copyright © 2007 pearson education, inc. slide 6-1
TRANSCRIPT
Copyright © 2007 Pearson Education, Inc. Slide 6-1
Copyright © 2007 Pearson Education, Inc. Slide 6-2
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
Copyright © 2007 Pearson Education, Inc. Slide 6-3
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.
Copyright © 2007 Pearson Education, Inc. Slide 6-4
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
Copyright © 2007 Pearson Education, Inc. Slide 6-5
6.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent
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
Copyright © 2007 Pearson Education, Inc. Slide 6-6
6.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent
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.
Copyright © 2007 Pearson Education, Inc. Slide 6-7
6.4 Example 1: Graph of a Parametric Equation and Its Rectangular Equivalent
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.)
Copyright © 2007 Pearson Education, Inc. Slide 6-8
6.4 Example 2: Graph of a Parametric Equation and Its Rectangular Equivalent
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
Copyright © 2007 Pearson Education, Inc. Slide 6-9
6.4 Example 2: Graph of a Parametric Equation and Its Rectangular Equivalent
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.
Copyright © 2007 Pearson Education, Inc. Slide 6-10
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).
Copyright © 2007 Pearson Education, Inc. Slide 6-11
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
Copyright © 2007 Pearson Education, Inc. Slide 6-12
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
Copyright © 2007 Pearson Education, Inc. Slide 6-13
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.