Copyright © 2007 Pearson Education, Inc. Slide 6-1

Copyright © 2007 Pearson Education, Inc. Slide 6-1


Chapter 6: Analytic Geometry

6.1 Circles and Parabolas

6.2 Ellipses and Hyperbolas

6.3 Summary of the Conic Sections

6.4 Parametric Equations


6.4 Parametric Equations

Parametric Equations of a Plane Curve

A plane curve is a set of points (x, y) such that x = f (t), y = g(t), and f and g are both defined on an interval I. The equations x = f (t) and y = g(t) are parametric equations with parameter t.


6.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent

Example For the plane curve defined by theparametric equations

graph the curve and then find an equivalent rectangular equation.

Analytic Solution Make a table of corresponding values of t, x, and y over the domain t and plot the points.

],3,3[ interval in the for ,32,2 ttytx

).3(9, :pointFirst 33)3(2)3(

9)3()3(e.g. 2

y

x



The arrow heads indicate

the direction the curve takes

as t increases.

9

9

3

7

4

2

531

101

101-

1-3-

49

2-3-

y

x

t



To find the equivalent rectangular form, eliminate the parameter t.

This is a horizontal parabola that opens to the right. Since t is in [–3, 3], x is in [0, 9] and y is in [–3, 9] . The rectangularequation is

22

2 )3(41

23

23

32

yy

tx

yt

ty

9].[0,in for ,)3(41 2 xyx

Use this equation because it leads to a unique solution.



Graphing Calculator Solution

Set the calculator in parametric mode where the variable is t and let X1T = t2 and Y1T = 2t + 3. (We have been in rectangular mode using variable x.)



Example Graph the plane curve defined by

Solution Get the equivalent rectangular form by

substitution of t. Since t is in [–2, 2], x is in [1, 9].

].2,2[in for ,4,52 2 ttytx

22

25

44

25

52

xty

xt

xt



116

)5(4

16)5(4

)5(164

25

4

22

22

22

22

xy

xy

xy

xy

This represents a complete ellipse. By definition, y 0. Therefore, the graph is the upper half of the ellipse only.


6.4 Graphing a Line Defined Parametrically

Example Graph the plane curve defined by x = t2,y = t2, and then find an equivalent rectangular form.

Solution x = t2 = y, so y = x. To be equivalent, however, the rectangular equation must be given as

y = x, x 0 (half the line y = x since t2 0).


6.4 Alternative Forms of Parametric Equations

• Parametric representations of a curve are not always unique.

• One simple parametric representation for y = f (x), with domain X, is

Example Give two parametric representations for

the parabola

Solution

.in for ),(, Xttfytx

.1)2( 2 xy

.in ,1 then ,2Let .2

.in ,1)2( then ,Let .12

2

ttytx

ttytx


6.4 Projectile Motion Application

• The path of a moving object with position (x, y) can be given by the functions where t represents time.

Example The motion of a projectile moving in a direction at a 45º angle with the horizontal (neglecting air resistance) is given by

where t is in seconds, 0 is the initial speed, x and y are in feet, and k > 0. Find the rectangular form of the equation.

),(),( tgytfx

],,0[in for ,1622

,22 2

00 ktttytx


Solution Solve the first equation for t and substitute

the result into the second equation.

6.4 Projectile Motion Application

22

0

2

000

20

00

32

12

216

12

222

1622

12

222

xxy

xxy

tty

xttx

A vertical parabola that opens downward.

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Copyright © 2007 Pearson Education, Inc. Slide 6-1