C. The Foci of a Hyperbola
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CHAPTER 8 Analytic Geometry and the Conic Sections
Figure 8.34
y
(x, y)
(c, 0) x
(Ϫc, 0)
(Ϫa, 0)
(a, 0)
EXAMPLE 6
ᮣ
As with the analytic definition of the ellipse, it can be shown that the constant k is
again equal to 2a (for horizontal hyperbolas). To find the equation of a hyperbola in
terms of a and b, we use an approach similar to that of the ellipse (see Appendix V),
y2
x2
and the result is identical to that seen earlier: 2 Ϫ 2 ϭ 1 where b2 ϭ c2 Ϫ a2
a
b
(see Figure 8.34).
We now have the ability to find the foci of any hyperbola —and can use this information in many significant applications. Since the location of the foci play such an important role, it is best to remember the relationship as c2 ϭ a2 ϩ b2 (called the foci
formula for hyperbolas), noting that for a hyperbola, c 7 a and c2 7 a2 (also c 7 b
and c2 7 b2). Be sure to note that for ellipses, the foci formula is c2 ϭ Ϳa2 Ϫ b2Ϳ since
a 7 c (horizontal ellipses) or b 7 c (vertical ellipses).
Graphing a Hyperbola and Identifying Its Foci by Completing the Square
For the hyperbola defined by 7x2 Ϫ 9y2 Ϫ 14x ϩ 72y Ϫ 200 ϭ 0, find the
coordinates of the center, vertices, foci, and the dimensions of the central
rectangle. Then sketch the graph, including the asymptotes.
Solution
ᮣ
given
7x2 Ϫ 9y2 Ϫ 14x ϩ 72y Ϫ 200 ϭ 0
2
2
group terms; add 200
7x Ϫ 14x Ϫ 9y ϩ 72y ϭ 200
factor out leading coefficients
71x2 Ϫ 2x ϩ ____ 2 Ϫ 91y2 Ϫ 8y ϩ ____ 2 ϭ 200
2
2
71x Ϫ 2x ϩ 12 Ϫ 91y Ϫ 8y ϩ 162 ϭ 200 ϩ 7 ϩ 1Ϫ1442 complete the square
c
c
c
S add 7 ϩ 1Ϫ1442
to right-hand side
c
adds Ϫ9 116 2 ϭ Ϫ144
adds 7112 ϭ 7
71x Ϫ 12 2 Ϫ 91y Ϫ 42 2 ϭ 63
1y Ϫ 42 2
1x Ϫ 12 2
Ϫ
ϭ1
9
7
1x Ϫ 12 2
1y Ϫ 42 2
Ϫ
ϭ1
32
1 172 2
factored form
divide by 63 and simplify
write denominators in squared form
This is a horizontal hyperbola with a ϭ 3 1a2 ϭ 92 and b ϭ 17 1b2 ϭ 72.
The center is at (1, 4), with vertices 1Ϫ2, 42 and (4, 4). Using the foci formula
c2 ϭ a2 ϩ b2 yields c2 ϭ 9 ϩ 7 ϭ 16, showing the foci are 1Ϫ3, 42 and (5, 4)
(4 units from center). The central rectangle is 2132 ϭ 6 by 2 17 Ϸ 5.29.
Drawing the rectangle and sketching the asymptotes results in the graph shown.
Horizontal hyperbola
y
10
Center at (1, 4)
Vertices at (Ϫ2, 4) and (4, 4)
(1, 4)
(Ϫ3, 4)
Ϫ10
(Ϫ2, 4)
Ϫ10
(4, 4)
Transverse axis: y ϭ 4
Conjugate axis: x ϭ 1
Location of foci: (Ϫ3, 4) and (5, 4)
(5, 4)
10
x
Width of rectangle
horizontal dimension and
distance between vertices
2a ϭ 2(3) ϭ 6
Length of rectangle
(vertical dimension)
2b ϭ 2(√7) ≈ 5.29
Now try Exercises 61 through 70
ᮣ
The focal chord for a horizontal hyperbola is a vertical line segment through the
focus with endpoints on the hyperbola. Similar to the focal chord of an ellipse, we can
use its length to find additional points on the graph of the hyperbola. The total length
2b2
is once again L ϭ
(for a horizontal hyperbola), meaning the distance from the foci
a
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Section 8.3 The Hyperbola
739
Figure 8.35
b2
.
10.2
a
2
For Example 6, a ϭ 3 and b ϭ 7, so the
vertical distance from focus to graph (in either
7
direction) is ϭ 2.3. From the left focus Ϫ9.4
9.4
3
1Ϫ3, 42 , we can now graph the additional
1Ϫ3, 4 ϩ 2.32 ϭ 1Ϫ3, 6.32 , and
points
1Ϫ3, 4 Ϫ 2.32 ϭ 1Ϫ3, 1.62 . From the right
Ϫ2.2
focus (5, 4), we obtain (5, 6.3) and (5, 1.6).
Graphical verification is provided in Figure 8.35.
Also see Exercise 80.
As with the ellipse, if any two of the values for a, b, and c are known, the relationship between them can be used to construct the equation of the hyperbola. See Exercises 71 through 78.
to the graph (along the focal chord) is
C. You’ve just seen how
we can locate the foci of a
hyperbola and use the foci and
other features to write its
equation
D. Applications Involving Foci
Applications involving the foci of a conic section can take many forms. As before, only
partial information about the hyperbola may be available, and we’ll determine a solution by manipulating a given equation or constructing an equation from given facts.
EXAMPLE 7
ᮣ
Applying the Properties of a Hyperbola—The Path of a Comet
Comets with a high velocity cannot be
captured by the Sun’s gravity, and are slung
around the Sun in a hyperbolic path with the
Sun at one focus. If the path illustrated by the
graph shown is modeled by the equation
2116x2 Ϫ 400y2 ϭ 846,400, how close did
the comet get to the Sun? Assume units are in
millions of miles and round to the nearest
million.
Solution
ᮣ
We are essentially asked to find the distance
between a vertex and focus. Begin by writing
the equation in standard form:
2116x2 Ϫ 400y2 ϭ 846,400 given
y2
x2
Ϫ
ϭ1
divide by 846,400
400
2116
write denominators in
y2
x2
Ϫ
ϭ1
squared form
2
2
20
46
This is a horizontal hyperbola with a ϭ 20 1a2 ϭ 4002 and b ϭ 46 1b2 ϭ 21162.
Use the foci formula to find c2 and c.
y
(0, 0)
x
c2 ϭ a2 ϩ b2
c2 ϭ 400 ϩ 2116
c2 ϭ 2516
c Ϸ Ϫ50 and c Ϸ 50
Since a ϭ 20 and ͿcͿ Ϸ 50, the comet came within about 50 Ϫ 20 ϭ 30 million
miles of the Sun.
Now try Exercises 81 through 84
ᮣ
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CHAPTER 8 Analytic Geometry and the Conic Sections
EXAMPLE 8
ᮣ
Applying the Properties of a Hyperbola — The Location of a Storm
Two amateur meteorologists, living 4 km
apart (4000 m), see a storm approaching. The
one farthest from the storm hears a loud clap
of thunder 9 sec after the one nearest.
Assuming the speed of sound is 340 m/sec,
determine an equation that models possible
locations for the storm at this time.
Solution
D. You’ve just seen how
we can solve applications
involving foci
ᮣ
Let M1 represent the meteorologist nearest
the storm and M2 the farthest. Since M2 heard
the thunder 9 sec after M1, M2 must be
9 # 340 ϭ 3060 m farther away from the
storm S. In other words, from our definition
of a hyperbola, we have Ϳd1 Ϫ d2Ϳ ϭ 3060.
The set of all points that satisfy this equation
will be on the graph of a hyperbola, and we’ll
use this fact to develop an equation model for
possible locations of the storm. Let’s place
the information on a coordinate grid. For
convenience, we’ll use the straight
y
S
line distance between M1 and M2 as
2
the x-axis, with the origin an equal
distance from each. With the
1
constant difference equal to 3060,
M2
M1
we have 2a ϭ 3060, a ϭ 1530 from
Ϫ3
Ϫ2
Ϫ1
1
2
3
x in 1000s
the definition of a hyperbola, giving
2
2
Ϫ1
y
x
Ϫ 2 ϭ 1. With c ϭ 2000 m
2
1530
b
Ϫ2
(the distance from the origin to M1
or M2), we find the value of b using
the equation c2 ϭ a2 ϩ b2: 20002 ϭ 15302 ϩ b2 or
b2 ϭ 120002 2 Ϫ 115302 2 ϭ 1,659,100 Ϸ 12882. The equation that models possible
y2
x2
Ϫ
Ϸ 1.
locations of the storm is
15302
12882
Now try Exercises 85 and 86
8.3 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. The line that passes through the vertices of a
hyperbola is called the
axis.
2. The center of a hyperbola is located
between the vertices.
3. The conjugate axis is
axis and contains the
of the hyperbola.
4. The center of the hyperbola defined by
to the
1x Ϫ 22 2
42
Ϫ
1y Ϫ 32 2
52
ϭ 1 is at
.
ᮣ
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Section 8.3 The Hyperbola
5. Compare/Contrast the two methods used to find the
asymptotes of a hyperbola. Include an example
illustrating both methods.
ᮣ
741
6. Explore/Explain why A1x Ϫ h2 2 Ϫ B1y Ϫ k2 2 ϭ F,
1A, B 7 02 results in a hyperbola regardless of
whether A ϭ B or A B. Illustrate with an
example.
DEVELOPING YOUR SKILLS
Graph each hyperbola. Label the center, vertices, and
any additional points used.
7.
y2
x2
Ϫ
ϭ1
9
4
8.
y2
x2
9.
Ϫ
ϭ1
4
9
11.
13.
15.
17.
12.
y2
x2
Ϫ
ϭ1
36
16
14.
y2
x2
Ϫ
ϭ1
9
1
16.
y2
x2
Ϫ
ϭ1
12
4
18.
2
19.
21.
2
y
x
Ϫ
ϭ1
9
9
2
27.
1y ϩ 12 2
Ϫ
1x Ϫ 22 2
y2
x2
ϭ 1 28.
Ϫ
ϭ1
4
25
4
9
1x Ϫ 32 2
1y ϩ 22 2
Ϫ
ϭ1
29.
36
49
30.
1x Ϫ 22 2
Ϫ
1y Ϫ 12 2
31.
1y ϩ 12 2
Ϫ
1x ϩ 52 2
y2
x2
Ϫ
ϭ1
1
4
32.
1y Ϫ 32 2
Ϫ
1x ϩ 22 2
y2
x2
Ϫ
ϭ1
9
18
33. 1x Ϫ 22 2 Ϫ 41y ϩ 12 2 ϭ 16
y2
x2
Ϫ
ϭ1
16
9
y2
x2
10.
Ϫ
ϭ1
25
16
y2
x2
Ϫ
ϭ1
49
16
20.
2
y
x
Ϫ
ϭ1
36
25
22.
Sketch a complete graph of each equation, including the
asymptotes. Be sure to identify the center and vertices.
y2
x2
Ϫ
ϭ1
25
9
y2
x2
Ϫ
ϭ1
81
16
9
7
16
4
9
5
ϭ1
ϭ1
ϭ1
2
2
34. 91x ϩ 12 2 Ϫ 1y Ϫ 32 2 ϭ 81
2
2
36. 91y Ϫ 42 2 Ϫ 51x Ϫ 32 2 ϭ 45
y
x
Ϫ
ϭ1
4
4
35. 21y ϩ 32 2 Ϫ 51x Ϫ 12 2 ϭ 50
y
x
Ϫ
ϭ1
16
4
37. 121x Ϫ 42 2 Ϫ 51y Ϫ 32 2 ϭ 60
38. 81x Ϫ 42 2 Ϫ 31y Ϫ 32 2 ϭ 24
For the graphs given, state the location of the vertices
and the equation of the transverse axis. Then identify
the location of the center and the equation of the
conjugate axis. Note the scale used on each axis.
23.
24.
y
10
Ϫ10
Ϫ10
25.
10 x
26.
y
41. 9y2 Ϫ 4x2 ϭ 36
42. 25y2 Ϫ 4x2 ϭ 100
43. 12x2 Ϫ 9y2 ϭ 72
44. 36x2 Ϫ 20y2 ϭ 180
Graph each hyperbola by writing the equation in
standard form. Label the center and vertices, and sketch
the asymptotes. Then use a graphing calculator to graph
each relation and locate four additional points whose
coordinates are rational.
45. 4x2 Ϫ y2 ϩ 40x Ϫ 4y ϩ 60 ϭ 0
Ϫ10
10
40. 16x2 Ϫ 25y2 ϭ 400
y
10
Ϫ10
10 x
39. 16x2 Ϫ 9y2 ϭ 144
46. x2 Ϫ 4y2 Ϫ 12x Ϫ 16y ϩ 16 ϭ 0
y
10
47. 4y2 Ϫ x2 Ϫ 24y Ϫ 4x ϩ 28 ϭ 0
48. Ϫ9x2 ϩ 4y2 Ϫ 18x Ϫ 24y Ϫ 9 ϭ 0
Ϫ10
10 x
Ϫ10
Ϫ10
10 x
Ϫ10
Classify each equation as that of a circle, ellipse, or
hyperbola. Justify your response (assume all are
nondegenerate).
49. Ϫ4x2 Ϫ 4y2 ϭ Ϫ24
50. 9y2 ϭ Ϫ4x2 ϩ 36
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CHAPTER 8 Analytic Geometry and the Conic Sections
51. x2 ϩ y2 ϭ 2x ϩ 4y ϩ 4
67. 4y2 Ϫ 16x2 Ϫ 24y Ϫ 28 ϭ 0
52. x2 ϭ y2 ϩ 6y Ϫ 7
68. 4y2 Ϫ 81x2 Ϫ 162x Ϫ 405 ϭ 0
53. 2x2 Ϫ 4y2 ϭ 8
69. 9x2 Ϫ 3y2 Ϫ 54x Ϫ 12y ϩ 33 ϭ 0
54. 36x2 ϩ 25y2 ϭ 900
70. 10x2 ϩ 60x Ϫ 5y2 ϩ 20y Ϫ 20 ϭ 0
55. x2 ϩ 5 ϭ 2y2
Find the equation of the hyperbola (in standard form)
that satisfies the following conditions:
56. x ϩ y2 ϭ 3x2 ϩ 9
57. 2x2 ϭ Ϫ2y2 ϩ x ϩ 20
71. vertices at (Ϫ6, 0) and (6, 0); foci at (Ϫ8, 0) and
(8, 0)
58. 2y ϩ 3 ϭ 6x ϩ 8
2
2
72. vertices at (Ϫ4, 0) and (4, 0); foci at (Ϫ6, 0) and
(6, 0)
59. 16x2 ϩ 5y2 Ϫ 3x ϩ 4y ϭ 538
60. 9x2 ϩ 9y2 Ϫ 9x ϩ 12y ϩ 4 ϭ 0
Use the definition of a hyperbola to find the distance
between the vertices and the dimensions of the rectangle
centered at (h, k). Figures are not drawn to scale. Note
that Exercises 63 and 64 are vertical hyperbolas.
61.
62.
y
(5, 2.25)
(a, 0) (5, 0)
(Ϫ5, 0)
(Ϫ15, 0)
64.
y
(0, Ϫb)
75.
(0, 13)
(6, 7.5)
(Ϫ9, 6.25)
x
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
y
(0, 10)
(0, b)
(a, 0) (15, 0) x
(Ϫa, 0)
(Ϫa, 0)
63.
74. foci at (Ϫ5, 2) and (7, 2); length of conjugate axis:
8 units
Use the characteristics of a hyperbola and the graph
given to write the related equation and state the location
of the foci (75 and 76) or the dimensions of the central
rectangle (77 and 78).
y
(Ϫ15, 6.75)
x
73. foci at 1Ϫ2, Ϫ3122 and 1Ϫ2, 3122; length of
conjugate axis: 6 units
76.
1 2 3 4 5 x
y
10
8
6
4
2
Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
2 4 6 8 10 x
(0, b)
(0, Ϫb)
(0, Ϫ10)
(0, Ϫ13)
Write each equation in standard form to find and list
the coordinates of the (a) center, (b) vertices, (c) foci,
and (d) dimensions of the central rectangle. Then
(e) sketch the graph, including the asymptotes.
x
77.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
78.
y
10
8
6
4
2
Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
2 4 6 8 10 x
65. 4x2 Ϫ 9y2 Ϫ 24x ϩ 72y Ϫ 144 ϭ 0
66. 4x2 Ϫ 36y2 Ϫ 40x ϩ 144y Ϫ 188 ϭ 0
ᮣ
WORKING WITH FORMULAS
36 ؊ 4x2
B ؊9
The “upper half” of a certain hyperbola is given by the equation shown. (a) Simplify the radicand, (b) state the
domain of the expression, and (c) enter the expression as Y1 on a graphing calculator and graph. What is the
equation for the “lower half” of this hyperbola?
79. Equation of a semi-hyperbola: y ؍
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Section 8.3 The Hyperbola
2m2
n
The focal chords of a hyperbola are line segments parallel to the conjugate axis with
10
8
endpoints on the hyperbola, and containing points f1 and f2 (see grid). The length of the
6
chord is given by the formula shown, where n is the distance from center to vertex and m is
4
f 2
the distance from center to one side of the central rectangle. Use the formula to find the
length of the focal chord for the hyperbola indicated, then compare the calculated value with Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2Ϫ2
Ϫ4
Ϫ6
the length estimated from the given graph:
80. Focal chord of a hyperbola: L ؍
1
1x Ϫ 22 2
4
ᮣ
Ϫ
1y Ϫ 12 2
5
ϭ 1.
y
f2
2 4 6 8 10 x
Ϫ8
Ϫ10
APPLICATIONS
81. Stunt pilots: At an air show, a stunt plane dives along
a hyperbolic path whose vertex is directly over the
grandstands. If the plane’s flight path can be modeled
by the hyperbola 25y2 Ϫ 1600x2 ϭ 40,000, what is
the minimum altitude of the plane as it passes over
the stands? Assume x and y are in yards.
82. Flying clubs: To test their skill as pilots, the members
of a flight club attempt to drop sandbags on a target
placed in an open field, by diving along a hyperbolic
path whose vertex is directly over the target area. If
the flight path of the plane flown by the club’s
president is modeled by 9y2 Ϫ 16x2 ϭ 14,400, what
is the minimum altitude of her plane as it passes over
the target? Assume x and y are in feet.
83. Charged particles: It has been shown that when
like particles with a common charge are hurled at
each other, they deflect and travel along paths that
are hyperbolic. Suppose the paths of two such
particles is modeled by the hyperbola
x2 Ϫ 9y2 ϭ 36. What is the minimum distance
between the particles as they approach each other?
Assume x and y are in microns.
84. Nuclear cooling towers: The natural draft cooling
towers for nuclear power stations are called
hyperboloids of one sheet. The perpendicular cross
sections of these hyperboloids form two branches
of a hyperbola. Suppose the central cross section of
one such tower is modeled by the hyperbola
1600x2 Ϫ 4001y Ϫ 502 2 ϭ 640,000. What is the
minimum distance between the sides of the tower?
Assume x and y are in feet.
85. Locating a ship using radar: Under certain
conditions, the properties of a hyperbola can be
used to help locate the position of a ship. Suppose
two radio stations are located 100 km apart along a
straight shoreline. A ship is sailing parallel to the
shore and is 60 km out to sea. The ship sends out a
distress call that is picked up by the closer station
in 0.4 milliseconds (msec — one-thousandth of a
second), while it takes 0.5 msec to reach the station
that is farther away. Radio waves travel at a speed
of approximately 300 km/msec. Use this
information to find the equation of a hyperbola that
will help you find the location of the ship, then find
the coordinates of the ship. (Hint: Draw the
hyperbola on a coordinate system with the radio
stations on the x-axis at the foci, then use the
definition of a hyperbola.)
86. Locating a plane using radar: Two radio stations
are located 80 km apart along a straight shoreline,
when a “mayday” call (a plea for immediate help)
is received from a plane that is about to ditch in the
ocean (attempt a water landing). The plane was
flying at low altitude, parallel to the shoreline, and
20 km out when it ran into trouble. The plane’s
distress call is picked up by the closer station in
0.1 msec, while it takes 0.3 msec to reach the other.
Use this information to construct the equation of a
hyperbola that will help you find the location of the
ditched plane, then find the coordinates of the
plane. Also see Exercise 85.
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CHAPTER 8 Analytic Geometry and the Conic Sections
EXTENDING THE CONCEPT
87. For a greater understanding as to why the branches
y2
x2
of a hyperbola are asymptotic, solve 2 Ϫ 2 ϭ 1
a
b
for y, then consider what happens as x S q (note
that x2 Ϫ k Ϸ x2 for large x).
88. Which has a greater area: (a) The central rectangle of
the hyperbola given by 1x Ϫ 52 2 Ϫ 1y ϩ 42 2 ϭ 57,
(b) the circle given by 1x Ϫ 52 2 ϩ 1y ϩ 42 2 ϭ 57,
or (c) the ellipse given by
91x Ϫ 52 2 ϩ 101y ϩ 42 2 ϭ 570?
89. It is possible for the plane to intersect only the vertex of the cone or to be tangent to the sides. These are called
degenerate cases of a conic section. Many times we’re unable to tell if the equation represents a degenerate case
until it’s written in standard form. Write the following equations in standard form and comment.
a. 4x2 Ϫ 32x Ϫ y2 ϩ 4y ϩ 60 ϭ 0
b. x2 Ϫ 4x ϩ 5y2 Ϫ 40y ϩ 84 ϭ 0
ᮣ
MAINTAINING YOUR SKILLS
90. (2.5) Graph the piecewise-defined function:
f 1x2 ϭ e
4Ϫx
5
2
Ϫ2 Յ x 6 3
xՆ3
91. (4.1) Use synthetic division and the remainder
theorem to determine if x ϭ 2 is a zero of
g1x2 ϭ x5 Ϫ 5x4 ϩ 4x3 ϩ 16x2 Ϫ 32x ϩ 16. If
yes, find its multiplicity.
92. (4.2) The number z ϭ 1 ϩ i 12 is a solution to two
out of the three equations given. Which two?
a. x4 ϩ 4 ϭ 0
b. x3 Ϫ 6x2 ϩ 11x Ϫ 12 ϭ 0
c. x2 Ϫ 2x ϩ 3 ϭ 0
8.4
93. (6.4) A government-approved company is licensed
to haul toxic waste. Each container of solid waste
weighs 800 lb and has a volume of 100 ft3. Each
container of liquid waste weighs 1000 lb and is
60 ft3 in volume. The revenue from hauling solid
waste is $300 per container, while the revenue from
liquid waste is $350 per container. The truck used by
this company has a weight capacity of 39.8 tons and
a volume capacity of 6960 ft3. What combination of
solid and liquid waste containers will produce the
maximum revenue?
The Analytic Parabola; More on Nonlinear Systems
LEARNING OBJECTIVES
In Section 8.4 you will see
how we can:
A. Graph parabolas with a
horizontal axis of
symmetry
B. Identify and use the
focus-directrix form of
the equation of a
parabola
C. Solve nonlinear systems
involving the conic
sections
D. Solve applications of the
analytic parabola
In previous coursework, you likely learned that the
Figure 8.36
graph of a quadratic function was a parabola. Parabolas Parabola
are actually the fourth and final member of the family
of conic sections, and as we saw in Section 8.1, the
Axis
graph can be obtained by observing the intersection of
Element
a plane and a cone. If the plane is parallel to the generator of the cone (shown as a dark line in Figure 8.36), the intersection of the plane with
one nappe forms a parabola. In this section we develop the general equation of a
parabola from its analytic definition, opening a new realm of applications that extends
far beyond those involving only zeroes and extreme values.
A. Parabolas with a Horizontal Axis
An introductory study of parabolas generally involves those with a vertical axis, defined by the equation y ϭ ax2 ϩ bx ϩ c. Unlike the previous conic sections, this equation has only one second-degree (squared) term in x and defines a function. As a
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College Algebra G&M—
8–39
Section 8.4 The Analytic Parabola; More on Nonlinear Systems
Figure 8.37
1. Opens upward
y
4. Axis of symmetry
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review, the primary characteristics are listed here and illustrated in Figure 8.37. See
Exercises 7 through 12.
Vertical Parabolas
For a second-degree equation of the form y ϭ ax2 ϩ bx ϩ c, the graph is a vertical
parabola with these characteristics:
1. opens upward if a 7 0, downward if a 6 0.
2. y-intercept: (0, c) (substitute 0 for x)
3. x-intercept(s): substitute 0 for y and solve.
Ϫb
4. axis of symmetry: x ϭ
2a
Ϫb 4ac Ϫ b2
5. vertex: 1h, k2 ϭ a ,
b
2a
4a
2. y-intercept
3. x-intercepts
x
5. Vertex
Horizontal Parabolas
Similar to our study of horizontal and vertical hyperbolas, the graph of a parabola can
open to the right or left, as well as up or down. After interchanging the variables x and
y in the standard equation, we obtain the parabola x ϭ ay2 ϩ by ϩ c, noting the resulting graph will be a reflection about the line y ϭ x. Here, the axis of symmetry is a horizontal line and factoring or the quadratic formula is used to find the y-intercepts (if they
exist). Note that although the graph is still a parabola—it is not the graph of a function.
Horizontal Parabolas
For a second-degree equation of the form x ϭ ay2 ϩ by ϩ c, the graph is a
horizontal parabola with these characteristics:
1. opens right if a 7 0, left if a 6 0.
2. x-intercept: (c, 0) (substitute 0 for y)
3. y-intercepts(s): substitute 0 for x and solve.
Ϫb
4. axis of symmetry: y ϭ
2a
2
4ac Ϫ b Ϫb
5. vertex: a
,
b
4a
2a
EXAMPLE 1
ᮣ
Graphing a Horizontal Parabola
Graph the relation whose equation is x ϭ y2 ϩ 3y Ϫ 4, then state the domain and
range of the relation.
y
Solution
ᮣ
10
Since the equation has a single squared term in
y, the graph will be a horizontal parabola. With
a 7 0 1a ϭ 12, the parabola opens to the right.
(0, 1)
The x-intercept is 1Ϫ4, 02. Factoring shows the
(Ϫ4, 0)
y-intercepts are y ϭ Ϫ4 and y ϭ 1. The axis of
Ϫ10
10
symmetry is y ϭ Ϫ3
(Ϫ6.25, Ϫ1.5)
y ϭ Ϫ1.5
2 ϭ Ϫ1.5, and substituting
this value into the original equation gives
(0, Ϫ4)
x ϭ Ϫ6.25. The coordinates of the vertex are
1Ϫ6.25, Ϫ1.52. Using horizontal and vertical
Ϫ10
boundary lines we find the domain for this
relation is x ʦ 3 Ϫ6.25, q 2 and the range is y ʦ 1Ϫq, q 2. The graph is shown.
Now try Exercises 13 through 18
x
ᮣ