AP CALCULUS AB

Chapter 1:Prerequisites for Calculus

Section 1.4:Parametric Equations

Relations Circles Ellipses Lines 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

. If is a closed interval, , the point

, is

parameter interval

initial poithe of the curve and the point ,

is the

t

n

te

t I

I a t b

f a g a f b g b

£ £

of the curve. When we give parametric equations

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

the curve. A grapher can draw a parametrize

rminal point

parametriz

d curve only over

ed

a closed

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

graphed does not.

Example Relations2Describe 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= = -

Section 1.4 – Parametric Equations To convert from a set of parametric equations to

the Cartesian equation of the curve, solve either equation for t and plug into the other equation.

xxy

xy

xy

xt

ttytx

99

39

3

9 3 :Ex

2

2

2

2

Section 1.4 – Parametric Equations You try: Describe the graph of the relation

determined by

, 2 , 0.x t y t t

Section 1.4 – Parametric Equations 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.

Section 1.4 – Parametric Equations Circles can be parameterized as

where a is the radius

Ellipses can be parameterized as

taytax sin and cos

222

2

222

2222

2222

1

sincos

sincos

sincos

ayx

a

tta

tata

tatayx

tbytax sin and cos

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.

Section 1.4 – Parametric Equations You try: Graph the

parametric curve

Find the Cartesian equation for a curve that presents the same parametric curve. What portion of the graph of the Cartesian equation is traced by the parametric curve? Indicate the direction in which the curve is traced and the initial and terminal points, if any.

2cos , 3sin , 0 2 .x t y t t

Lines and Other Curves

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

Section 1.4 – Parametric Equations To parameterize a line segment with endpoints

let This line goes through the pointwhen t= 0.To find a and b, make the line reach when t=1.

2211 , and , yxyx. and 11 btyyatxx

11, yx

22 , yx

10 ,2 and 73 So

1 7

121 134

2 3

1,4 to2,3 fromsegment line :Ex

ttytx

ba

ba

btyatx

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

= = + = +

=

Section 1.4 – Parametric Equations You try: Draw and identify

the graph of the parametric curve determined by

2 , 3 1, 0 1.x t y t t

Section 1.4 – Parametric Equations You try: Find a parameterization for the line

segment with endpoints (2, -3) and (1, 4).