E. Applications of Polynomials and Polynomial Modeling
Tải bản đầy đủ  0trang
cob19545_ch04_411429.qxd
8/19/10
11:12 PM
Page 422
College Algebra Graphs & Models—
422
4–42
CHAPTER 4 Polynomial and Rational Functions
evident, the number of apparent turning points, any anticipated behavior, and so on.
However, due to the endbehavior 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
(endbehavior, 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
ᮣ
cob19545_ch04_411429.qxd
11/26/10
7:44 AM
Page 423
College Algebra Graphs & Models—
4–43
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 yintercept 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?
DEVELOPING YOUR SKILLS
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 endbehavior 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 endbehavior and yintercept 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
cob19545_ch04_411429.qxd
8/19/10
11:13 PM
Page 424
College Algebra Graphs & Models—
424
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 endbehavior, and
yintercept 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, endbehavior, 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 yintercepts, zeroes of multiplicity, and
a few midinterval points to roundout 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
cob19545_ch04_411429.qxd
8/19/10
11:14 PM
Page 425
College Algebra Graphs & Models—
4–45
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
yintercept 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 yintercept 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.
cob19545_ch04_411429.qxd
8/19/10
11:14 PM
Page 426
College Algebra Graphs & Models—
426
ᮣ
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 9yr 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
6month 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
6month 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 twostage 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,
cob19545_ch04_411429.qxd
8/19/10
11:15 PM
Page 427
College Algebra Graphs & Models—
4–47
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
that an adequate
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 endbehavior 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 endbehavior 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
cob19545_ch04_411429.qxd
11/29/10
10:40 AM
Page 428
College Algebra Graphs & Models—
428
ᮣ
4–48
CHAPTER 4 Polynomial and Rational Functions
MAINTAINING YOUR SKILLS
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
MIDCHAPTER 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 thirddegree
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, endbehavior,
yintercept, 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 harddeck during a 5sec 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 harddeck 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 harddeck at
t ϭ 2 and t ϭ 4.
cob19545_ch04_411429.qxd
8/19/10
11:16 PM
Page 429
College Algebra Graphs & Models—
4–49
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.
cob19545_ch04_430445.qxd
8/19/10
11:29 PM
Page 430
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
cob19545_ch04_430445.qxd
8/19/10
11:30 PM
Page 431
College Algebra G&M—
4–51
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 piecewisedefined 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
ᮣ