5.2
Before
Now
Why?
Evaluate and Graph
Polynomial Functions
You evaluated and graphed linear and quadratic functions.
You will evaluate and graph other polynomial functions.
So you can model skateboarding participation, as in Ex. 55.
Key Vocabulary
Recall that a monomial is a number, a variable, or a product of numbers
and variables. A polynomial is a monomial or a sum of monomials.
• polynomial
• polynomial function A polynomial function is a function of the form
• synthetic
f(x) 5 anx n 1 an 2 1x n 2 1 1 . . . 1 a1x 1 a0
substitution
• end behavior
where a Þ 0, the exponents are all whole numbers, and the coefficients are all
n
real numbers. For this function, an is the leading coefficient, n is the degree, and
a0 is the constant term. A polynomial function is in standard form if its terms
are written in descending order of exponents from left to right.
Common Polynomial Functions
Degree
Type
Standard form
Example
0
Constant
f (x) 5 a0
f (x) 5 214
1
Linear
f (x) 5 a1x 1 a0
f (x) 5 5x 2 7
2
Quadratic
f (x) 5 a2 x 1 a1x 1 a0
f (x) 5 2x 2 1 x 2 9
3
Cubic
f(x) 5 a3x 3 1 a2 x 2 1 a1x 1 a0
f (x) 5 x 3 2 x 2 1 3x
4
Quartic
f(x) 5 a4x 4 1 a3x 3 1 a2 x 2 1 a1x 1 a0
f (x) 5 x 4 1 2x 2 1
EXAMPLE 1
2
Identify polynomial functions
Decide whether the function is a polynomial function. If so, write it in
standard form and state its degree, type, and leading coefficient.
1x2 1 3
a. h(x) 5 x 4 2 }
4
b. g(x) 5 7x 2 Ï 3 1 πx 2
c. f(x) 5 5x 2 1 3x21 2 x
d. k(x) 5 x 1 2x 2 0.6x5
}
Solution
a. The function is a polynomial function that is already written in
standard form. It has degree 4 (quartic) and a leading coefficient of 1.
}
b. The function is a polynomial function written as g(x) 5 πx2 1 7x 2 Ï 3 in
standard form. It has degree 2 (quadratic) and a leading coefficient of π.
c. The function is not a polynomial function because the term 3x21 has an
exponent that is not a whole number.
d. The function is not a polynomial function because the term 2x does not
have a variable base and an exponent that is a whole number.
5.2 Evaluate and Graph Polynomial Functions
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EXAMPLE 2
Evaluate by direct substitution
Use direct substitution to evaluate f (x) 5 2x 4 2 5x 3 2 4x 1 8 when x 5 3.
✓
f(x) 5 2x 4 2 5x 3 2 4x 1 8
Write original function.
f(3) 5 2(3)4 2 5(3) 3 2 4(3) 1 8
Substitute 3 for x.
5 162 2 135 2 12 1 8
Evaluate powers and multiply.
5 23
Simplify.
GUIDED PRACTICE
for Examples 1 and 2
Decide whether the function is a polynomial function. If so, write it in
standard form and state its degree, type, and leading coefficient.
2. p(x) 5 9x4 2 5x22 1 4
1. f(x) 5 13 2 2x
3. h(x) 5 6x2 1 π 2 3x
Use direct substitution to evaluate the polynomial function for the given
value of x.
4. f(x) 5 x 4 1 2x 3 1 3x 2 2 7; x 5 22
5. g(x) 5 x 3 2 5x 2 1 6x 1 1; x 5 4
SYNTHETIC SUBSTITUTION Another way to evaluate a polynomial function is
to use synthetic substitution. This method, shown in the next example, involves
fewer operations than direct substitution.
EXAMPLE 3
Evaluate by synthetic substitution
Use synthetic substitution to evaluate f(x) from Example 2 when x 5 3.
Solution
AVOID ERRORS
The row of coefficients
for f (x) must include a
coefficient of 0 for the
“missing” x2-term.
STEP 1 Write the coefficients of f (x) in order of descending exponents. Write
the value at which f(x) is being evaluated to the left.
x-value
3
2
25
0
24
8
coefficients
STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by
the x-value. Write the product under the second coefficient. Add.
3
2
25
0
24
8
6
2
1
STEP 3 Multiply the previous sum by the x-value. Write the product under the
third coefficient. Add. Repeat for all of the remaining coefficients. The
final sum is the value of f(x) at the given x-value.
3
2
2
25
0
24
8
6
3
9
15
1
3
5
23
c Synthetic substitution gives f(3) 5 23, which matches the result in Example 2.
338
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Chapter 5 Polynomials and Polynomial Functions
10/17/05 10:55:51 AM
END BEHAVIOR The end behavior of a function’s graph is the behavior of the
graph as x approaches positive infinity (1`) or negative infinity (2`). For the
graph of a polynomial function, the end behavior is determined by the function’s
degree and the sign of its leading coefficient.
For Your Notebook
KEY CONCEPT
End Behavior of Polynomial Functions
READING
The expression
“x → 1`” is read as
“x approaches positive
infinity.”
Degree: odd
Leading coefficient: positive
Degree: odd
Leading coefficient: negative
f(x) → 1` f(x) → 1`
as x → 1` as x → 2`
y
x
f(x) → 2`
as x → 2`
x
Degree: even
Leading coefficient: positive
f(x) → 1`
as x → 2`
f (x) → 1`
as x → 1`
y
EXAMPLE 4
f (x) → 2`
as x → 1`
Degree: even
Leading coefficient: negative
x
★
y
y
f (x) → 2`
as x → 2`
x
f (x) → 2`
as x → 1`
Standardized Test Practice
What is true about the degree and leading coefficient
of the polynomial function whose graph is shown?
4
y
A Degree is odd; leading coefficient is positive
B Degree is odd; leading coefficient is negative
3x
C Degree is even; leading coefficient is positive
D Degree is even; leading coefficient is negative
From the graph, f(x) → 2` as x → 2` and f(x) → 2` as x → 1`. So, the degree is
even and the leading coefficient is negative.
c The correct answer is D. A B C D
✓
GUIDED PRACTICE
for Examples 3 and 4
Use synthetic substitution to evaluate the polynomial
function for the given value of x.
y
3
6. f(x) 5 5x 3 1 3x 2 2 x 1 7; x 5 2
7. g(x) 5 22x4 2 x 3 1 4x 2 5; x 5 21
1
x
8. Describe the degree and leading coefficient of the
polynomial function whose graph is shown.
5.2 Evaluate and Graph Polynomial Functions
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GRAPHING POLYNOMIAL FUNCTIONS To graph a polynomial function, first plot
points to determine the shape of the graph’s middle portion. Then use what you
know about end behavior to sketch the ends of the graph.
EXAMPLE 5
Graph polynomial functions
Graph (a) f(x) 5 2x 3 1 x 2 1 3x 2 3 and (b) f (x) 5 x 4 2 x 3 2 4x 2 1 4.
Solution
a. To graph the function, make a table of values and
y
plot the corresponding points. Connect the points
with a smooth curve and check the end behavior.
(22, 3)
1 (1, 0)
3
x
23
22
21
0
1
2
3
y
24
3
24
23
0
21
212
x
(2, 21)
(21, 24)
(0, 23)
The degree is odd and leading coefficient is negative.
So, f (x) → 1` as x → 2` and f(x) → 2` as x → 1`.
b. To graph the function, make a table of values and
y (0, 4)
plot the corresponding points. Connect the points
with a smooth curve and check the end behavior.
x
23
22
21
0
1
2
3
y
76
12
2
4
0
24
22
(21, 2)
(1, 0)
The degree is even and leading coefficient is positive.
So, f (x) → 1` as x → 2` and f(x) → 1` as x → 1`.
"MHFCSB
EXAMPLE 6
1
3
x
(2, 24)
at classzone.com
Solve a multi-step problem
PHYSICAL SCIENCE The energy E (in foot-pounds) in each square foot of a wave is
given by the model E 5 0.0029s4 where s is the wind speed (in knots). Graph the
model. Use the graph to estimate the wind speed needed to generate a wave
with 1000 foot-pounds of energy per square foot.
Solution
Wave Energy
Make a table of values. The model
only deals with positive values of s.
s
0
10
20
30
40
E
0
29
464
2349
7424
STEP 2 Plot the points and connect them with
a smooth curve. Because the leading
coefficient is positive and the degree is
even, the graph rises to the right.
Energy per square foot
(foot-pounds)
STEP 1
E
3000
2000
1000
0
(24, 1000)
0
10 20 24 30 40 s
Wind speed (knots)
STEP 3 Examine the graph to see that s < 24 when E 5 1000.
c The wind speed needed to generate the wave is about 24 knots.
340
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Chapter 5 Polynomials and Polynomial Functions
10/17/05 10:55:53 AM
✓
GUIDED PRACTICE
for Examples 5 and 6
Graph the polynomial function.
9. f(x) 5 x4 1 6x 2 2 3
10. f(x) 5 2x 3 1 x 2 1 x 2 1
11. f(x) 5 4 2 2x 3
12. WHAT IF? If wind speed is measured in miles per hour, the model in
Example 6 becomes E 5 0.0051s4. Graph this model. What wind speed is
needed to generate a wave with 2000 foot-pounds of energy per square foot?
5.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 21, 27, and 57
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 24, 37, 50, 52, and 59
5 MULTIPLE REPRESENTATIONS
Ex. 56
SKILL PRACTICE
1. VOCABULARY Identify the degree, type, leading coefficient, and constant
term of the polynomial function f(x) 5 6 1 2x2 2 5x4.
2. ★ WRITING Explain what is meant by the end behavior of a polynomial
function.
EXAMPLE 1
POLYNOMIAL FUNCTIONS Decide whether the function is a polynomial function.
on p. 337
for Exs. 3–8
If so, write it in standard form and state its degree, type, and leading coefficient.
3. f(x) 5 8 2 x 2
}
6. h(x) 5 x 3Ï 10 1 5x22 1 1
}
4. f(x) 5 6x 1 8x4 2 3
5. g(x) 5 πx4 1 Ï 6
5 x 3 1 3x 2 10
7. h(x) 5 2}
2
2
8. g(x) 5 8x 3 2 4x 2 1 }
x
EXAMPLE 2
DIRECT SUBSTITUTION Use direct substitution to evaluate the polynomial
on p. 338
for Exs. 9–14
function for the given value of x.
EXAMPLE 3
on p. 338
for Exs. 15–23
9. f(x) 5 5x 3 2 2x2 1 10x 2 15; x 5 21
10. f(x) 5 8x 1 5x4 2 3x2 2 x 3 ; x 5 2
11. g(x) 5 4x 3 2 2x5 ; x 5 23
12. h(x) 5 6x 3 2 25x 1 20; x 5 5
1 x4 2 3 x 3 1 10; x 5 24
13. h(x) 5 x 1 }
}
4
2
14. g(x) 5 4x5 1 6x 3 1 x 2 2 10x 1 5; x 5 22
SYNTHETIC SUBSTITUTION Use synthetic substitution to evaluate the
polynomial function for the given value of x.
15. f(x) 5 5x 3 2 2x2 2 8x 1 16; x 5 3
16. f(x) 5 8x4 1 12x 3 1 6x 2 2 5x 1 9; x 5 22
17. g(x) 5 x 3 1 8x2 2 7x 1 35; x 5 26
18. h(x) 5 28x 3 1 14x 2 35; x 5 4
19. f(x) 5 22x4 1 3x 3 2 8x 1 13; x 5 2
20. g(x) 5 6x5 1 10x 3 2 27; x 5 23
21. h(x) 5 27x 3 1 11x 2 1 4x; x 5 3
22. f(x) 5 x4 1 3x 2 20; x 5 4
23. ERROR ANALYSIS Describe and correct
the error in evaluating the polynomial
function f(x) 5 24x4 1 9x2 2 21x 1 7
when x 5 22.
22
24
24
9
221
7
8
234
110
17
255
117
5.2 Evaluate and Graph Polynomial Functions
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EXAMPLE 4
24. ★ MULTIPLE CHOICE The graph of a polynomial function is shown.
y
What is true about the function’s degree and leading coefficient?
on p. 339
for Exs. 24–27
2
A The degree is odd and the leading coefficient is positive.
1
B The degree is odd and the leading coefficient is negative.
x
C The degree is even and the leading coefficient is positive.
D The degree is even and the leading coefficient is negative.
USING END BEHAVIOR Describe the degree and leading coefficient of the
polynomial function whose graph is shown.
25.
26.
y
4
27.
y
y
1
1
1
x
2
x
1
x
DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the
polynomial function by completing these statements: f(x) → ? as x → 2`
and f (x) → ? as x → 1`.
28. f(x) 5 10x4
29. f(x) 5 2x6 1 4x 3 2 3x
30. f(x) 5 22x 3 1 7x 2 4
31. f(x) 5 x 7 1 3x4 2 x2
32. f(x) 5 3x10 2 16x
33. f(x) 5 26x5 1 14x2 1 20
34. f(x) 5 0.2x 3 2 x 1 45
35. f(x) 5 5x8 1 8x 7
36. f(x) 5 2x 273 1 500x271
37. ★ OPEN-ENDED MATH Write a polynomial function f of degree 5 such that
the end behavior of the graph of f is given by f(x) → 1` as x → 2` and
f(x) → 2` as x → 1`. Then graph the function to verify your answer.
EXAMPLE 5
GRAPHING POLYNOMIALS Graph the polynomial function.
on p. 340
for Exs. 38–50
38. f(x) 5 x 3
39. f(x) 5 2x4
40. f(x) 5 x5 1 3
41. f(x) 5 x4 2 2
42. f(x) 5 2x 3 1 5
43. f(x) 5 x 3 2 5x
44. f(x) 5 2x4 1 8x
45. f(x) 5 x5 1 x
46. f(x) 5 2x 3 1 3x2 2 2x 1 5
47. f(x) 5 x5 1 x2 2 4
48. f(x) 5 x4 2 5x2 1 6
49. f(x) 5 2x4 1 3x 3 2 x 1 1
50. ★ MULTIPLE CHOICE Which function is
represented by the graph shown?
2
1 x3 1 1
A f(x) 5 }
3
1 x3 1 1
B f(x) 5 2}
3
1 x3 2 1
C f(x) 5 }
1 x3 2 1
D f(x) 5 2}
3
y
1
x
3
51. VISUAL THINKING Suppose f(x) → 1` as x → 2` and f (x) → 2` as x → 1`.
Describe the end behavior of g(x) 5 2f(x).
52. ★ SHORT RESPONSE A cubic polynomial function f has leading coefficient 2
and constant term 25. If f (1) 5 0 and f (2) 5 3, what is f(25)? Explain how you
found your answer.
342
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5 WORKED-OUT SOLUTIONS
Chapter 5 Polynomials
on p. WS1 and Polynomial Functions
★
5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
10/17/05 10:55:55 AM
53. CHALLENGE Let f(x) 5 x 3 and g(x) 5 x 3 2 2x2 1 4x.
a. Copy and complete the table.
f (x)
g(x)
x
f (x)
g(x)
}
b. Use the numbers in the table to complete
f (x)
this statement: As x → 1`, } → ? .
g(x)
10
?
?
?
20
?
?
?
c. Explain how the result from part (b)
50
?
?
?
100
?
?
?
200
?
?
?
shows that the functions f and g have the
same end behavior as x → 1`.
PROBLEM SOLVING
EXAMPLE 6
54. DIAMONDS The weight of an ideal round-cut diamond
can be modeled by
on p. 340
for Exs. 54–59
w 5 0.0071d3 2 0.090d2 1 0.48d
where w is the diamond’s weight (in carats) and d is
its diameter (in millimeters). According to the model,
what is the weight of a diamond with a diameter of
15 millimeters?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
55. SKATEBOARDING From 1992 to 2003, the number of people in the United
States who participated in skateboarding can be modeled by
S 5 20.0076t 4 1 0.14t 3 2 0.62t 2 1 0.52t 1 5.5
where S is the number of participants (in millions) and t is the number
of years since 1992. Graph the model. Then use the graph to estimate the
first year that the number of skateboarding participants was greater than
8 million.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
56.
MULTIPLE REPRESENTATIONS From 1987 to 2003, the number of indoor
movie screens M in the United States can be modeled by
M 5 211.0t 3 1 267t 2 2 592t 1 21,600
where t is the number of years since 1987.
a. Classifying a Function State the degree and type of the function.
b. Making a Table Make a table of values for the function.
c. Sketching a Graph Use your table to graph the function.
57. SNOWBOARDING From 1992 to 2003, the number of people
in the United States who participated in snowboarding can
be modeled by
S 5 0.0013t 4 2 0.021t 3 1 0.084t 2 1 0.037t 1 1.2
where S is the number of participants (in millions) and t
is the number of years since 1992. Graph the model. Use
the graph to estimate the first year that the number of
snowboarding participants was greater than 2 million.
5.2 Evaluate and Graph Polynomial Functions
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58. MULTI-STEP PROBLEM From 1980 to 2002, the number of quarterly
periodicals P published in the United States can be modeled by
P 5 0.138t 4 2 6.24t 3 1 86.8t 2 2 239t 1 1450
where t is the number of years since 1980.
a. Describe the end behavior of the graph of the model.
b. Graph the model on the domain 0 ≤ t ≤ 22.
c. Use the model to predict the number of quarterly periodicals in the year
2010. Is it appropriate to use the model to make this prediction? Explain.
59. ★ EXTENDED RESPONSE The weight of Sarus crane chicks S and hooded
crane chicks H (both in grams) during the 10 days following hatching
can be modeled by the functions
S 5 20.122t 3 1 3.49t 2 2 14.6t 1 136
H 5 20.115t 3 1 3.71t 2 2 20.6t 1 124
where t is the number of days after hatching.
a. Calculate According to the models, what is the difference in weight
between 5-day-old Sarus crane chicks and hooded crane chicks?
b. Graph Sketch the graphs of the two models.
c. Apply A biologist finds that the weight of a crane chick after 3 days
is 130 grams. What species of crane is the chick more likely to be?
Explain how you found your answer.
60. CHALLENGE The weight y (in pounds) of a rainbow trout can be modeled
by y 5 0.000304x3 where x is the length of the trout (in inches).
a. Write a function that relates the weight y and length x of a rainbow trout
if y is measured in kilograms and x is measured in centimeters. Use the
fact that 1 kilogram ø 2.20 pounds and 1 centimeter ø 0.394 inch.
b. Graph the original function and the function from part (a) in the same
coordinate plane. What type of transformation can you apply to the
graph of y 5 0.000304x 3 to produce the graph from part (a)?
MIXED REVIEW
Solve the equation or inequality.
61. 2b 1 11 5 15 2 6b (p. 18)
62. 2.7n 1 4.3 5 12.94 (p. 18)
63. 27 < 6y 2 1 < 5 (p. 41)
64. x2 2 14x 1 48 5 0 (p. 252)
65. 224q2 2 90q 5 21 (p. 259)
66. z2 1 5z < 36 (p. 300)
The variables x and y vary directly. Write an equation that relates x and y. Then
find the value of x when y 5 23. (p. 107)
PREVIEW
Prepare for
Lesson 5.3
in Exs. 73–78.
344
n2pe-0502.indd 344
67. x 5 4, y 5 12
68. x 5 3, y 5 221
69. x 5 10, y 5 24
70. x 5 0.8, y 5 0.2
71. x 5 20.45, y 5 20.35
72. x 5 26.5, y 5 3.9
Write the quadratic function in standard form. (p. 245)
73. y 5 (x 1 3)(x 2 7)
74. y 5 8(x 2 4)(x 1 2)
75. y 5 23(x 2 5)2 2 25
76. y 5 2.5(x 2 6)2 1 9.3
1 (x 2 4)2
77. y 5 }
2
5 (x 1 4)(x 1 9)
78. y 5 2}
3
PRACTICE
for Lesson
5.2, p. 1014
Chapter 5 EXTRA
Polynomials
and Polynomial
Functions
ONLINE QUIZ at classzone.com
10/17/05 10:55:57 AM
Use after Lesson 5.2
classzone.com
Keystrokes
5.2 Set a Good Viewing Window
QUESTION
What is a good viewing window for a polynomial function?
When you graph a function with a graphing calculator, you should choose a
viewing window that displays the important characteristics of the graph.
EXAMPLE
Graph a polynomial function
Graph f(x) 5 0.2x 3 2 5x 2 1 38x 2 97.
STEP 1 Graph the function
Graph the function in the
standard viewing window.
210 ≤ x ≤ 10, 210 ≤ y ≤ 10
STEP 2 Adjust horizontally
Adjust the horizontal scale so
that the end behavior of the
graph as x → 1` is visible.
210 ≤ x ≤ 20, 210 ≤ y ≤ 10
STEP 3 Adjust vertically
Adjust the vertical scale so that the
turning points and end behavior
of the graph as x → 2` are visible.
210 ≤ x ≤ 20, 220 ≤ y ≤ 10
PRACTICE
Find intervals for x and y that describe a good viewing window for the graph
of the polynomial function.
1. f (x) 5 x 3 1 4x 2 2 8x 1 11
2. f (x) 5 2x 3 1 36x2 2 10
3. f (x) 5 x4 2 4x 2 1 2
4. f (x) 5 2x4 2 2x 3 1 3x 2 2 4x 1 5
5. f (x) 5 2x4 1 3x 3 1 15x
6. f (x) 5 2x4 2 7x 3 1 x 2 8
7. f (x) 5 2x5 1 9x 3 2 12x 1 18
8. f (x) 5 x5 2 7x4 1 25x 3 2 40x2 1 13x
9. REASONING Let g(x) 5 f (x) 1 c where f (x) and g(x) are polynomial functions
and c is a positive constant. How is a good viewing window for the graph of
f(x) related to a good viewing window for the graph of g(x)?
10. BASEBALL From 1994 to 2003, the average salary S (in thousands of dollars)
for major league baseball players can be modeled by
S(x) 5 24.10x3 1 67.4x2 2 121x 1 1170
where x is the number of years since 1994. Find intervals for the horizontal
and vertical axes that describe a good viewing window for the graph of S.
5.2 Evaluate and Graph Polynomial Functions
345