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D. Application of the Analytic Parabola

# D. Application of the Analytic Parabola

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Section 8.4 The Analytic Parabola; More on Nonlinear Systems

8.4 EXERCISES

CONCEPTS AND VOCABULARY

Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.

1. The equation x ϭ ay2 ϩ by ϩ c is that of a(n)

parabola, opening to the

if

a 7 0 and to the left if

.

2. If point P is on the graph of a parabola with

directrix D, the distance from P to line D is equal

to the distance between P and the

.

3. Given y2 ϭ 4px, the focus is at

equation of the directrix is

4. Given x2 ϭ Ϫ16y, the value of p is

the coordinates of the focus are

and the

.

5. Discuss/Explain how to find the vertex, directrix,

and focus from the equation 1x Ϫ h2 2 ϭ 4p1y Ϫ k2.

and

.

6. If a horizontal parabola has a vertex of 12, Ϫ3)

with a 7 0, what can you say about the

y-intercepts? Will the graph always have an

x-intercept? Explain.

Find the x- and y-intercepts (if they exist) and the vertex of

the parabola. Then sketch the graph by using symmetry

and a few additional points or completing the square and

shifting a parent function. Scale the axes as needed to

comfortably fit the graph and state the domain and range.

7. y ϭ x2 Ϫ 2x Ϫ 3

8. y ϭ x2 ϩ 6x ϩ 5

9. y ϭ Ϫ2x2 ϩ 8x ϩ 10 10. y ϭ Ϫ3x2 Ϫ 12x ϩ 15

11. y ϭ 2x2 ϩ 5x Ϫ 7

12. y ϭ 2x2 Ϫ 7x ϩ 3

Find the x- and y-intercepts (if they exist) and the vertex

of the graph. Then sketch the graph using symmetry

and a few additional points (scale the axes as needed).

Finally, state the domain and range of the relation.

31. y ϭ 1x Ϫ 22 2 ϩ 3

32. y ϭ 1x ϩ 22 2 Ϫ 4

35. x ϭ 21y Ϫ 32 2 ϩ 1

36. x ϭ Ϫ21y ϩ 32 2 Ϫ 5

33. x ϭ 1y Ϫ 32 2 ϩ 2

34. x ϭ 1y ϩ 12 2 Ϫ 4

Find the vertex, focus, and directrix for the parabolas

defined by the equations given, then use this

information to sketch a complete graph (illustrate and

name these features). For Exercises 49 to 60, also

include the focal chord.

37. x2 ϭ 8y

38. x2 ϭ 16y

39. x2 ϭ Ϫ24y

40. x2 ϭ Ϫ20y

41. x2 ϭ 6y

42. x2 ϭ 18y

13. x ϭ y2 Ϫ 2y Ϫ 3

14. x ϭ y2 Ϫ 4y Ϫ 12

43. y2 ϭ Ϫ4x

44. y2 ϭ Ϫ12x

15. x ϭ Ϫy2 ϩ 6y ϩ 7

16. x ϭ Ϫy2 ϩ 8y Ϫ 12

45. y2 ϭ 18x

46. y2 ϭ 20x

17. x ϭ Ϫy2 ϩ 8y Ϫ 16

18. x ϭ Ϫy2 ϩ 6y Ϫ 9

47. y2 ϭ Ϫ10x

48. y2 ϭ Ϫ14x

Sketch by completing the square and using symmetry

and shifts of a basic function. Be sure to find the x- and

y-intercepts (if they exist) and the vertex of the graph,

then state the domain and range of the relation.

49. x2 Ϫ 8x Ϫ 8y ϩ 16 ϭ 0

50. x2 Ϫ 10x Ϫ 12y ϩ 25 ϭ 0

51. x2 Ϫ 14x Ϫ 24y ϩ 1 ϭ 0

19. x ϭ y2 Ϫ 6y

20. x ϭ y2 Ϫ 8y

52. x2 Ϫ 10x Ϫ 12y ϩ 1 ϭ 0

21. x ϭ y2 Ϫ 4

22. x ϭ y2 Ϫ 9

53. 3x2 Ϫ 24x Ϫ 12y ϩ 12 ϭ 0

23. x ϭ Ϫy2 ϩ 2y Ϫ 1

24. x ϭ Ϫy2 ϩ 4y Ϫ 4

54. 2x2 Ϫ 8x Ϫ 16y Ϫ 24 ϭ 0

25. x ϭ y2 ϩ y Ϫ 6

26. x ϭ y2 ϩ 4y Ϫ 5

55. y2 Ϫ 12y Ϫ 20x ϩ 36 ϭ 0

27. x ϭ y2 Ϫ 10y ϩ 4

28. x ϭ y2 ϩ 12y Ϫ 5

56. y2 Ϫ 6y Ϫ 16x ϩ 9 ϭ 0

29. x ϭ 3 Ϫ 8y Ϫ 2y2

30. x ϭ 2 Ϫ 12y ϩ 3y2

57. y2 Ϫ 6y ϩ 4x ϩ 1 ϭ 0

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CHAPTER 8 Analytic Geometry and the Conic Sections

58. y2 Ϫ 2y ϩ 8x ϩ 9 ϭ 0

75.

76.

y

y

2

6

59. 2y2 Ϫ 20y ϩ 8x ϩ 2 ϭ 0

(4, 0)

Ϫ4

4

60. 3y Ϫ 18y ϩ 12x ϩ 3 ϭ 0

2

(Ϫ4, 2)

Ϫ2

For Exercises 61–72, find the equation of the parabola in

standard form that satisfies the conditions given.

Ϫ4

2

Ϫ2

4

6

8

Ϫ4

2

x

y ϭ Ϫ6

Ϫ2

61. focus: (0, 2)

directrix: y ϭ Ϫ2

62. focus: (0, Ϫ3)

directrix: y ϭ 3

63. focus: (4, 0)

directrix: x ϭ Ϫ4

64. focus: (Ϫ3, 0)

directrix: x ϭ 3

65. focus: (0, Ϫ5)

directrix: y ϭ 5

66. focus: (5, 0)

directrix: x ϭ Ϫ5

77. e

x2 ϩ y2 ϭ 25

2x2 Ϫ 3y2 ϭ 5

78. e

y2 Ϫ x2 ϭ 12

x2 ϩ y2 ϭ 20

67. vertex: (2, Ϫ2)

focus: (Ϫ1, Ϫ2)

68. vertex: (4, 1)

focus: (1, 1)

79. e

80. e

69. vertex: (4, Ϫ7)

focus: (4, Ϫ4)

70. vertex: (Ϫ3, Ϫ4)

focus: (Ϫ3, Ϫ1)

x2 Ϫ y ϭ 4

y2 Ϫ x2 ϭ 16

2x2 Ϫ 3y2 ϭ 38

x2 ϩ 5y ϭ 35

5x2 Ϫ 2y2 ϭ 75

2x2 ϩ 3y2 ϭ 125

82. e

71. focus: (3, 4)

directrix: y ϭ 0

72. focus: (Ϫ1, 2)

directrix: x ϭ Ϫ5

81. e

3x2 Ϫ 7y2 ϭ 20

4x2 ϩ 9y2 ϭ 45

Solve using substitution or elimination, then verify

your solutions by graphing the system on a graphing

calculator.

Solve the following systems using a graphing calculator.

Round approximate solutions to three decimal places.

For the graphs in Exercises 73–76, only two of the

following four features are displayed: vertex, focus,

directrix, and endpoints of the focal chord. Find the

remaining two features and the equation of the parabola.

73.

74.

y

4

(1, 4)

6

yϭ5

4

Ϫ2

2

4

6

x

2

Ϫ2

Ϫ4

x ϭ Ϫ3

(1, Ϫ4)

Ϫ4

1x Ϫ 22 2 ϩ y2 ϭ 20

83. • x2

ϩyϭ8

4

84. e

y

2

(2, 2)

2

4

6

10 x

Ϫ2

(Ϫ2, 2)

Ϫ6

2

8

10 x

Ϫ2

85. e

86. e

4x2 Ϫ 1y Ϫ 122 2 ϭ 441

x2 ϩ 1y Ϫ 122 2 ϭ 1764

1x Ϫ 102 2 ϩ 1y Ϫ 102 2 ϭ 144

1x ϩ 42 2 ϩ y2 ϭ 144

31x ϩ 242 2 ϩ y2 ϭ 196

4x ϩ 4y ϭ Ϫ31

WORKING WITH FORMULAS

87. The area of a right parabolic segment: A ‫ ؍‬23 ab

y

A right parabolic segment is that part of a parabola formed by a line perpendicular to its

axis, which cuts the parabola. The area of this segment is given by the formula shown, where

b is the length of the chord cutting the parabola and a is the perpendicular distance from

the vertex to this chord. What is the area of the parabolic segment shown in the figure?

88. The arc length of a right parabolic segment:

b2

4a ؉ 2b2 ؉ 16a2

1

2b2 ؉ 16a2 ؉

lna

b

2

8a

b

Although a fairly simple concept, finding the length of the parabolic arc traversed by a

projectile requires a good deal of computation. To find the length of the arc ABC

shown, we use the formula given where a is the maximum height attained by the

projectile, b is the horizontal distance it traveled, and “ln” represents the natural log

function. Suppose a baseball thrown from centerfield reaches a maximum height of

20 ft and traverses an arc length of 340 ft. Will the ball reach the catcher 310 ft away

without bouncing?

10

8

(Ϫ3, 4) 6

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

2 4 6 8 10 x

B

a

A

C

b

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APPLICATIONS

cross section of a typical car

headlight can be modeled by an

equation similar to 25x ϭ 16y2,

where x and y are in inches and

x ʦ 30, 4 4 . Use this information

to graph the relation for the

indicated domain.

90. Parabolic flashlights: The cross section of a

typical flashlight reflector can be modeled by an

equation similar to 4x ϭ y2, where x and y are in

centimeters and x ʦ 3 0, 2.254 . Use this information

to graph the relation for the indicated domain.

technicians at professional sports

as they move along the sidelines. If a

two-dimensional cross section of the

receiver is modeled by the equation

y2 ϭ 54x, and is 36 in. in diameter,

how deep is the parabolic receiver?

What is the location of the focus?

[Hint: Graph the parabola on the

coordinate grid (scale the axes).]

Exercise 91

y

x

92. Parabolic sound receivers: Private investigators

will often use a smaller and less expensive

parabolic receiver (see Exercise 91) to gather

information for their clients. If a two-dimensional

cross section of the receiver is modeled by the

equation y2 ϭ 24x, and the receiver is 12 in. in

diameter, how deep is the parabolic dish? What is

the location of the focus?

known as S.E.T.I. (Search for Extra-Terrestrial

Intelligence) involves a group of scientists using

possible intelligent species in outer space. The radio

telescopes are actually parabolic dishes that vary in

size from a few feet to hundreds of feet in diameter.

If a particular radio telescope is 100 ft in diameter

753

Section 8.4 The Analytic Parabola; More on Nonlinear Systems

and has a cross

section modeled

by the equation

x2 ϭ 167y, how deep

is the parabolic dish?

What is the location

of the focus? [Hint:

Graph the parabola

on the coordinate

grid (scale the axes).]

y

94. Solar furnace:

x

Another form of

technology that uses a

parabolic dish is

called a solar furnace.

In general, the rays of

the Sun are reflected

by the dish and

concentrated at the

focus, producing extremely high temperatures.

Suppose the dish of one of these parabolic reflectors

has a 30-ft diameter and a cross section modeled by

the equation x2 ϭ 50y. How deep is the parabolic

dish? What is the location of the focus?

95. Commercial flashlights: The reflector of a large,

commercial flashlight has the shape of a parabolic

dish, with a diameter of 10 cm and a depth of 5 cm.

What equation will the engineers and technicians

use for the manufacture of the dish? How far from

the vertex (the lowest point of the dish) will the bulb

be placed? (Hint: Analyze the information using a

coordinate system.)

96. Industrial spotlights: The reflector of an industrial

spotlight has the shape of a parabolic dish with a

diameter of 120 cm. What is the depth of the dish if

the correct placement of the bulb is 11.25 cm above

the vertex (the lowest point of the dish)? What

equation will the engineers and technicians use for

the manufacture of the dish? (Hint: Analyze the

information using a coordinate system.)

EXTENDING THE CONCEPT

97. In a study of quadratic graphs from the equation

y ϭ ax2 ϩ bx ϩ c, no mention is made of a

parabola’s focus and directrix. Generally, when

a Ն 1, the focus of a parabola is very near its

vertex. Complete the square of the function

y ϭ 2x2 Ϫ 8x and write the result in the form

1x Ϫ h2 2 ϭ 4p1y Ϫ k2 . What is the value of p?

What are the coordinates of the vertex?

98. Like the ellipse and hyperbola, the focal chord of a

parabola (also called the latus rectum) can be used

to help sketch its graph. From our earlier work, we

know the endpoints of the focal chord are 2p units

from the focus. Write the equation

Ϫ12y ϩ 15 ϭ x2 Ϫ 6x in the form

4p1y Ϯ k2 ϭ 1x Ϯ h2 2, and use the endpoints of the

focal chord to help graph the parabola.

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CHAPTER 8 Analytic Geometry and the Conic Sections

99. (6.2) Construct a system of three equations in three

variables using the equation y ϭ ax2 ϩ bx ϩ c and

the points 1Ϫ3, 32, (0, 6), and 11, Ϫ12. Then use a

matrix equation to find the equation of the parabola

containing these points.

100. (3.1/4.2) Find all roots (real and complex) of the

equation x6 Ϫ 64 ϭ 0. (Hint: Begin by factoring as

the difference of two perfect squares.)

101. (2.1) What are the characteristics of an even function?

What are the characteristics of an odd function?

102. (4.2/4.3) Use the function

f 1x2 ϭ x5 ϩ 2x4 ϩ 17x3 ϩ 34x2 Ϫ 18x Ϫ 36

to comment and give illustrations of the tools

available for working with polynomials:

(a) synthetic division, (b) rational roots theorem,

(c) the remainder and factor theorems, (d) the tests

for x ϭ Ϫ1 and x ϭ 1, (e) the upper/lower bounds

property, (f) Descartes’ rule of signs, and (g) roots

of multiplicity (bounces, crosses, alternating

intervals).

MAKING CONNECTIONS

Making Connections: Graphically, Symbolically, Numerically, and Verbally

Eight graphs (a) through (h) are given. Match the characteristics shown in 1 through 16 to one of the eight graphs.

y

(a)

y

(b)

5

Ϫ5

Ϫ5

5 x

Ϫ5

5 x

y

Ϫ5

5 x

Ϫ5

5 x

Ϫ5

1. ____ 1x Ϫ 12 2 ϩ 1y Ϫ 12 2 ϭ 16

2. ____ y ϭ

Ϫ5

5 x

1

1x Ϫ 12 2

4

3. ____ foci at 1Ϫ1, 1 Ϯ 252

y

(h)

5

Ϫ5

5 x

Ϫ5

y

(g)

5

Ϫ5

5

Ϫ5

y

(f)

5

y

(d)

5

Ϫ5

Ϫ5

(e)

y

(c)

5

5 x

5

Ϫ5

Ϫ5

5 x

Ϫ5

9. ____ vertices at (Ϫ3, 1) and (5, 1)

10. ____ 41y Ϫ 12 2 Ϫ 1x ϩ 12 2 ϭ 16

11. ____ center at (0, Ϫ2)

4. ____ transverse axis y ϭ 1

12. ____ focus at (1, 1)

1

5. ____ x ϭ Ϫ 1y ϩ 12 2 ϩ 3

2

13. ____ 41x ϩ 22 2 ϩ 1y ϩ 12 2 ϭ 16

7. ____ 41x ϩ 12 2 Ϫ 1y Ϫ 12 2 ϭ 16

15. ____ axis of symmetry: y ϭ Ϫ1

6. ____ domain: x ʦ 3Ϫ3, 5 4 , range: y ʦ 3 Ϫ3, 54

8. ____ x ϩ 1y Ϫ 22 ϭ 9

2

2

14. ____ 1x Ϫ 12 2 ϩ 41y Ϫ 12 2 ϭ 16

16. ____ domain: x ʦ 3Ϫ4, 04 , range: y ʦ 3Ϫ5, 34

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Summary and Concept Review

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SUMMARY AND CONCEPT REVIEW

SECTION 8.1

A Brief Introduction to Analytical Geometry

KEY CONCEPTS

• The midpoint and distance formulas play important roles in the study of analytical geometry:

x2 ϩ x1 y2 ϩ y1

midpoint: 1x, y2 ϭ a

,

b distance: d ϭ 21x2 Ϫ x1 2 2 ϩ 1y2 Ϫ y1 2 2

2

2

• The perpendicular distance from a point to a line is the length of the line segment perpendicular to the given line

with the given point and the point of intersection as endpoints.

• Using these tools, we can verify or construct relationships between points, lines, and curves in the plane; verify

properties of geometric figures; prove theorems from Euclidean geometry; and construct relationships that define

the conic sections.

EXERCISES

1. Verify the closed figure with vertices (Ϫ3, Ϫ4), (Ϫ5, 4), (3, 6), and (5, Ϫ2) is a square.

2. Find the equation of the circle that circumscribes the square in Exercise 1.

3. A theorem from Euclidean geometry states: If any two points are equidistant from the endpoints of a line segment,

they are on the perpendicular bisector of the segment. Determine if the line through (Ϫ3, 6) and (6, Ϫ9) is the

perpendicular bisector of the segment through (Ϫ5, Ϫ2) and (5, 4).

4. Four points are given here. Verify that the distance from each point to the line y ϭ Ϫ1 is the same as the distance

from the given point to the fixed point (0, 1): (Ϫ6, 9), (Ϫ2, 1), (4, 4), and (8, 16).

SECTION 8.2

The Circle and the Ellipse

KEY CONCEPTS

• The equation of a circle centered at (h, k) with radius r is 1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2.

1x Ϫ h2 2

1y Ϫ k2 2

ϩ

ϭ 1, showing the horizontal and

• Dividing both sides by r2, we obtain the standard form

r2

r2

vertical distance from center to graph is r.

1x Ϫ h2 2

1y Ϫ k2 2

The

equation

of

an

ellipse

in

standard

form

is

ϩ

ϭ 1. The center of the ellipse is (h, k), with

a2

b2

horizontal distance a and vertical distance b from center to graph.

y

• Given two fixed points f1 and f2 in a plane (called the foci), an ellipse is the set of all

(x, y)

points (x, y) such that the distance from the first focus to (x, y), plus the distance from

d

d2

1

the second focus to (x, y), remains constant.

(a, 0)

(Ϫa, 0)

x

• For an ellipse, the distance from center to vertex is greater than the distance c from

(Ϫc, 0)

(c, 0)

center to one focus.

d1 ϩ d2 ϭ k

• To find the foci of a horizontal ellipse, use: a2 ϭ b2 ϩ c2 (since a 7 c), or c2 ϭ |a2 Ϫ b2 |.

EXERCISES

Sketch the graph of each equation in Exercises 5 through 9.

5. x2 ϩ y2 ϭ 16

6. x2 ϩ 4y2 ϭ 36

7. 9x2 ϩ y2 Ϫ 18x Ϫ 27 ϭ 0

2

2

1x ϩ 32

1y Ϫ 22

8. x2 ϩ y2 ϩ 6x ϩ 4y ϩ 12 ϭ 0

9.

ϩ

ϭ1

16

9

10. Find the equation of the ellipse with minor axis of length 6 and foci at (Ϫ4, 0) and (4, 0). Then graph the equation

on a graphing calculator using a “friendly” window and use the TRACE feature to locate four additional points on the

graph with coordinates that are rational.

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CHAPTER 8 Analytic Geometry and the Conic Sections

11. Find the equation of the ellipse with vertices at (a) (Ϫ13, 0) and (13, 0), foci at (Ϫ12, 0) and (12, 0); (b) foci at

(0, Ϫ16) and (0, 16), major axis: 40 units.

12. Write the equation in standard form and sketch the graph, noting all of the characteristic features of the ellipse.

4x2 ϩ 25y2 Ϫ 16x Ϫ 50y Ϫ 59 ϭ 0

SECTION 8.3

The Hyperbola

KEY CONCEPTS

1x Ϫ h2 2

1y Ϫ k2 2

Ϫ

ϭ 1. The center of the hyperbola

a2

b2

is (h, k) with horizontal distance a from center to vertices, and vertical distance b from center to the midpoint of

y

the sides of the central rectangle.

(x, y)

• Given two fixed points f1 and f2 in a plane (called the foci), a hyperbola is the set of all

d1

points (x, y) such that the distance from one focus to point (x, y), less the distance from

d

2

(Ϫa, 0)

(a, 0)

the other focus to (x, y), remains a positive constant: |d1 Ϫ d2 | ϭ k.

(Ϫc, 0)

(c, 0) x

• For a hyperbola, the distance from center to one vertex is less than the distance from center

to the focus c.

• To find the foci of a hyperbola: c2 ϭ a2 ϩ b2 (since c 7 a).

|d1 Ϫ d2| ϭ k

• The equation of a horizontal hyperbola in standard form is

EXERCISES

Sketch the graph of each equation in Exercises 13 through 17, indicating the center, vertices, and asymptotes.

1y Ϫ 32 2

1x ϩ 22 2

1y Ϫ 12 2

1x ϩ 22 2

13. 4y2 Ϫ 25x2 ϭ 100

14.

15.

Ϫ

ϭ1

Ϫ

ϭ1

16

9

9

4

16. 9y2 Ϫ x2 Ϫ 18y Ϫ 72 ϭ 0

17. x2 Ϫ 4y2 Ϫ 12x Ϫ 8y ϩ 16 ϭ 0

4

18. Find the equation of the hyperbola with vertices at (Ϫ3, 0) and (3, 0), and asymptotes of y ϭ Ϯ x. Then graph the

3

equation on a graphing calculator using a “friendly” window and use the TRACE feature to locate two additional

points with rational coordinates.

19. Find the equation of the hyperbola with (a) vertices at (Ϯ15, 0), foci at (Ϯ17, 0), and (b) foci at (0, Ϯ5) with

vertical dimension of central rectangle 8 units.

20. Write the equation in standard form and sketch the graph, noting all of the characteristic features of the hyperbola.

4x2 Ϫ 9y2 Ϫ 40x ϩ 36y ϩ 28 ϭ 0

SECTION 8.4

The Analytic Parabola; More on Nonlinear Systems

KEY CONCEPTS

• Horizontal parabolas have equations of the form x ϭ ay2 ϩ by ϩ c; a

0.

• A horizontal parabola will open to the right if a 7 0, and to the left if a 6 0. The axis of symmetry is y ϭ

Ϫb

or by completing the square and writing

2a

2

the equation in shifted form: 1x Ϫ h2 ϭ a1y Ϫ k2 .

• Given a fixed point f (called the focus) and fixed line D (called the directrix) in the plane, a

parabola is the set of all points (x, y) such that the distance from f to (x, y) is equal to the

distance from (x, y) to line D.

The

equation x2 ϭ 4py describes a vertical parabola, opening upward if p 7 0, and opening

downward if p 6 0.

• The equation y2 ϭ 4px describes a horizontal parabola, opening to the right if p 7 0, and

opening to the left if p 6 0.

with the vertex (h, k) found by evaluating at y ϭ

Ϫb

,

2a

y

d 1 ϭ d2

f

d1

(x, y)

d2

Vertex

x

D

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Practice Test

757

• p is the distance from the vertex to the focus (or from the vertex to the directrix).

• The focal chord of a parabola is a line segment that contains the focus and is parallel the directrix, with its

endpoints on the graph. It has a total length of Ϳ4pͿ, meaning the distance from the focus to a point on the graph

(as described) is Ϳ2pͿ. It is commonly used to assist in drawing a graph of the parabola.

EXERCISES

For Exercises 21 and 22, find the vertex and x- and y-intercepts if they exist. Then sketch the graph using symmetry

and a few points or by completing the square and shifting a parent function.

21. x ϭ y2 Ϫ 4

22. x ϭ y2 ϩ y Ϫ 6

For Exercises 23 and 24, find the vertex, focus, and directrix for each parabola. Then sketch the graph using this

information and the focal chord. Also graph the directrix.

23. x2 ϭ Ϫ20y

24. x2 Ϫ 8x Ϫ 8y ϩ 16 ϭ 0

25. Identify the conic sections in the system, then solve. Check solutions using the intersection-of-graphs method and

a graphing calculator.

e

1x ϩ 72 2 ϩ 42 ϭ 20

y2 Ϫ 7 ϭ x

PRACTICE TEST

By inspection only (no graphing, completing the square,

etc.), match each equation to its correct description.

1. x ϩ y Ϫ 6x ϩ 4y ϩ 9 ϭ 0

2

2

2. 4y2 ϩ x2 Ϫ 4x ϩ 8y ϩ 20 ϭ 0

3. y Ϫ x2 Ϫ 4x ϩ 20 ϭ 0

4. x2 Ϫ 4y2 Ϫ 4x ϩ 12y ϩ 20 ϭ 0

a. Parabola b. Hyperbola c. Circle

d. Ellipse

Graph each conic section, and label the center,

vertices, foci, focal chords, asymptotes, and other

important features where applicable.

5. 1x Ϫ 42 2 ϩ 1y ϩ 32 2 ϭ 9

1x Ϫ 22 2

1y ϩ 32 2

ϩ

ϭ1

6.

16

1

1x ϩ 32 2

1y Ϫ 42 2

Ϫ

ϭ1

7.

9

4

8. x2 ϩ y2 Ϫ 10x ϩ 4y ϩ 4 ϭ 0

9. 9x2 ϩ 4y2 ϩ 18x Ϫ 24y ϩ 9 ϭ 0

10. 9x2 Ϫ 4y2 ϩ 18x Ϫ 24y Ϫ 63 ϭ 0

11. x ϭ 1y ϩ 32 2 Ϫ 2

12. y2 Ϫ 6y Ϫ 12x Ϫ 15 ϭ 0

Solve each nonlinear system using any method.

2y2 Ϫ x2 ϭ 4

4x2 Ϫ y2 ϭ 16

b. e 2

yϪxϭ2

x ϩ y2 ϭ 8

14. A support bracket on the frame of a large ship is a

steel right triangle with a hypotenuse of 25 ft and a

perimeter of 60 ft. Find the lengths of the other sides

using a system of nonlinear equations.

13. a. e

15. Find an equation for the circle whose center is at

(Ϫ2, 5) and whose graph goes through the point

(0, 3).

16. Find the equation of the ellipse (in standard form)

with vertices at (Ϫ4, 0) and (4, 0) with foci located

at (Ϫ2, 0) and (2, 0). Then use a graphing calculator

to determine where this ellipse and the circle

x2 ϩ y2 ϭ 13 intersect.

17. The orbit of Mars around the Sun is elliptical, with

the Sun at one focus. When the orbit is expressed as

a central ellipse on the coordinate grid, its equation

y2

x2

ϩ

ϭ 1, with a and b in

is

1141.652 2

1141.032 2

millions of miles. Use this information to find the

aphelion of Mars (distance from the Sun at its

farthest point), and the perihelion of Mars (distance

from the Sun at its closest point).

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CHAPTER 8 Analytic Geometry and the Conic Sections

Determine the equation of each relation and state its domain and range. For the parabola and the ellipse, also

give the location of the foci.

18.

19.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

(Ϫ4, 1)

1 2 3 4 5 x

(1, Ϫ4)

20.

y

10

8

6

4

2

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

y

(1, 6)

(Ϫ3, 6)

(6, 1)

(Ϫ6, 0)

2 4 6 8 10 x

10

8

6

4

2

(0, 0)

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2

Ϫ4

Ϫ6

(Ϫ3, Ϫ6)Ϫ8

Ϫ10

(1, Ϫ4)

2 4 6 8 10 x

CALCULATOR EXPLORATION AND DISCOVERY

Elongation and Eccentricity

Technically speaking, a circle is an ellipse with both foci at the center. As the distance between foci increases, the

ellipse becomes more elongated. We saw other instances of elongation in stretches and compressions of parabolic

graphs, and in hyperbolic graphs where the asymptotic slopes varied depending on the values a and b. The measure

c

used to quantify this elongation is called the eccentricity e, and is determined by the ratio e ϭ . For this Exploration and

a

Discovery, we’ll use the repeat graph feature of a graphing calculator to explore the eccentricity of the graph of a conic.

The “repeat graph” feature enables you to graph a family of curves by enclosing changes in a parameter in braces “{ }.”

For instance, entering 5Ϫ2, Ϫ1, 0, 1, 26 X ϩ 3 as Y1 on the Y= screen will automatically graph these five lines:

y ϭ Ϫ2x ϩ 3, y ϭ Ϫx ϩ 3, y ϭ 3,

y ϭ x ϩ 3, and y ϭ 2x ϩ 3.

We’ll use this feature to graph a family of ellipses, observing the result and calculating the eccentricity for each

y2

x2

curve in the family. The standard form is 2 ϩ 2 ϭ 1, which we’ll solve for y and enter as Y1 and Y2. After

a

b

x2

simplification the result is y ϭ Ϯ b 1 Ϫ 2 , but for this investigation we’ll use the constant b ϭ 2 and vary the

B

a

x2

parameter a using the values a ϭ 2, 4, 6, and 8. The result is y ϭ 2 1 Ϫ

. Note from Figure 8.43 that

B

54, 16, 36, 646

we’ve set Y2 ϭ ϪY1 to graph the lower half of the ellipse. Using the “friendly window” shown (Figure 8.44) gives

the result shown in Figure 8.45, where we see the ellipse is increasingly elongated in the horizontal direction (note

when a ϭ 2 the result is a circle since a ϭ b).

Figure 8.45

Figure 8.43

Figure 8.44

6.2

Ϫ9.4

9.4

Ϫ6.2

Using a ϭ 2, 4, 6, and 8 with b ϭ 2 in the foci formula c ϭ 2a2 Ϫ b2 gives c ϭ 0, 2 13, 4 12, and 2 115,

respectively, with these eccentricities:

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Strengthening Core Skills

759

2115

0 2 13 412

eϭ ,

,

. While difficult to see in radical form, we find that the eccentricity of an ellipse always

, and

2 4

6

8

satisfies the inequality 0 6 e 6 1 (excluding the circle ϭ ellipse case). To two decimal places, the values are e ϭ 0,

0.87, 0.94, and 0.97, respectively.

c

As a final note, it’s interesting how the e ϭ definition of eccentricity relates to our everyday use of the word

a

“eccentric.” A normal or “noneccentric” person is thought to be well-rounded, and sure enough e ϭ 0 produces a

well-rounded figure—a circle. A person who is highly eccentric is thought to be far from the norm, deviating greatly

from the center, and greater values of e produce very elongated ellipses.

Exercise 1: Perform a similar exploration using a family of hyperbolas. What do you notice about the eccentricity?

Exercise 2: Perform a similar exploration using a family of parabolas. What do you notice about the eccentricity?

STRENGTHENING CORE SKILLS

Ellipses and Hyperbolas with Rational/Irrational Values of a and b

Using the process known as completing the square, we were able to convert from the polynomial form of a conic section

to the standard form. However, for some equations, values of a and b are somewhat difficult to identify, since the

coefficients are not factors. Consider the equation 20x2 ϩ 120x ϩ 27y2 Ϫ 54y ϩ 192 ϭ 0, the equation of an ellipse.

20x2 ϩ 120x ϩ 27y2 Ϫ 54y ϩ 192 ϭ 0

201x2 ϩ 6x ϩ ___ 2 ϩ 271y2 Ϫ 2y ϩ ___ 2 ϭ Ϫ192

201x2 ϩ 6x ϩ 92 ϩ 271y2 Ϫ 2y ϩ 12 ϭ Ϫ192 ϩ 27 ϩ 180

201x ϩ 32 2 ϩ 271y Ϫ 12 2 ϭ 15

41x ϩ 32 2

91y Ϫ 12 2

ϩ

ϭ1

3

5

original equation

subtract 192

complete the square in x and y

factor and simplify

standard form

Unfortunately, we cannot easily identify the values of a and b, since the coefficients of each binomial square are

not “1.” In these cases, we can write the equation in standard form by using a simple property of fractions—the

numerator and denominator of any fraction can be divided by the same quantity to obtain an equivalent fraction.

1y Ϫ 12 2

1x ϩ 32 2

ϩ

ϭ 1.

Although the result may look odd, it can nevertheless be applied here, giving a result of

3

5

4

9

We can now identify a and b by writing these denominators in squared form, which gives the following expression:

1x ϩ 32 2

1y Ϫ 12 2

15

13

ϩ

ϭ 1. The values of a and b are now easily seen as a ϭ

Ϸ 0.866 and b ϭ

Ϸ 0.745.

2

2

2

3

13

15

a

b

a

b

2

3

Use this idea to complete the following exercises.

Exercise 1: Write the equation in standard form, then identify the values of a and b and use them to graph the ellipse.

41x ϩ 32 2

49

ϩ

251y Ϫ 12 2

36

ϭ1

Exercise 2: Write the equation in standard form, then identify the values of a and b and use them to graph the hyperbola.

91x ϩ 32 2

80

Ϫ

41y Ϫ 12 2

81

ϭ1

Exercise 3: Identify the values of a and b by writing the equation 100x2 Ϫ 400x Ϫ 18y2 Ϫ 108y ϩ 230 ϭ 0 in

standard form.

Exercise 4: Identify the values of a and b by writing the equation 28x2 Ϫ 56x ϩ 48y2 ϩ 192y ϩ 195 ϭ 0 in

standard form.

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CHAPTER 8 Analytic Geometry and the Conic Sections

CUMULATIVE REVIEW CHAPTERS R–8

Solve each equation.

1. x Ϫ 2x ϩ 4x Ϫ 8 ϭ 0

2. 2Ϳn ϩ 4Ϳ ϩ 3 ϭ 13

3

2

3. 2x Ϫ 3 ϩ 5 ϭ x

3

4. x2 ϩ 8 ϭ 0

5. x2 Ϫ 6x ϩ 13 ϭ 0

1

6. 4 # 2xϩ1 ϭ

8

7. 3xϪ2 ϭ 7

8. ln x ϭ 2

Solve each system of equations with a graphing

calculator. Use a matrix equation for Exercise 22, and

the intersection-of-graphs method for Exercise 23.

9. log x ϩ log 1x Ϫ 32 ϭ 1

Graph each relation. Include vertices, x- and

y-intercepts, asymptotes, and other features.

2

10. y ϭ x ϩ 2

3

11. y ϭ Ϳ x Ϫ 2 Ϳ ϩ 3

1

ϩ2

12. y ϭ

xϪ1

13. y ϭ 1x Ϫ 3 ϩ 1

14. a. g1x2 ϭ 1x Ϫ 321x ϩ 121x ϩ 42

b. f 1x2 ϭ x4 ϩ x3 Ϫ 13x2 Ϫ x ϩ 12

xϪ2

x2 Ϫ 9

17. f 1x2 ϭ log2 1x ϩ 12

15. h1x2 ϭ

y

21. Determine the following for

10

the indicated graph (write all

8

6

(Ϫ1, 4) 4

2

(a) the domain, (b) the range,

(Ϫ4, 0)

(2, 0)

x

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ2 2 4 6 8 10

(c) interval(s) where f(x) is

Ϫ4

increasing or decreasing,

Ϫ6

Ϫ8

(d) interval(s) where f(x) is

Ϫ10

constant, (e) location of any

maximum or minimum value(s), (f) interval(s) where

f(x) is positive, and (g) interval(s) where f(x) is

negative.

16. q1x2 ϭ 2x ϩ 3

18. x ϭ y2 ϩ 4y ϩ 7

19. x2 ϩ y2 ϩ 10x Ϫ 4y ϩ 20 ϭ 0

20. 41x Ϫ 12 2 Ϫ 361y ϩ 22 2 ϭ 144

4x ϩ 3y ϭ 13

22. • Ϫ9y ϩ 5z ϭ 19

x Ϫ 4z ϭ Ϫ4

23. e

x2 ϩ y2 ϭ 25

64x2 ϩ 12y2 ϭ 768

24. If a person invests \$5000 at 9% compounded

quarterly, how long would it take for the money to

grow to \$12,000?

25. A radiator contains 10 L of liquid that is 40%

antifreeze. How much should be drained off and

replaced with pure antifreeze for a 60% mixture?

Solve each equation using a graphing calculator.

26.

1 3

1

x Ϫ 4x ϩ 3 ϭ x2 Ϫ 5

8

4

27. 0x ϩ 4 0 ϭ 8 Ϫ 0x 0

28. e2x Ϫ 3ex ϭ 4

29. 3xϪ1 ϭ 22Ϫx

30.

xϩ3

Ն3

xϪ4

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College Algebra Graphs & Models—

CHAPTER CONNECTIONS

in Algebra

CHAPTER OUTLINE

9.1 Sequences and Series 762

9.2 Arithmetic Sequences 773

9.3 Geometric Sequences 782

9.4 Mathematical Induction 796

9.5 Counting Techniques 804

For a corporation of any size, decisions made

by upper management often depend on a large

number of factors, with the desired outcome

attainable in many different ways. For instance,

consider a legal firm that specializes in family

law, with a support staff of 15 employees—6

paralegals and 9 legal assistants. Due to recent

changes in the law, the firm wants to send some

combination of five support staff to a conference

dedicated to the new changes. In Chapter 9,

we’ll see how counting techniques and probability

can be used to determine the various ways such

a group can be randomly formed, even if certain

constraints are imposed. This application

appears as Exercise 34 in Section 9.6.

9.6 Introduction to Probability 816

Check out these other real-world connections:

9.7 The Binomial Theorem 829

Determining the Effects of Inflation

(Section 9.1, Exercise 86)

Calculating Possible Movements of a Computer

Animation (Section 9.2, Exercise 77)

Counting the Number of Possible Area

Codes and Phone Numbers

(Section 9.5, Exercises 84 and 85)

Tracking and Improving Customer Service

Using Probability (Section 9.6, Exercise 53)

761

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