segments and many other curves can be defined parametrically.

Example Lines and Other Curves

If the parametrization of a curve is , 2, 0 2 graph the curve.

Find the initial and terminal points and indicate the direction in which the curve

is traced. Find a Cartesian equation for a cu

x t y t t= = + £ £

rve that contains the parametrized

curve. What portion of the Cartesian equation is traced by the parametrized curve?

, 2, 0 2x t y t t= = + £ £

Initial point (0,2) Terminal point (2,4)

Curve is traced from left to right

( )( ) ( )

If and 2, then by substitution 2.

The graph of the Cartesian equation is a line through 0,2 with 1.

The segment of that line from 0,2 to 2, 4 is traced by the parametrized curve.

x t y t y x

m

= = + = +

=