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E. Applications of Polynomials and Polynomial Modeling

# E. Applications of Polynomials and Polynomial Modeling

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CHAPTER 4 Polynomial and Rational Functions

evident, the number of apparent turning points, any anticipated behavior, and so on.

However, due to the end-behavior of polynomial models, great care must be exercised

when these models are used to make projections beyond the limits of the given data.

EXAMPLE 11

Solution

The Earth’s atmosphere consists of several layers that

are defined in terms of altitude and the characteristics

of the air in each layer. In order, these are the

Troposphere (0–12 km), Stratosphere (12–50 km),

Mesosphere (50–80 km), and Thermosphere

(80–100 km). Due to their chemical and physical

characteristics, the air temperature within each layer

and from layer to layer varies a great deal. The data

in the table gives the temperature in °C at an altitude

of h kilometers (km). Use the data to:

a. Draw a scatterplot and decide on an appropriate

form of regression, then find the regression

equation.

b. Use the regression equation to find the

temperature at altitudes of 32.6 km and 63.6 km.

c. As the space shuttle rockets into orbit, a

temperature reading of Ϫ75°C is taken. What are

the possible altitudes for the shuttle at this point?

Altitude

(km)

Temperature

(؇C)

0

20

4

Ϫ20

8

Ϫ45

12

Ϫ55

20

Ϫ57

30

Ϫ43

40

Ϫ16

50

Ϫ2

60

Ϫ14

70

Ϫ54

80

Ϫ91

90

Ϫ93

100

Ϫ45

a. The scatterplot is shown in Figure 4.28. Using the characteristics exhibited

(end-behavior, three turning points), it appears a quartic regression (degree 4)

is appropriate and the equation is shown in Figure 4.29.

Figure 4.28

Figure 4.29

40

Ϫ10

110

Ϫ110

b. At altitudes of 32.6 km and 63.6 km, the temperature is very near Ϫ30.0°C

(Figure 4.30).

c. Setting Y2 ϭ Ϫ75, we note the line intersects the graph in two places, one

indicating an altitude of about 76.4 km, and the other an altitude of about

96.1 km (Figure 4.31).

Figure 4.31

Figure 4.30

40

Ϫ10

E. You’ve just seen how

to solve applications of

polynomials and polynomial

modeling

110

Ϫ110

Now try Exercises 89 through 92

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423

Section 4.3 Graphing Polynomial Functions

4.3 EXERCISES

CONCEPTS AND VOCABULARY

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

1. For a polynomial with factors of the form 1x Ϫ c2 m,

c is called a

of multiplicity

.

3. The graphs of Y1 ϭ 1x Ϫ 22 2 and Y2 ϭ 1x Ϫ 22 4

both

at x ϭ 2, but the graph of Y2 is

than the graph of Y1 at this point.

5. In your own words, explain/discuss how to find the

degree and y-intercept of a function that is given in

factored form. Use f 1x2 ϭ 1x ϩ 12 3 1x Ϫ 22 1x ϩ 42 2

to illustrate.

2. A polynomial function of degree n has

zeroes and at most

“turning points.”

4. Since x4 7 0 for all x, the ends of its graph will

always point in the

direction. Since

x3 7 0 when x 7 0 and x3 6 0 when x 6 0,

the ends of its graph will always point in the

direction.

6. Name all of the “tools” at your disposal that play

a role in the graphing of polynomial functions.

Which tools are indispensable and always used?

Which tools are used only as the situation merits?

Determine whether each graph is the graph of a

polynomial function. If yes, state the least possible

degree of the function. If no, state why.

7.

y

f(x)

12

10

8

6

4

2

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ2

Ϫ4

Ϫ6

Ϫ8

8.

y

1 2 3 4 5 x

13. f (x)

1 2 3 4 5 x

14. g(x)

y

10

8

6

4

2

h(x)

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

State the end-behavior of the functions shown.

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

y

g(x)

5

4

3

2

1

Ϫ5Ϫ4 Ϫ3Ϫ2Ϫ1

Ϫ1

10.

30

24

18

12

6

1 2 3 4 5 x

y

q(x)

1 2 3 4 5 x

12.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

f(x)

1 2 3 4 5 x

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ6

Ϫ12

Ϫ18

Ϫ24

Ϫ30

1 2 3 4 5 x

16. h(x)

y

p(x)

Ϫ2

Ϫ3

Ϫ4

Ϫ5

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

5

4

3

2

1

Ϫ5Ϫ4 Ϫ3Ϫ2Ϫ1

Ϫ1

1 2 3 4 5 x

Ϫ2

Ϫ3

Ϫ4

Ϫ5

11.

y

g(x)

30

24

18

12

6

1 2 3 4 5 x

15. H(x)

9.

y

f(x)

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ6

Ϫ12

Ϫ18

Ϫ24

Ϫ30

y

H(x)

1 2 3 4 5 x

h(x)

325

260

195

130

65

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ65

2 4 6 8 10 x

Ϫ130

Ϫ195

Ϫ260

Ϫ325

State the end-behavior and y-intercept of the functions

given. Do not graph.

17. f 1x2 ϭ x3 ϩ 6x2 Ϫ 5x Ϫ 2

18. g1x2 ϭ x4 Ϫ 4x3 Ϫ 2x2 ϩ 16x Ϫ 12

19. p1x2 ϭ Ϫ2x4 ϩ x3 ϩ 7x2 Ϫ x Ϫ 6

20. q1x2 ϭ Ϫ2x3 Ϫ 18x2 ϩ 7x ϩ 3

21. Y1 ϭ Ϫ3x5 ϩ x3 ϩ 7x2 Ϫ 6

22. Y2 ϭ Ϫx6 Ϫ 4x5 ϩ 4x3 ϩ 16x Ϫ 12

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For each polynomial graph, (a) state whether the degree

of the function is even or odd; (b) use the graph to name

the zeroes of f, then state whether their multiplicity is

even or odd; (c) state the minimum possible degree of f

and write one possible function for f in factored form;

and (d) estimate the domain and range. Assume all

zeroes are real.

23.

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

1 2 3 4 5 x

40

32

24

16

8

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ8

Ϫ16

Ϫ24

Ϫ32

Ϫ40

c.

e.

y

1 2 3 4 5 x

State the degree of each function, the end-behavior, and

y-intercept of its graph.

29. f 1x2 ϭ 1x Ϫ 321x ϩ 12 3 1x Ϫ 22 2

30. g1x2 ϭ 1x ϩ 22 1x Ϫ 42 1x ϩ 12

2

31. Y1 ϭ Ϫ1x ϩ 12 2 1x Ϫ 22 12x Ϫ 32 1x ϩ 42

32. Y2 ϭ Ϫ1x ϩ 12 1x Ϫ 22 3 15x Ϫ 32

33. r1x2 ϭ 1x2 ϩ 321x ϩ 42 3 1x Ϫ 12

34. s1x2 ϭ 1x ϩ 22 2 1x Ϫ 12 2 1x2 ϩ 52

35. h1x2 ϭ 1x2 ϩ 22 1x Ϫ 12 2 11 Ϫ x2

36. H1x2 ϭ 1x ϩ 22 2 12 Ϫ x2 1x2 ϩ 42

Every function in Exercises 37 through 42 has the zeroes

x ‫ ؍‬؊1, x ‫ ؍‬؊3, and x ‫ ؍‬2. Match each to its

corresponding graph using degree, end-behavior, and

the multiplicity of each zero.

37. f 1x2 ϭ 1x ϩ 12 2 1x ϩ 32 1x Ϫ 22

38. F1x2 ϭ 1x ϩ 12 1x ϩ 32 2 1x Ϫ 22

39. g1x2 ϭ 1x ϩ 121x ϩ 321x Ϫ 22 3

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ4

1 2 3 4 5 x

1 2 3 4 5 x

Ϫ8

Ϫ12

Ϫ16

Ϫ20

y

25

20

15

10

5

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ5

Ϫ10

Ϫ15

Ϫ20

Ϫ25

1 2 3 4 5 x

20

16

12

8

4

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ4

Ϫ8

Ϫ12

Ϫ16

Ϫ20

y

20

16

12

8

4

1 2 3 4 5 x

y

28.

y

10

8

6

4

2

42. Y2 ϭ 1x ϩ 12 3 1x ϩ 32 1x Ϫ 22 2

y

a.

b.

60

48

36

24

12

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ12

Ϫ24

Ϫ36

Ϫ48

Ϫ60

1 2 3 4 5 x

41. Y1 ϭ 1x ϩ 12 2 1x ϩ 32 1x Ϫ 22 2

y

26.

y

40. G1x2 ϭ 1x ϩ 12 3 1x ϩ 32 1x Ϫ 22

10

8

6

4

2

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

1 2 3 4 5 x

30

24

18

12

6

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ6

Ϫ12

Ϫ18

Ϫ24

Ϫ30

27.

24.

y

10

8

6

4

2

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ2

Ϫ4

Ϫ6

Ϫ8

Ϫ10

25.

4–44

CHAPTER 4 Polynomial and Rational Functions

y

y

40

32

24

16

8

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ8

Ϫ16

Ϫ24

Ϫ32

Ϫ40

1 2 3 4 5 x

80

64

48

32

16

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ16

Ϫ32

Ϫ48

Ϫ64

Ϫ80

d.

f.

1 2 3 4 5 x

1 2 3 4 5 x

y

40

32

24

16

8

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ8

Ϫ16

Ϫ24

Ϫ32

Ϫ40

1 2 3 4 5 x

Sketch the graph of each function using the degree, endbehavior, x- and y-intercepts, zeroes of multiplicity, and

a few midinterval points to round-out the graph.

Connect all points with a smooth, continuous curve.

43. f 1x2 ϭ 1x ϩ 321x ϩ 121x Ϫ 22

44. g1x2 ϭ 1x ϩ 221x Ϫ 421x Ϫ 12

45. p1x2 ϭ Ϫ1x ϩ 12 2 1x Ϫ 32

46. q1x2 ϭ Ϫ1x ϩ 221x Ϫ 22 2

47. Y1 ϭ 1x ϩ 12 2 13x Ϫ 22 1x ϩ 32

48. Y2 ϭ 1x ϩ 22 1x Ϫ 12 2 15x Ϫ 22

49. r1x2 ϭ Ϫ1x ϩ 12 2 1x Ϫ 22 2 1x Ϫ 12

50. s1x2 ϭ Ϫ1x Ϫ 32 1x Ϫ 12 2 1x ϩ 12 2

51. f 1x2 ϭ 12x ϩ 321x Ϫ 12 3

52. g1x2 ϭ 13x Ϫ 421x ϩ 12 3

53. h1x2 ϭ 1x ϩ 12 3 1x Ϫ 32 1x Ϫ 22

54. H1x2 ϭ 1x ϩ 321x ϩ 12 2 1x Ϫ 22 2

55. Y3 ϭ 1x ϩ 12 3 1x Ϫ 12 2 1x Ϫ 22

56. Y4 ϭ 1x Ϫ 32 1x Ϫ 12 3 1x ϩ 12 2

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Use the Guidelines for Graphing Polynomial Functions to

graph the polynomials.

57. y ϭ x3 ϩ 3x2 Ϫ 4

58. y ϭ x3 Ϫ 13x ϩ 12

59. f 1x2 ϭ x3 Ϫ 3x2 Ϫ 6x ϩ 8

60. g1x2 ϭ x3 ϩ 2x2 Ϫ 5x Ϫ 6

61. h1x2 ϭ Ϫx3 Ϫ x2 ϩ 5x Ϫ 3

62. H1x2 ϭ Ϫx3 Ϫ x2 ϩ 8x ϩ 12

63. p1x2 ϭ Ϫx4 ϩ 10x2 Ϫ 9

64. q1x2 ϭ Ϫx ϩ 13x Ϫ 36

4

2

65. r1x2 ϭ x4 Ϫ 9x2 Ϫ 4x ϩ 12

66. s1x2 ϭ x Ϫ 5x ϩ 20x Ϫ 16

4

3

67. Y1 ϭ x4 Ϫ 6x3 ϩ 8x2 ϩ 6x Ϫ 9

68. Y2 ϭ x4 Ϫ 4x3 Ϫ 3x2 ϩ 10x ϩ 8

69. Y3 ϭ 3x4 ϩ 2x3 Ϫ 36x2 ϩ 24x ϩ 32

70. Y4 ϭ 2x4 Ϫ 3x3 Ϫ 15x2 ϩ 32x Ϫ 12

71. F1x2 ϭ 2x ϩ 3x Ϫ 9x

4

3

2

72. G1x2 ϭ 3x4 ϩ 2x3 Ϫ 8x2

73. f 1x2 ϭ x5 ϩ 4x4 Ϫ 16x2 Ϫ 16x

74. g1x2 ϭ x5 Ϫ 3x4 ϩ x3 Ϫ 3x2

425

Section 4.3 Graphing Polynomial Functions

75. h1x2 ϭ x6 Ϫ 2x5 Ϫ 4x4 ϩ 8x3

76. H1x2 ϭ x6 ϩ 3x5 Ϫ 4x4

In preparation for future course work, it becomes

helpful to recognize the most common square roots

in mathematics: 12 Ϸ 1.414, 13 Ϸ 1.732, and

16 Ϸ 2.449. Graph the following polynomials on a

graphing calculator, and use the calculator to locate the

maximum/minimum values and all zeroes. Use the zeroes

to write the polynomial in factored form, then verify the

y-intercept from the factored form and polynomial form.

77. h1x2 ϭ x5 ϩ 4x4 Ϫ 9x Ϫ 36

78. H1x2 ϭ x5 ϩ 5x4 Ϫ 4x Ϫ 20

79. f 1x2 ϭ 2x5 ϩ 5x4 Ϫ 10x3 Ϫ 25x2 ϩ 12x ϩ 30

80. g1x2 ϭ 3x5 ϩ 2x4 Ϫ 24x3 Ϫ 16x2 ϩ 36x ϩ 24

Use the graph of each function to construct its equation in

factored form and in polynomial form. Be sure to check

the y-intercept and adjust the lead coefficient if necessary.

81.

y

7

6

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

1 2 3 4 5 x

82.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

WORKING WITH FORMULAS

83. Roots tests for a quartic polynomial

ax4 ؉ bx3 ؉ cx2 ؉ dx ؉ e:

b2 ؊ 2ac

1r1 2 2 ؉ 1r2 2 2 ؉ 1r3 2 2 ؉ 1r4 2 2 ‫؍‬

a2

In the Chapter 3 Reinforcing Basic Concepts

feature, we used relationships between the roots of

a quadratic equation and its coefficients to verify

the roots without having to substitute. Similar

root/coefficient relationships exist for cubic and

quartic polynomials, but the method soon becomes

too time consuming (see Exercise 94). There is

actually a little known formula for checking the

roots of a quartic polynomial (and others) that is

much more efficient. Given that r1, r2, r3, and r4

are the roots of the polynomial, the sum of the

b2 Ϫ 2ac

.

a2

Note that if a ϭ 1, the formula reduces to b2 Ϫ 2c.

(a) Use this test to verify that x ϭ Ϫ3, Ϫ1, 2, and 4

are the roots of x4 Ϫ 2x3 Ϫ 13x2 ϩ 14x ϩ 24 ϭ 0,

then (b) use these roots and the factored form to write

the equation in polynomial form to confirm results.

squares of the roots must be equal to

84. It is worth noting that the root test in Exercise 83

still applies when the roots are irrational and/or

complex. Use this test to verify that

x ϭ Ϫ 13, 13, 1 ϩ 2i, and 1 Ϫ 2i are the

solutions to x4 Ϫ 2x3 ϩ 2x2 ϩ 6x Ϫ 15 ϭ 0, then

use these zeroes and the factored form to write the

equation in polynomial form to confirm results.

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CHAPTER 4 Polynomial and Rational Functions

4–46

APPLICATIONS

85. Traffic volume: Between the hours of 6:00 A.M.

and 6.00 P.M., the volume of traffic at a busy

intersection can be modeled by the polynomial

v1t2 ϭ Ϫt4 ϩ 25t3 Ϫ 192t2 ϩ 432t, where v(t)

represents the number of vehicles above/below

average, and t is number of hours past 6:00 A.M.

(6:00 A.M. S 02. (a) Use the remainder theorem to

find the volume of traffic during rush hour

(8:00 A.M.), lunch time (12 noon), and the trip

home (5:00 P.M.). (b) Use the rational zeroes

theorem to find the times when the volume of

traffic is at its average 3 v1t2 ϭ 04 . (c) Use this

information to graph v(t), then use the graph to

estimate the maximum and minimum flow of traffic

and the time at which each occurs.

86. Insect population: The population of a certain

insect varies dramatically with the weather, with

springlike temperatures causing a population

boom and extreme weather (summer heat and

winter cold) adversely affecting the population.

This phenomena can be modeled by the polynomial

p1m2 ϭ Ϫm4 ϩ 26m3 Ϫ 217m2 ϩ 588m, where p(m)

represents the number of live insects (in hundreds

of thousands) in month m 1m ʦ 10, 1 4 S Jan2 .

(a) Use the remainder theorem to find the

population of insects during the cool of spring

(March) and the fair weather of fall (October).

(b) Use the rational zeroes theorem to find the

times when the population of insects becomes

dormant 3 p1m2 ϭ 04 . (c) Use this information to

graph p(m), then use the graph to estimate the

maximum and minimum population of insects, and

the month at which each occurs.

87. Balance of payments: The graph shown represents

the balance of payments (surplus versus deficit) for

a large county over a 9-yr period. Use it to answer

the following:

Balance

a. What is the minimum

10 (10,000s)

8

possible degree

(9.5, ~6)

6

4

polynomial that can

2

Year

model this graph?

Ϫ2 1 2 3 4 5 6 7 8 9 10

b. How many years did this Ϫ4

Ϫ6

Ϫ8

county run a deficit?

Ϫ10

c. Construct an equation

model in factored form and in polynomial

form, adjusting the lead coefficient as needed.

How large was the deficit in year 8?

88. Water supply: The graph shown represents the

water level in a reservoir (above and below normal)

that supplies water to a metropolitan area, over a

6-month period. Use it to answer the following:

Level

a. What is the

10

(inches)

8

minimum possible

6

degree polynomial

4

2

Month

that can model this

1

2

3

4

5

6

Ϫ2

graph?

Ϫ4

Ϫ6

b. How many months

Ϫ8

Ϫ10

was the water level

below normal in this

6-month period?

c. At the beginning of this period 1m ϭ 02, the

water level was 36 in. above normal, due to a

long period of rain. Use this fact to help

construct an equation model in factored form

and in polynomial form, adjusting the lead

coefficient as needed. Use the equation to

determine the water level in months three

and five.

89. In order to

Time

Volume

determine if the

(6:00

A.M. S 0)

(vehicles/min)

number of lanes

0

0

on a certain

highway should

2

222

be increased, the

4

100

flow of traffic (in

6

114

vehicles per min)

8

360

is carefully

10

550

monitored from

11

429

6:00 A.M. to

6:00 P.M. The

data collected are shown the table (6:00 A.M.

corresponds to t ϭ 0). (a) Draw a scatterplot and

decide on an appropriate form of regression, then

find the regression equation and graph the function

and scatterplot on the same screen. (b) Use the

regression equation and its graph to find the

maximum flow of traffic for the morning and

evening rush hours. (c) During what time(s) of day

is the flow rate 350 vehicles per hour?

90. The Goddard Memorial Rocket

Velocity

Club is testing a new two-stage Time

(sec)

(ft/sec)

rocket. Using a specialized

0

0

tracking device, the velocity of

1

441

the rocket is monitored every

second for the first 4.5 sec of

2

484

flight, with the data collected

3

459

in the table shown. (a) Draw a

4

696

scatterplot and decide on an

appropriate form of regression, then find the

regression equation and graph the function and

scatterplot on the same screen. (b) Use the

regression equation and its graph to find how many

seconds elapsed before the first stage burned out,

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Section 4.3 Graphing Polynomial Functions

and the rocket’s velocity at this time, then

(c) determine how many seconds elapsed (after

liftoff) until the second stage ignited. (d) At a

velocity of 1000 ft/sec, the fuel was exhausted and

the return chutes deployed. How many seconds

after liftoff did this occur?

91. A posh restaurant

Time

Customer

in a thriving

(10 A.M. S 0)

count

neighborhood opens

0

0

at 10 A.M. for the

2

79

lunch crowd, and

closes at 9 P.M. as the

4

41

dinner crowd leaves.

6

43

In order to ensure

9

122

11

3

number of cooks and

servers are available,

their hourly customer count is monitored each day

for 1 month with the data averaged and compiled in

the table shown. (a) Draw a scatterplot and decide

on an appropriate form of regression, then find the

regression equation and graph the function and

scatterplot on the same screen. (b) Use the

regression equation and its graph to find what time

the restaurant reaches its morning peak and its

evening peak. (c) At what time is business slowest,

and how many customers are in the restaurant at

that time? (d) Between what times is the restaurant

serving 100 customers or more?

427

92. Using the wind to

Wind velocity

Power

generate power is

(mph)

(W)

becoming more and

20

419

more prevalent. While

25

623

most people are aware

30

635

that a wind turbine

generates more power

35

593

with a stronger wind,

40

639

many are not aware that

the generators are built with a stall mechanism to

protect the generator, blades, and infrastructure in

very high winds. This affects the actual power output

as the generator operates near its threshold. The

power output [in watts (W)] of a certain generator is

shown in the table for wind velocity v in miles per

hour. (a) Draw a scatterplot and decide on an

appropriate form of regression, then find the

regression equation and graph the function and

scatterplot on the same screen. (b) Use the regression

equation and its graph to find the maximum safe

power output for this generator, and the wind speed

at which this occurs. (c) What is the power output in

a 23 mph wind? (d) If the manufacturer stipulates

that the turbine will experience automatic shutdown

when power output exceeds 900 W, what is the

greatest wind speed this turbine can tolerate?

EXTENDING THE CONCEPT

93. As discussed in this section, the study of endbehavior looks at what happens to the graph of a

function as ͿxͿ S q. Notice that as ͿxͿ S q, both 1x

and x12 approach zero. This fact can be used to

study the end-behavior of polynomial graphs.

a. For f 1x2 ϭ x3 ϩ x2 Ϫ 3x ϩ 6, factoring out x3

gives the expression

1

6

3

f 1x2 ϭ x3a1 ϩ Ϫ 2 ϩ 3 b. What happens

x

x

x

to the value of the expression as x S q?

As x S Ϫq ?

b. Factor out x4 from g1x2 ϭ x4 ϩ 3x3 Ϫ 4x2 ϩ

5x Ϫ 1. What happens to the value of the

expression as x S q? As x S Ϫq? How does

this affirm the end-behavior must be up/up?

94. If u, v, w, and z represent the roots of the quartic

polynomial ax4 ϩ bx3 ϩ cx2 ϩ dx ϩ e ϭ 0, then

the following relationships are true:

(a) u ϩ v ϩ w ϩ z ϭ Ϫb, (b) u1v ϩ z2 ϩ

v1w ϩ z2 ϩ w1u ϩ z2 ϭ c, (c) u1vw ϩ wz2 ϩ

v1uz ϩ wz2 ϭ Ϫd, and (d) u # v # w # z ϭ e. Use

these tests to verify that x ϭ Ϫ3, Ϫ1, 2, 4 are the

solutions to x4 Ϫ 2x3 Ϫ 13x2 ϩ 14x ϩ 24 ϭ 0,

then use these zeroes and the factored form to write

the equation in polynomial form to confirm results.

95. For what value of c will three of the four real roots

of x4 ϩ 5x3 ϩ x2 Ϫ 21x ϩ c ϭ 0 be shared by the

polynomial x3 ϩ 2x2 Ϫ 5x Ϫ 6 ϭ 0?

Show the following equations have no rational roots.

96. x5 Ϫ x4 Ϫ x3 ϩ x2 Ϫ 2x ϩ 3 ϭ 0

97. x5 Ϫ 2x4 Ϫ x3 ϩ 2x2 Ϫ 3x ϩ 4 ϭ 0

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4–48

CHAPTER 4 Polynomial and Rational Functions

98. (3.6) Given f 1x2 ϭ x2 Ϫ 2x and g1x2 ϭ 1x , find the

compositions h1x2 ϭ 1 f ‫ ؠ‬g21x2 and H1x2 ϭ

1g ‫ ؠ‬f 21x2, then state the domain of each.

99. (3.1) By direct substitution, verify that x ϭ 1 Ϫ 2i

is a solution to x2 Ϫ 2x ϩ 5 ϭ 0 and name the

second solution.

100. (R.3/R.6) Solve each of the following equations.

a. Ϫ12x ϩ 52 Ϫ 16 Ϫ x2 ϩ 3 ϭ x Ϫ 31x ϩ 22

b. 1x ϩ 1 ϩ 3 ϭ 12x ϩ 2

21

2

ϩ4

ϩ5ϭ 2

c.

xϪ3

x Ϫ9

101. (1.3) Determine if the relation shown is a function.

If not, explain how the definition of a function is

violated.

feline

canine

dromedary

equestrian

bovine

cow

horse

cat

camel

dog

MID-CHAPTER CHECK

1. Compute 1x3 ϩ 8x2 ϩ 7x Ϫ 142 Ϭ 1x ϩ 22 using

long division and write the result in two ways:

(a) dividend ϭ 1quotient21divisor2 ϩ remainder and

remainder

dividend

(b)

.

ϭ 1quotient2 ϩ

divisor

divisor

2. Given that x Ϫ 2 is a factor of f 1x2 ϭ 2x4 Ϫ x3 Ϫ

8x2 ϩ x ϩ 6, use the rational zeroes theorem to

write f (x) in completely factored form.

3. Use the remainder theorem to evaluate f 1Ϫ22, given

f 1x2 ϭ Ϫ3x4 ϩ 7x2 Ϫ 8x ϩ 11.

4. Use the factor theorem to find a third-degree

polynomial having x ϭ Ϫ2 and x ϭ 1 ϩ i as roots.

5. Use the intermediate value theorem to show that

g1x2 ϭ x3 Ϫ 6x Ϫ 4 has a root in the interval (2, 3).

6. Use the rational zeroes theorem, tests for Ϫ1 and 1,

synthetic division, and the remainder theorem to

write f 1x2 ϭ x4 ϩ 5x3 Ϫ 20x Ϫ 16 in completely

factored form.

7. Find all the zeroes of h, real and complex:

h1x2 ϭ x4 ϩ 3x3 ϩ 10x2 ϩ 6x Ϫ 20.

8. Sketch the graph of p using its degree, end-behavior,

y-intercept, zeroes of multiplicity, and any

midinterval points needed, given

p1x2 ϭ 1x ϩ 12 2 1x Ϫ 121x Ϫ 32.

9. Use the Guidelines for Graphing to draw the graph

of q1x2 ϭ x3 ϩ 5x2 ϩ 2x Ϫ 8.

10. When fighter pilots train for dogfighting, a “harddeck” is usually established below which no

competitive activity can take place. The polynomial

graph given shows Maverick’s altitude above and

below this hard-deck during a 5-sec interval.

Altitude

a. What is the minimum

A (100s of feet)

15

possible degree

12

9

polynomial that could

6

form this graph? Why?

3

Seconds

b. How many seconds

(total) was Maverick

below the hard-deck for

these 5 sec of the

exercise?

Ϫ3

Ϫ6

Ϫ9

Ϫ12

Ϫ15

1 2 3 4 5 6 7 8 9 10 t

c. At the beginning of this time interval (t ϭ 0),

Maverick’s altitude was 1500 ft above the harddeck. Use this fact and the graph given to help

construct an equation model in factored form and

in polynomial form, adjusting the lead coefficient

if needed. Use the equation to determine

Maverick’s altitude in relation to the hard-deck at

t ϭ 2 and t ϭ 4.

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Reinforcing Basic Concepts

429

REINFORCING BASIC CONCEPTS

Approximating Real Zeroes

Consider the equation x4 ϩ x3 ϩ x Ϫ 6 ϭ 0. Using the rational zeroes theorem, the possible rational zeroes are

5Ϯ1, Ϯ6, Ϯ2, Ϯ36. The tests for 1 and Ϫ1 indicate that neither is a zero: f 112 ϭ Ϫ3 and f 1Ϫ12 ϭ Ϫ7. Descartes’

rule of signs reveals there must be one positive real zero since the coefficients of f 1x2 change sign one time:

f 1x2 ϭ x4 ϩ x3 ϩ x Ϫ 6, and one negative real zero since f 1Ϫx2 also changes sign one time: f 1Ϫx2 ϭ

x4 Ϫ x3 Ϫ x Ϫ 6. The remaining two zeroes must be complex. Using x ϭ 2 with synthetic division shows 2 is not a

zero, but the coefficients in the quotient row are all positive, so 2 is an upper bound:

2|

1

1

1

2

3

0

6

6

1

12

13

Ϫ6

26

20

|

coefficients of f (x)

q (x)

Using x ϭ Ϫ2 shows that Ϫ2 is a zero and a lower bound for all other zeroes (quotient row alternates in sign):

Ϫ2|

1

1

1

Ϫ2

Ϫ1

0

2

2

1

Ϫ4

Ϫ3

Ϫ6

6

|0

coefficients of f (x)

q1(x)

This means the remaining real zero must be a positive irrational

number less than 2 (all other possible rational zeroes were eliminated). The

x

f(x)

Conclusion

quotient polynomial q1 1x2 ϭ x3 Ϫ x2 ϩ 2x Ϫ 3 is not factorable, yet we’re

1

Ϫ3

— Zero is here,

left with the challenge of finding this final zero. While there are many

1.5

3.94

use x ϭ 1.25

advanced techniques available for approximating irrational zeroes, at this

next

2

20

level either technology or a technique called bisection is commonly used.

The bisection method combines the intermediate value theorem with

x

f(x)

Conclusion

successively smaller intervals of the input variable, to narrow down the

Ϫ3

1

location of the irrational zero. Although “bisection” implies halving the

Zero is here,

1.25 Ϫ0.36

interval each time, any number within the interval will do. The bisection

— use x ϭ 1.30

1.5

3.94

method may be most efficient using a succession of short input/output

next

tables as shown, with the number of tables increased if greater accuracy is

x

f(x)

Conclusion

desired. Since f 112 ϭ Ϫ3 and f 122 ϭ 20, the intermediate value theorem

tells us the zero must be in the interval [1, 2]. We begin our search here,

1.25 Ϫ0.36

— Zero is here,

rounding noninteger outputs to the nearest 100th. As a visual aid, positive

1.30

0.35

use x ϭ 1.275

outputs are in blue, negative outputs in red.

1.5

3.94

next

A reasonable estimate for the zero appears to be x ϭ 1.275. Evaluating

the function at this point gives f 11.2752 Ϸ Ϫ0.0097, which is very close

to zero. Naturally, a closer approximation is obtained using the capabilities of a graphing calculator. To seven decimal

places the zero is x Ϸ 1.2756822.

Exercise 1: Use the intermediate value theorem to show that f 1x2 ϭ x3 Ϫ 3x ϩ 1 has a zero in the interval [1, 2], then

use bisection to locate the zero to three decimal place accuracy.

Exercise 2: The function f 1x2 ϭ x4 ϩ 3x Ϫ 15 has two real zeroes in the interval 3 Ϫ5, 54 . Use the intermediate value

theorem to locate the zeroes, then use bisection to find the zeroes accurate to three decimal places.

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College Algebra G&M—

4.4

Graphing Rational Functions

LEARNING OBJECTIVES

In Section 4.4 you will see

how we can:

A. Locate the vertical

B.

C.

D.

E.

asymptotes and find the

domain of a rational

function

Apply the concept of

“roots of multiplicity” to

rational functions and

graphs

Find horizontal

asymptotes of rational

functions

Graph general rational

functions

Solve applications of

rational functions

Our first exposure to rational functions occurred in Section 2.4, where we looked at the

1

1

attributes and properties of the basic rational functions f 1x2 ϭ and g1x2 ϭ 2 shown

x

x

in Figures 4.32 and 4.33.

Figure 4.33

1

g1x2 ϭ 2

x

Figure 4.32

1

f1x2 ϭ

x

y

y

5

5

4

4

3

3

2

2

1

1

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

1

2

3

4

5

x

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

Ϫ2

Ϫ2

Ϫ3

Ϫ3

Ϫ4

Ϫ4

Ϫ5

Ϫ5

1

2

3

4

5

x

Much of what we learned about these functions can be generalized and applied to the

general rational functions that follow. For convenience and emphasis, the definition of

a rational function is repeated here.

Rational Functions

A rational function V(x) is one of the form

V1x2 ϭ

p1x2

d1x2

,

where p and d are polynomials and d1x2 0.

The domain of V(x) is all real numbers, except the zeroes of d.

Our study begins by taking a closer look at the zeroes of d(x) that are excluded from the

domain, and what happens to the graph of a rational function at or near these zeroes.

These observations will form a key component of graphing general rational functions.

A. Rational Functions and Vertical Asymptotes

The graphs shown in Figures 4.34 through 4.37 illustrate that rational graphs come in

many shapes, often in “pieces,” and exhibit asymptotic behavior.

WORTHY OF NOTE

In Section 2.5, we studied special

p1x2

cases of d1x2 , where p and d shared

a common factor, creating a “hole”

in the graph. Rational functions of

this form will be investigated further

in Section 4.5. In this section, we’ll

assume the functions are given in

simplest form (the numerator and

denominator have no common

factors).

430

Figure 4.34

2x

g1x2 ϭ 2

x Ϫ1

Figure 4.35

3

w1x2 ϭ 2

x ϩ1

y

y

5

5

4

4

3

3

2

2

1

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

1

1

2

3

4

5

x

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

Ϫ2

Ϫ2

Ϫ3

Ϫ3

Ϫ4

Ϫ4

Ϫ5

Ϫ5

1

2

3

4

5

x

4–50

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College Algebra G&M—

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431

Section 4.4 Graphing Rational Functions

Figure 4.36

x2

h1x2 ϭ 2

x Ϫ 2x Ϫ 3

Figure 4.37

v 1x2 ϭ

x2 Ϫ 4

xϩ1

y

y

10

10

8

8

6

6

4

4

2

2

Ϫ10Ϫ8 Ϫ6 Ϫ4 Ϫ2

Ϫ2

Ϫ10Ϫ8 Ϫ6 Ϫ4 Ϫ2

Ϫ2

2

4

6

8 10

x

Ϫ4

Ϫ4

Ϫ6

Ϫ6

Ϫ8

Ϫ8

Ϫ10

Ϫ10

2

4

6

8 10

x

For the functions f 1x2 ϭ 1x and g1x2 ϭ x12, a vertical asymptote occurred at the zero of

each denominator. This actually applies to all rational functions in simplified form. For

V1x2 ϭ p1x2

d1x2 , if c is a zero of d(x), the function can be evaluated at every point near c,

but not at c. This creates a break or discontinuity in the graph of V resulting in the asymptotic behavior.

Vertical Asymptotes of a Rational Function

Given V1x2 ϭ

p1x2

is a rational function in simplest form,

d1x2

vertical asymptotes will occur at the real zeroes of d.

Breaks created by vertical asymptotes are said to be nonremovable, because

there is no way to repair the break, even if a piecewise-defined function were used. See

Example 5, Section 2.6.

EXAMPLE 1

Finding Vertical Asymptotes and the Domain of a Rational Function

Locate the vertical asymptote(s) of each function given, then state its domain.

2x

3

a. g1x2 ϭ 2

b. w1x2 ϭ 2

x Ϫ1

x ϩ1

2

x

x2 Ϫ 4

c. h1x2 ϭ 2

d. v1x2 ϭ

xϩ1

x Ϫ 2x Ϫ 3

Solution

a. Setting the denominator equal to zero gives x2 Ϫ 1 ϭ 0, so vertical

asymptotes will occur at x ϭ Ϫ1 and x ϭ 1. The domain of g is

x ʦ 1Ϫq, Ϫ12 ´ 1Ϫ1, 12 ´ 11, q 2 . See Figure 4.34.

b. Since the equation x2 ϩ 1 ϭ 0 has no real zeroes, there are no vertical

asymptotes and the domain of w is unrestricted: x ʦ R. See Figure 4.35.

c. Solving x2 Ϫ 2x Ϫ 3 ϭ 0 gives 1x ϩ 121x Ϫ 32 ϭ 0, with solutions x ϭ Ϫ1

and x ϭ 3. There are vertical asymptotes at x ϭ Ϫ1 and x ϭ 3, and the

domain of h is x ʦ 1Ϫq, Ϫ12 ´ 1Ϫ1, 32 ´ 13, q 2 . See Figure 4.36.

d. Solving x ϩ 1 ϭ 0 gives x ϭ Ϫ1, and a vertical asymptote will occur at

x ϭ Ϫ1. The domain of v is x ʦ 1Ϫq, Ϫ12 ´ 1Ϫ1, q 2 . See Figure 4.37.

Now try Exercises 7 through 14

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