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C. The Foci of a Hyperbola

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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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—



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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







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C. The Foci of a Hyperbola

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