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.
DEVELOPING YOUR SKILLS
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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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
89. Parabolic car headlights: The
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.
91. Parabolic sound receivers: Sound
technicians at professional sports
events often use parabolic receivers
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?
93. Parabolic radio wave receivers: The program
known as S.E.T.I. (Search for Extra-Terrestrial
Intelligence) involves a group of scientists using
radio telescopes to look for radio signals from
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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MAINTAINING YOUR SKILLS
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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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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College Algebra G&M—
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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
answers in interval notation):
(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
Additional Topics
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