Solution Manual Of Algebra And Trigonometry 9th Edition By SULLIVAN

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Algebra And Trigonometry 9th edition By SULLIVAN

CHAPTER 1 Equations and

Inequalities

1.1 Assess Your Understanding

‘

Are You Prepared?’

1. The fact that is because of the

Property. (pp. 9–13)

2. The fact that implies that is a result of the

Property. (pp. 9–13)

3x = 0 x = 0

21x + 32 = 2x + 6 3. The domain of the variable in the expression is

. (p. 21)

x

x – 4

Concepts and Vocabulary

4. True or False Multiplying both sides of an equation by any

number results in an equivalent equation.

5. An equation that is satisfied for every value of the variable for

which both sides are defined is called a(n) .

6. An equation of the form is called a(n)

equation or a(n) equation .

ax + b = 0

7. True or False The solution of the equation

is

8. True or False Some equations have no solution.

3

8

.

3x - 8 = 0

Skill Building

In Problems 9–16, mentally solve each equation.

13. 14. 15. 16.

2

3 x =

9

2

1

3 x =

5

12

2x - 3 = 0 3x + 4 = 0

In Problems 17–64, solve each equation.

17. 3x + 4 = x 18. 2x + 9 = 5x 19. 2t - 6 = 3 - t
18. 5y + 6 = -18 - y 21. 6 - x = 2x + 9 22. 3 - 2x = 2 - x
19. 3 + 2n = 4n + 7 24. 6 - 2m = 3m + 1 25. 213 + 2x2 = 31x - 42
20. 312 - x2 = 2x - 1 27. 8x - 13x + 22 = 3x - 10 28. 7 - 12x - 12 = 10
21. 30. 31.

1

2 x - 5 =

3

4 x

1

3 x = 2 -

2

3 x 3

2 x + 2 =

1

2 -

1

2 x

32. 33. 34.

1

2 -

1

3 p =

4

3

2

3 p =

1

2 p +

1

3

1 -

1

2 x = 6

35. 36. 37.

x + 1

3 +

x + 2

7 0.9t = 0.4 + 0.1t 0.9t = 1 + t = 2

38. 39. 40.

4

y - 5 =

5

2y

2

y +

4

y = 3

2x + 1

3 + 16 = 3x

41. 42. 43. 1x + 721x - 12 = 1x + 122 3

x -

1

3 =

1

6

1

2 +

2

x =

3

4

47. 48. 49.

x

x - 2 + 3 =

2

x - 2

z1z2 + 12 = 3 + z3 w14 - w22 = 8 - w3

44. 1x + 221x - 32 = 1x + 322 45. x12x - 32 = 12x + 121x - 42 46. x11 + 2x2 = 12x - 121x - 22
45. 51. 52.

x

x2 - 9 +

4

x + 3 =

3

x2 - 9

2x

x2 - 4 =

4

x2 - 4 -

3

x + 2

2x

x + 3 = -6

x + 3 - 2

53. 54. 55.

5

2x - 3 =

3

x + 5

3x

x - 1 = 2

x

x + 2 =

3

2

56. 57. 58.

8w + 5

10w - 7 =

4w - 3

5w + 7 59. 60. 61.

2

y + 3 +

3

y - 4 =

5

y + 6

-4

2x + 3 +

1

x - 1 =

1

12x + 321x - 12

4

x - 2 = -3

x + 5 +

7

1x + 521x - 22

62. 63. 64.

x + 1

x2 + 2x

-

x + 4

x2 + x

= -3

x2 + 3x + 2

x

x2 - 1 -

x + 3

x2 - x

= -3

x2 + x

5

5z - 11 +

4

2z - 3 = -3

5 - z

In Problems 65–68, use a calculator to solve each equation. Round the solution to two decimal places.

65. 66. 6.2x -

19.1

83.72 3.2x + = 0.195

21.3

65.871 = 19.23

67. 68. 18.63x -

21.2

2.6 =

14

2.32 14.72 - 21.58x = x - 20

18

2.11 x + 2.4

75. Find the number a for which is a solution of the

equation

76. Find the number b for which is a solution of the

equation

x + 2b = x - 4 + 2bx

x = 2

x + 2a = 16 + ax - 6a

x = 4

69. 70. 71.

x

a +

x

b ax - b = c, a Z 0 1 - ax = b, a Z 0 = c, a Z 0, b Z 0, a Z -b

Applications and Extensions

In Problems 69–74, solve each equation. The letters a, b, and c are constants.

72. 73. 74.

b + c

x + a =

b - c

x - a

, c Z 0, a Z 0

1

x - a +

1

x + a =

2

x - 1

a

x +

b

x = c, c Z 0

Problems 77–82 list some formulas that occur in applications. Solve each formula for the indicated variable.

77. Electricity
78. Finance
79. Mechanics
80. Chemistry
81. Mathematics
82. Mechanics
83. Finance A total of $20,000 is to be invested, some in bonds

and some in certificates of deposit (CDs). If the amount

invested in bonds is to exceed that in CDs by $3000, how

much will be invested in each type of investment?

84. Finance A total of $10,000 is to be divided between Sean

and George, with George to receive $3000 less than Sean.

How much will each receive?

85. Computing Hourly Wages Sandra, who is paid time-and-ahalf

for hours worked in excess of 40 hours, had gross

weekly wages of $442 for 48 hours worked. What is her

regular hourly rate?

86. Computing Hourly Wages Leigh is paid time-and-a-half

for hours worked in excess of 40 hours and double-time for

hours worked on Sunday. If Leigh had gross weekly wages

of $456 for working 50 hours, 4 of which were on Sunday,

what is her regular hourly rate?

87. Computing Grades Going into the final exam, which will

count as two tests, Brooke has test scores of 80, 83, 71, 61,

and 95.What score does Brooke need on the final in order

to have an average score of 80?

v = -gt + v0 for t

S =

a

1 - r

for r

PV = nRT for T

F =

mv2

R

for R

A = P11 + rt2 for r

1

R =

1

R1

+

1

R2

for R

88. Computing Grades Going into the final exam, which will

count as two-thirds of the final grade, Mike has test scores of

86, 80, 84, and 90.What score does Mike need on the final in

order to earn a B, which requires an average score of 80?

What does he need to earn an A, which requires an average

of 90?

89. Business: Discount Pricing A builder of tract homes

reduced the price of a model by 15%. If the new price is

$425,000, what was its original price? How much can be

saved by purchasing the model?

90. Business: Discount Pricing A car dealer, at a year-end

clearance, reduces the list price of last year’s models by

15%. If a certain four-door model has a discounted price of

$8000, what was its list price? How much can be saved by

purchasing last year’s model?

91. Business: Marking up the Price of Books A college book

store marks up the price that it pays the publisher for a book

by 35%. If the selling price of a book is $92.00, how much

did the bookstore pay for this book 92. Personal Finance: Cost of a Car The suggested list price of

a new car is $18,000. The dealer’s cost is 85% of list. How

much will you pay if the dealer is willing to accept $100 over

cost for the car?

93. Business: Theater Attendance The manager of the Coral

Theater wants to know whether the majority of its patrons

are adults or children. One day in July, 5200 tickets were

sold and the receipts totaled $29,961.The adult admission is

$7.50, and the children’s admission is $4.50. How many adult

patrons were there?

94. Business: Discount Pricing A wool suit, discounted by

30% for a clearance sale, has a price tag of $399.What was

the suit’s original price?

95. Geometry The perimeter of a rectangle is 60 feet. Find its

length and width if the length is 8 feet longer than the width.

96. Geometry The perimeter of a rectangle is 42 meters. Find

its length and width if the length is twice the width.

97. Sharing the Cost of a Pizza Judy and Tom agree to share

the cost of an $18 pizza based on how much each ate. If Tom

ate the amount that Judy ate, how much should each pay?

[Hint: Some pizza may be left.]

2

3

92 CHAPTER 1 Equations and Inequalities

Tom’s portion

Judy’s portion

98. What Is Wrong? One step in the following list contains an

error. Identify it and explain what is wrong.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

1 = 0 (8)

x + 5 = x + 4

1x - 221x + 52 = 1x - 221x + 42

x2 + 3x - 10 = x2 + 2x - 8

x2 + 3x = x2 + 2x + 2

3x = 2x + 2

3x - 2x = 2

x = 2

99. The equation

has no solution, yet when we go through the process of

solving it we obtain Write a brief paragraph to

explain what causes this to happen.

100. Make up an equation that has no solution and give it to a

fellow student to solve. Ask the fellow student to write a

critique of your equation.

1.2   Linear Functions

Assess Your Understanding

‘Are You Prepared?’

.

1. Factor: (pp. 49–55)
2. Factor: (pp. 49–55)
3. The solution set of the equation is

. (p. 13)

1x - 3213x + 52 = 0

2x2 - x - 3

x2 - 5x - 6 4. True or False (pp. 23–24)

5. Complete the square of . Factor the new

expression. (p. 56)

Concepts and Vocabulary

6. The quantity is called the of a

quadratic equation. If it is , the equation has no

real solution.

7. True or False Quadratic equations always have two real

solutions.

b2 - 4ac 8. True or False If the discriminant of a quadratic equation is

positive, then the equation has two solutions that are

negatives of one another.

Skill Building

In Problems 9–28, solve each equation by factoring.

9. x2 - 9x = 0 10. x2 + 4x = 0 11. x2 - 25 = 0 12. x2 - 9 = 0
10. z2 + z - 6 = 0 14. v2 + 7v + 6 = 0 15. 2x2 - 5x - 3 = 0 16. 3x2 + 5x + 2 = 0
11. 3t2 - 48 = 0 18. 2y2 - 50 = 0 19. x1x - 82 + 12 = 0 20. x1x + 42 = 12
12. 4x2 + 9 = 12x 22. 25x2 + 16 = 40x 23. 61p2 - 12 = 5p 24. 212u2 - 4u2 + 3 = 0
13. 6x - 5 =

6

x

26. x +

12

x = 7 27.

41x - 22

x - 3 +

3

x = -3

x1x - 32

28.

5

x + 4 = 4 +

3

x - 2

In Problems 29–34, solve each equation by the Square Root Method.

29. x2 = 25 30. x2 = 36 31. 1x - 122 = 4
30. 1x + 222 = 1 33. 12y + 322 = 9 34. 13z - 222 = 4

In Problems 35–40, solve each equation by completing the square.

35. 36. 37. x2 -

1

2 x -

3

16 x2 + 4x = 21 x2 - 6x = 13 = 0

38. 39. 3x2 + x - 40. 2x2 - 3x - 1 = 0

1

2 x2 + = 0

2

3 x -

1

3 = 0

In Problems 41–64, find the real solutions, if any, of each equation. Use the quadratic formula.

41. x2 - 4x + 2 = 0 42. x2 + 4x + 2 = 0 43. x2 - 4x - 1 = 0
42. x2 + 6x + 1 = 0 45. 2x2 - 5x + 3 = 0 46. 2x2 + 5x + 3 = 0
43. 2x2 = 1 - 2x 51. 4x2 = 9x 52. 5x = 4x2
44. 54. 55.

3

4 x2 -

1

4 x -

1

2 9t2 - 6t + 1 = 0 4u2 - 6u + 9 = 0 = 0 56. 57. 58.

3

5

x2 - x =

1

5

5

3

x2 - x =

1

3

2

3 x2 - x - 3 = 0

59. 60. 61. 4 -

1

x -

2

x 2x1x + 22 = 3 3x1x + 22 = 1 2 = 0

62. 63. 64.

2x

x - 3 +

1

x = 4

3x

x - 2 +

1

x 4 + = 4

1

x -

1

x2 = 0

In Problems 65–70, find the real solutions, if any, of each equation. Use the quadratic formula and a calculator. Express any solutions

rounded to two decimal places.

65. x2 - 4.1x + 2.2 = 0 66. x2 + 3.9x + 1.8 = 0 67. x2 + 23 x - 3 = 0
66. x2 + 22 x - 2 = 0 69. px2 - x - p = 0 70. px2 + px - 2 = 0

In Problems 71–76, use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real

solution, or no real solution, without solving the equation.

71. 2x2 - 6x + 7 = 0 72. x2 + 4x + 7 = 0 73. 9x2 - 30x + 25 = 0
72. 25x2 - 20x + 4 = 0 75. 3x2 + 5x - 8 = 0 76. 2x2 - 3x - 7 = 0

Mixed Practice

In Problems 77–90, find the real solutions, if any, of each equation. Use any method.

77. x2 - 5 = 0 78. x2 - 6 = 0 79. 16x2 - 8x + 1 = 0
78. 9x2 - 12x + 4 = 0 81. 10x2 - 19x - 15 = 0 82. 6x2 + 7x - 20 = 0
79. 84. 85. x2 + 22 x =

1

2

2 + z = 6z2 2 = y + 6y2

86. 87. x2 + x = 4 88. x2 + x = 1

1

2 x2 = 22 x + 1

89. 90.

3x

x + 2 +

1

x - 1 =

4 - 7x

x2 + x - 2

x

x - 2 +

2

x + 1 =

7x + 1

x2 - x - 2

91. Pythagorean Theorem How many right triangles have a

hypotenuse that measures meters and legs that

measure meters and meters? What are the

dimensions of the triangle(s)?

92. Pythagorean Theorem How many right triangles have a

hypotenuse that measures inches and legs that

measure inches and x inches? What are the

dimensions of the triangle(s)?

93. Dimensions of a Window The area of the opening of a

rectangular window is to be 143 square feet. If the length is

to be 2 feet more than the width, what are the dimensions?

94. Dimensions of a Window The area of a rectangular window

is to be 306 square centimeters. If the length exceeds the

width by 1 centimeter, what are the dimensions?

95. Geometry Find the dimensions of a rectangle whose

perimeter is 26 meters and whose area is 40 square meters.

96. Watering a Field An adjustable water sprinkler that sprays

water in a circular pattern is placed at the center of a square

field whose area is 1250 square feet (see the figure).What is

3x + 13

4x + 5

2x - 5 x + 7

2x + 3

the shortest radius setting that can be used if the field is to

be completely enclosed within the circle?

97. Constructing a Box An open box is to be constructed from

a square piece of sheet metal by removing a square of side

1 foot from each corner and turning up the edges. If the box

is to hold 4 cubic feet, what should be the dimensions of the

sheet metal?

98. Constructing a Box Rework Problem 97 if the piece of

sheet metal is a rectangle whose length is twice its width.

99. Physics A ball is thrown vertically upward from the

top of a building 96 feet tall with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from

the ground after t seconds is

(a) After how many seconds does the ball strike the

ground?

(b) After how many seconds will the ball pass the top of the

building on its way down?

100. Physics An object is propelled vertically upward with an

initial velocity of 20 meters per second. The distance s (in

meters) of the object from the ground after t seconds is

(a) When will the object be 15 meters above the ground?

(b) When will it strike the ground?

(c) Will the object reach a height of 100 meters?

101. Reducing the Size of a Candy Bar A jumbo chocolate bar

with a rectangular shape measures 12 centimeters in length,

7 centimeters in width, and 3 centimeters in thickness. Due

to escalating costs of cocoa, management decides to reduce

the volume of the bar by 10%.To accomplish this reduction,

management decides that the new bar should have the same

3 centimeter thickness, but the length and width of each

should be reduced an equal number of centimeters. What

should be the dimensions of the new candy bar?

s = -4.9t2 + 20t.

s = 96 + 80t - 16t2.

of 3 inches, how wide will the border be? (1 cubic yard

27 cubic feet)

=

10 ft

x

102. Reducing the Size of a Candy Bar Rework Problem 101 if

the reduction is to be 20%.

103. Constructing a Border around a Pool A circular pool

measures 10 feet across. One cubic yard of concrete is to be

used to create a circular border of uniform width around

the pool. If the border is to have a depth of 3 inches, how

wide will the border be? (1 cubic yard 27 cubic feet) See

the illustration.

=

104. Constructing a Border around a Pool Rework Problem 103

if the depth of the border is 4 inches.

105. Constructing a Border around a Garden A landscaper,

who just completed a rectangular flower garden measuring

6 feet by 10 feet, orders 1 cubic yard of premixed cement,

all of which is to be used to create a border of uniform width

around the garden. If the border is to have a depth

traditional

4:3

LCD

16:9

37" 37"

108. Comparing TVs Refer to Problem 107. Find the screen

area of a traditional 50-inch TV and compare it with a

50-inch Plasma TV whose screen is in a 16 : 9 format.Which

screen is larger?

109. The sum of the consecutive integers is given by

the formula How many consecutive integers,

starting with 1, must be added to get a sum of 666?

110. Geometry If a polygon of n sides has diagonals,

how many sides will a polygon with 65 diagonals have? Is

there a polygon with 80 diagonals?

111. Show that the sum of the roots of a quadratic equation is
112. Show that the product of the roots of a quadratic equation

is

113. Find k such that the equation has a

repeated real solution.

kx2 + x + k = 0

c

a

.

-

b

a

.

1

2 n1n - 32

1

2 n1n + 12.

1, 2, 3, Á , n

114. Find k such that the equation has a

repeated real solution.

115. Show that the real solutions of the equation

are the negatives of the real solutions of

the equation Assume that

116. Show that the real solutions of the equation

are the reciprocals of the real solutions

of the equation cx2 + bx + a = 0.Assume that b2 - 4ac Ú 0.

Explaining Concepts: Discussion and Writing

117. Which of the following pairs of equations are equivalent?

Explain.

(a) (b)

(c)

118. Describe three ways that you might solve a quadratic equation.

State your preferred method; explain why you chose it.

119. Explain the benefits of evaluating the discriminant of a

quadratic equation before attempting to solve it.

1x - 121x - 22 = 1x - 122; x - 2 = x - 1

x2 = 9; x = 3 x = 29; x = 3

120. Create three quadratic equations: one having two distinct

solutions, one having no real solution, and one having exactly

one real solution.

121. The word quadratic seems to imply four (quad), yet a quadratic

equation is an equation that involves a polynomial of

degree 2. Investigate the origin of the term quadratic as it is

used in the expression quadratic equation.Write a brief essay

on your findings.

1.3 Complex Numbers,Quardatic Equations in the Complex Number System

1.3 Assess Your Understanding

1. Name the integers and the rational numbers in the set

b -3, 0, 12, (pp. 4–5)

6

5

, pr.

2. True or False Rational numbers and irrational numbers are

in the set of real numbers. (pp. 4–5)

3. Rationalize the denominator of (p. 45)

Concepts and Vocabulary

4. In the complex number the number 5 is called the

part; the number 2 is called the

part; the number i is called the .

5. The equation x2 = -4 has the solution set .

5 + 2i, 6. True or False The conjugate of is

7. True or False All real numbers are complex numbers.
8. True or False If is a solution of a quadratic equation

with real coefficients, then -2 + 3i is also a solution.

Skill Building

9. 12 - 3i2 + 16 + 8i2 10. 14 + 5i2 + 1-8 + 2i2 11. 1-3 + 2i2 - 14 - 4i2 12. 13 - 4i2 - 1-3 - 4i2
10. 12 - 5i2 - 18 + 6i2 14. 1-8 + 4i2 - 12 - 2i2 15. 312 - 6i2 16. -412 + 8i2
11. 2i12 - 3i2 18. 3i1-3 + 4i2 19. 13 - 4i212 + i2 20. 15 + 3i212 - i2
12. 1-6 + i21-6 - i2 22. 1-3 + i213 + i2 23.

10

3 - 4i

24.

13

5 - 12i

In Problems 9–46, write each expression in the standard form a + bi.

25.

2 + i

i

26.

2 - i

-2i

27.

6 - i

1 + i

28.

2 + 3i

1 - i

29. a

1

2 +

23

2 ib

2

30. a

23

2 -

1

2 ib

2

31. 11 + i22 32. 11 - i22 Nu 112 CHAPTER 1 Equations and Inequalities

In Problems 53–72, solve each equation in the complex number system.

85. Electrical Circuits The impedance Z, in ohms, of a circuit

element is defined as the ratio of the phasor voltage V, in

volts, across the element to the phasor current I, in amperes,

through the elements.That is, . If the voltage across a

circuit element is 18 i volts and the current through the element

is 3 4i amperes,determine the impedance.

86. Parallel Circuits In an ac circuit with two parallel pathways,

the total impedance Z, in ohms, satisfies the formula

where is the impedance of Z1 the first pathway

1

Z =

1

Z1

+

1

Z2

,

-

+

Z =

V

I

and is the impedance of the second pathway. Determine

the total impedance if the impedances of the two pathways

are ohms and ohms.

87. Use to show that and
88. Use z = a + bi to show that z = z.

z = a + bi z + z = 2a z - z = 2bi.

Z1 = 2 + i Z2 = 4 - 3i

Z2

91. Explain to a friend how you would add two complex

numbers and how you would multiply two complex numbers.

Explain any differences in the two explanations.

92. Write a brief paragraph that compares the method used to

rationalize the denominator of a radical expression and the

method used to write the quotient of two complex numbers

in standard form.

93. Use an Internet search engine to investigate the origins of

complex numbers. Write a paragraph describing what you

find and present it to the class.

94. Explain how the method of multiplying two complex

numbers is related to multiplying two binomials.

95. What Went Wrong? A student multiplied and

as follows:

The instructor marked the problem incorrect.Why?

= 9

= 281

2-9 # 2-9 = 2(-9)(-9)

2-9 2-9

47. 2-4 48. 2-9 49. 2-25
48. 2-64 51. 413 + 4i214i - 32 52. 414 + 3i213i - 42
49. i711 + i22 44. 2i411 + i22 45. i6 + i4 + i2 + 1 46. i7 + i5 + i3 + i
50. x2 + 4 = 0 54. x2 - 4 = 0 55. x2 - 16 = 0 56. x2 + 25 = 0
51. x2 - 6x + 13 = 0 58. x2 + 4x + 8 = 0 59. x2 - 6x + 10 = 0 60. x2 - 2x + 5 = 0
52. 8x2 - 4x + 1 = 0 62. 10x2 + 6x + 1 = 0 63. 5x2 + 1 = 2x 64. 13x2 + 1 = 6x
53. x2 + x + 1 = 0 66. x2 - x + 1 = 0 67. x3 - 8 = 0 68. x3 + 27 = 0
54. z + z 82. w - w 83. zz 84. z - w
55. x4 = 16 70. x4 = 1 71. x4 + 13x2 + 36 = 0 72. x4 + 3x2 - 4 = 0

In Problems 47–52, perform the indicated operations and express your answer in the form a + bi.

In Problems 73–78, without solving, determine the character of the solutions of each equation in the complex number system.

73. 3x2 - 3x + 4 = 0 74. 2x2 - 4x + 1 = 0 75. 2x2 + 3x = 4
74. x2 + 6 = 2x 77. 9x2 - 12x + 4 = 0 78. 4x2 + 12x + 9 = 0

In Problems 81–84, z = 3 - 4i and w = 8 + 3i.Write each expression in the standard form a + bi.

Applications and Extensions

33. i23 34. i14 35. i-15 36. i-23 37. i6 - 5
34. 4 + i3 39. 6i3 - 4i5 40. 4i3 - 2i2 + 1 41. 11 + i23 42. 13i24 + 1
35. is a solution of a quadratic equation with real

coefficients. Find the other solution.

2 + 3i 80. is a solution of a quadratic equation with real

coefficients. Find the other solution.

4 - i

6

5

5-3, 06; r

89. Use and to show that
90. Use z = a + bi and w = c + di to show that z # w = z # w.

z + w = z + w.

z = a + bi w = c + di

Explaining Concepts: Discussion and Writing

91. Explain to a friend how you would add two complex

numbers and how you would multiply two complex numbers.

Explain any differences in the two explanations.

92. Write a brief paragraph that compares the method used to

rationalize the denominator of a radical expression and the

method used to write the quotient of two complex numbers

in standard form.

93. Use an Internet search engine to investigate the origins of

complex numbers. Write a paragraph describing what you

find and present it to the class.

94. Explain how the method of multiplying two complex

numbers is related to multiplying two binomials.

95. What Went Wrong? A student multiplied and

as follows:

The instructor marked the problem incorrect.Why?

CHAPTER1.4 Radial  Equations,Equation Quardatics in Form,Factorable Equations

1.4 Assess Your Understanding

1. True or False The principal square root of any nonnegative

real number is always nonnegative. (pp. 23–24)

2. (pp. 73–75)
3. Factor 6x3 - 2x2 (pp. 49–55)

Concepts and Vocabulary

4. When an apparent solution does not satisfy the original

equation, it is called a(n) solution.

5. If u is an expression involving x, the equation

is called a(n) equation

.

au2 + bu + c = 0, a Z 0,

6. True or False Radical equations sometimes have extraneous
7. 22t - 1 = 1 8. 23t + 4 = 2 9. 23t + 4 = -6
8. 25t + 3 = -2 11. 23 1 - 2x - 3 = 0 12. 23 1 - 2x - 1 = 0
9. 24 5x - 4 = 2 14. 25 2x - 3 = -1 15. 25 x2 + 2x = -1
10. 24 x2 + 16 = 25 17. x = 81x 18. x = 31x
11. 215 - 2x = x 20. 212 - x = x 21. x = 22x - 1
12. x = 22-x - 1 23. 4x2 - x - 4 = x + 2 24. 43 - x + x2 = x - 2
13. 3 + 23x + 1 = x 26. 2 + 212 - 2x = x 27. 22x + 3 - 2x + 1 = 1

Skill Building

In Problems 7–40, find the real solutions of each equation.

28. 23x + 7 + 2x + 2 = 1 29. 23x + 1 - 2x - 1 = 2 30. 23x - 5 - 2x + 7 = 2

" 4Numb 31. 23 - 21x = 1x 32. 210 + 31x = 1x 33. 13x + 121>2 = 4

34. 13x - 521>2 = 2 35. 15x - 221>3 = 2 36. 12x + 121>3 = -1
35. x4 - 5x2 + 4 = 0 42. x4 - 10x2 + 25 = 0 43. 3x4 - 2x2 - 1 = 0
36. 2x4 - 5x2 - 12 = 0 45. x6 + 7x3 - 8 = 0 46. x6 - 7x3 - 8 = 0
37. 1x + 222 + 71x + 22 + 12 = 0 48. 12x + 522 - 12x + 52 - 6 = 0 49. 13x + 422 - 613x + 42 + 9 = 0
38. 12 - x22 + 12 - x2 - 20 = 0 51. 21s + 122 - 51s + 12 = 3 52. 311 - y22 + 511 - y2 + 2 = 0
39. x - 4x1x = 0 54. x + 81x = 0 55. x + 1x = 20
40. x + 1x = 6 57. t1>2 - 2t1>4 + 1 = 0 58. z1>2 - 4z1>4 + 4 = 0
41. 4x1>2 - 9x1>4 + 4 = 0 60. x1>2 - 3x1>4 + 2 = 0 61. 44

5x2 - 6 = x

62. 44

4 - 5x2 = x 63. x2 + 3x + 4x2 + 3x = 6 64. x2 - 3x - 4x2 - 3x = 2

65.

1

1x + 122 =

1

x + 1 + 2 66.

1

1x - 122 +

1

x - 1 = 12 67. 3x-2 - 7x-1 - 6 = 0

68. 2x-2 - 3x-1 - 4 = 0 69. 2x2>3 - 5x1>3 - 3 = 0 70. 3x4>3 + 5x2>3 - 2 = 0
69. a

v

v + 1

b

2

+

2v

v + 1 = 8 72. a

y

y - 1

b

2

= 6a

y

y - 1

b + 7

37. 1x2 + 921>2 = 5 38. 1x2 - 1621>2 = 9 39. x3>2 - 3x1>2 = 0 40. x3>4 - 9x1>4 = 0

In Problems 73–88, find the real solutions of each equation by factoring.

73. x3 - 9x = 0 74. x4 - x2 = 0 75. 4x3 = 3x2 76. x5 = 4x3
74. x3 + x2 - 20x = 0 78. x3 + 6x2 - 7x = 0 79. x3 + x2 - x - 1 = 0
75. x3 + 4x2 - x - 4 = 0 81. x3 - 3x2 - 4x + 12 = 0 82. x3 - 3x2 - x + 3 = 0
76. 2x3 + 4 = x2 + 8x 84. 3x3 + 4x2 = 27x + 36 85. 5x3 + 45x = 2x2 + 18
77. 3x3 + 12x = 5x2 + 20 87. x1x2 - 3x21>3 + 21x2 - 3x24>3 = 0 88. 3x1x2 + 2x21>2 - 21x2 + 2x23>2 = 0
78. x - 4x1>2 + 2 = 0 90. x2>3 + 4x1>3 + 2 = 0 91. x4 + 23 x2 - 3 = 0
79. x4 + 22 x2 - 2 = 0 93. p11 + t22 = p + 1 + t 94. p11 + r22 = 2 + p11 + r2

Mixed Practice

95. 96. If k =

x + 3

x - 4

If k = and k2 - 3k = 28, find x.

Applications

97. Physics: Using Sound to Measure Distance The distance to the surface of

the water in a well can sometimes be found by dropping an object into the

well and measuring the time elapsed until a sound is heard. If is the time

(measured in seconds) that it takes for the object to strike the water, then

will obey the equation where s is the distance (measured in feet). It

follows that Suppose that is the time that it takes for the sound of

the impact to reach your ears. Because sound waves are known to travel at a

speed of approximately 1100 feet per second, the time to travel the

distance s will be See t2 = the illustration. Now is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard.We

have the equation

Find the distance to the water’s surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

98. Crushing Load A civil engineer relates the thickness T, in inches, and height H, in feet, of a square wooden pillar to its crushing

load L, in tons, using the model . If a square wooden pillar is 4 inches thick and 10 feet high, what is its crushing load?

99. Foucault’s Pendulum The period of a pendulum is the time it takes the pendulum to make one full swing back and forth. The

period T, in seconds, is given by the formula ,where l is the length, in feet, of the pendulum. In 1851, Jean Bernard Leon

Foucault demonstrated the axial rotation of Earth using a large pendulum that he hung in the Panthéon in Paris. The period of

Foucault’s pendulum was approximately 16.5 seconds.What was its length?

Explaining Concepts: Discussion and Writing

100. Make up a radical equation that has no solution.
101. Make up a radical equation that has an extraneous solution.
102. Discuss the step in the solving process for radical equations

that leads to the possibility of extraneous solutions.Why is

there no such possibility for linear and quadratic equations?

103. What Went Wrong? On an exam, Jane solved the equation

and wrote that the solution set was

Jane received 3 out of 5 points for the problem.

Jane asks you why she received 3 out of 5 points. Provide an

explanation.

{-1, 3}.

22x + 3 - x = 0

Explaining Concepts: Discussion and Writing

‘Are You Prepared?’Answers

1. True -2 3. 2x213x – 12

• 5 SOLVING INEQUALITIES

   S1. SECTION 1.5 Solving Inequalities 127

In Problems 17–22, an inequality is given. Write the inequality obtained by:

(a) Adding 3 to each side of the given inequality.

(b) Subtracting 5 from each side of the given inequality.

(c) Multiplying each side of the given inequality by 3.

(d) Multiplying each side of the given inequality by -2.

3. If each side of an inequality is multiplied by a(n)

number, then the sense of the inequality symbol is reversed.

4. A(n) , denoted consists of all real

numbers x for which

5. The state that the sense, or

direction, of an inequality remains the same if each side is

multiplied by a positive number, while the direction is

reversed if each side is multiplied by a negative number.

a … x … b.

3a, b4,

In Problems 6–9, assume that and

6. True or False
7. True or False
8. True or False
9. True or False
10. True or False The square of any real number is always

nonnegative.

a

c 6

b

c

ac 7 bc

a - c 6 b - c

a + c 6 b + c

a 6 b c 6 0.

Concepts and Vocabulary

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. Graph the inequality: x Ú -2. (pp. 17–26) 2. True or False -5 7 -3 (pp. 17–26)

1.5 Assess Your Understanding

Skill Building

In Problems 11–16, express the graph shown in blue using interval notation. Also express each as an inequality involving x.

11. 12. 13.
12. 15. 16.

–1 0 1 2 3 –2 –1 0 1 2 –1 0 1 2 3

–2 –1 0 1 2 –1 0 1 2 3 –1 0 1 2 3

17. 3 6 5 18. 2 7 1 19. 4 7 -3 20. -3 7 -5 21. 2x + 1 6 2 22. 1 - 2x 7 5

In Problems 23–30, write each inequality using interval notation, and illustrate each inequality using the real number line.

23. 0 … x … 4 24. -1 6 x 6 5 25. 4 … x 6 6 26. -2 6 x 6 0
24. x Ú 4 28. x … 5 29. x 6 -4 30. x 7 1
25. 32, 54 32. 11, 22 33. 1-3, -22 34. 30, 12
26. 34, q2 36. 1-q, 24 37. 1-q, -32 38. 1-8, q2

In Problems 31–38, write each interval as an inequality involving x, and illustrate each inequality using the real number line.

In Problems 39–52, fill in the blank with the correct inequality symbol.

39. If x 6 5, then x - 5 0. 40. If x 6 -4, then x + 4 0.
40. If x 7 -4, then x + 4 0. 42. If x 7 6, then x - 6 0.
41. If x Ú -4, then 3x -12. 44. If x … 3, then 2x 6.
42. If x 7 6, then -2x -12. 46. If x 7 -2, then -4x 8.
43. If x Ú 5, then -4x -20. 48. If x … -4, then -3x 12.
44. If 2x 7 6, then x 3. 50. If 3x … 12, then x 4.
45. If - then x -6.

1

2 x … 3, 52. If - then x -4.

1

4 x 7 1, 128 CHAPTER 1 Equations and Inequalities

In Problems 53–88, solve each inequality. Express your answer using set notation or interval notation. Graph the solution set.

89. If then
90. If then
91. If then
92. If then
93. If then
94. If then
95. If then
96. If then
97. If then
98. If then
99. What is the domain of the variable in the expression
100. What is the domain of the variable in the expression
101. A young adult may be defined as someone older than 21,

but less than 30 years of age. Express this statement using

inequalities.

102. Middle-aged may be defined as being 40 or more and less

than 60. Express this statement using inequalities.

103. Life Expectancy The Social Security Administration

determined that an average 30-year-old male in 2005 could

expect to live at least 46.60 more years and an average

28 + 2x ?

23x + 6 ?

0 6 2x 6 6, a 6 x2 6 b.

6 6 3x 6 12, a 6 x2 6 b.

a 6

1

x - 6 2 6 x 6 4, 6 b.

a 6

1

x + 4 -3 6 x 6 0, 6 b.

-3 6 x 6 3, a 6 1 - 2x 6 b.

0 6 x 6 4, a 6 2x + 3 6 b.

a 6

1

2 -4 6 x 6 0, x 6 b.

2 6 x 6 3, a 6 -4x 6 b.

-3 6 x 6 2, a 6 x - 6 6 b.

-1 6 x 6 1, a 6 x + 4 6 b. 30-year-old female in 2005 could expect to live at least 51.03

more years.

(a) To what age can an average 30-year-old male expect to

live? Express your answer as an inequality.

(b) To what age can an average 30-year-old female expect

to live? Express your answer as an inequality.

(c) Who can expect to live longer, a male or a female? By

how many years?

Source: Social Security Administration,Period Life Table, 2005

53. x + 1 6 5 54. x - 6 6 1 55. 1 - 2x … 3
54. 2 - 3x … 5 57. 3x - 7 7 2 58. 2x + 5 7 1
55. 3x - 1 Ú 3 + x 60. 2x - 2 Ú 3 + x 61. -21x + 32 6 8
56. -311 - x2 6 12 63. 4 - 311 - x2 … 3 64. 8 - 412 - x2 … -2x

65.

1

2 1x - 42 7 x + 8 66. 3x + 4 7

1

3 1x - 22 67.

x

2 Ú 1 -

x

4

68.

x

3 Ú 2 +

x

6

69. 0 … 2x - 6 … 4 70. 4 … 2x + 2 … 10
70. -5 … 4 - 3x … 2 72. -3 … 3 - 2x … 9 73. -3 6

2x - 1

4 6 0

74. 0 6

3x + 2

2 6 4 75. 1 6 1 -

1

2 x 6 4 76. 0 6 1 -

1

3 x 6 1

77. 1x + 221x - 32 7 1x - 121x + 12 78. 1x - 121x + 12 7 1x - 321x + 42 79. x14x + 32 … 12x + 122
78. x19x - 52 … 13x - 122 81.

1

2 …

x + 1

3 6

3

4

82.

1

3 6

x + 1

2 …

2

3

83. 14x + 22-1 6 0 84. 12x - 12-1 7 0 85. 0 6

2

x 6

3

5

86. 0 6

4

x 6

2

3

87. 0 6 12x - 42-1 6

1

2

88. 0 6 13x + 62-1 6

1

3

Applications and Extensions

In Problems 89–98, find a and b.

JAN

2005

JULY

2051 JAN

2056

104. General Chemistry For a certain ideal gas, the volume V

(in cubic centimeters) equals 20 times the temperature T (in

degrees Celsius). If the temperature varies from 80° to 120° C

inclusive, what is the corresponding range of the volume of

the gas?

*105. Real Estate A real estate agent agrees to sell an apartment

complex according to the following commission

schedule: $45,000 plus 25% of the selling price in excess of

$900,000. Assuming that the complex will sell at some price

between $900,000 and $1,100,000 inclusive, over what range SECTION 1.5 Solving Inequalities 129

does the agent’s commission vary? How does the commission

vary as a percent of selling price?

106. Sales Commission A used car salesperson is paid a

commission of $25 plus 40% of the selling price in excess of

owner’s cost. The owner claims that used cars typically sell

for at least owner’s cost plus $200 and at most owner’s cost

plus $3000. For each sale made, over what range can the

salesperson expect the commission to vary?

107. Federal Tax Withholding The percentage method of

withholding for federal income tax (2010) states that a

single person whose weekly wages, after subtracting

withholding allowances, are over $693, but not over $1302,

shall have $82.35 plus 25% of the excess over $693 withheld.

Over what range does the amount withheld vary if the

weekly wages vary from $700 to $900 inclusive?

Source: Employer’s Tax Guide. Internal Revenue Service, 2010.

108. Exercising Sue wants to lose weight. For healthy weight

loss, the American College of Sports Medicine (ACSM)

recommends 200 to 300 minutes of exercise per week. For

the first six days of the week, Sue exercised 40, 45, 0, 50, 25,

and 35 minutes. How long should Sue exercise on the

seventh day in order to stay within the ACSM guidelines?

109. Electricity Rates Commonwealth Edison Company’s charge

for electricity in January 2010 was 9.44¢ per kilowatt-hour.

In addition, each monthly bill contains a customer charge of

$12.55. If last year’s bills ranged from a low of $76.27 to a

high of $248.55, over what range did usage vary (in

kilowatt-hours)?

Source: Commonwealth Edison Co., Chicago, Illinois, 2010.

110. Water Bills The Village of Oak Lawn charges homeowners

$37.62 per quarter-year plus $3.86 per 1000 gallons for water

usage in excess of 10,000 gallons. In 2010 one homeowner’s

quarterly bill ranged from a high of $122.54 to a low of

$68.50. Over what range did water usage vary?

Source: Village of Oak Lawn, Illinois, January 2010.

111. Markup of a New Car The markup over dealer’s cost of a

new car ranges from 12% to 18%. If the sticker price is

$18,000, over what range will the dealer’s cost vary?

112. IQ Tests A standard intelligence test has an average score

of 100.According to statistical theory, of the people who take

the test, the 2.5% with the highest scores will have scores of

more than above the average, where (sigma, a

number called the standard deviation) depends on the nature

of the test. If for this test and there is (in principle) no

upper limit to the score possible on the test,write the interval

of possible test scores of the people in the top 2.5%.

s = 12

1.96s s

What do I need to get a B?

68

87

82

89

!4 !2 0

113. Computing Grades In your Economics 101 class, you have

scores of 68, 82, 87, and 89 on the first four of five tests. To

get a grade of B, the average of the first five test scores must

be greater than or equal to 80 and less than 90.

(a) Solve an inequality to find the range of the score that

you need on the last test to get a B.

(b) What score do you need if the fifth test counts double?

114. “Light” Foods For food products to be labeled “light,” the

U.S. Food and Drug Administration requires that the

altered product must either contain one-third or fewer

calories than the regular product or it must contain one-half

or less fat than the regular product. If a serving of Miracle

Whip® Light contains 20 calories and 1.5 grams of fat, then

what must be true about either the number of calories or

the grams of fat in a serving of regular Miracle Whip®?

115. Arithmetic Mean If show that The

number is called the arithmetic mean of a and b.

116. Refer to Problem 115. Show that the arithmetic mean of a

and b is equidistant from a and b.

117. Geometric Mean If show that

The number is called the geometric mean of a and b.

118. Refer to Problems 115 and 117. Show that the geometric

mean of a and b is less than the arithmetic mean of a and b.

119. Harmonic Mean For let h be defined by

Show that The number h is called the harmonic

mean of a and b.

120. Refer to Problems 115, 117, and 119. Show that the harmonic

mean of a and b equals the geometric mean squared divided

by the arithmetic mean.

121. Another Reciprocal Property Prove that if ,

then 0 6

1

b 6

1

a

.

0 6 a 6 b

a 6 h 6 b.

1

h =

1

2 a

1

a +

1

b

b

0 6 a 6 b,

1ab

0 6 a 6 b, a 6 1ab 6 b.

a + b

2

a 6

a + b

2 a 6 b, 6 b.

122. Make up an inequality that has no solution. Make up one

that has exactly one solution.

123. The inequality has no real solution.Explain why.
124. Do you prefer to use inequality notation or interval notation

to express the solution to an inequality? Give your reasons.

Are there particular circumstances when you prefer one to

the other? Cite examples.

x2 + 1 6 -5

125. How would you explain to a fellow student the underlying

reason for the multiplication properties for inequalities

(page 122) that is, the sense or direction of an inequality

remains the same if each side is multiplied by a positive real

number, whereas the direction is reversed if each side is

multiplied by a negative real number.

Explaining Concepts: Discussion and Writing

‘Are You Prepared?’Answers

1. False

• EQUATIONS AND INEQUALITIES INVOLVING ABSOLUTE VALUE

3. 6 Assess Your Understanding

Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. ƒ -2ƒ = (pp. 17–26) 2. True or False ƒxƒ Ú 0 for any real number x. (pp.17–26)

ƒxƒ = 5 5 6. 5. True or False The equation has no solution.

6. True or False The inequality has the set of real

numbers as its solution set.

ƒxƒ Ú -2

ƒxƒ = -2

Concepts and Vocabulary

3. The solution set of the equation is
4. The solution set of the inequality is

{xƒ }.

ƒxƒ 6 5

ƒxƒ = 5 5 6. 5. True or False The equation has no solution.

6. True or False The inequality has the set of real

numbers as its solution set.

Ƒxƒ

SECTION 1.6 Equations and Inequalities Involving Absolute Value 133

In Problems 7–34, solve each equation.

Skill Building

7. ƒ2xƒ = 6 8. ƒ3xƒ = 12 9. ƒ2x + 3ƒ = 5 10. ƒ3x - 1ƒ = 2
8. ƒ1 - 4t ƒ + 8 = 13 12. ƒ1 - 2zƒ + 6 = 9 13. ƒ -2xƒ = ƒ8ƒ 14. ƒ -xƒ = ƒ1ƒ
9. ƒ -2ƒx = 4 16. ƒ3ƒx = 9 17.

2

3 ƒxƒ = 9 18.

3

4 ƒxƒ = 9

19. `

x

3 +

2

5

` = 2 20. `

x

2 -

1

3

` = 1 21. ƒu - 2ƒ = -

1

2

22. ƒ2 - vƒ = -1
23. 4 - ƒ2xƒ = 3 24. 5 - `

1

2 x ` = 3 25. ƒx2 - 9ƒ = 0 26. ƒx2 - 16ƒ = 0

27. ƒx2 - 2xƒ = 3 28. ƒx2 + xƒ = 12 29. ƒx2 + x - 1ƒ = 1 30. ƒx2 + 3x - 2ƒ = 2
28. `

3x - 2

2x - 3

` = 2 32. `

2x + 1

3x + 4

` = 1 33. ƒx2 + 3xƒ = ƒx2 - 2xƒ 34. ƒx2 - 2xƒ = ƒx2 + 6xƒ

35. ƒ2xƒ 6 8 36. ƒ3xƒ 6 15 37. ƒ3xƒ 7 12 38. ƒ2xƒ 7 6
36. ƒx - 2ƒ + 2 6 3 40. ƒx + 4ƒ + 3 6 5 41. ƒ3t - 2ƒ … 4 42. ƒ2u + 5ƒ … 7
37. ƒ2x - 3ƒ Ú 2 44. ƒ3x + 4ƒ Ú 2 45. ƒ1 - 4xƒ - 7 6 -2 46. ƒ1 - 2xƒ - 4 6 -1
38. ƒ1 - 2xƒ 7 3 48. ƒ2 - 3xƒ 7 1 49. ƒ -4xƒ + ƒ -5ƒ … 1 50. ƒ -xƒ - ƒ4ƒ … 2
39. ƒ -2xƒ 7 ƒ -3ƒ 52. ƒ -x - 2ƒ Ú 1 53. - ƒ2x - 1ƒ Ú -3 54. - ƒ1 - 2xƒ Ú -3
40. ƒ2xƒ 6 -1 56. ƒ3xƒ Ú 0 57. ƒ5xƒ Ú -1 58. ƒ6xƒ 6 -2
41. `

2x + 3

3 -

1

2

` 6 1 60. 3 - ƒx + 1ƒ 6

1

2

61. 5 + ƒx - 1ƒ 7

1

2

62. `

2x - 3

2 +

1

3

` 7 1

In Problems 35–62, solve each inequality. Express your answer using set notation or interval notation. Graph the solution set.

63. Body Temperature “Normal” human body temperature is

98.6°F. If a temperature x that differs from normal by at least

1.5° is considered unhealthy, write the condition for an unhealthy

temperature x as an inequality involving an absolute

value, and solve for x.

value, and solve the inequality for x to determine the interval

in which the actual average is likely to fall.

Note: In statistics, this interval is called a 99% confidence

interval.

66. Speed of Sound According to data from the Hill Aerospace

Museum (Hill Air Force Base, Utah), the speed of

sound varies depending on altitude, barometric pressure,

and temperature. For example, at 20,000 feet, 13.75 inches of

mercury, and –12.3°F, the speed of sound is about 707 miles

per hour, but the speed can vary from this result by as much

as 55 miles per hour as conditions change.

(a) Express this situation as an inequality involving an

absolute value.

(b) Using x for the speed of sound, solve for x to find an

interval for the speed of sound.

67. Express the fact that x differs from 3 by less than as an

inequality involving an absolute value. Solve for x.

68. Express the fact that x differs from by less than 1 as an

inequality involving an absolute value. Solve for x.

69. Express the fact that x differs from by more than 2 as an

inequality involving an absolute value. Solve for x.

70. Express the fact that x differs from 2 by more than 3 as an

inequality involving an absolute value. Solve for x.

-3

-4

1

2

Applications and Extensions

64. Household Voltage In the United States, normal household

voltage is 110 volts. However, it is not uncommon for

actual voltage to differ from normal voltage by at most 5 volts.

Express this situation as an inequality involving an absolute

value. Use x as the actual voltage and solve for x.

65. Reading Books A Gallup poll conducted May 20–22,

2005, found that Americans read an average of 13.4 books

per year. Gallup is 99% confident that the result from this

poll is off by fewer than 1.35 books from the actual average

1. x. Express this situation as an inequality involving absolute

In Problems 71–76, find a and b.

71. If then
72. If then
73. If then
74. If then
75. If then
76. If then
77. Show that: if and , then

[Hint: .]

78. Show that
79. Prove the triangle inequality

[Hint: Expand and use the result of

Problem 78.]

ƒa + bƒ2 = 1a + b22,

ƒa + bƒ … ƒaƒ + ƒbƒ.

a … ƒaƒ.

b - a = 11b - 1a211b + 1a2

a 7 0, b 7 0, 1a 6 1b a 6 b.

a …

1

x + 5 ƒx + 1ƒ … 3, … b.

a …

1

x - 10 ƒx - 2ƒ … 7, … b.

ƒx - 3ƒ … 1, a … 3x + 1 … b.

ƒx + 4ƒ … 2, a … 2x - 3 … b.

ƒx + 2ƒ 6 5, a 6 x - 2 6 b.

ƒx - 1ƒ 6 3, a 6 x + 4 6 b.

80. Prove that

[Hint: Apply the triangle inequality from Problem 79 to

]

81. If show that the solution set of the inequality

consists of all numbers x for which

82. If show that the solution set of the inequality

consists of all numbers x for which

x 6 -1a or x 7 1a

x2 7 a

a 7 0,

-1a 6 x 6 1a

x2 6 a

a 7 0,

ƒaƒ = ƒ1a - b2 + bƒ.

ƒa - bƒ Ú ƒaƒ - ƒbƒ.

93. The equation has no solution. Explain why.
94. The inequality has all real numbers as solutions.

Explain why.

ƒxƒ 7 -0.5

ƒxƒ = -2 95. The inequality has as solution set Explain

why.

ƒxƒ 7 0 5x ƒ x Z 06.

Explaining Concepts: Discussion and Writing

1. 2 True

1-7  PROBLEM SOLVING:INTEREST,MIXTURE,UNIFORM MOTION,CONTANT RATE JOB APPLICATIONS

• Assess Your Understanding

2. Concepts and Vocabulary
3. The process of using variables to represent unknown quantities
4. and then finding relationships that involve these variables
5. is referred to as .
6. The money paid for the use of money is .
7. Objects that move at a constant speed are said to be in
8. .
9. True or False The amount charged for the use of principal
10. for a given period of time is called the rate of interest.
11. True or False If an object moves at an average speed
12. the distance d covered in time t is given by the formula
13. Suppose that you want to mix two coffees in order to obtain
14. 100 pounds of a blend. If x represents the number of pounds
15. of coffee A, write an algebraic expression that represents
16. the number of pounds of coffee B.

SECTION 1.7 Problem Solving: Interest,Mixture, Uniform Motion,Constant Rate Job Applications 141

In Problems 7–16, translate each sentence into a mathematical equation. Be sure to identify the meaning of all symbols.

23. Business: Mixing Nuts A nut store normally sells cashews

for $9.00 per pound and almonds for $3.50 per pound. But at

the end of the month the almonds had not sold well, so, in

order to sell 60 pounds of almonds, the manager decided to

mix the 60 pounds of almonds with some cashews and sell

the mixture for $7.50 per pound. How many pounds of

cashews should be mixed with the almonds to ensure no

change in the profit?

24. Business: Mixing Candy A candy store sells boxes of

candy containing caramels and cremes. Each box sells for

$12.50 and holds 30 pieces of candy (all pieces are the same

size). If the caramels cost $0.25 to produce and the cremes

cost $0.45 to produce, how many of each should be in a box

to make a profit of $3?

25. Physics: Uniform Motion A motorboat can maintain a

constant speed of 16 miles per hour relative to the water.

The boat makes a trip upstream to a certain point in 20 minutes;

the return trip takes 15 minutes.What is the speed of

the current? See the figure.

Applications and Extensions

7. Geometry The area of a circle is the product of the number

and the square of the radius.

8. Geometry The circumference of a circle is the product of

the number and twice the radius.

9. Geometry The area of a square is the square of the length

of a side.

10. Geometry The perimeter of a square is four times the

length of a side.

11. Physics Force equals the product of mass and acceleration.
12. Physics Pressure is force per unit area.
13. Physics Work equals force times distance.
14. Physics Kinetic energy is one-half the product of the mass

and the square of the velocity.

15. Business The total variable cost of manufacturing x dishwashers

is $150 per dishwasher times the number of dishwashers

manufactured.

16. Business The total revenue derived from selling x dishwashers

is $250 per dishwasher times the number of dishwashers

sold.

17. Financial Planning Betsy, a recent retiree, requires $6000

per year in extra income. She has $50,000 to invest and can

invest in B-rated bonds paying 15% per year or in a certificate

of deposit (CD) paying 7% per year. How much money

should be invested in each to realize exactly $6000 in

interest per year?

18. Financial Planning After 2 years, Betsy (see Problem 17)

finds that she will now require $7000 per year. Assuming

that the remaining information is the same, how should the

money be reinvested?

19. Banking A bank loaned out $12,000, part of it at the rate of

8% per year and the rest at the rate of 18% per year. If the interest

received totaled $1000, how much was loaned at 8%?

20. Banking Wendy, a loan officer at a bank, has $1,000,000 to

lend and is required to obtain an average return of 18% per

year. If she can lend at the rate of 19% or at the rate of 16%,

how much can she lend at the 16% rate and still meet her

requirement?

21. Blending Teas The manager of a store that specializes in

selling tea decides to experiment with a new blend. She will

mix some Earl Grey tea that sells for $5 per pound with some

Orange Pekoe tea that sells for $3 per pound to get 100 pounds

of the new blend. The selling price of the new blend is to be

$4.50 per pound, and there is to be no difference in revenue

from selling the new blend versus selling the other types.How

many pounds of the Earl Grey tea and Orange Pekoe tea are

required?

22. Business: Blending Coffee A coffee manufacturer wants

to market a new blend of coffee that sells for $3.90 per

pound by mixing two coffees that sell for $2.75 and $5 per

pound, respectively.What amounts of each coffee should be

blended to obtain the desired mixture?

[Hint: Assume that the total weight of the desired blend is

100 pounds.]

p

p

26. Physics: Uniform Motion A motorboat heads upstream on

a river that has a current of 3 miles per hour. The trip upstream

takes 5 hours, and the return trip takes 2.5 hours.What

is the speed of the motorboat? (Assume that the motorboat

maintains a constant speed relative to the water.)

27. Physics: Uniform Motion A motorboat maintained a

constant speed of 15 miles per hour relative to the water in

going 10 miles upstream and then returning. The total time

for the trip was 1.5 hours. Use this information to find the

speed of the current.

28. Physics: Uniform Motion Two cars enter the Florida

Turnpike at Commercial Boulevard at 8:00 AM, each

heading for Wildwood. One car’s average speed is 10 miles

per hour more than the other’s. The faster car arrives at

Wildwood at 11:00 AM, hour before the other car. What

was the average speed of each car? How far did each travel?

29. Moving Walkways The speed of a moving walkway is

typically about 2.5 feet per second. Walking on such a

moving walkway, it takes Karen a total of 40 seconds to

travel 50 feet with the movement of the walkway and then

back again against the movement of the walkway.What is

Karen’s normal walking speed?

Source: Answers.com

1

2 1 0 2 0 3 0 4 0

4 0 3 0 2 0 1 0

SOUTH

TE

DB

142 CHAPTER 1 Equations and Inequalities

30. Moving Walkways The Gare Montparnasse train station

in Paris has a high-speed version of a moving walkway. If he

walks while riding this moving walkway, Jean Claude can

travel 200 meters in 30 seconds less time than if he stands

still on the moving walkway. If Jean Claude walks at a

normal rate of 1.5 meters per second, what is the speed of

the Gare Montparnasse walkway?

Source: Answers.com

31. Tennis A regulation doubles tennis court has an area of

2808 square feet. If it is 6 feet longer than twice its width,

determine the dimensions of the court.

Source: United States Tennis Association

32. Laser Printers It takes an HP LaserJet 1300 laser printer

10 minutes longer to complete a 600-page print job by itself

than it takes an HP LaserJet 2420 to complete the same job

by itself. Together the two printers can complete the job in

12 minutes. How long does it take each printer to complete

the print job alone? What is the speed of each printer?

Source: Hewlett-Packard

33. Working Together on a Job Trent can deliver his newspapers

in 30 minutes. It takes Lois 20 minutes to do the same

route. How long would it take them to deliver the

newspapers if they work together?

34. Working Together on a Job Patrice, by himself, can paint

four rooms in 10 hours. If he hires April to help, they can do

the same job together in 6 hours. If he lets April work alone,

how long will it take her to paint four rooms?

35. Enclosing a Garden A gardener has 46 feet of fencing to

be used to enclose a rectangular garden that has a border

2 feet wide surrounding it. See the figure.

(a) If the length of the garden is to be twice its width, what

will be the dimensions of the garden?

(b) What is the area of the garden?

(c) If the length and width of the garden are to be the same,

what would be the dimensions of the garden?

(d) What would be the area of the square garden?

38. Computing Business Expense Therese, an outside salesperson,

uses her car for both business and pleasure. Last

year, she traveled 30,000 miles, using 900 gallons of gasoline.

Her car gets 40 miles per gallon on the highway and 25 in

the city. She can deduct all highway travel, but no city travel,

on her taxes. How many miles should Therese be allowed as

a business expense?

39. Mixing Water and Antifreeze How much water should be

added to 1 gallon of pure antifreeze to obtain a solution that

is 60% antifreeze?

40. Mixing Water and Antifreeze The cooling system of a

certain foreign-made car has a capacity of 15 liters. If the

system is filled with a mixture that is 40% antifreeze, how

much of this mixture should be drained and replaced by

pure antifreeze so that the system is filled with a solution

that is 60% antifreeze?

41. Chemistry: Salt Solutions How much water must be

evaporated from 32 ounces of a 4% salt solution to make a

6% salt solution?

42. Chemistry: Salt Solutions How much water must be

evaporated from 240 gallons of a 3% salt solution to produce

a 5% salt solution?

43. Purity of Gold The purity of gold is measured in karats,

with pure gold being 24 karats. Other purities of gold are

expressed as proportional parts of pure gold.Thus, 18-karat

gold is or 75% pure gold; 12-karat gold is or 50%

pure gold; and so on. How much 12-karat gold should be

mixed with pure gold to obtain 60 grams of 16-karat gold?

44. Chemistry: Sugar Molecules A sugar molecule has twice

as many atoms of hydrogen as it does oxygen and one more

atom of carbon than oxygen. If a sugar molecule has a

total of 45 atoms, how many are oxygen? How many are

hydrogen?

45. Running a Race Mike can run the mile in 6 minutes, and

Dan can run the mile in 9 minutes. If Mike gives Dan a head

start of 1 minute, how far from the start will Mike pass Dan?

How long does it take? See the figure.

12

24

,

18

24

,

2 ft

2 ft

36. Construction A pond is enclosed by a wooden deck that is

3 feet wide.The fence surrounding the deck is 100 feet long.

(a) If the pond is square, what are its dimensions?

(b) If the pond is rectangular and the length of the pond is

to be three times its width, what are its dimensions?

(c) If the pond is circular, what is its diameter?

(d) Which pond has the most area?

37. Football A tight end can run the 100-yard dash in 12 seconds.

A defensive back can do it in 10 seconds. The tight end

catches a pass at his own 20-yard line with the defensive

back at the 15-yard line. (See the figure.) If no other players

are nearby, at what yard line will the defensive back catch

up to the tight end?

[Hint: At time the defensive back is 5 yards behind

the tight end.]

t = 0,

Start

mi 1–4

mi 3–4

mi 1–2

Dan Mike Chapter Review 143

CHAPTER REVIEW

Things to Know

Quadratic formula (pp. 97 and 110)

If then

If there are no real solutions.

Discriminant (pp. 97 and 110)

If there are two distinct real solutions.

If there is one repeated real solution.

If there are no real solutions, but there are two distinct complex solutions that are not real; the complex solutions

are conjugates of each other.

b2 - 4ac 6 0,

b2 - 4ac = 0,

b2 - 4ac 7 0,

b2 - 4ac 6 0,

x =

-b ; 4b2 - 4ac

2a

ax2 + bx + c = 0, a Z 0,

46. Range of an Airplane An air rescue plane averages

300 miles per hour in still air. It carries enough fuel for 5

hours of flying time. If, upon takeoff, it encounters a head

wind of how far can it fly and return safely?

(Assume that the wind remains constant.)

47. Emptying Oil Tankers An oil tanker can be emptied by

the main pump in 4 hours.An auxiliary pump can empty the

tanker in 9 hours. If the main pump is started at 9 AM, when

should the auxiliary pump be started so that the tanker is

emptied by noon?

48. Cement Mix A 20-pound bag of Economy brand cement

mix contains 25% cement and 75% sand. How much

pure cement must be added to produce a cement mix that is

40% cement?

49. Emptying a Tub A bathroom tub will fill in 15 minutes

with both faucets open and the stopper in place.With both

faucets closed and the stopper removed, the tub will empty

in 20 minutes. How long will it take for the tub to fill if both

faucets are open and the stopper is removed?

50. Using Two Pumps A 5-horsepower (hp) pump can empty a

pool in 5 hours.A smaller, 2-hp pump empties the same pool

in 8 hours. The pumps are used together to begin emptying

this pool.After two hours, the 2-hp pump breaks down.How

long will it take the larger pump to empty the pool?

51. A Biathlon Suppose that you have entered an 87-mile

biathlon that consists of a run and a bicycle race. During

30 mi/hr,

39 oz.

7 in.

55. Critical Thinking You are the manager of a clothing store

and have just purchased 100 dress shirts for $20.00 each.

After 1 month of selling the shirts at the regular price, you

plan to have a sale giving 40% off the original selling price.

However, you still want to make a profit of $4 on each shirt

at the sale price.What should you price the shirts at initially

to ensure this? If, instead of 40% off at the sale, you give

50% off, by how much is your profit reduced?

56. Critical Thinking Make up a word problem that requires

solving a linear equation as part of its solution.Exchange problems

with a friend.Write a critique of your friend’s problem.

57. Critical Thinking Without solving, explain what is wrong

with the following mixture problem: How many liters of

25% ethanol should be added to 20 liters of 48% ethanol to

obtain a solution of 58% ethanol? Now go through an algebraic

solution.What happens?

58. Computing Average Speed In going from Chicago to

Atlanta, a car averages 45 miles per hour, and in going from

Atlanta to Miami, it averages 55 miles per hour. If Atlanta is

halfway between Chicago and Miami, what is the average

speed from Chicago to Miami? Discuss an intuitive solution.

Write a paragraph defending your intuitive solution.

Then solve the problem algebraically. Is your intuitive solution

the same as the algebraic one? If not, find the flaw.

59. Speed of a Plane On a recent flight from Phoenix to Kansas

City, a distance of 919 nautical miles, the plane arrived

20 minutes early.On leaving the aircraft, I asked the captain,

“What was our tail wind?”He replied, “I don’t know, but our

ground speed was 550 knots.” How can you determine if

enough information is provided to find the tail wind? If possible,

find the tail wind. (1 knot = 1 nautical mile per hour)

Explaining Concepts: Discussion and Writing

your run, your average speed is 6 miles per hour, and during

your bicycle race, your average speed is 25 miles per hour.

You finish the race in 5 hours.What is the distance of the

run? What is the distance of the bicycle race?

52. Cyclists Two cyclists leave a city at the same time, one

going east and the other going west. The westbound cyclist

bikes 5 mph faster than the eastbound cyclist. After 6 hours

they are 246 miles apart. How fast is each cyclist riding?

53. Comparing Olympic Heroes In the 1984 Olympics,
54. Lewis of the United States won the gold medal in the 100-

meter race with a time of 9.99 seconds. In the 1896

Olympics,Thomas Burke, also of the United States, won the

gold medal in the 100-meter race in 12.0 seconds. If they ran

in the same race repeating their respective times, by how

many meters would Lewis beat Burke?

54. Constructing a Coffee Can A 39-ounce can of Hills Bros.®

coffee requires 188.5 square inches of aluminum. If its

height is 7 inches, what is its radius? [Hint:The surface area

S of a right cylinder is where r is the

radius and h is the height.]

S = 2pr2 + 2prh, Interval notation (p. 120)

All real numbers

Properties of inequalities

Addition property (p. 121) If then

If then

Multiplication properties (p. 122) (a) If and if then (b) If and if then

If and if then If and if then

Reciprocal properties (p. 123) If then If then

If then If then

Absolute value

If then or (p. 130)

If then (p. 131)

If ƒuƒ Ú a, a 7 0, then u … -a or u Ú a (p. 132)

ƒuƒ … a, a 7 0, -a … u … a

ƒuƒ = a, a 7 0, u = -a u = a

a 6 0

1

a a 7 0 6 0,

1

a 7 0,

1

a a 6 0, 6 0

1

a a 7 0, 7 0

a 6 b c 6 0, ac 7 bc a 7 b c 6 0, ac 6 bc

a 6 b c 7 0, ac 6 bc a 7 b c 7 0, ac 7 bc

a 7 b, a + c 7 b + c

a 6 b, a + c 6 b + c

1-q, q2

1a, b2 5x ƒ a 6 x 6 b6 1-q, a2 5x ƒ x 6 a6

1a, b4 5x ƒ a 6 x … b6 1-q, a4 5x ƒ x … a6

3a, b2 5x ƒ a … x 6 b6 1a, q2 5x ƒ x 7 a6

1. 3a, b4 5x ƒ a … x … b6 3a, q2 5x ƒ x Ú a6

‘

CHAPTER 2 GRAPHS

2.1 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. On the real number line the origin is assigned the number

. (p. 17)

2. If and 5 are the coordinates of two points on the real

number line, the distance between these points is .

(pp. 19–20)

3. If 3 and 4 are the legs of a right triangle, the hypotenuse is

. (p. 30)

4. Use the converse of the Pythagorean Theorem to show that

a triangle whose sides are of lengths 11, 60, and 61 is a right

triangle. (pp. 30–31)

-3

5. The area A of a triangle whose base is b and whose altitude is

h is A ! . (p. 31)

6. True or False Two triangles are congruent if two angles

and the included side of one equals two angles and the

included side of the other (pp. 32–33).

Concepts and Vocabulary

7. If are the coordinates of a point P in the xy-plane,

then x is called the of P and y is the

of P.

8. The coordinate axes divide the xy-plane into four sections

called .

9. If three distinct points P, Q, and R all lie on a line and if

then Q is called the

of the line segment from P to R.

d1P, Q2 = d1Q, R2,

1x, y2 10. True or False The distance between two points is sometimes

a negative number.

11. True or False The point lies in quadrant IV of the

Cartesian plane.

12. True or False The midpoint of a line segment is found by

averaging the x-coordinates and averaging the y-coordinates

of the endpoints.

In Problems 13 and 14, plot each point in the xy-plane.Tell in which quadrant or on what coordinate axis each point lies.

13. (a)

(b)

(c) C = 1-2, -22

B = 16, 02

A = 1-3, 22 14. (a)

(b)

(c) C = 1-3, 42

B = 1-3, -42

A = 11, 42

Skill Building

(d)

(e)

(f) F = 16, -32

E = 10, -32

D = 16, 52 (d)

(e)

(f) F = 1-3, 02SECTION 2.1 The Distance and Midpoint Formulas 155

15. Plot the points and . Describe the set of all points of the form where y is a real number.
16. Plot the points , and . Describe the set of all points of the form where x is a real number.

In Problems 17–28, find the distance d1P1 , P22 between the points P1 and P2 .

10, 32, 11, 32, 1-2, 32, 15, 32 1-4, 32 1x, 32,

12, 02, 12, -32, 12, 42, 12, 12, 12, -12 12, y2,

17. 18. 19. 20.

P1 = (0, 0)

P2 = (2, 1)

x

y

–2

–1

2

2

P2 = (–2, 1) P1 = (0, 0)

x

y

–2

–1

2

2

–1

x

y

2

–2 2

P2 ! (–2, 2)

P1 ! (1, 1)

x

y

–2 –1 2

P 2 1 = (–1, 1)

P2 = (2, 2)

21. P1 = 13, -42; P2 = 15, 42 22. P1 = 1-1, 02; P2 = 12, 42
22. P1 = 1-3, 22; P2 = 16, 02 24. P1 = 12, -32; P2 = 14, 22
23. P1 = 14, -32; P2 = 16, 42 26. P1 = 1-4, -32; P2 = 16, 22
24. P1 = 1a, b2; P2 = 10, 02 28. P1 = 1a, a2; P2 = 10, 02
25. A = 1-2, 52; B = 11, 32; C = 1-1, 02 30. A = 1-2, 52; B = 112, 32; C = 110, -112
26. A = 1-5, 32; B = 16, 02; C = 15, 52 32. A = 1-6, 32; B = 13, -52; C = 1-1, 52
27. A = 14, -32; B = 10, -32; C = 14, 22 34. A = 14, -32; B = 14, 12; C = 12, 12
28. P1 = 13, -42; P2 = 15, 42 36. P1 = 1-2, 02; P2 = 12, 42
29. P1 = 1-3, 22; P2 = 16, 02 38. P1 = 12, -32; P2 = 14, 22
30. P1 = 14, -32; P2 = 16, 12 40. P1 = 1-4, -32; P2 = 12, 22
31. P1 = 1a, b2; P2 = 10, 02 42. P1 = 1a, a2; P2 = 10, 02

In Problems 29–34, plot each point and form the triangle ABC.Verify that the triangle is a right triangle. Find its area.

In Problems 35–42, find the midpoint of the line segment joining the points P1 and P2 .

43. If the point is shifted 3 units to the right and 2 units

down, what are its new coordinates?

44. If the point is shifted 2 units to the left and 4 units

up, what are its new coordinates?

45. Find all points having an x-coordinate of 3 whose distance

from the point is 13.

(a) By using the Pythagorean Theorem.

(b) By using the distance formula.

46. Find all points having a y-coordinate of whose distance

from the point is 17.

(a) By using the Pythagorean Theorem.

(b) By using the distance formula.

47. Find all points on the x-axis that are 6 units from the point

.

48. Find all points on the y-axis that are 6 units from the point

.

49. The midpoint of the line segment from P1 to P2 is . If

, what is ?

50. The midpoint of the line segment from to is . If

what is ?

51. Geometry The medians of a triangle are the line segments

from each vertex to the midpoint of the opposite side (see

the figure). Find the lengths of the medians of the triangle

with vertices at A = 10, 02, B = 16, 02, and C = 14, 42.

P2 = 17, -22, P1

P1 P2 15, -42

P1 = 1-3, 62 P2

1-1, 42

14, -32

14, -32

11, 22

-6

1-2, -12

1-1, 62

12, 52

52. Geometry An equilateral triangle is one in which all three

sides are of equal length. If two vertices of an equilateral

triangle are and find the third vertex. How

many of these triangles are possible?

10, 42 10, 02,

Applications and Extensions

Median

A B

C

Midpoint

53. Geometry Find the midpoint of each diagonal of a square

with side of length s. Draw the conclusion that the diagonals

of a square intersect at their midpoints.

[Hint: Use (0, 0), (0, s), (s, 0), and (s, s) as the vertices of the

square.]

54. Geometry Verify that the points (0, 0), (a, 0), and

are the vertices of an equilateral triangle. Then

show that the midpoints of the three sides are the vertices of

a second equilateral triangle (refer to Problem 52).

a

a

2

,

23 a

2

b

s s

s156 CHAPTER 2 Graphs

65. Drafting Error When a draftsman draws three lines that

are to intersect at one point, the lines may not intersect as

intended and subsequently will form an error triangle. If

this error triangle is long and thin, one estimate for the

location of the desired point is the midpoint of the shortest

side.The figure shows one such error triangle.

Source: www.uwgb.edu/dutchs/STRUCTGE/sl00.htm

In Problems 55–58, find the length of each side of the triangle determined by the three points and . State whether the triangle

is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of

equal length.)

P1 , P2 , P3

55.

56.

57.

58.

59. Baseball A major league baseball “diamond” is actually a

square, 90 feet on a side (see the figure).What is the distance

directly from home plate to second base (the diagonal of the

square)?

P1 = 17, 22; P2 = 1-4, 02; P3 = 14, 62

P1 = 1-2, -12; P2 = 10, 72; P3 = 13, 22

P1 = 1-1, 42; P2 = 16, 22; P3 = 14, -52

P1 = 12, 12; P2 = 1-4, 12; P3 = 1-4, -32

Pitching

rubber

Home plate

1st base

2nd base

3rd base

90 ft

90 ft

60. Little League Baseball The layout of a Little League

playing field is a square, 60 feet on a side. How far is it

directly from home plate to second base (the diagonal of the

square)?

Source: Little League Baseball, Official Regulations and

Playing Rules, 2010.

61. Baseball Refer to Problem 59. Overlay a rectangular

coordinate system on a major league baseball diamond so

that the origin is at home plate, the positive x-axis lies

in the direction from home plate to first base, and the

positive y-axis lies in the direction from home plate to

third base.

(a) What are the coordinates of first base, second base, and

third base? Use feet as the unit of measurement.

(b) If the right fielder is located at how far is it

from the right fielder to second base?

(c) If the center fielder is located at how far is it

from the center fielder to third base?

62. Little League Baseball Refer to Problem 60. Overlay a

rectangular coordinate system on a Little League baseball

diamond so that the origin is at home plate, the positive

x-axis lies in the direction from home plate to first base, and

the positive y-axis lies in the direction from home plate to

third base.

(a) What are the coordinates of first base, second base, and

third base? Use feet as the unit of measurement.

(b) If the right fielder is located at how far is it

from the right fielder to second base?

(c) If the center fielder is located at , how far is it

from the center fielder to third base?

63. Distance between Moving Objects A Dodge Neon and a

Mack truck leave an intersection at the same time.The Neon

heads east at an average speed of 30 miles per hour, while

the truck heads south at an average speed of 40 miles per

hour. Find an expression for their distance apart d (in miles)

at the end of t hours.

1220, 2202

1180, 202,

1300, 3002,

1310, 152,

15 mph

East

100 ft

x

y

(1.4, 1.3)

(2.7, 1.7)

(2.6, 1.5)

1.7

1.5

1.3

1.4 2.6 2.7

Wal-Mart Stores, Inc.

Net sales (in $ billions)

2002 2004

375

204

2003

Year

2005 2006 2007 2008

350

300

250

200

Net sales ($ billions)

150

50

0

100

64. Distance of a Moving Object from a Fixed Point A hot-air

balloon, headed due east at an average speed of 15 miles per

hour and at a constant altitude of 100 feet, passes over an

intersection (see the figure).Find an expression for the distance

d (measured in feet) from the balloon to the intersection

t seconds later.

(a) Find an estimate for the desired intersection point.

(b) Find the length of the median for the midpoint found in

part (a). See Problem 51.

66. Net Sales The figure illustrates how net sales of Wal-Mart

Stores, Inc., have grown from 2002 through 2008. Use the

midpoint formula to estimate the net sales of Wal-Mart

Stores, Inc., in 2005. How does your result compare to the

reported value of $282 billion?

Source:Wal-Mart Stores, Inc., 2008 Annual Report67. Poverty Threshold Poverty thresholds are determined by

the U.S. Census Bureau. A poverty threshold represents the

minimum annual household income for a family not to be

considered poor. In 1998, the poverty threshold for a family

of four with two children under the age of 18 years was

$16,530. In 2008, the poverty threshold for a family of four

with two children under the age of 18 years was $21,834.

Assuming poverty thresholds increase in a straight-line

fashion, use the midpoint formula to estimate the poverty

threshold of a family of four with two children under the

age of 18 in 2003. How does your result compare to the

actual poverty threshold in 2003 of $18,660?

Source: U.S. Census Bureau

2.2 GRAPGH OF VARIABLE IN TWO VARIABLES IN,TERCEPTS,SYMMETRY

2.2 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. Solve the equation 21x + 32 - 1 = - (pp. 82–85) 2. Solve the equation x2 - 9 = 0. (pp. 93–94)
2. Concepts and Vocabulary
3. The points, if any, at which a graph crosses or touches the

coordinate axes are called .

4. The x-intercepts of the graph of an equation are those

x-values for which .

5. If for every point on the graph of an equation the

point is also on the graph, then the graph is

symmetric with respect to the .

6. If the graph of an equation is symmetric with respect to the

y-axis and is an x-intercept of this graph, then is

also an x-intercept.

-4

1-x, y2

1x, y2

7. If the graph of an equation is symmetric with respect to the

origin and is a point on the graph, then

is also a point on the graph.

8. True or False To find the y-intercepts of the graph of an

equation, let and solve for y.

9. True or False The y-coordinate of a point at which the

graph crosses or touches the x-axis is an x-intercept.

10. True or False If a graph is symmetric with respect to the

x-axis, then it cannot be symmetric with respect to theSECTION 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 165

In Problems 17–28, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

Skill Building

In Problems 11–16, determine which of the given points are on the graph of the equation.

11. Equation:

Points: 10, 02; 11, 12; 1-1, 02

y = x4 - 1x 12. Equation:

Points: 10, 02; 11, 12; 11, -12

y = x3 - 21x 13. Equation:

Points: 10, 32; 13, 02; 1-3, 02

y2 = x2 + 9

14. Equation:

Points: 11, 22; 10, 12; 1-1, 02

y3 = x + 1 15. Equation:

Points: 10, 22; 1-2, 22; A22, 22B

x2 + y2 = 4 16. Equation:

Points: 10, 12; 12, 02; a2,

1

2

b

x2 + 4y2 = 4

17. y = x + 2 18. y = x - 6 19. y = 2x + 8 20. y = 3x - 9
18. y = x2 - 1 22. y = x2 - 9 23. y = -x2 + 4 24. y = -x2 + 1
19. 2x + 3y = 6 26. 5x + 2y = 10 27. 9x2 + 4y = 36 28. 4x2 + y = 4

In Problems 29–38, plot each point. Then plot the point that is symmetric to it with respect to (a) the x-axis; (b) the y-axis; (c) the origin.

29. 13, 42 30. 15, 32 31. 1-2, 12 32. 14, -22 33. 15, -22
30. 1-1, -12 35. 1-3, -42 36. 14, 02 37. 10, -32 38. 1-3, 02

In Problems 39–50, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respect to

the x-axis, the y-axis, or the origin.

39. 40. 41. 42.
40. 44. 45. 46.
41. 48. 49. 50.

In Problems 51–54, draw a complete graph so that it has the type of symmetry indicated.

51. y-axis 52. x-axis 53. Origin 54. y-axis

x

y

3

–3 3

–3

x

y

3

3

–3

–3

y

6

!6

!3 3 x

40

!40

x

!6 6

y

3

–3 x

y

3

–3

3

!3 x

y

3

!3

x

y

#

1

!1

!# #––

! 2 #––

2 x

y

3

4

!4

!3

y

3

!3

!3 3 x

y

3

!3

!3 3 x

!8

8

!2 2

!4

4

!4 4

–9 9

–9

9

x

y

(0, –9)

(2, –5)

x

y

–5 5

–5

5

(–4, 0)

(0, 2)

(5, 3)

x

y

–4

4

"––

2

(0, 0) (", 0)

( , 2)

y

–3 3

–2

4

(0, 4)

(0, 0)

(2, 2)

x166 CHAPTER 2 Graphs

In Problems 55–70, list the intercepts and test for symmetry.

55. y2 = x + 4 56. y2 = x + 9 57. y = 13

x 58. y = 15 x

59. x2 + y - 9 = 0 60. x2 - y - 4 = 0 61. 9x2 + 4y2 = 36 62. 4x2 + y2 = 4
60. y = x3 - 27 64. y = x4 - 1 65. y = x2 - 3x - 4 66. y = x2 + 4
61. y =

3x

x2 + 9 68. y =

x2 - 4

2x

69. y = -x3

x2 - 9

70. y =

x4 + 1

2x5

In Problems 71–74, draw a quick sketch of each equation.

71. y = x3 72. x = y2 73. y = 1x 74. y =

1

x

75. If 13, b2 is a point on the graph of y = 4x + 1, what is b? 76. If 1-2, b2 is a point on the graph of 2x + 3y = 2, what is b?
76. If 1a, 42 is a point on the graph of y = x2 + 3x, what is a? 78. If 1a, -52 is a point on the graph of y = x2 + 6x, what is a?
77. Given that the point (1, 2) is on the graph of an equation

that is symmetric with respect to the origin, what other point

is on the graph?

80. If the graph of an equation is symmetric with respect to the

y-axis and 6 is an x-intercept of this graph, name another

x-intercept.

81. If the graph of an equation is symmetric with respect to the

origin and is an x-intercept of this graph, name another

x-intercept.

82. If the graph of an equation is symmetric with respect to the

x-axis and 2 is a y-intercept, name another y-intercept.

83. Microphones In studios and on stages, cardioid microphones

are often preferred for the richness they add to voices

and for their ability to reduce the level of sound from the

sides and rear of the microphone. Suppose one such cardioid

pattern is given by the equation

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the x-axis, y-axis, and

origin.

Source: www.notaviva.com

1x2 + y2 - x22 = x2 + y2.

-4

84. Solar Energy The solar electric generating systems at

Kramer Junction, California, use parabolic troughs to heat a

heat-transfer fluid to a high temperature. This fluid is used

to generate steam that drives a power conversion system to

produce electricity. For troughs 7.5 feet wide, an equation

for the cross-section is 16y2 = 120x - 225.

Applications and Extensions

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the x-axis, y-axis, and

origin.

Source: U.S. Department of Energy

85. (a) Graph , and ,

noting which graphs are the same.

(b) Explain why the graphs of and are the

same.

(c) Explain why the graphs of and are

not the same.

(d) Explain why the graphs of and are not

the same.

86. Explain what is meant by a complete graph.
87. Draw a graph of an equation that contains two x-intercepts;

at one the graph crosses the x-axis, and at the other the

graph touches the x-axis.

88. Make up an equation with the intercepts , and

. Compare your equation with a friend’s equation.

Comment on any similarities.

10, 12

12, 02, 14, 02

y = 3x2 y = x

y = x y = 11x22

y = 3x2 y = ƒxƒ

y = 3x2 , y = x, y = ƒxƒ y = 11x22 89. Draw a graph that contains the points

, and . Compare your graph with those of other

students. Are most of the graphs almost straight lines? How

many are “curved”? Discuss the various ways that these

points might be connected.

90. An equation is being tested for symmetry with respect to

the x-axis, the y-axis, and the origin. Explain why, if two of

these symmetries are present, the remaining one must also

be present.

91. Draw a graph that contains the points , ,

and (0, 2) that is symmetric with respect to the y-axis.

Compare your graph with those of other students; comment

on any similarities. Can a graph contain these points and

be symmetric with respect to the x-axis? the origin? Why

or why not?

(-2, 5) (-1, 3)

11, 32 13, 52

1-2, -12, 10, 12,

1. Explaining Concepts: Discussion and Writing

Interactive Exercises

92. y-axis Symmetry Open the y-axis symmetry applet. Move

point A around the Cartesian Plane with your mouse. How

are the coordinates of point A and the coordinates of point B

related?

93. x-axis Symmetry Open the x-axis symmetry applet. Move

point A around the Cartesian Plane with your mouse. How

are the coordinates of point A and the coordinates of point B

related?

94. Origin Symmetry Open the origin symmetry applet. Move

point A around the Cartesian Plane with your mouse. How

are the coordinates of point A and the coordinates of point B

related?

• LINES

3. 3 Assess Your Understanding

4. In Problems 37–44, find an equation of the line L.
5. x
6. y
7. –2 2
8. (0, 0)
9. 2 (2, 1)
10. –1
11. L
12. x
13. y
14. –2 2
15. (–2, 1)
16. (0, 0)
17. 2
18. –1 L x
19. y
20. –2 2
21. (–1, 3)
22. (1, 1)
23. 3
24. –1
25. L
26. x
27. y
28. –2 2
29. (–1, 1)
30. (2, 2)
31. 2
32. –1
33. L
34. 38. 39. 40.
35. 42. 43. 44.
36. y = 2x
37. (3, 3)
38. L is parallel to y = 2x
39. L
40. x
41. y
42. 3
43. 3
44. –1
45. y = –x
46. (1, 2)
47. L
48. L is parallel to y = –x
49. x
50. y
51. 3
52. 3
53. –1
54. y = 2x
55. L is perpendicular
56. to y = 2x
57. (1, 2)
58. L
59. x
60. y
61. 3
62. 3
63. –1
64. y = –x
65. x
66. y
67. (–1, 1)
68. 1
69. 3
70. –3
71. L
72. L is perpendicular
73. to y = –x
74. x
75. y
76. –2 2
77. (0, 0)
78. 2 (2, 1)
79. –1
80. x
81. y
82. –2 2
83. (–2, 1)
84. (0, 0)
85. 2
86. –1
87. 178 CHAPTER 2 Graphs
88. Skill Building
89. In Problems 11–14, (a) find the slope of the line and (b) interpret the slope.
90. 12. 13. 14.
91. 12, 32; 14, 02 16. 14, 22; 13, 42 17. 1-2, 32; 12, 12 18. 1-1, 12; 12, 32
92. 1-3, -12; 12, -12 20. 14, 22; 1-5, 22 21. 1-1, 22; 1-1, -22 22. 12, 02; 12, 22
93. x
94. y
95. –2 2
96. (–2, 2) (1, 1) 2
97. –1
98. x
99. y
100. –2 2
101. (–1, 1)
102. (2, 2)
103. 2
104. –1
105. In Problems 15–22, plot each pair of points and determine the slope of the line containing them. Graph the line.
106. In Problems 23–30, graph the line containing the point P and having slope m.
107. P = 11, 22; m = 3 24. P = 12, 12; m = 4 25. P = 12, 42; m = -
108. 3
109. 4
110. P = 11, 32; m = -
111. 2
112. 5
113. P = 1-1, 32; m = 0 28. P = 12, -42; m = 0 29. P = 10, 32; slope undefined 30. P = 1-2, 02; slope undefined
114. In Problems 31–36, the slope and a point on a line are given. Use this information to locate three additional points on the line. Answers
115. may vary.
116. [Hint: It is not necessary to find the equation of the line. See Example 3.]
117. Slope 4; point 11, 22 32. Slope 2; point 1-2, 32 33. Slope - point 12, -42
118. 3
119. 2
120. ;
121. Slope point 1-3, 22
122. 4
123. 3
124. ; Slope -2; point 1-2, -32 36. Slope -1; point 14, 12
125. Concepts and Vocabulary
126. The slope of a vertical line is ; the slope of a
127. horizontal line is .
128. For the line , the x-intercept is and the
129. y-intercept is .
130. A horizontal line is given by an equation of the form
131. , where b is the .
132. True or False Vertical lines have an undefined slope.
133. True or False The slope of the line is 3.
134. True or False The point 11, 22 is on the line 2x + y = 4.
135. 2y = 3x + 5
136. 2x + 3y = 6
137. Two nonvertical lines have slopes and respectively.
138. The lines are parallel if and the
139. are unequal; the lines are perpendicular if .
140. The lines and are parallel if
141. .
142. The lines and are perpendicular if
143. .
144. True or False Perpendicular lines have slopes that are
145. reciprocals of one another.
146. a =
147. y = 2x - 1 y = ax + 2
148. a =
149. y = 2x + 3 y = ax + 5
150. m1 m2 ,
151. SECTION 2.3 Lines 179
152. In Problems 45–70, find an equation for the line with the given properties. Express your answer using either the general form or the
153. slope–intercept form of the equation of a line, whichever you prefer.
154. Slope = 3; containing the point 1-2, 32 46. Slope = 2; containing the point 14, -32
155. Slope = - containing the point 11, -12
156. 2
157. 3
158. ; Slope = containing the point 13, 12
159. 1
160. 2
161. ;
162. Containing the points 11, 32 and 1-1, 22 50. Containing the points 1-3, 42 and 12, 52
163. Slope = -3; y-intercept = 3 52. Slope = -2; y-intercept = -2
164. x-intercept = 2; y-intercept = -1 54. x-intercept = -4; y-intercept = 4
165. Slope undefined; containing the point 12, 42 56. Slope undefined; containing the point 13, 82
166. Horizontal; containing the point 1-3, 22 58. Vertical; containing the point 14, -52
167. Parallel to the line y = 2x; containing the point 1-1, 22 60. Parallel to the line y = -3x; containing the point 1-1, 22
168. Parallel to the line 2x - y = -2; containing the point 10, 02 62. Parallel to the line x - 2y = -5; containing the point 10, 02
169. Parallel to the line x = 5; containing the point 14, 22 64. Parallel to the line y = 5; containing the point 14, 22
170. Perpendicular to the line containing the point
171. 11, -22
172. y =
173. 1
174. 2 x + 4; Perpendicular to the line containing the point
175. 11, -22
176. y = 2x - 3;
177. Perpendicular to the line containing the point
178. 1-3, 02
179. 2x + y = 2; Perpendicular to the line containing the
180. point 10, 42
181. x - 2y = -5;
182. Perpendicular to the line x = 8; containing the point 13, 42 70. Perpendicular to the line y = 8; containing the point 13, 42
183. In Problems 71–90, find the slope and y-intercept of each line. Graph the line.
184. y = 2x + 3 72. y = -3x + 4 73.
185. 1
186. 2 y = x - 1
187. 1
188. 3 x + y = 2 y =
189. 1
190. 2 x + 2
191. y = 2x +
192. 1
193. 2
194. x + 2y = 4 78. -x + 3y = 6 79. 2x - 3y = 6 80. 3x + 2y = 6
195. x + y = 1 82. x - y = 2 83. x = -4 84. y = -1 85. y = 5
196. x = 2 87. y - x = 0 88. x + y = 0 89. 2y - 3x = 0 90. 3x + 2y = 0
197. In Problems 91–100, (a) find the intercepts of the graph of each equation and (b) graph the equation.
198. !4
199. 4
200. !6 6
201. !2
202. 2
203. !3 3
204. 2x + 3y = 6 92. 3x - 2y = 6 93. -4x + 5y = 40
205. 6x - 4y = 24 95. 7x + 2y = 21 96. 5x + 3y = 18
207. 1
208. 2 x +
209. 1
210. 3 y = 1
211. In Problems 103–106, the equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither.
212. Find an equation of the x-axis. 102. Find an equation of the y-axis.
214. y = 2x + 4
215. y = 2x - 3
216. y = -2x + 4
217. y =
218. 1
219. 2 x - 3
220. y = -4x + 2
221. y = 4x + 5
222. y = -
223. 1
224. 2 x + 2
225. y = -2x + 3
226. In Problems 107–110, write an equation of each line. Express your answer using either the general form or the slope–intercept form of
227. the equation of a line, whichever you prefer.
228. 108. 109. 110.
229. !2
230. 2
231. !3 3
232. !2
233. 2
234. !3 3
235. x -
236. 2
237. 3 y = 4 0.2x - 0.5y = 1 100. -0.3x + 0.4y = 1.2

y

Ramp

Platform

30"

x

(a) Write a linear equation that relates the height y of the

ramp above the floor to the horizontal distance x from

the platform.

(b) Find and interpret the x-intercept of the graph of your

equation.

(c) Design requirements stipulate that the maximum run

be 30 feet and that the maximum slope be a drop of

1 inch for each 12 inches of run.Will this ramp meet the

requirements? Explain.

(d) What slopes could be used to obtain the 30-inch rise and

still meet design requirements?

Source: www.adaptiveaccess.com/wood_ramps.php

124. Cigarette Use A report in the Child Trends DataBase

indicated that, in 1996, 22.2% of twelfth grade students

reported daily use of cigarettes. In 2006, 12.2% of twelfth

grade students reported daily use of cigarettes.

(a) Write a linear equation that relates the percent y of

twelfth grade students who smoke cigarettes daily to

the number x of years after 1996.

180 CHAPTER 2 Graphs

Applications and Extensions

111. Geometry Use slopes to show that the triangle whose

vertices are , , and is a right triangle.

112. Geometry Use slopes to show that the quadrilateral

whose vertices are , , , and is a

parallelogram.

113. Geometry Use slopes to show that the quadrilateral

whose vertices are , , , and is a

rectangle.

114. Geometry Use slopes and the distance formula to show

that the quadrilateral whose vertices are , , ,

and is a square.

115. Truck Rentals A truck rental company rents a moving

truck for one day by charging $29 plus $0.20 per mile.Write a

linear equation that relates the cost C, in dollars, of renting

the truck to the number x of miles driven.What is the cost

of renting the truck if the truck is driven 110 miles?

230 miles?

116. Cost Equation The fixed costs of operating a business are

the costs incurred regardless of the level of production.

Fixed costs include rent, fixed salaries, and costs of leasing

machinery.The variable costs of operating a business are the

costs that change with the level of output. Variable costs

include raw materials, hourly wages, and electricity. Suppose

that a manufacturer of jeans has fixed daily costs of $500

and variable costs of $8 for each pair of jeans manufactured.

Write a linear equation that relates the daily cost C, in

dollars, of manufacturing the jeans to the number x of jeans

manufactured.What is the cost of manufacturing 400 pairs

of jeans? 740 pairs?

117. Cost of Driving a Car The annual fixed costs for owning a

small sedan are $1289, assuming the car is completely paid

for.The cost to drive the car is approximately $0.15 per mile.

Write a linear equation that relates the cost C and the

number x of miles driven annually.

Source: www.pacebus.com

118. Wages of a Car Salesperson Dan receives $375 per week

for selling new and used cars at a car dealership in Oak Lawn,

Illinois. In addition, he receives 5% of the profit on any sales

that he generates. Write a linear equation that represents

Dan’s weekly salary S when he has sales that generate a profit

of x dollars.

119. Electricity Rates in Illinois Commonwealth Edison

Company supplies electricity to residential customers for a

monthly customer charge of $10.55 plus 9.44 cents per

kilowatt-hour for up to 600 kilowatt-hours.

13, -12

10, 02 11, 32 14, 22

1-1, 02 12, 32 11, -22 14, 12

11, -12 14, 12 12, 22 15, 42

1-2, 52 11, 32 1-1, 02

(a) Write a linear equation that relates the monthly charge

C, in dollars, to the number x of kilowatt-hours used in a

month, .

(b) Graph this equation.

(c) What is themonthly charge for using 200 kilowatt-hours?

(d) What is themonthly charge for using 500 kilowatt-hours?

(e) Interpret the slope of the line.

Source: Commonwealth Edison Company, January, 2010.

120. Electricity Rates in Florida Florida Power & Light

Company supplies electricity to residential customers for a

monthly customer charge of $5.69 plus 8.48 cents per

kilowatt-hour for up to 1000 kilowatt-hours.

(a) Write a linear equation that relates the monthly charge

C, in dollars, to the number x of kilowatt-hours used in a

month, .

(b) Graph this equation.

(c) What is themonthly charge for using 200 kilowatt-hours?

(d) What is themonthly charge for using 500 kilowatt-hours?

(e) Interpret the slope of the line.

Source: Florida Power & Light Company, February, 2010.

121. Measuring Temperature The relationship between Celsius

(°C) and Fahrenheit (°F) degrees of measuring temperature

is linear. Find a linear equation relating °C and °F if 0°C

corresponds to 32°F and 100°C corresponds to 212°F. Use

the equation to find the Celsius measure of 70°F.

122. Measuring Temperature The Kelvin (K) scale for measuring

temperature is obtained by adding 273 to the Celsius

temperature.

(a) Write a linear equation relating K and °C.

(b) Write a linear equation relating K and °F (see

Problem 121).

123. Access Ramp A wooden access ramp is being built to

reach a platform that sits 30 inches above the floor. The

ramp drops 2 inches for every 25-inch run.

0 … x … 1000

0 … x … 600SECTION 2.3 Lines 181

(b) Find the intercepts of the graph of your equation.

(c) Do the intercepts have any meaningful interpretation?

(d) Use your equation to predict the percent for the year

2016. Is this result reasonable?

Source: www.childtrendsdatabank.org

125. Product Promotion A cereal company finds that the number

of people who will buy one of its products in the first

month that it is introduced is linearly related to the amount

of money it spends on advertising. If it spends $40,000 on

advertising, then 100,000 boxes of cereal will be sold, and if

it spends $60,000, then 200,000 boxes will be sold.

(a) Write a linear equation that relates the amount A spent

on advertising to the number x of boxes the company

aims to sell.

(b) How much advertising is needed to sell 300,000 boxes of

cereal?

(c) Interpret the slope.

126. Show that the line containing the points and ,

, is perpendicular to the line . Also show that the

midpoint of and lies on the line .

127. The equation defines a family of lines, one line

for each value of C. On one set of coordinate axes, graph the

members of the family when , and .Can

you draw a conclusion from the graph about each member of

the family?

128. Prove that if two nonvertical lines have slopes whose product

is then the lines are perpendicular. [Hint: Refer to

Figure 47 and use the converse of the PythagoreanTheorem.]

-1

C = -4, C = 0 C = 2

2x - y = C

1a, b2 1b, a2 y = x

a Z b y = x

1a, b2 1b, a2

Explaining Concepts: Discussion and Writing

129. Which of the following equations might have the graph

shown? (More than one answer is possible.)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

130. Which of the following equations might have the graph

shown? (More than one answer is possible.)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

131. The figure shows the graph of two parallel lines.Which of the

following pairs of equations might have such a graph?

(a)

(b)

(c)

(d)

(e)

132. The figure shows the graph of two perpendicular lines.

Which of the following pairs of equations might have such a

graph?

(a)

(b)

(c)

(d)

(e)

2y + x = -2

2x + y = -2

x + 2y = -1

y - 2x = 2

2y + x = -2

2y - x = 2

2y + x = 0

y - 2x = 0

y + 2x = -1

y - 2x = 2

x + 2y = -1

x + 2y = 2

2x - 2y = -4

x - y = -2

x - y = 1

x - y = -2

x + y = -1

x + y = 2

x + 2y = 7

x - 2y = 3

y = x + 4

y = -

1

2 x + 10

y = -2x - 1

x - y = -1

x - y = 1

3x + 4y = 12

2x - 3y = 6

2x + 3y = 6

y = -3x + 3

y = 2x + 3

y = 3x - 5

x - y = -1

x - y = 1

3x - 4y = -12

-2x + 3y = 6

2x + 3y = 6 y

x

y

x

133. mis for Slope The accepted symbol used to denote the slope

of a line is the letterm.Investigate the origin of this symbolism.

Begin by consulting a French dictionary and looking up the

French word monter.Write a brief essay on your findings.

134. Grade of a Road The term grade is used to describe the

inclination of a road.How does this term relate to the notion

of slope of a line? Is a 4% grade very steep? Investigate the

grades of some mountainous roads and determine their

slopes. Write a brief essay on your findings.

x

y

x

y

Steep

7% Grade

135. Carpentry Carpenters use the term pitch to describe the

steepness of staircases and roofs. How does pitch relate to

slope? Investigate typical pitches used for stairs and for

roofs. Write a brief essay on your findings.

136. Can the equation of every line be written in slope–intercept

form? Why?

137. Does every line have exactly one x-intercept and one

y-intercept? Are there any lines that have no intercepts?

138. What can you say about two lines that have equal slopes and

equal y-intercepts?

139. What can you say about two lines with the same x-intercept

and the same y-intercept? Assume that the x-intercept is

not 0.

140. If two distinct lines have the same slope, but different

x-intercepts, can they have the same y-intercept?

141. If two distinct lines have the same y-intercept, but different

slopes, can they have the same x-intercept?

142. Which form of the equation of a line do you prefer to use?

Justify your position with an example that shows that your

choice is better than another. Have reasons.

143. What Went Wrong? A student is asked to find the slope of

the line joining and . He states that the

slope is . Is he correct? If not, what went wrong?

3

2

1. (-3, 2) (1, -4)
2. Interactive Exercises

Ask your instructor if the applet below is of interest to you.

144. Slope Open the slope applet. Move point B around the Cartesian plane with your mouse.

(a) Move B to the point whose coordinates are 2, 7 .What is the slope of the line?

(b) Move B to the point whose coordinates are 3, 6 .What is the slope of the line?

(c) Move B to the point whose coordinates are 4, 5 .What is the slope of the line?

(d) Move B to the point whose coordinates are 4, 4 .What is the slope of the line?

(e) Move B to the point whose coordinates are 4, 1 .What is the slope of the line?

(f) Move B to the point whose coordinates are .What is the slope of the line?

(g) Slowly move B to a point whose x-coordinate is 1.What happens to the value of the slope as the x-coordinate approaches 1?

(h) What can be said about a line whose slope is positive? What can be said about a line whose slope is negative? What can be said

about a line whose slope is 0?

(i) Consider the results of parts (a)–(c).What can be said about the steepness of a line with positive slope as its slope increases?

(j) Move B to the point whose coordinates are 3, 5 .What is the slope of the line? Move B to the point whose coordinates are

1. 15, 62.What is the slope of the line? Move B to the point whose coordinates are 1-1, 32.What is the slope of the line?

2.4 CIRCLES

2.4 Assess Your Understanding

Are You Prepared? Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. To complete the square of , you would (add/

subtract) the number . (p. 56)

x2 + 10x 2. Use the Square Root Method to solve the equation

1x - 222 = 9. (pp. 94–95)

Concepts and Vocabulary

3. True or False Every equation of the form

has a circle as its graph.

4. For a circle, the is the distance from the center to any

point on the circle.

x2 + y2 + ax + by + c = 0

5. True or False The radius of the circle is 3.
6. True or False The center of the circle

is (3, -2).

1x + 322 + 1y - 222 = 13

Skill Building

In Problems 7–10, find the center and radius of each circle. Write the standard form of the equation.

7. 8. 9. 10.
8. r = 2; 1h, k2 = 10, 02 12. r = 3; 1h, k2 = 10, 02 13. r = 2; 1h, k2 = 10, 22 14. r = 3; 1h, k2 = 11, 02
9. r = 5; 1h, k2 = 14, -32 16. r = 4; 1h, k2 = 12, -32 17. r = 4; 1h, k2 = 1-2, 12 18. r = 7; 1h, k2 = 1-5, -22

x

y

(0, 1) (2, 1)

x

y

(1, 2)

(1, 0) x

y

(1, 2)

(4, 2)

x

y

(0, 1)

(2, 3)

In Problems 11–20, write the standard form of the equation and the general form of the equation of each circle of radius r and

center 1h, k2.Graph each circle.

19. r =

1

2

; 1h, k2 = a

1

2

, 0b 20. r =

1

2

; 1h, k2 = a0, -

1

2

b

In Problems 21–34, (a) find the center 1h, k2 and radius r of each circle; (b) graph each circle; (c) find the intercepts, if any.

21. x2 + y2 = 4 22. x2 + 1y - 122 = 1 23. 21x - 322 + 2y2 = 8
22. 31x + 122 + 31y - 122 = 6 25. x2 + y2 - 2x - 4y - 4 = 0 26. x2 + y2 + 4x + 2y - 20 = 0
23. x2 + y2 + 4x - 4y - 1 = 0 28. x2 + y2 - 6x + 2y + 9 = 0 29. x2 + y2 - x + 2y + 1 = 0
24. x2 + y2 + x + y -

1

2 = 0 31. 2x2 + 2y2 - 12x + 8y - 24 = 0 32. 2x2 + 2y2 + 8x + 7 = 0

33. 2x2 + 8x + 2y2 = 0 34. 3x2 + 3y2 - 12y = 0

In Problems 35–42, find the standard form of the equation of each circle.

35. Center at the origin and containing the point 1-2, 32 36. Center 11, 02 and containing the point 1-3, 22
36. Center 12, 32 and tangent to the x-axis 38. Center 1-3, 12 and tangent to the y-axis
37. With endpoints of a diameter at 11, 42 and 1-3, 22 40. With endpoints of a diameter at 14, 32 and 10, 12
38. Center 1-1, 32 and tangent to the line y " 2 42. Center 14, -22 and tangent to the line x " 1

In Problems 43–46, match each graph with the correct equation.

(a) 1x - 322 + 1y + 322 = 9 (b) 1x + 122 + 1y - 222 = 4 (c) 1x - 122 + 1y + 222 = 4 (d) 1x + 322 + 1y - 322 = 9

43. 44. 45. 46.

!4

4

!6 6

!6

6

!9 9

!4

4

!6 6

!6

6

!9 9

186 CHAPTER 2 Graphs SECTION 2.4 Circles 187

48. Find the area of the blue shaded region in the figure, assuming

the quadrilateral inside the circle is a square.

49. Ferris Wheel The original Ferris wheel was built in 1893

by Pittsburgh,Pennsylvania, bridge builder George W.Ferris.

The Ferris wheel was originally built for the 1893 World’s

Fair in Chicago, but was also later reconstructed for the 1904

World’s Fair in St. Louis. It had a maximum height of

264 feet and a wheel diameter of 250 feet. Find an equation

for the wheel if the center of the wheel is on the y-axis.

Source: inventors.about.com

50. Ferris Wheel In 2008, the Singapore Flyer opened as the

world’s largest Ferris wheel. It has a maximum height of

165 meters and a diameter of 150 meters,with one full rotation

taking approximately 30 minutes. Find an equation for the

wheel if the center of the wheel is on the y-axis.

Source:Wikipedia

47. Find the area of the square in the figure. 51. Weather Satellites Earth is represented on a map of a portion

of the solar system so that its surface is the circle with

equation . A weather satellite

circles 0.6 unit above Earth with the center of its circular

orbit at the center of Earth. Find the equation for the orbit

of the satellite on this map.

x2 + y2 + 2x + 4y - 4091 = 0

Applications and Extensions

x 2 # y2 " y 9

x

y x 2 # y2 " 36

x

r

r

y

x

52. The tangent line to a circle may be defined as the line that

intersects the circle in a single point, called the point of

tangency. See the figure.

If the equation of the circle is and the equation

of the tangent line is , show that:

(a)

[Hint: The quadratic equation

has exactly one solution.]

(b) The point of tangency is .

(c) The tangent line is perpendicular to the line containing

the center of the circle and the point of tangency.

53. The Greek Method The Greek method for finding the

equation of the tangent line to a circle uses the fact that at

any point on a circle the lines containing the center and the

tangent line are perpendicular (see Problem 52). Use this

method to find an equation of the tangent line to the circle

at the point .

54. Use the Greek method described in Problem 53 to find

an equation of the tangent line to the circle

at the point .

55. Refer to Problem 52.The line is tangent to

a circle at .The line is tangent to the same

circle at . Find the center of the circle.

56. Find an equation of the line containing the centers of the

two circles

and

57. If a circle of radius 2 is made to roll along the x-axis, what is

an equation for the path of the center of the circle?

58. If the circumference of a circle is 6p, what is its radius?

x2 + y2 + 6x + 4y + 9 = 0

x2 + y2 - 4x + 6y + 4 = 0

13, -12

10, 22 y = 2x - 7

x - 2y + 4 = 0

x2 + y2 - 4x + 6y + 4 = 0 13, 212 - 32

x2 + y2 = 9 11, 2122

¢ -r2

m

b

,

r2

b

!

x2 + 1mx + b22 = r2

r211 + m22 = b2

y = mx + b

x2 + y2 = r2 Explaining Concepts: Discussion and Writing

59. Which of the following equations might have the graph

shown? (More than one answer is possible.)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h) x2 + y2 - 4x - 4y = 4

x2 + y2 - 9x - 4y = 0

x2 + y2 + 4x - 2y = 0

x2 + y2 - 4x - 9y = 0

1x + 222 + 1y - 222 = 8

1x - 222 + 1y - 322 = 13

1x - 222 + 1y - 222 = 8

1x - 222 + 1y + 322 = 13

60. Which of the following equations might have the graph

shown? (More than one answer is possible.)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h) x2 + y2 - 9x - 10 = 0

x2 + y2 + 9x + 10 = 0

x2 + y2 + 10x - 2y = 1

x2 + y2 + 10x + 16 = 0

1x + 222 + y2 = 4

x2 + 1y - 222 = 3

1x + 222 + y2 = 3

y 1x - 222 + y2 = 3

x

y

x

61. Explain how the center and radius of a circle can be used to graph the circle.
62. What Went Wrong? A student stated that the center and radius of the graph whose equation is are

(3, -2) and 4, respectively.Why is this incorrect?

(x + 3)2 + (y - 2)2 = 16

Interactive Exercises

Ask your instructor if the applets below are of interest to you.

63. Center of a Circle Open the “Circle: the role of the center”

applet. Place the cursor on the center of the circle and hold

the mouse button. Drag the center around the Cartesian

plane and note how the equation of the circle changes.

(a) What is the radius of the circle?

(b) Draw a circle whose center is at .What is the equation

of the circle?

(c) Draw a circle whose center is at . What is the

equation of the circle?

(d) Draw a circle whose center is at .What is the

equation of the circle?

(e) Draw a circle whose center is at . What is the

equation of the circle?

(f) Write a few sentences explaining the role the center of

the circle plays in the equation of the circle.

64. Radius of a Circle Open the “Circle: the role of the radius”

applet. Place the cursor on point B, press and hold the

mouse button. Drag B around the Cartesian plane.

11, -32

1-1, -32

1-1, 32

11, 32

(a) What is the center of the circle?

(b) Move B to a point in the Cartesian plane directly above

the center such that the radius of the circle is 5.

(c) Move B to a point in the Cartesian plane such that the

radius of the circle is 4.

(d) Move B to a point in the Cartesian plane such that the

radius of the circle is 3.

(e) Find the coordinates of two points with integer coordinates

in the fourth quadrant on the circle that result in

a circle of radius 5 with center equal to that found in

part (a).

(f) Use the concept of symmetry about the center, vertical

line through the center of the circle, and horizontal line

through the center of the circle to find three other

points with integer coordinates in the other three

quadrants that lie on the circle of radius five with center

equal to that found in part (a).

‘Are You Prepared?’Answers

1. add; 25 5-1, 56

2.5 VARIATION

2.5 Assess Your Understanding

Concepts and Vocabulary

1. If x and y are two quantities, then y is directly proportional

to x if there is a nonzero number k such that .

2. True or False If y varies directly with x, then where

k is a constant.

Skill Building

In Problems 3–14, write a general formula to describe each variation.

3. y varies directly with x; y = 2 when x = 10 4. v varies directly with t; v = 16 when t = 2
4. A varies directly with x2; A = 4p when x = 2 6. V varies directly with x3; V = 36p when x = 3
5. F varies inversely with d2; F = 10 when d = 5 8. y varies inversely with 1x; y = 4 when x = 9
6. z varies directly with the sum of the squares of x and y;

z = 5 when x = 3 and y = 4

10. T varies jointly with the cube root of x and the square of d;

T = 18 when x = 8 and d = 3 15. Geometry The volume V of a sphere varies directly with the

cube of its radius r. The constant of proportionality is .

16. Geometry The square of the length of the hypotenuse c of

a right triangle varies jointly with the sum of the squares of

the lengths of its legs a and b.The constant of proportionality

is 1.

17. Geometry The area A of a triangle varies jointly with the

lengths of the base b and the height h. The constant of

proportionality is

18. Geometry The perimeter p of a rectangle varies jointly

with the sum of the lengths of its sides l and .The constant

of proportionality is 2.

19. Physics: Newton’s Law The force F (in newtons) of attraction

between two bodies varies jointly with their masses m

and M (in kilograms) and inversely with the square of the

distance d (in meters) between them. The constant of

proportionality is .

20. Physics: Simple Pendulum The period of a pendulum is the

time required for one oscillation; the pendulum is usually

referred to as simple when the angle made to the vertical is

less than 5°.The period T of a simple pendulum (in seconds)

varies directly with the square root of its length l (in feet).

The constant of proportionality is

21. Mortgage Payments The monthly payment p on a

mortgage varies directly with the amount borrowed B. If the

monthly payment on a 30-year mortgage is $6.49 for every

$1000 borrowed, find a linear equation that relates the

monthly payment p to the amount borrowed B for a

mortgage with the same terms. Then find the monthly

payment p when the amount borrowed B is $145,000.

22. Mortgage Payments The monthly payment p on a mortgage

varies directly with the amount borrowed B. If the monthly

payment on a 15-year mortgage is $8.99 for every $1000

borrowed, find a linear equation that relates the monthly

payment p to the amount borrowed B for a mortgage with the

same terms. Then find the monthly payment p when the

amount borrowed B is $175,000.

23. Physics: Falling Objects The distance s that an object falls

is directly proportional to the square of the time t of the

fall. If an object falls 16 feet in 1 second, how far will it fall

in 3 seconds? How long will it take an object to fall 64 feet?

24. Physics: Falling Objects The velocity of a falling object

is directly proportional to the time t of the fall. If, after

2 seconds, the velocity of the object is 64 feet per second,

what will its velocity be after 3 seconds?

v

2p

232

.

G = 6.67 * 10-11

w

1

2

.

4p

3

25. Physics: Stretching a Spring The elongation E of a spring

balance varies directly with the applied weight W (see the

figure). If E = 3 when W = 20, find E when W = 15.

Applications and Extensions

In Problems 15–20, write an equation that relates the quantities.

11. M varies directly with the square of d and inversely with the square root of x; when and
12. z varies directly with the sum of the cube of x and the square of y; when and
13. The square of T varies directly with the cube of a and inversely with the square of d; when and
14. The cube of z varies directly with the sum of the squares of x and y; z = 2 when x = 9 and y = 4

T = 2 a = 2 d = 4

z = 1 x = 2 y = 3

M = 24 x = 9 d = 4

E

W

26. Physics: Vibrating String The rate of vibration of a string

under constant tension varies inversely with the length of

the string. If a string is 48 inches long and vibrates 256 times

per second, what is the length of a string that vibrates 576

times per second?

27. Revenue Equation At the corner Shell station, the revenue

R varies directly with the number g of gallons of gasoline

sold. If the revenue is $47.40 when the number of gallons

sold is 12, find a linear equation that relates revenue R to the

number g of gallons of gasoline. Then find the revenue R

when the number of gallons of gasoline sold is 10.5.

28. Cost Equation The cost C of roasted almonds varies

directly with the number A of pounds of almonds purchased.

If the cost is $23.75 when the number of pounds of roasted

almonds purchased is 5, find a linear equation that relates the

cost C to the number A of pounds of almonds purchased.

Then find the cost C when the number of pounds of almonds

purchased is 3.5.

29. Demand Suppose that the demand D for candy at the

movie theater is inversely related to the price p.

(a) When the price of candy is $2.75 per bag, the theater

sells 156 bags of candy. Express the demand for candy in

terms of its price.

(b) Determine the number of bags of candy that will be

sold if the price is raised to $3 a bag.

30. Driving to School The time t that it takes to get to school

varies inversely with your average speed s.

(a) Suppose that it takes you 40 minutes to get to school

when your average speed is 30 miles per hour.

Express the driving time to school in terms of average

speed.

(b) Suppose that your average speed to school is 40 miles

per hour. How long will it take you to get to school?

31. Pressure The volume of a gas V held at a constant

temperature in a closed container varies inversely with

its pressure P. If the volume of a gas is 600 cubic

centimeters when the pressure is 150 millimeters

of mercury (mm Hg), find the volume when the pressure

is 200 mm Hg.

1cm32

192 CHAPTER 2 Graphs SECTION 2.5 Variation 193

32. Resistance The current i in a circuit is inversely proportional

to its resistance Z measured in ohms. Suppose that

when the current in a circuit is 30 amperes the resistance is

8 ohms. Find the current in the same circuit when the resistance

is 10 ohms.

33. Weight The weight of an object above the surface of Earth

varies inversely with the square of the distance from the

center of Earth. If Maria weighs 125 pounds when she is on

the surface of Earth (3960 miles from the center), determine

Maria’s weight if she is at the top of Mount McKinley

(3.8 miles from the surface of Earth).

34. Intensity of Light The intensity I of light (measured in

foot-candles) varies inversely with the square of the distance

from the bulb. Suppose that the intensity of a 100-watt light

bulb at a distance of 2 meters is 0.075 foot-candle.Determine

the intensity of the bulb at a distance of 5 meters.

35. Geometry The volume V of a right circular cylinder varies

jointly with the square of its radius r and its height h. The

constant of proportionality is . See the figure. Write an

equation for V.

p

surface and the square of the velocity of the wind. If the force

on an area of 20 square feet is 11 pounds when the wind

velocity is 22 miles per hour, find the force on a surface area

of 47.125 square feet when the wind velocity is 36.5 miles

per hour.

39. Horsepower The horsepower (hp) that a shaft can safely

transmit varies jointly with its speed (in revolutions per

minute, rpm) and the cube of its diameter. If a shaft of a

certain material 2 inches in diameter can transmit 36 hp at

75 rpm, what diameter must the shaft have in order to

transmit 45 hp at 125 rpm?

40. Chemistry: Gas Laws The volume V of an ideal gas varies

directly with the temperature T and inversely with the

pressure P. Write an equation relating V, T, and P using k as

the constant of proportionality. If a cylinder contains oxygen

at a temperature of 300 K and a pressure of 15 atmospheres

in a volume of 100 liters, what is the constant of proportionality

k? If a piston is lowered into the cylinder, decreasing

the volume occupied by the gas to 80 liters and raising the

temperature to 310 K, what is the gas pressure?

41. Physics: Kinetic Energy The kinetic energy K of a moving

object varies jointly with its mass m and the square of its

velocity . If an object weighing 25 kilograms and moving

with a velocity of 10 meters per second has a kinetic energy

of 1250 joules, find its kinetic energy when the velocity is

15 meters per second.

42. Electrical Resistance of a Wire The electrical resistance

of a wire varies directly with the length of the wire and

inversely with the square of the diameter of the wire. If a

wire 432 feet long and 4 millimeters in diameter has a

resistance of 1.24 ohms, find the length of a wire of the

same material whose resistance is 1.44 ohms and whose

diameter is 3 millimeters.

43. Measuring the Stress of Materials The stress in the

material of a pipe subject to internal pressure varies jointly

with the internal pressure and the internal diameter of the

pipe and inversely with the thickness of the pipe. The stress

is 100 pounds per square inch when the diameter is 5 inches,

the thickness is 0.75 inch, and the internal pressure is

25 pounds per square inch. Find the stress when the internal

pressure is 40 pounds per square inch if the diameter is

8 inches and the thickness is 0.50 inch.

44. Safe Load for a Beam The maximum safe load for a

horizontal rectangular beam varies jointly with the width of

the beam and the square of the thickness of the beam and

inversely with its length. If an 8-foot beam will support up to

750 pounds when the beam is 4 inches wide and 2 inches

thick, what is the maximum safe load in a similar beam

10 feet long, 6 inches wide, and 2 inches thick?

v

h

r

r

h

36. Geometry The volume V of a right circular cone varies

jointly with the square of its radius r and its height h. The

constant of proportionality is See the figure. Write an

equation for V.

p

3

.

37. Weight of a Body The weight of a body above the surface

of Earth varies inversely with the square of the distance

from the center of Earth. If a certain body weighs 55 pounds

when it is 3960 miles from the center of Earth, how much

will it weigh when it is 3965 miles from the center?

38. Force of the Wind on a Window The force exerted by the

wind on a plane surface varies jointly with the area of the

Explaining Concepts: Discussion and Writing

45. In the early 17th century, Johannes Kepler discovered that

the square of the period T of the revolution of a planet

around the Sun varies directly with the cube of its mean

distance a from the Sun. Go to the library and research this

law and Kepler’s other two laws. Write a brief paper about

these laws and Kepler’s place in history.

46. Using a situation that has not been discussed in the text,

write a real-world problem that you think involves two

variables that vary directly. Exchange your problem with

another student’s to solve and critique.

47. Using a situation that has not been discussed in the text,

write a real-world problem that you think involves two

variables that vary inversely. Exchange your problem with

another student’s to solve and critique.

48. Using a situation that has not been discussed in the text,

write a real-world problem that you think involves three

variables that vary jointly. Exchange your problem with

another student’s to solve and critique.

CHAPTER REVIEW

Things to Know

Formulas

Distance formula (p. 151)

Midpoint formula (p. 154)

Slope (p. 167) if undefined if

Parallel lines (p. 175) Equal slopes and different y-intercepts

Perpendicular lines (p. 176) Product of slopes is

Direct variation (p. 189)

Inverse variation (p. 189)

Equations of Lines and Circles

Vertical line (p. 171) ; a is the x-intercept

Horizontal line (p. 172) ; b is the y-intercept

Point–slope formof the equation of a line (p. 171) is the slope of the line, is a point on the line

Slope–intercept form of the equation of a line (p. 172) is the slope of the line, b is the y-intercept

General form of the equation of a line (p. 174) not both 0

Standard form of the equation of a circle (p. 182) r is the radius of the circle, is the center

of the circle

Equation of the unit circle (p. 183)

General form of the equation of a circle (p. 184) x2 + y2 + ax + by + c = 0, with restrictions on a, b, and c

Review Exercises

In Problems 1–6, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

1. 2.
2. 4.
3. 6.
4. Graph y = x2 + 4 by plotting points.

14, -42; 14, 82 1-3, 42; 12, 42

11, -12; 1-2, 32 1-2, 22; 11, 42

10, 02; 14, 22 10, 02; 1-4, 62

8. List the intercepts of the graph below.

In Problems 9–16, list the intercepts and test for symmetry with respect to the x-axis, the y-axis, and the origin.

9. 2x = 3y2 10. y = 5x 11. x2 + 4y2 = 16 12. 9x2 - y2 = 9

In Problems 17–20, find the standard form of the equation of the circle whose center and radius are given.

17. 1h, k2 = 1-2, 32; r = 4 18. 1h, k2 = 13, 42; r = 4 19. 1h, k2 = 1-1, -22; r = 1 20. 1h, k2 = 12, -42; r = 3

In Problems 37–40, find the slope and y-intercept of each line. Graph the line, labeling any intercepts.

37. 4x - 5y = -20 38. 3x + 4y = 12 39.

1

2 x -

1

3 y = -

1

6

40. -

3

4 x +

1

2 y = 0

In Problems 41–44, find the intercepts and graph each line.

41. 2x - 3y = 12 42. x - 2y = 8 43.

1

2

x +

1

3

y = 2 44.

1

3

x -

1

4

y = 1

In Problems 21–26, find the center and radius of each circle. Graph each circle. Find the intercepts, if any, of each circle.

21. x2 + 1y - 122 = 4 22. 1x + 222 + y2 = 9 23. x2 + y2 - 2x + 4y - 4 = 0

In Problems 27–36, find an equation of the line having the given characteristics. Express your answer using either the general form or the

slope–intercept form of the equation of a line, whichever you prefer.

27. Slope = -2; containing the point 13, -12 28. Slope = 0; containing the point 1-5, 42
28. Vertical; containing the point 1-3, 42 30. x-intercept = 2; containing the point 14, -52
29. y-intercept = -2; containing the point 15, -32 32. Containing the points 13, -42 and 12, 12
30. Parallel to the line 2x - 3y = -4; containing the point 1-5, 32
31. Parallel to the line x + y = 2; containing the point 11, -32
32. Perpendicular to the line x + y = 2; containing the point 14, -32
33. Perpendicular to the line 3x - y = -4; containing the point 1-2, 42 196 CHAPTER 2 Graphs
34. Sketch a graph of .
35. Sketch a graph of .
36. Graph the line with slope containing the point .
37. Show that the points , and

are the vertices of an isosceles triangle.

49. Show that the points , and

are the vertices of a right triangle in two ways:

(a) By using the converse of the Pythagorean Theorem

(b) By using the slopes of the lines joining the vertices

50. The endpoints of the diameter of a circle are and

. Find the center and radius of the circle. Write the

standard equation of this circle.

51. Show that the points , and

lie on a line by using slopes.

A = 12, 52, B = 16, 12 C = 18, -12

15, -62

1-3, 22

C = 18, 52

A = 1-2, 02, B = 1-4, 42

C = 1-2, 32

A = 13, 42, B = 11, 12

11, 22

2

3

y = 2x

y = x3 52. Mortgage Payments Themonthly payment p on amortgage

varies directly with the amount borrowed B. If the monthly

payment on a 30-year mortgage is $854.00 when $130,000 is

borrowed, find an equation that relates the monthly payment

p to the amount borrowed B for a mortgage with the same

terms. Then find the monthly payment p when the amount

borrowed B is $165,000.

53. Revenue Function At the corner Esso station, the revenue

R varies directly with the number g of gallons of gasoline

sold. If the revenue is $46.67 when the number of gallons sold

is 13, find an equation that relates revenue R to the number g

of gallons of gasoline. Then find the revenue R when the

number of gallons of gasoline sold is 11.2.

54. Weight of a Body The weight of a body varies inversely with

the square of its distance from the center of Earth.Assuming

that the radius of Earth is 3960 miles, how much would a

man weigh at an altitude of 1 mile above Earth’s surface if

he weighs 200 pounds on Earth’s surface?

55. Kepler’s Third Law of Planetary Motion Kepler’s Third Law of Planetary Motion states that the square of the period of

revolution T of a planet varies directly with the cube of its mean distance a from the Sun. If the mean distance of Earth from the

Sun is 93 million miles, what is the mean distance of the planet Mercury from the Sun, given that Mercury has a “year”of 88 days?

Mean distance Mean distance

T = 88 days

Mercury

T = 365 days

Sun Earth

56. Create four problems that you might be asked to do given the two points and . Each problem should involve a

different concept. Be sure that your directions are clearly stated.

57. Describe each of the following graphs in the -plane. Give justification.

(a) (b) (c) (d) (e)

58. Suppose that you have a rectangular field that requires watering.Your watering system consists of an arm of variable length that

rotates so that the watering pattern is a circle. Decide where to position the arm and what length it should be so that the entire

field is watered most efficiently.When does it become desirable to use more than one arm?

[Hint: Use a rectangular coordinate system positioned as shown in the figures. Write equations for the circle(s) swept out by the

watering arm(s).]

x = 0 y = 0 x + y = 0 xy = 0 x2 + y2 = 0

xy

1-3, 42 16, 12

Square field

y

x

Rectangular field, one arm

y

x

Rectangular field, two arms

y

x In Problems 1–3, use and .

1. Find the distance from to .
2. Find the midpoint of the line segment joining and .
3. (a) Find the slope of the line containing and .

(b) Interpret this slope.

4. Graph by plotting points.
5. Sketch the graph of .
6. List the intercepts and test for symmetry: .
7. Write the slope–intercept form of the line with slope -2

containing the point . Graph the line.

8. Write the general form of the circle with center and

radius 5.

14, -32

13, -42

x2 + y = 9

y2 = x

y = x2 - 9

P1 P2

P1 P2

P1 P2

P1 = 1-1, 32 P2 = 15, -12 9. Find the center and radius of the circle

. Graph this circle.

10. For the line , find a line parallel to it containing

the point . Also find a line perpendicular to it

containing the point .

11. Resistance due to a Conductor The resistance (in ohms)

of a circular conductor varies directly with the length of the

conductor and inversely with the square of the radius of the

conductor. If 50 feet of wire with a radius of

has a resistance of 10 ohms, what would be the resistance

of 100 feet of the same wire if the radius is increased to

7 * 10-3 inch?

6 * 10-3 inch

10, 32

11, -12

2x + 3y = 6

x2 + y2 + 4x - 2y - 4 = 0

CHAPTER PROJECT

Internet-based Project

Predicting Olympic Performance Measurements of human performance

over time sometimes follow a strong linear relationship

for reasonably short periods. In 2004 the Summer Olympic

Games returned to Greece, the home of both the ancient

Olympics and the first modern Olympics.The following data represent

the winning times (in hours) for men and women in the

Olympic marathon.

Year Men Women

1984 2.16 2.41

1988 2.18 2.43

1992 2.22 2.54

1996 2.21 2.43

2000 2.17 2.39

Source: www.hickoksports.com/history/olmtandf.shtml

CHAPTER TEST

The Chapter Test Prep Videos are step-by-step test solutions available in the

Video Resources DVD, in , or on this text’s Channel. Flip

back to the Student Resources page to see the exact web address for this

text’s YouTube channel.

In Problems 1–8, find the real solution(s) of each equation.

1. 2.
2. 4.
3. 6.
4. 8.

In Problems 9 and 10, solve each equation in the complex number

system.

9. 10.

In Problems 11–14, solve each inequality. Graph the solution set.

11. 12.
12. ƒx - 2ƒ … 1 14. ƒ2 + xƒ 7 3

2x - 3 … 7 -1 6 x + 4 6 5

x2 = -9 x2 - 2x + 5 = 0

ƒx - 2ƒ = 1 3x2 + 4x = 2

x2 + 2x + 5 = 0 22x + 1 = 3

2x2 - 5x - 3 = 0 x2 - 2x - 2 = 0

3x - 5 = 0 x2 - x - 12 = 0

15. Find the distance between the points and

.Find the midpoint of the line segment from P to Q.

16. Which of the following points are on the graph of

(a) (b) (c)

17. Sketch the graph of .
18. Find the equation of the line containing the points

and . Express your answer in slope–intercept form.

19. Find the equation of the line perpendicular to the line

and containing the point . Express your

answer in slope–intercept form and graph the line.

20. Graph the equation x2 + y2 - 4x + 8y - 5 = 0.

y = 2x + 1 13, 52

12, -22

1-1, 42

y = x3

1-2, -12 12, 32 13, 12

y = x3 - 3x + 1?

Q = 14, -22

P = 1-1, 32

CUMULATIVE REVIEW

197 1. Treating year as the independent variable and the winning

value as the dependent variable, find linear equations

relating these variables (separately for men and women)

using the data for the years 1992 and 1996. Compare the

equations and comment on any similarities or differences.

2. Interpret the slopes in your equations from part 1. Do the

y-intercepts have a reasonable interpretation? Why or why

not?

3. Use your equations to predict the winning time in the 2004

Olympics. Compare your predictions to the actual results

(2.18 hours for men and 2.44 hours for women). How well

did your equations do in predicting the winning times?

4. Repeat parts 1 to 3 using the data for the years 1996 and
5. How do your results compare?
6. Would your equations be useful in predicting the winning

marathon times in the 2104 Summer Olympics? Why or

why not?

6. Pick your favorite Winter Olympics event and find the

winning value (that is distance, time, or the like) in two

Winter Olympics prior to 2006. Repeat parts 1 to 3 using

your selected event and years and compare to the actual

results of the 2006 Winter Olympics in Torino, Italy.

CHAPTER 3 Functions and

Their Graphs

3.1 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. The inequality can be written in interval

notation as . (pp. 120–121)

2. If the value of the expression is

. (pp. 20–23)

3x2 - 5x +

1

x

x = -2,

-1 6 x 6 3 3. The domain of the variable in the expression is

. (pp. 20–23)

4. Solve the inequality: Graph the solution se

Concepts and Vocabulary

5. If is a function defined by the equation then x

is called the variable and y is the

variable.

6. The set of all images of the elements in the domain of a

function is called the .

7. If the domain of is all real numbers in the interval

and the domain of g is all real numbers in the interval

the domain of is all real numbers in the interval

.

8. The domain of consists of numbers x for which

g1x2 0 that are in the domains of both and .

f

g

3-2, 54, f + g

f 30, 74

f y = f1x2, 9. If and then

10. True or False Every relation is a function.
11. True or False The domain of consists of the

numbers x that are in the domains of both and g.

12. True or False The independent variable is sometimes

referred to as the argument of the function.

13. True or False If no domain is specified for a function then

the domain of is taken to be the set of real numbers.

14. True or False The domain of the function

is 5x ƒ x Z ;26.

f1x2 =

x2 - 4

x SECTION 3.1 Functions 211

In Problems 15–26, determine whether each relation represents a function. For each function, state the domain and range.

16.

Skill Building

Elvis

Person

Colleen

Kaleigh

Marissa

Jan. 8

Birthday

Mar. 15

Sept. 17

15.

Bob

Father

John

Chuck

Beth

Daughter

Diane

Linda

Marcia

17. 18.

20 Hours

Hours Worked

30 Hours

40 Hours

$200

Salary

$300

$350

$425

Less than 9th grade

9th-12th grade

High School Graduate

Some College

College Graduate

Level of Education

$18,120

$23,251

$36,055

$45,810

$67,165

Average Income

19. 512, 62, 1-3, 62, 14, 92, 12, 1026 20. 51-2, 52, 1-1, 32, 13, 72, 14, 1226 21. 511, 32, 12, 32, 13, 32, 14, 326
20. 510, -22, 11, 32, 12, 32, 13, 726 23. 51-2, 42, 1-2, 62, 10, 32, 13, 726 24. 51-4, 42, 1-3, 32, 1-2, 22, 1-1, 12, 1-4, 026
21. 51-2, 42, 1-1, 12, 10, 02, 11, 126 26. 51-2, 162, 1-1, 42, 10, 32, 11, 426
22. y = x2 28. y = x3 29. y =

1

x

30. y = ƒxƒ
31. y2 = 4 - x2 32. y = ;21 - 2x 33. x = y2 34. x + y2 = 1
32. y = 2x2 - 3x + 4 36. y =

3x - 1

x + 2

37. 2x2 + 3y2 = 1 38. x2 - 4y2 = 1

In Problems 27–38, determine whether the equation defines y as a function of x.

39. f1x2 = 3x2 + 2x - 4 40. f1x2 = -2x2 + x - 1 41. f1x2 =

x

x2 + 1

42. f1x2 =

x2 - 1

x + 4

43. f1x2 = ƒxƒ + 4 44. f1x2 = 2x2 + x 45. f1x2 =

2x + 1

3x - 5

46. f1x2 = 1 -

1

1x + 222

In Problems 39–46, find the following for each function:

(a) f102 (b) f112 (c) f1-12 (d) f1-x2 (e) -f1x2 (f) f1x + 12 (g) f12x2 (h) f1x + h2

47. f1x2 = -5x + 4 48. f1x2 = x2 + 2 49. f1x2 =

x

x2 + 1

50. f1x2 =

x2

x2 + 1

51. g1x2 =

x

x2 - 16

52. h1x2 =

2x

x2 - 4

53. F1x2 =

x - 2

x3 + x

54. G1x2 =

x + 4

x3 - 4x

55. h1x2 = 23x - 12 56. G1x2 = 21 - x 57. f1x2 =

4

2x - 9

58. f1x2 =

x

2x - 4

59. p1x2 = A

2

x - 1

60. q1x2 = 2-x - 2 61. P(t) =

2t - 4

3t - 21

62. h(z) =

2z + 3

z - 2

In Problems 47–62, find the domain of each function.

63. f1x2 = 3x + 4; g1x2 = 2x - 3 64. f1x2 = 2x + 1; g1x2 = 3x - 2
64. f1x2 = x - 1; g1x2 = 2x2 66. f1x2 = 2x2 + 3; g1x2 = 4x3 + 1

In Problems 63–72, for the given functions and g, find the following. For parts (a)–(d), also find the domain.

(a) (b) (c) (d)

(e) (f) (g) (h) a

f

g

1f + g2132 1f - g2142 1f # g2122 b112

a

f

g

1f + g21x2 1f - g21x2 1f # g21x2 b1x2

f 212 CHAPTER 3 Functions and Their Graphs

67. f1x2 = 1x; g1x2 = 3x - 5 68. f1x2 = ƒxƒ; g1x2 = x
68. f1x2 = 1 +

1

x

; g1x2 =

1

x

70. f1x2 = 2x - 1; g1x2 = 24 - x
71. f1x2 =

2x + 3

3x - 2

; g1x2 =

4x

3x - 2

72. f1x2 = 2x + 1; g1x2 =

2

x

73. Given and find the

function g.

1f + g21x2 = 6 -

1

2 f1x2 = 3x + 1 x, 74. Given and a find the function g.

f

g

b1x2 =

x + 1

x2 - x

f1x2 = ,

1

x

In Problems 75–82, find the difference quotient of that is, find for each function. Be sure to simplify.

f1x + h2 - f1x2

h

f; , h Z 0,

83. If and what is the

value of A?

84. If and what is the value

of B?

85. If and what is the value of A?
86. If and what is the value of B?
87. If and what is the value of A?

Where is not defined?

88. If and is undefined, what are

the values of A and B?

89. Geometry Express the area A of a rectangle as a function

of the length x if the length of the rectangle is twice its

width.

90. Geometry Express the area A of an isosceles right triangle

as a function of the length x of one of the two equal sides.

91. Constructing Functions Express the gross salary G of a

person who earns $10 per hour as a function of the number

x of hours worked.

92. Constructing Functions Tiffany, a commissioned salesperson,

earns $100 base pay plus $10 per item sold. Express her

gross salary G as a function of the number x of items sold.

93. Population as a Function of Age The function

represents the population P (in millions) of Americans that

are a years of age or older.

(a) Identify the dependent and independent variables.

(b) Evaluate P 20 . Provide a verbal explanation of the

meaning of P 20 .

(c) Evaluate P 0 . Provide a verbal explanation of the

meaning of P102.

1 2

1 2

1 2

P1a2 = 0.015a2 - 4.962a + 290.580

f1x2 = f112

x - B

x - A

, f122 = 0

f

f1x2 = f142 = 0,

2x - A

x - 3

f122 =

1

2

f1x2 = ,

2x - B

3x + 4

f1x2 = f102 = 2,

3x + 8

2x - A

f1x2 = 3x2 - Bx + 4 f1-12 = 12,

f1x2 = 2x3 + Ax2 + 4x - 5 f122 = 5, 94. Number of Rooms The function

represents the number N of housing units (in millions) that

have r rooms, where r is an integer and

(a) Identify the dependent and independent variables.

(b) Evaluate N(3). Provide a verbal explanation of the

meaning of N(3).

95. Effect of Gravity on Earth If a rock falls from a height of

20 meters on Earth, the height H (in meters) after x seconds

is approximately

(a) What is the height of the rock when

(b) When is the height of the rock 15 meters? When is it

10 meters? When is it 5 meters?

(c) When does the rock strike the ground?

96. Effect of Gravity on Jupiter If a rock falls from a height of

20 meters on the planet Jupiter, its height H (in meters)

after x seconds is approximately

(a) What is the height of the rock when

(b) When is the height of the rock 15 meters? When is it

10 meters? When is it 5 meters?

(c) When does the rock strike the ground?

x = 1.1 seconds? x = 1.2 seconds?

x = 1 second?

H1x2 = 20 - 13x2

x = 1.1 seconds? x = 1.2 seconds? x = 1.3 seconds?

x = 1 second?

H1x2 = 20 - 4.9x2

2 … r … 9.

N1r2 = -1.44r2 + 14.52r - 14.96

Applications and Extensions

75. f1x2 = 4x + 3 76. f1x2 = -3x + 1 77. f1x2 = x2 - x + 4 78. f1x2 = 3x2 - 2x + 6
76. f(x) =

1

x2 80. f1x2 =

1

x + 3

81.

[Hint: Rationalize the

numerator.]

f(x) = 2x 82. f(x) = 2x + 1

SECTION 3.1 Functions 213

97. Cost of Trans-Atlantic Travel A Boeing 747 crosses the

Atlantic Ocean (3000 miles) with an airspeed of 500 miles

per hour. The cost C (in dollars) per passenger is given by

where x is the ground speed

(a) What is the cost per passenger for quiescent (no wind)

conditions?

(b) What is the cost per passenger with a head wind of

50 miles per hour?

(c) What is the cost per passenger with a tail wind of

100 miles per hour?

(d) What is the cost per passenger with a head wind of

100 miles per hour?

98. Cross-sectionalArea The cross-sectional area of a beamcut

from a log with radius 1 foot is given by the function

where x represents the length, in feet,

of half the base of the beam. See the figure. Determine the

cross-sectional area of the beam if the length of half the

base of the beam is as follows:

(a) One-third of a foot

(b) One-half of a foot

(c) Two-thirds of a foot

A1x2 = 4x21 - x2 ,

1airspeed ; wind2.

C1x2 = 100 +

x

10 +

36,000

x

100. Crimes Suppose that represents the number of

violent crimes committed in year x and represents the

number of property crimes committed in year x. Determine

a function T that represents the combined total of violent

crimes and property crimes in year x.

101. Health Care Suppose that P x represents the percentage

of income spent on health care in year x and represents

income in year x. Determine a function H that represents

total health care expenditures in year x.

102. Income Tax Suppose that represents the income of

an individual in year x before taxes and represents the

individual’s tax bill in year x. Determine a function N that

represents the individual’s net income (income after taxes)

in year x.

103. Profit Function Suppose that the revenueR, in dollars, from

selling x cell phones, in hundreds, is

The cost C, in dollars, of selling x cell phones is

(a) Find the profit function,

(b) Find the profit if hundred cell phones are sold.

(c) Interpret P(15).

104. Profit Function Suppose that the revenue R, in dollars,

from selling x clocks is The cost C, in dollars, of

selling x clocks is

(a) Find the profit function,

(b) Find the profit if clocks are sold.

(c) Interpret P(30).

105. Some functions have the property that

for all real numbers a and b. Which of the

following functions have this property?

(a) (b)

(c) (d) G1x2 =

1

x

F1x2 = 5x - 2

h1x2 = 2x g1x2 = x2

f1a2 + f1b2

f f1a + b2 =

x = 30

P1x2 = R1x2 - C1x2.

C1x2 = 0.1x2 + 7x + 400.

R1x2 = 30x.

x = 15

P1x2 = R1x2 - C1x2.

C1x2 = 0.05x3 - 2x2 + 65x + 500.

R1x2 = -1.2x2 + 220x.

T1x2

I1x2

I1x2

1 2

P1x2

V1x2

A(x ) " 4x#1 ! x 2

x

1

99. Economics The participation rate is the number of people

in the labor force divided by the civilian population (excludes

military). Let represent the size of the labor force in

year x and represent the civilian population in year x.

Determine a function that represents the participation rate

R as a function of x.

P1x2

L1x2

106. Are the functions and the

same? Explain.

107. Investigate when, historically, the use of the function notation

y = f1x2 first appeared.

g1x2 =

x2 - 1

x + 1

f1x2 = x - 1 108. Find a function H that multiplies a number x by 3, then

subtracts the cube of x and divides the result by your age.

Explaining Concepts: Discussion and Writing

‘Are You Prepared?’Answers

1. 1-1, 32 2. 21.5 3. 5x ƒ x Z -46 4. 5x ƒ x 6 -16

!1 0 1

3.2 THE GRAPH OF A FUNCTION

3.2 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. The intercepts of the equation are

______________. (pp. 159–160)

x2 + 4y2 = 16 2. True or False The point is on the graph of the

equation x = 2y - 2. (pp. 157–159)

Concepts and Vocabulary

3. A set of points in the xy-plane is the graph of a function if

and only if every line intersects the graph in at

most one point.

4. If the point is a point on the graph of then
5. Find a so that the point is on the graph of

f1x2 = ax2 + 4.

1-1, 22

f1 2 = .

15, -32 f,

6. True or False A function can have more than one

y-intercept.

7. True or False The graph of a function always

crosses the y-axis.

8. True or False The y-intercept of the graph of the function

y = f1x2, whose domain is all real numbers, is f102.

Skill Building

9. Use the given graph of the function f to answer parts (a)–(n). 10. Use the given graph of the function f to answer parts (a)–(n).

(a) Find and

(b) Find and

(c) Is positive or negative?

(d) Is positive or negative?

(e) For what values of x is

(f) For what values of x is

(g) What is the domain of

(h) What is the range of

(i) What are the x-intercepts?

(j) What is the y-intercept?

(k) How often does the line intersect the graph?

(l) How often does the line intersect the graph?

(m) For what values of x does

(n) For what values of x does f1x2 = -2?

f1x2 = 3?

x = 5

y =

1

2

f?

f?

f1x2 7 0?

f1x2 = 0?

f1-42

f132

f162 f1112.

f102 f1-62. (a) Find and

(b) Find and

(c) Is positive or negative?

(d) Is positive or negative?

(e) For what values of x is

(f) For what values of x is

(g) What is the domain of

(h) What is the range of

(i) What are the x-intercepts?

(j) What is the y-intercept?

(k) How often does the line intersect the graph?

(l) How often does the line intersect the graph?

(m) For what value of x does

(n) For what value of x does f1x2 = -2? SECTION 3.2 The Graph of a Function 219

In Problems 11–22, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find:

(a) The domain and range

(b) The intercepts, if any

(c) Any symmetry with respect to the x-axis, the y-axis, or the origin

11. 12. 13. 14.
12. 16. 17. 18.
13. 20. 21. 22.

x

y

3

3

!3

!3 x

y

3

3

!3

!3 %

1

!1

!% %––

!2 %––

2

x

y

! %––

2

–%– %

2

!% x

y

1

!1

x

y

3

3

!3

!3 x

y

3

3

!3

!3 x

y

3

3

!3

!3

(4, 3)

x

y

4

4

!4

!4

(!1, 2) (1, 2)

x

y

3

3

!3

!3 x

y

3

3

!3

!3

x

y

9

!3

!1 3

(2, !3)

6

3

x

y

4

!3 3

) ( , 5 1–2

In Problems 23–28, answer the questions about the given function.

23.

(a) Is the point on the graph of

(b) If what is What point is on the graph of

(c) If what is x?What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of

24.

(a) Is the point on the graph of

(b) If what is What point is on the graph of

(c) If what is x? What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of

25.

(a) Is the point on the graph of

(b) If what is What point is on the graph of

(c) If what is x? What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of

26.

(a) Is the point a1, on the graph of f?

3

5

b

f1x2 =

x2 + 2

x + 4

f.

f.

f?

f?

f1x2 = 2,

x = 4, f1x2? f?

13, 142 f?

f1x2 =

x + 2

x - 6

f.

f.

f?

f?

f1x2 = -2,

x = -2, f1x2? f?

1-1, 22 f?

f1x2 = -3x2 + 5x

f.

f.

f?

f?

f1x2 = -1,

x = -2, f1x2? f?

1-1, 22 f?

f1x2 = 2x2 - x - 1 (b) If what is What point is on the graph of

(c) If what is x? What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of

27.

(a) Is the point on the graph of

(b) If what is What point is on the graph

of

(c) If what is x? What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of

28.

(a) Is the point on the graph of

(b) If what is What point is on the graph of

(c) If what is x? What point(s) are on the graph

of

(d) What is the domain of

(e) List the x-intercepts, if any, of the graph of

(f) List the y-intercept, if there is one, of the graph of f.

f.

f?

f?

f1x2 = 1,

x = 4, f1x2? f?

a f?

1

2

, -

2

3

b

f1x2 =

2x

x - 2

f.

f.

f?

f?

f1x2 = 1,

f?

x = 2, f1x2?

1-1, 12 f?

f1x2 =

2x2

x4 + 1

f.

f.

f?

f?

f1x2 =

1

2

,

x = 0, f1x2? f? 220 CHAPTER 3 Functions and Their Graphs

29. Free-throw Shots According to physicist Peter Brancazio,

the key to a successful foul shot in basketball lies in the arc

of the shot. Brancazio determined the optimal angle of the

arc from the free-throw line to be 45 degrees. The arc also

depends on the velocity with which the ball is shot. If a

player shoots a foul shot, releasing the ball at a 45-degree

angle from a position 6 feet above the floor, then the path of

the ball can be modeled by the function

where h is the height of the ball above the floor, x is the

forward distance of the ball in front of the foul line, and is the

initial velocity with which the ball is shot in feet per second.

Suppose a player shoots a ball with an initial velocity of

28 feet per second.

(a) Determine the height of the ball after it has traveled

8 feet in front of the foul line.

(b) Determine the height of the ball after it has traveled

12 feet in front of the foul line.

(c) Find additional points and graph the path of the

basketball.

(d) The center of the hoop is 10 feet above the floor and

15 feet in front of the foul line.Will the ball go through

the hoop? Why or why not? If not, with what initial

velocity must the ball be shot in order for the ball to go

through the hoop?

Source: The Physics of Foul Shots, Discover, Vol. 21, No. 10,

October 2000

30. Granny Shots The last player in the NBA to use an underhand

foul shot (a “granny” shot) was Hall of Fame forward

Rick Barry who retired in 1980. Barry believes that current

NBA players could increase their free-throw percentage if

they were to use an underhand shot. Since underhand shots

are released from a lower position, the angle of the shot must

be increased. If a player shoots an underhand foul shot,

releasing the ball at a 70-degree angle from a position

3.5 feet above the floor, then the path of the ball can be

modeled by the function ,

where h is the height of the ball above the floor, x is the

forward distance of the ball in front of the foul line, and

is the initial velocity with which the ball is shot in feet per

second.

(a) The center of the hoop is 10 feet above the floor and

15 feet in front of the foul line. Determine the initial

velocity with which the ball must be shot in order for

the ball to go through the hoop.

(b) Write the function for the path of the ball using the

velocity found in part (a).

(c) Determine the height of the ball after it has traveled

9 feet in front of the foul line.

(d) Find additional points and graph the path of the

basketball.

Source:The Physics of Foul Shots,Discover,Vol. 21,No. 10,

October 2000

31. Motion of a Golf Ball A golf ball is hit with an initial

velocity of 130 feet per second at an inclination of 45° to

the horizontal. In physics, it is established that the height h

v

h1x2 = -

136x2

v2 + 2.7x + 3.5

v

h1x2 = - x + 6

44x2

v2 +

of the golf ball is given by the function

where x is the horizontal distance that the golf ball has

traveled.

h1x2 = -32x2

1302 + x

Applications and Extensions

(a) Determine the height of the golf ball after it has

traveled 100 feet.

(b) What is the height after it has traveled 300 feet?

(c) What is the height after it has traveled 500 feet?

(d) How far was the golf ball hit?

(e) Use a graphing utility to graph the function

(f) Use a graphing utility to determine the distance that the

ball has traveled when the height of the ball is 90 feet.

(g) Create a TABLE with and

To the nearest 25 feet, how far does the ball travel

before it reaches a maximum height? What is the

maximum height?

(h) Adjust the value of until you determine the

distance, to within 1 foot, that the ball travels before it

reaches a maximum height.

32. Cross-sectional Area The cross-sectional area of a beam

cut from a log with radius 1 foot is given by the function

where x represents the length, in

feet, of half the base of the beam. See the figure.

A1x2 = 4x21 - x2 ,

¢Tbl

TblStart = 0 ¢Tbl = 25.

h = h1x2.

A(x ) " 4x#1 ! x 2

x

1

(a) Find the domain of A.

(b) Use a graphing utility to graph the function

(c) Create a TABLE with and for

Which value of x maximizes the crosssectional

area? What should be the length of the base of

the beam to maximize the cross-sectional area?

0 … x … 1.

TblStart = 0 ¢Tbl = 0.1

A = A1x2. SECTION 3.2 The Graph of a Function 221

33. Cost of Trans-Atlantic Travel A Boeing 747 crosses the

Atlantic Ocean (3000 miles) with an airspeed of 500 miles

per hour.The cost C (in dollars) per passenger is given by

where x is the ground speed

(a) Use a graphing utility to graph the function

(b) Create a TABLE with and

(c) To the nearest 50 miles per hour, what ground speed

minimizes the cost per passenger?

34. Effect of Elevation on Weight If an object weighs m pounds

at sea level, then its weight W (in pounds) at a height of

h miles above sea level is given approximately by

(a) If Amy weighs 120 pounds at sea level, how much will

she weigh on Pike’s Peak, which is 14,110 feet above sea

level?

(b) Use a graphing utility to graph the function

Use m = 120 pounds.

W = W1h2.

W1h2 = ma

4000

4000 + h

b

2

TblStart = 0 ¢Tbl = 50.

C = C1x2.

1airspeed ; wind2.

C1x2 = 100 +

x

10 +

36,000

x

(c) Create a Table with and to see

how the weight W varies as h changes from 0 to 5 miles.

(d) At what height will Amy weigh 119.95 pounds?

(e) Does your answer to part (d) seem reasonable? Explain.

35. The graph of two functions, f and g, is illustrated. Use the

graph to answer parts (a)–(f).

TblStart = 0 ¢Tbl = 0.5

x

y!f(x)

y

2 4

(2, 1) (4, 1) (6, 1)

(2, 2)

(6, 0)

(4, "3)

(3, "2) (5, "2)

2

"4

"2

y!g(x)

(a) (b)

(c) (d)

(e) (f) a

f

g

1f # g2122 b142

1f - g2162 1g - f2162

1f + g2122 1f + g2142

36. Describe how you would proceed to find the domain and range of a function if you were given its graph. How would your strategy

change if you were given the equation defining the function instead of its graph?

37. How many x-intercepts can the graph of a function have? How many y-intercepts can the graph of a function have?
38. Is a graph that consists of a single point the graph of a function? Can you write the equation of such a function?
39. Match each of the following functions with the graph that best describes the situation.

(a) The cost of building a house as a function of its square footage

(b) The height of an egg dropped from a 300-foot building as a function of time

(c) The height of a human as a function of time

(d) The demand for Big Macs as a function of price

(e) The height of a child on a swing as a function of time

Explaining Concepts: Discussion and Writing

(I)

x

(II) (III)

x

(IV)

x

(V)

x

y y y y y

x

40. Match each of the following functions with the graph that best describes the situation.

(a) The temperature of a bowl of soup as a function of time

(b) The number of hours of daylight per day over a 2-year period

(c) The population of Florida as a function of time

(d) The distance travelled by a car going at a constant velocity as a function of time

(e) The height of a golf ball hit with a 7-iron as a function of time

(I) (II) (III) (IV) (V)

x

y

x

y

x

y

x

y

x

y 41. Consider the following scenario: Barbara decides to take a

walk. She leaves home, walks 2 blocks in 5 minutes at a

constant speed, and realizes that she forgot to lock the door.

So Barbara runs home in 1 minute.While at her doorstep, it

takes her 1minute to find her keys and lock the door.Barbara

walks 5 blocks in 15 minutes and then decides to jog home. It

takes her 7 minutes to get home. Draw a graph of Barbara’s

distance fromhome (in blocks) as a function of time.

42. Consider the following scenario: Jayne enjoys riding her

bicycle through the woods. At the forest preserve, she gets

on her bicycle and rides up a 2000-foot incline in 10 minutes.

She then travels down the incline in 3 minutes. The next

5000 feet is level terrain and she covers the distance in

20 minutes. She rests for 15 minutes. Jayne then travels

10,000 feet in 30 minutes. Draw a graph of Jayne’s distance

traveled (in feet) as a function of time.

43. The following sketch represents the distance d (in miles)

that Kevin was from home as a function of time t (in hours).

Answer the questions based on the graph. In parts (a)–(g),

how many hours elapsed and how far was Kevin from home

during this time?

(c) From to

(d) From to

(e) From to

(f) From to

(g) From to

(h) What is the farthest distance that Kevin was from home?

(i) How many times did Kevin return home?

44. The following sketch represents the speed (in miles per

hour) of Michael’s car as a function of time t (in minutes).

v

t = 4.2 t = 5.3

t = 3.9 t = 4.2

t = 3 t = 3.9

t = 2.8 t = 3

t = 2.5 t = 2.8

d(t )

(2, 3) (2.5, 3)

(2.8, 0) (3, 0)

(4.2, 2.8)

(5.3, 0)

(3.9, 2.8)

t

v (t )

(2, 30) (4, 30)

(4.2, 0) (6, 0) (9.1, 0)

(7.4, 50)

(8, 38)

(7.6, 38)

(7, 50)

t

(a) From to

(b) From t = 2 to t = 2.5

t = 0 t = 2

(a) Over what interval of time wasMichael traveling fastest?

(b) Over what interval(s) of time wasMichael’s speed zero?

(c) What was Michael’s speed between 0 and 2 minutes?

(d) What was Michael’s speed between 4.2 and 6 minutes?

(e) What was Michael’s speed between 7 and 7.4 minutes?

(f) When was Michael’s speed constant?

45. Draw the graph of a function whose domain is

and whose range is

What point(s) in the rectangle

cannot be on the graph?

Compare your graph with those of other students. What

differences do you see?

46. Is there a function whose graph is symmetric with respect to

the x-axis? Explain.

-3 … x … 8, -1 … y … 2

5y ƒ -1 … y … 2, y Z 06.

5x ƒ -3 … x … 8, x Z 56

1. 1-4, 02, 14, 02, 10, -22, 10, 22 False

3.3 PROPERTIES OF FUNCTIONS

3.3 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. The interval can be written as the inequality .

(pp. 120–121)

2. The slope of the line containing the points and

is . (pp. 167–169)

3. Test the equation for symmetry with respect to

the x-axis, the y-axis, and the origin. (pp. 160–162)

y = 5x2 - 1

13, 82

1-2, 32

12, 52 4. Write the point–slope form of the line with slope 5 containing

the point (p. 171)

5. The intercepts of the equation are .

(pp. 159–160)

y = x2 - 9

13, -22.

6. A function is on an open interval I if, for any

choice of and in I,with we have

7. A(n) function is one for which for

every x in the domain of a(n) function is one for

which for every x in the domain of

8. True or False A function is decreasing on an open

interval I if, for any choice of and in I, with we

have f1x12 7 f1x22.

x1 x2 x1 6 x2 ,

f

f1-x2 = -f1x2 f.

f; f

f f1-x2 = f1x2

x1 x2 x1 6 x2 , f1x12 6 f1x22.

f 9. True or False A function has a local maximum at c if

there is an open interval I containing c so that for all in I,

10. True or False Even functions have graphs that are

symmetric with respect to the origin.

f1x2 … f1c2.

x

f

Concepts and Vocabulary

In Problems 11–20, use the graph of the function given.

11. Is increasing on the interval
12. Is decreasing on the interval
13. Is increasing on the interval
14. Is decreasing on the interval
15. List the interval(s) on which is increasing.
16. List the interval(s) on which is decreasing.
17. Is there a local maximum value at 2? If yes, what is it?
18. Is there a local maximum value at 5? If yes, what is it?
19. List the number(s) at which has a local maximum.What are the local maximum values?
20. List the number(s) at which f has a local minimum.What are the local minimum values?

SECTION 3.3 Properties of Functions 231

x

y

4

4

!4

( (0, 3) !4, 2)

(2, 0)

(4, 2)

(!2, 0) x

y

3

3

!3

(!3, 3) (3, 3)

(0, 2)

(!1, 0) (1, 0) x

y

3

3

!3

(0, 1)

x

y

3

3

!3 (1, 0)

x

y

2

!2

!" ! "

(!", !1) (", !1)

2 "

2 "

(0, 1)

x

y

3

3

–3

–3

(–3, 2)

(–1, 2) (3, 1)

(1, –1) (2, –1) ) ( , 0 1–3

) (0,1–2

x

y

3

3

!2

!3

(!3, !2)

(!2, 1)

(!2.3, 0)

(2, 2)

(0, 1) (3, 0)

x

y

4

4

!4

(0, 3)

(!2, 0) (2, 0) x

y

3

3

!3

(0, 2)

(!1, 0) (1, 0)

x

y

2

!2

!! !

(!!, !1) (!, !1)

2 !

2 !

!

In Problems 21–28, the graph of a function is given. Use the graph to find:

(a) The intercepts, if any

(b) The domain and range

(c) The intervals on which it is increasing, decreasing, or constant

(d) Whether it is even, odd, or neither

21. 22. 23. 24.
22. 26. 27. 28.

In Problems 29–32, the graph of a function is given. Use the graph to find:

(a) The numbers, if any, at which has a local maximum value. What are the local maximum values?

(b) The numbers, if any, at which has a local minimum value. What are the local minimum values?

29. 30. 31. 32.

f

f

f

In Problems 33–44, determine algebraically whether each function is even, odd, or neither.

33. f1x2 = 4x3 34. f1x2 = 2x4 - x2 35. g1x2 = -3x2 - 5 36. h1x2 = 3x3 + 5
34. 38. 39. 40. f1x2 = 23 F1x2 = 13 G1x2 = 1x f1x2 = x + ƒxƒ 2x2 + 1 x
35. 42. 43. 44. F1x2 =

2x

ƒxƒ

h1x2 = -x3

3x2 - 9

h1x2 =

x

x2 - 1

g1x2 =

1

x2

In Problems 45–52, for each graph of a function find the absolute maximum and the absolute minimum, if they exist.

45. 46. 47. 48.

y = f(x),

(5, 1)

(3, 3)

(2, 2)

(1, 4)

1 3 5 x

y

4

2

(0, 1)

(1, 3)

(2, 4)

1 3 x

y

4

2

(5, 0)

(4, 4)

(1, 1)

(0, 2)

1 3 5 x

y

4

2

(4, 3)

(3, 4)

(1, 1)

(0, 3)

1 3 5 x

y

4

2

, 1

! ––

"2

! ––

2

( , "1)

( )

! ––

" 2

! ––

2

x

y

2

"2

"! !

x

y

"! !

, 1

! ––

"2

! ––

2

( , "1)

( )

! ––

" 2

! ––

2

1

"1232 CHAPTER 3 Functions and Their Graphs

(0, 0)

(2, 3)

(3, 2)

1 3 x

y

4

2

(2, 4)

(0, 2)

1 3 x

y

4

2

(0, 2)

(1, 3)

(2, 0)

(3, 1)

1 3 x

y

2

53. f1x2 = x3 - 3x + 2 1-2, 22 54. f1x2 = x3 - 3x2 + 5 1-1, 32
54. 50. 51. 52.

In Problems 53–60, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values

and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places.

Mixed Practice

71.

(a) Determine whether g is even, odd, or neither.

(b) There is a local minimum value of at 3.

Determine the local maximum value.

72.

(a) Determine whether f is even, odd, or neither.

(b) There is a local maximum value of 16 at 2. Determine

the local minimum value.

73.

(a) Determine whether F is even, odd, or neither.

(b) There is a local maximum value of 24 at .

Determine a second localmaximumvalue.

(c) Suppose the area under the graph of F between

and that is bounded below by the x-axis is 47.4

square units. Using the result from part (a), determine

the area under the graph of F between and

bounded below by the x-axis.

74.

(a) Determine whether G is even, odd, or neither.

(b) There is a local maximum value of 400 at

Determine a second localmaximumvalue.

(c) Suppose the area under the graph of G between

and that is bounded below by the x-axis

is 1612.8 square units. Using the result from part (a),

determine the area under the graph of G between

x = -6 and x = 0 bounded below by the x-axis.

x = 0 x = 6

x = 4.

G(x) = -x4 + 32x2 + 144

x = 0

x = -3

x = 3

x = 0

x = 2

F(x) = -x4 + 8x2 + 8

f(x) = -x3 + 12x

-54

g(x) = x3 - 27x

61. Find the average rate of change of

(a) From 0 to 2

(b) From 1 to 3

(c) From 1 to 4

62. Find the average rate of change of

(a) From 0 to 2

(b) From 1 to 3

(c) From to 1

63. Find the average rate of change of

(a) From to

(b) From to 1

(c) From 1 to 3

64. Find the average rate of change of

(a) From to 1

(b) From 0 to 2

(c) From 2 to 5

65.

(a) Find the average rate of change from 1 to 3.

(b) Find an equation of the secant line containing

and

66.

(a) Find the average rate of change from 2 to 5.

(b) Find an equation of the secant line containing

and

67.

(a) Find the average rate of change from to 1.

(b) Find an equation of the secant line containing

and

68.

(a) Find the average rate of change from to 2.

(b) Find an equation of the secant line containing

and

69.

(a) Find the average rate of change from 2 to 4.

(b) Find an equation of the secant line containing

and

70.

(a) Find the average rate of change from 0 to 3.

(b) Find an equation of the secant line containing

and 13, h1322.

10, h1022

h1x2 = -2x2 + x

14, h1422.

12, h1222

h1x2 = x2 - 2x

1-1, g1-122 12, g1222.

-1

g1x2 = x2 + 1

1-2, g1-222 11, g1122.

-2

g1x2 = x2 - 2

15, f1522.

12, f1222

f1x2 = -4x + 1

13, f1322.

11, f1122

f1x2 = 5x - 2

-1

h1x2 = x2 - 2x + 3

-1

-3 -2

g1x2 = x3 - 2x + 1

-1

f1x2 = -x3 + 1

f1x2 = -2x2 + 4

55. f1x2 = x5 - x3 1-2, 22 56. f1x2 = x4 - x2 1-2, 22
56. f1x2 = -0.2x3 - 0.6x2 + 4x - 6 1-6, 42 58. f1x2 = -0.4x3 + 0.6x2 + 3x - 2 1-4, 52
57. f1x2 = 0.25x4 + 0.3x3 - 0.9x2 + 3 1-3, 22 60. f1x2 = -0.4x4 - 0.5x3 + 0.8x2 - 2 1-3, 22

(2, 0)

(3, 2)

(4, 1)

1 3 x

y

2SECTION 3.3 Properties of Functions 233

75. Minimum Average Cost The average cost per hour in

dollars, , of producing x riding lawn mowers can be

modeled by the function

(a) Use a graphing utility to graph .

(b) Determine the number of riding lawn mowers to

produce in order to minimize average cost.

(c) What is the minimum average cost?

76. Medicine Concentration The concentration C of a

medication in the bloodstream t hours after being

administered is modeled by the function

(a) After how many hours will the concentration be

highest?

(b) A woman nursing a child must wait until the concentration

is below 0.5 before she can feed him. After taking

the medication, how long must she wait before feeding

her child?

77. E-coli Growth A strain of E-coli Beu 397-recA441 is

placed into a nutrient broth at 30° Celsius and allowed to

grow. The data shown in the table are collected. The

population is measured in grams and the time in hours.

Since population P depends on time t and each input

corresponds to exactly one output, we can say that

population is a function of time; so represents the

population at time t.

(a) Find the average rate of change of the population from

0 to 2.5 hours.

(b) Find the average rate of change of the population from

4.5 to 6 hours.

(c) What is happening to the average rate of change as time

passes?

P1t2

C1t2 = -0.002x4 + 0.039t3 - 0.285t2 + 0.766t + 0.085

C = C1x2

C1x2 = 0.3x2 + 21x - 251 +

2500

x

C

corresponds to exactly one output, the percentage of

returns filed electronically is a function of the filing year; so

represents the percentage of returns filed electronically

for filing year y.

(a) Find the average rate of change of the percentage of

e-filed returns from 2000 to 2002.

(b) Find the average rate of change of the percentage of

e-filed returns from 2004 to 2006.

(c) Find the average rate of change of the percentage of

e-filed returns from 2006 to 2008.

(d) What is happening to the average rate of change as time

passes?

P1y2

Applications and Extensions

Population

(grams), P

Time

(hours), t

0

2.5

3.5

4.5

6

0.09

0.18

0.26

0.35

0.50

78. e-Filing Tax Returns The Internal Revenue Service

Restructuring and Reform Act (RRA) was signed into law

by President Bill Clinton in 1998. A major objective of

the RRA was to promote electronic filing of tax returns.

The data in the table that follows, show the percentage of

individual income tax returns filed electronically for filing

years 2000–2008. Since the percentage P of returns filed

electronically depends on the filing year y and each input

Year Percentage of returns e-filed

2000

2001

2002

2003

2004

2005

2006

2007

2008

27.9

31.1

35.9

40.6

47.0

51.8

54.5

58.0

59.8

SOURCE: Internal Revenue Service

79. For the function compute each average rate of

change:

(a) From 0 to 1

(b) From 0 to 0.5

(c) From 0 to 0.1

(d) From 0 to 0.01

(e) From 0 to 0.001

(f) Use a graphing utility to graph each of the secant lines

along with .

(g) What do you think is happening to the secant lines?

(h) What is happening to the slopes of the secant lines? Is

there some number that they are getting closer to?

What is that number?

80. For the function compute each average rate of

change:

(a) From 1 to 2

(b) From 1 to 1.5

(c) From 1 to 1.1

(d) From 1 to 1.01

(e) From 1 to 1.001

(f) Use a graphing utility to graph each of the secant lines

along with .

(g) What do you think is happening to the secant lines?

(h) What is happening to the slopes of the secant lines? Is

there some number that they are getting closer to?

What is that number?

f

f1x2 = x2,

f

f1x2 = x2, Problems 81–88 require the following discussion of a secant line. The slope of the secant line containing the two points and

on the graph of a function may be given as

In calculus, this expression is called the difference quotient of

(a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer.

(b) Find for 0.1, and 0.01 at What value does approach as h approaches 0?

(c) Find the equation for the secant line at with

(d) Use a graphing utility to graph f and the secant line found in part (c) on the same viewing window.

x = 1 h = 0.01.

msec h = 0.5, x = 1. msec

f.

msec =

f1x + h2 - f1x2

1x + h2 - x =

f1x + h2 - f1x2

h h Z 0

1x + h, f1x + h22 y = f1x2

1x, f1x22

81. f1x2 = 2x + 5 82. f1x2 = -3x + 2 83. f1x2 = x2 + 2x 84. f1x2 = 2x2 + x
82. 86. 87. 88. f1x2 =

1

x2 f1x2 =

1

x

f1x2 = 2x2 - 3x + 1 f1x2 = -x2 + 3x - 2

89. Draw the graph of a function that has the following

properties: domain: all real numbers; range: all real numbers;

intercepts: and a local maximum value of

is at a local minimum value of is at 2. Compare your

graph with those of others. Comment on any differences.

90. Redo Problem 89 with the following additional information:

increasing on decreasing on

Again compare your graph with others and comment on any

differences.

91. How many x-intercepts can a function defined on an interval

have if it is increasing on that interval? Explain.

92. Suppose that a friend of yours does not understand the idea

of increasing and decreasing functions. Provide an explanation,

complete with graphs, that clarifies the idea.

1-q, -12, 12, q2; 1-1, 22.

-1; -6

10, -32 13, 02; -2

93. Can a function be both even and odd? Explain.
94. Using a graphing utility, graph on the interval

Use MAXIMUM to find the local maximum values on

Comment on the result provided by the calculator.

95. A function f has a positive average rate of change on the

interval 2, 5 . Is f increasing on 2, 5 ? Explain.

96. Show that a constant function has an average rate

of change of 0. Compute the average rate of change of

on the interval Explain how this can

happen.

y = 24 - x2 3-2, 24.

f(x) = b

3 4 3 4

1-3, 32.

y = 5 1-3, 32.

Explaining Concepts: Discussion and Writing

1. 2 6 x 6 5 1 3. symmetric with respect to the y-axis 4. y + 2 = 51x - 32 5. 1-3, 02, 13, 02, 10, -92

3.4 LIBRARY OF FUNCTIONS PIECEWISE- DEFINED FUNCTIONS

3.4 Assess Your Understanding

‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

1. Sketch the graph of (p. 163)
2. Sketch the graph of y = (pp. 164–165)

1

x

.

y = 1x. 3. List the intercepts of the equation y = x3 - 8. (pp. 159–160)

Concepts and Vocabulary

4. The function is decreasing on the interval

.

5. When functions are defined by more than one equation,

they are called functions.

6. True or False The cube function is odd and is increasing

on the interval 1-q, q2.

f1x2 = x2 7. True or False The cube root function is odd and is

decreasing on the interval

8. True or False The domain and the range of the reciprocal

function are the set of all real numbers.

Skill Building

In Problems 9–16, match each graph to its function.

1. Constant function B. Identity function C. Square function D. Cube function
2. Square root function F. Reciprocal function G. Absolute value function H. Cube root function

In Problems 17–24, sketch the graph of each function. Be sure to label three points on the graph.

17. f1x2 = x 18. f1x2 = x2 19. f1x2 = x3 20. f1x2 = 1x
18. 22. 23. f1x2 = 1 24. f1x2 = 3 3 f1x2 = f1x2 = ƒxƒ x

1

x

25. If

find: (a) (b) (c)

26. If

find: (a) f1-22 (b) f1-12 (c) f102

f1x2 = c

-3x if x 6 -1

0 if x = -1

2x2 + 1 if x 7 -1

f1-22 f102 f122

f1x2 = c

x2 if x 6 0

2 if x = 0

2x + 1 if x 7 0

9. 10. 11. 12.
10. 14. 15. 16.
11. If

find: (a) (b) (c) (d)

28. If

find: (a) f1-12 (b) f102 (c) f112 (d) f(3)

242 CHAPTER 3 Functions and Their Graphs

In Problems 41–44, the graph of a piecewise-defined function is given. Write a definition for each function.

41. 42. 43. 44.
42. If find

(a) f11.22 (b) f11.62 (c) f1-1.82

f1x2 = int12x2,

Applications and Extensions

47. Cell Phone Service Sprint PCS offers a monthly cellular

phone plan for $39.99. It includes 450 anytime minutes and

charges $0.45 per minute for additional minutes.The following

function is used to compute the monthly cost for a

subscriber:

where x is the number of anytime minutes used. Compute

the monthly cost of the cellular phone for use of the

following number of anytime minutes:

(a) 200 (b) 465 (c) 451

Source: Sprint PCS

48. Parking at O’Hare International Airport The short-term

(no more than 24 hours) parking fee F (in dollars) for parking

x hours at O’Hare International Airport’s main parking

garage can be modeled by the function

Determine the fee for parking in the short-term parking

garage for

(a) 2 hours (b) 7 hours (c) 15 hours

(d) 8 hours and 24 minutes

Source: O’Hare International Airport

F1x2 = c

3 if 0 6 x … 3

5 int1x + 12 + 1 if 3 6 x 6 9

50 if 9 … x … 24

C1x2 = b 39.99 if 0 … x … 450

0.45x - 162.51 if x 7 450

49. Cost of Natural Gas In April 2009, Peoples Energy had

the following rate schedule for natural gas usage in singlefamily

residences:

Monthly service charge $15.95

Per therm service charge

1st 50 therms $0.33606/therm

Over 50 therms $0.10580/therm

Gas charge $0.3940/therm

(a) What is the charge for using 50 therms in a month?

(b) What is the charge for using 500 therms in a month?

(c) Develop a model that relates the monthly charge C for

x therms of gas.

(d) Graph the function found in part (c).

Source: Peoples Energy, Chicago, Illinois, 2009

50. Cost of Natural Gas In April 2009, Nicor Gas had the

following rate schedule for natural gas usage in singlefamily

residences:

Monthly customer charge $8.40

Distribution charge

1st 20 therms $0.1473/therm

Next 30 therms $0.0579/therm

Over 50 therms $0.0519/therm

Gas supply charge $0.43/therm

(a) What is the charge for using 40 therms in a month?

(b) What is the charge for using 150 therms in a month?

(c) Develop a model that gives the monthly charge C for x

therms of gas.

(d) Graph the function found in part (c).

Source: Nicor Gas,Aurora, Illinois, 2009

46. If find

(a) f11.22 (b) f11.62 (c) f1-1.82

f1x2 = inta

x

2

b,

x

y

2

!2

(!1, 1)

(0, 0)

(2, 1)

2 x

y

2

!2 (0, 0) 2

(2, 1)

(!1, !1)

x

y

2

!2 (2, 0)

(1, 1)

(!1, 1)

(0, 0)

x

y

!2

(2, 2)

(1, 1)

(!1, 0)

(0, 2)

2

In Problems 29–40:

(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function.

(d) Based on the graph, find the range. (e) Is f continuous on its domain?

29. f1x2 = b 2x if x Z 0

1 if x = 0

30. f1x2 = b 3x if x Z 0

4 if x = 0

31. f1x2 = b -2x + 3 if x 6 1

3x - 2 if x Ú 1

32. f1x2 = b x + 3 if x 6 -2

-2x - 3 if x Ú -2

33. f1x2 = c

x + 3 if -2 … x 6 1

5 if x = 1

-x + 2 if x 7 1

34. f1x2 = c

2x + 5 if -3 … x 6 0

-3 if x = 0

-5x if x 7 0

35. f1x2 = b 1 + x if x 6 0

x2 if x Ú 0

36. f1x2 = c

1

x

if x 6 0

13 x if x Ú 0

37. f1x2 = b ƒx ƒ if -2 … x 6 0

x3 if x 7 0

38. f1x2 = b 2 - x if -3 … x 6 1

2x if x 7 1

39. f1x2 = 2 int1x2 40. f1x2 = int12x2SECTION 3.4 Library of Functions; Piecewise-defined Functions 243
40. Federal Income Tax Two 2009 Tax Rate Schedules are given in the accompanying table. If x equals taxable income and y equals

the tax due, construct a function y = f1x2 for Schedule X.

REVISED 2009 TAX RATE SCHEDULES

Schedule X—Single Schedule Y-1—Married Filing jointly or qualifying Widow(er)

If Taxable

Income Is

Over

But Not

Over

The Tax

Is This

Amount

Plus

This

%

Of the

Excess

Over

If Taxable

Income

Is Over

But Not

Over

The Tax

Is This

Amount

Plus

This

%

Of The

Excess

Over

$0 $8,350 – # 10% $0 $0 $16,700 – # 10% $0

8,350 33,950 $835.00 # 15% 8,350 16,700 67,900 $1,670.00 # 15% 16,700

33,950 82,250 4,675.00 # 25% 33,950 67,900 137,050 9,350.00 # 25% 67,900

82,250 171,550 16,750.00 # 28% 82,250 137,050 208,850 26,637.50 # 28% 137,050

171,550 372,950 41,754.00 # 33% 171,550 208,850 372,950 46,741.50 # 33% 208,850

372,950 – 108,216.00 # 35% 372,950 372,950 – 100,894.50 # 35% 372,950

Source: Internal Revenue Service

52. Federal Income Tax Refer to the revised 2009 tax rate

schedules. If x equals taxable income and y equals the tax

due, construct a function for Schedule Y-1.

53. Cost of Transporting Goods A trucking company transports

goods between Chicago and New York, a distance of

960 miles.The company’s policy is to charge, for each pound,

$0.50 per mile for the first 100 miles, $0.40 per mile for the

next 300 miles, $0.25 per mile for the next 400 miles, and no

charge for the remaining 160 miles.

(a) Graph the relationship between the cost of transportation

in dollars and mileage over the entire 960-mile

route.

(b) Find the cost as a function of mileage for hauls between

100 and 400 miles from Chicago.

(c) Find the cost as a function of mileage for hauls between

400 and 800 miles from Chicago.

54. Car Rental Costs An economy car rented in Florida

from National Car Rental® on a weekly basis costs $95

per week. Extra days cost $24 per day until the day rate

exceeds the weekly rate, in which case the weekly rate

applies. Also, any part of a day used counts as a full day.

Find the cost C of renting an economy car as a function of

the number x of days used, where Graph this

function.

55. Minimum Payments for Credit Cards Holders of credit

cards issued by banks, department stores, oil companies, and

so on, receive bills each month that state minimum amounts

that must be paid by a certain due date. The minimum due

depends on the total amount owed. One such credit card

company uses the following rules: For a bill of less than $10,

the entire amount is due. For a bill of at least $10 but less

than $500, the minimum due is $10. A minimum of $30 is

due on a bill of at least $500 but less than $1000, a minimum

of $50 is due on a bill of at least $1000 but less than $1500,

and a minimum of $70 is due on bills of $1500 or more. Find

the function that describes the minimum payment due on

a bill of x dollars. Graph

56. Interest Payments for Credit Cards Refer to Problem 55.

The card holder may pay any amount between the minimum

due and the total owed. The organization issuing the card

f.

f

7 … x … 14.

y = f1x2

charges the card holder interest of 1.5% per month for

the first $1000 owed and 1% per month on any unpaid

balance over $1000. Find the function g that gives the amount

of interest charged per month on a balance of x dollars.

Graph g.

57. Wind Chill The wind chill factor represents the

equivalent air temperature at a standard wind speed that

would produce the same heat loss as the given temperature

and wind speed. One formula for computing the equivalent

temperature is

where represents the wind speed (in meters per second)

and t represents the air temperature (°C). Compute the

wind chill for the following:

(a) An air temperature of 10°C and a wind speed of 1 meter

per second

(b) An air temperature of 10°C and a wind speed of

(c) An air temperature of 10°C and a wind speed of

(d) An air temperature of 10°C and a wind speed of

(e) Explain the physical meaning of the equation

corresponding to

(f) Explain the physical meaning of the equation

corresponding to

58. Wind Chill Redo Problem 57(a)–(d) for an air temperature

of

59. First-class Mail In 2009 the U.S. Postal Service charged

$1.17 postage for first-class mail retail flats (such as an 8.5

by 11 envelope) weighing up to 1 ounce, plus $0.17 for each

additional ounce up to 13 ounces. First-class rates do not

apply to flats weighing more than 13 ounces. Develop a

model that relates C, the first-class postage charged, for a

flat weighing x ounces. Graph the function.

Source: United States Postal Service

–

–

-10°C.

v 7 20.

0 … v 6 1.79.

25 m/sec

15 m/sec

5 m/sec

1m/sec2

v

W = d

t 0 … v 6 1.79

33 -

110.45 + 101v - v2133 - t2

22.04

1.79 … v … 20

33 - 1.5958133 - t2 v 7 20Explaining Concepts: Discussion and Writing

In Problems 60–67, use a graphing utility.

60. Exploration Graph Then on the same screen

graph followed by followed by

What pattern do you observe? Can you predict

the graph of Of

61. Exploration Graph Then on the same screen

graph followed by followed by

What pattern do you observe? Can you

predict the graph of Of

62. Exploration Graph Then on the same screen

graph followed by followed by

What pattern do you observe? Can you predict the graph of

Of

63. Exploration Graph Then on the same screen

graph What pattern do you observe? Now try

and What do you conclude?

64. Exploration Graph Then on the same screen

graph What pattern do you observe? Now try

and What y = 2x + 1 y = 21-x2 + 1. do you conclude?

y = 1-x.

y = 1x.

y = ƒxƒ y = - ƒxƒ.

y = -x2.

y = x2.

y = y = 5ƒxƒ?

1

4 ƒxƒ?

y =

1

2 y = 2ƒxƒ, y = 4ƒxƒ, ƒxƒ.

y = ƒxƒ.

y = 1x + 422? y = 1x - 522?

y = 1x + 222.

y = 1x - 222, y = 1x - 422,

y = x2.

y = x2 - 4? y = x2 + 5?

y = x2 - 2.

y = x2 + 2, y = x2 + 4,

y = x2. 65. Exploration Graph Then on the same screen graph

Could you have predicted the result?

66. Exploration Graph and on the

same screen. What do you notice is the same about each

graph? What do you notice that is different?

67. Exploration Graph and on the

same screen. What do you notice is the same about each

graph? What do you notice that is different?

68. Consider the equation

Is this a function? What is its domain? What is its range?

What is its y-intercept, if any? What are its x-intercepts, if

any? Is it even, odd, or neither? How would you describe its

graph?

69. Define some functions that pass through and

and are increasing for Begin your list with

and Can you propose a general

result about such functions?

3.5 GRAPHING TECHNIQUES;TRANSFORMATIONS

3.5 Assess Your Understanding

Concepts and Vocabulary

1. Suppose that the graph of a function is known. Then the

graph of may be obtained by a(n)

shift of the graph of to the a distance of 2 units.

2. Suppose that the graph of a function is known. Then the

graph of may be obtained by a reflection about

the -axis of the graph of the function

3. Suppose that the graph of a function g is known. The graph

of may be obtained by a shift of

the graph of g a distance of 2 units.

y = g1x2 + 2

y = f1x2.

y = f1-x2

f

f

y = f1x - 22

f 4. True or False The graph of is the reflection

about the x-axis of the graph of

5. True or False To obtain the graph of shift

the graph of horizontally to the right 2 units.

6. True or False To obtain the graph of shiftSkill Building

In Problems 7–18, match each graph to one of the following functions:

1. B. C. D.
2. F. G. H.
3. J. K. L.
4. 8. 9. 10.

y = 2x2 y = -2x2 y = 2ƒxƒ y = -2ƒxƒ

y = 1x - 222 y = -1x + 222 y = ƒx - 2ƒ y = - ƒx + 2ƒ

y = x2 + 2 y = -x2 + 2 y = ƒxƒ + 2 y = - ƒxƒ + 2

x

y

3

3

!3 x

y

3

!3 3

x

y

1

!3 3

x254 CHAPTER 3 Functions and Their Graphs

11. 12. 13. 14.

x

y

3

!3 3

!3

x

y

5

!3 3

!1

x

y

3

!3 3

!3

x

y

8

!6 6

!4

x

y

4

!4 4

!4

x

y

3

!3 3

!3

x

y

4

!4 4

!4

x

y

3

!3 3

!3

15. 16. 17. 18.

In Problems 19–26, write the function whose graph is the graph of y = x3, but is:

19. Shifted to the right 4 units 20. Shifted to the left 4 units

In Problems 27–30, find the function that is finally graphed after each of the following transformations is applied to the graph of

y = 1x in the order stated.

27. (1) Shift up 2 units

(2) Reflect about the x-axis

(3) Reflect about the y-axis

28. (1) Reflect about the x-axis

(2) Shift right 3 units

(3) Shift down 2 units

21. Shifted up 4 units 22. Shifted down 4 units
22. Reflected about the y-axis 24. Reflected about the x-axis
23. Vertically stretched by a factor of 4 26. Horizontally stretched by a factor of 4
24. (1) Reflect about the x-axis

(2) Shift up 2 units

(3) Shift left 3 units

31. If is a point on the graph of which of the

following points must be on the graph of

(a) (b)

(c) 13, -62 (d) 1-3, 62

16, 32 16, -32

y = -f1x2?

13, 62 y = f1x2,

33. If is a point on the graph of which of the

following points must be on the graph of

(a) (b)

(c) (d) a

1

2

11, 62 , 3b

a1, 12, 32

3

2

b

y = 2f1x2?

11, 32 y = f1x2, 34. If is a point on the graph of which of the

following points must be on the graph of ?

(a) (b)

(c) 12, 22 (d) 14, 42

14, 12 18, 22

y = f12x2

14, 22 y = f1x2,

35. Suppose that the x-intercepts of the graph of are

and 3.

(a) What are the x-intercepts of the graph of

(b) What are the x-intercepts of the graph of

(c) What are the x-intercepts of the graph of

(d) What are the x-intercepts of the graph of y = f1-x2?

y = 4f1x2?

y = f1x - 22?

y = f1x + 22?

-5

y = f1x2 36. Suppose that the x-intercepts of the graph of are

and 1.

(a) What are the x-intercepts of the graph of

(b) What are the x-intercepts of the graph of

(c) What are the x-intercepts of the graph of

(d) What are the x-intercepts of the graph of y = f1-x2?

y = 2f1x2?

y = f1x - 32?

y = f1x + 42?

-8

y = f1x2

37. Suppose that the function is increasing on the

interval

(a) Over what interval is the graph of

increasing?

(b) Over what interval is the graph of

increasing?

(c) What can be said about the graph of

(d) What can be said about the graph of y = f1-x2?

y = -f1x2?

y = f1x - 52

y = f1x + 22

1-1, 52.

y = f1x2 38. Suppose that the function is decreasing on the

interval

(a) Over what interval is the graph of

decreasing?

(b) Over what interval is the graph of

decreasing?

(c) What can be said about the graph of

(d) What can be said about the graph of y = f1-x2?

y = -f1x2?

y = f1x - 52

y = f1x + 22

1-2, 72.

y = f1x2

30. (1) Shift up 2 units

(2) Reflect about the y-axis

(3) Shift left 3 units

32. If is a point on the graph of which of the

following points must be on the graph of

(a) (b)

(c) 13, -62 (d) 1-3, 62

16, 32 16, -32

y = f1-x2?

13, 62 y = f1x2, SECTION 3.5 Graphing Techniques:Transformations 255

In Problems 39–62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph

of the basic function (for example, ) and show all stages. Be sure to show at least three key points. Find the domain and the range

of each function.

y = x2

39. f1x2 = x2 - 1 40. f1x2 = x2 + 4 41. g1x2 = x3 + 1
40. g1x2 = x3 - 1 43. h1x2 = 2x - 2 44. h1x2 = 2x + 1
41. f1x2 = 1x - 123 + 2 46. f1x2 = 1x + 223 - 3 47. g1x2 = 41x
42. g1x2 =

1

2 1x 49. f1x2 = -13 x 50. f1x2 = -1x

51. f1x2 = 21x + 122 - 3 52. f1x2 = 31x - 222 + 1 53. g1x2 = 22x - 2 + 1
52. g1x2 = 3ƒx + 1ƒ - 3 55. h1x2 = 1-x - 2 56. h1x2 =

4

x + 2

57. f1x2 = -1x + 123 - 1 58. f1x2 = -42x - 1 59. g1x2 = 2ƒ1 - xƒ
58. g1x2 = 422 - x 61. h1x2 = 2 int1x - 12 62. h1x2 = int1-x2

In Problems 63–66, the graph of a function is illustrated. Use the graph of as the first step toward graphing each of the following

functions:

(a) (b) (c) (d)

(e) Q1x2 = (f) g1x2 = f1-x2 (g) h1x2 = f12x2

1

2 f1x2

F1x2 = f1x2 + 3 G1x2 = f1x + 22 P1x2 = -f1x2 H1x2 = f1x + 12 - 2

f f

63. 64. 65. 66.
64. The equation defines a family of parabolas,

one parabola for each value of c. On one set of coordinate

axes, graph the members of the family for and

76. Repeat Problem 75 for the family of parabolas
77. Thermostat Control Energy conservation experts estimate

that homeowners can save 5% to 10% on winter heating

bills by programming their thermostats 5 to 10 degrees

lower while sleeping. In the given graph, the temperature T

(in degrees Fahrenheit) of a home is given as a function of

time t (in hours after midnight) over a 24-hour period.

y = x2 + c.

c = -2.

c = 0, c = 3,

y = 1x - c22

x

y

!4 2

(!4, !2)

(4, 0)

(0, 2) (2, 2)

4

!2

x

y

!4 !2 2

(!4, !2)

(!2, !2) (4, !2)

4

(2, 2)

4

2

!2

!" "

1

!1

x

y

"–2

( , 1) "–

2

"–2

!

(! , !1) "–

2

!" "

1

!1

x

y

(!", !1) (", !1)

"–2

"–2

!

Mixed Practice

In Problems 67–74, complete the square of each quadratic expression. Then graph each function using the technique of shifting.

(If necessary, refer to Chapter R, Section R.5 to review completing the square.)

67. 68. 69. 70.
68. f1x2 = 2x2 - 12x + 19 72. f1x2 = 3x2 + 6x + 1 73. f1x2 = -3x2 - 12x - 17 74. f1x2 = -2x2 - 12x - 13

f1x2 = x2 + 2x f1x2 = x2 - 6x f1x2 = x2 - 8x + 1 f1x2 = x2 + 4x + 2

Applications and Extensions

t

T

0 4

Time (hours after midnight)

Temperature (°F )

56

60

64

68

72

76

80

8 12 16 20 24256 CHAPTER 3 Functions and Their Graphs

(a) At what temperature is the thermostat set during

daytime hours? At what temperature is the thermostat

set overnight?

(b) The homeowner reprograms the thermostat to

Explain how this affects the temperature

in the house. Graph this new function.

(c) The homeowner reprograms the thermostat to

Explain how this affects the temperature

in the house. Graph this new function.

Source: Roger Albright, 547 Ways to Be Fuel Smart, 2000

78. Digital Music Revenues The total projected worldwide

digital music revenues R, in millions of dollars, for the years

2005 through 2010 can be estimated by the function

where x is the number of years after 2005.

(a) Find and and explain what each value

represents.

(b) Find

(c) Find and and explain what each value

represents.

(d) In the model r, what does x represent?

(e) Would there be an advantage in using the model r when

estimating the projected revenues for a given year

instead of the model R?

Source: eMarketer.com, May 2006

79. Temperature Measurements The relationship between

the Celsius (°C) and Fahrenheit (°F) scales for measuring

temperature is given by the equation

The relationship between the Celsius (°C) and Kelvin (K)

scales is Graph the equation

using degrees Fahrenheit on the y-axis and degrees Celsius

on the x-axis. Use the techniques introduced in this section

to obtain the graph showing the relationship between

Kelvin and Fahrenheit temperatures.

80. Period of a Pendulum The period T (in seconds) of a

simple pendulum is a function of its length l (in feet)

defined by the equation

where feet per second per second is the acceleration

of gravity.

g L 32.2

T = 2pA

l

g

F =

9

5 K = C + 273. C + 32

F =

9

5 C + 32

r152, r182, r1102

r = R1x - 52.

R102, R132, R152

R1x2 = 170.7x2 + 1373x + 1080

y = T1t + 12.

y = T1t2 - 2.

(a) Use a graphing utility to graph the function

(b) Now graph the functions

and T = T1l + 32.

T = T1l + 12, T = T1l + 22,

T = T1l2.

(c) Discuss how adding to the length l changes the period T.

(d) Now graph the functions and

(e) Discuss how multiplying the length l by factors of 2, 3,

and 4 changes the period T.

81. Cigar Company Profits The daily profits of a cigar company

from selling x cigars are given by

The government wishes to impose a tax on cigars (sometimes

called a sin tax) that gives the company the option of

either paying a flat tax of $10,000 per day or a tax of 10%

on profits.As chief financial officer (CFO) of the company,

you need to decide which tax is the better option for the

company.

(a) On the same screen, graph and

(b) Based on the graph, which option would you select?

Why?

(c) Using the terminology learned in this section, describe

each graph in terms of the graph of

(d) Suppose that the government offered the options of a

flat tax of $4800 or a tax of 10% on profits.Which would

you select? Why?

82. The graph of a function is illustrated in the figure.

(a) Draw the graph of

(b) Draw the graph of y = f1ƒxƒ2.

y = ƒf1x2ƒ.

f

p1x2.

Y2 = 11 - 0.102p1x2.

Y1 = p1x2 - 10,000

p1x2 = -0.05x2 + 100x - 2000

T = T14l2.

T = T12l2, T = T13l2,

!3 3

(1, 1)

(2, 0)

(!1, !1)

(!2, !1)

2

!2

x

y

!3 3

(1, 1)

(2, 0)

(!1, !1)

(!2, 0)

!2

2

x

y

83. The graph of a function is illustrated in the figure.

(a) Draw the graph of

(b) Draw the graph of y = f1ƒxƒ2.

y = ƒf1x2ƒ.

f

84. Suppose 1, 3 is a point on the graph of .

(a) What point is on the graph of

(b) What point is on the graph of

(c) What point is on the graph of

85. Suppose is a point on the graph of .

(a) What point is on the graph of

(b) What point is on the graph of

(c) What point is on the graph of y = g13x + 92?

y = -3g1x - 42 + 3?

y = g1x + 12 - 3?

1-3, 52 y = g1x2

y = f12x + 32?

y = -2f1x - 22 + 1?

y = f1x + 32 - 5?

1 2 y = f1x2Suppose that the graph of a function is known. Explain

how the graph of differs from the graph of

87. Suppose that the graph of a function f is known. Explain

how the graph of differs from the graph of

y = f1x - 22.

y = f1x2 - 2

y = f14x2.

y = 4f1x2

f 88. The area under the curve bounded below by the

x-axis and on the right by is square units. Using the

ideas presented in this section, what do you think is the area

under the curve of bounded below by the x-axis

and on the left by x = -4? Justify your answer.

y = 1-x

16

3

x = 4

y = 1x

Explaining Concepts: Discussion and Writing

Ask your instructor if the applets below are of interest to you.

(ii) What x-coordinate is required on the graph of

, if the y-coordinate is to be 2?

(iii) What x-coordinate is required on the graph of

, if the y-coordinate is to be 3?

(iv) What x-coordinate is required on the graph of

, if the y-coordinate is to be 1?

(v) What x-coordinate is required on the graph of

, if the y-coordinate is to be 2?

(vi) What x-coordinate is required on the graph of

, if the y-coordinate is to be 3?

93. Reflection about the y-axis Open the reflection about the

y-axis applet. Move your mouse to grab the slide and change

the value of a from 1 to .

94. Reflection about the x-axis Open the reflection about the

x-axis applet. Move your mouse to grab the slide and change

the value of a from 1 to -1.

-1

g1x2 = A

1

2

x

g1x2 = A

1

2

x

g1x2 = A

1

2

x

g1x2 = 22x

g1x2 = 22x

Interactive Exercises: Exploring Transformations

89. Vertical Shifts Open the vertical shift applet. Use your

mouse to grab the slider and change the value of k. Note the

role k plays in the graph of where

90. Horizontal Shifts Open the horizontal shift applet. Use

your mouse to grab the slider and change the value of h.

Note the role h plays in the graph of ,

where

91. Vertical Stretches Open the vertical stretch applet. Use

your mouse to grab the slider and change the value of a.

Note the role a plays in the graph of where

92. Horizontal Stretches Open the horizontal stretch applet.

(a) Use your mouse to grab the slider and change the value

of a. Note the role a plays in the graph of

where What

happens to the points on the graph of g when

What happens to the points on the graph

when

(b) To further understand the concept of horizontal compressions,

fill in the spreadsheet to the right of the graph

as follows:

• What x-coordinate is required on the graph of

3.6 MATHEMATICAL MODELS;BUILDING FUNCTIONS

3.6 Assess Your Understanding

1. Let be a point on the graph of

(a) Express the distance d from P to the origin as a function

of x.

(b) What is d if

(c) What is d if

(d) Use a graphing utility to graph

(e) For what values of x is d smallest?

2. Let be a point on the graph of

(a) Express the distance d from P to the point as a

function of x.

(b) What is d if

(c) What is d if

(d) Use a graphing utility to graph

(e) For what values of x is d smallest?

3. Let be a point on the graph of

(a) Express the distance d from P to the point as a

function of x.

(b) Use a graphing utility to graph

(c) For what values of x is d smallest?

4. Let be a point on the graph of

(a) Express the distance d from P to the origin as a function

of x.

(b) Use a graphing utility to graph

(c) For what values of x is d smallest?

5. Aright triangle has one vertex on the graph of

at another at the origin, and the third on the positive

y-axis at as shown in the figure. Express the area A of

the triangle as a function of x.

10, y2,

1x, y2,

y = x3, x 7 0,

d = d1x2.

y =

1

x

P = 1x, y2 .

d = d1x2.

11, 02

P = 1x, y2 y = 1x.

d = d1x2.

x = -1?

x = 0?

10, -12

P = 1x, y2 y = x2 - 8.

d = d1x2.

x = 1?

x = 0?

P = 1x, y2 y = x2 - 8.

6. A right triangle has one vertex on the graph of

at another at the origin, and the

third on the positive x-axis at Express the area A of

the triangle as a function of x.

7. A rectangle has one corner in quadrant I on the graph

of another at the origin, a third on the

positive y-axis, and the fourth on the positive x-axis. See

the figure.

y = 16 - x2,

1x, 02.

y = 9 - x2, x 7 0, 1x, y2,

Applications and Extensions

xd

N

S

W E

x

y

2

P ! (x, y)

x 2 " y 2 ! 4

#2

2

#2

SECTION 3.6 Mathematical Models: Building Functions 261

r

x

y

2

y ! 4 # x 2 P ! (x, y )

#2

(a) Express the area A of the rectangle as a function

of x.

(b) What is the domain of A?

(c) Graph For what value of x is A largest?

8. A rectangle is inscribed in a semicircle of radius 2. See the

figure. Let be the point in quadrant I that is a

vertex of the rectangle and is on the circle.

P = 1x, y2

A = A1x2.

12. Geometry A wire 10 meters long is to be cut into two

pieces. One piece will be shaped as an equilateral triangle,

and the other piece will be shaped as a circle.

(a) Express the total areaAenclosed by the pieces ofwire as a

function of the length x of a side of the equilateral triangle.

(b) What is the domain of A?

(c) Graph For what value of x is A smallest?

13. A wire of length x is bent into the shape of a circle.

(a) Express the circumference C of the circle as a function

of x.

(b) Express the area A of the circle as a function of x.

14. A wire of length x is bent into the shape of a square.

(a) Express the perimeter p of the square as a function

of x.

(b) Express the area A of the square as a function of x.

15. Geometry A semicircle of radius r is inscribed in a rectangle

so that the diameter of the semicircle is the length of the

rectangle. See the figure.

A = A1x2.

(a) Express the area A of the rectangle as a function

of x.

(b) Express the perimeter p of the rectangle as a function

of x.

(c) Graph For what value of x is A largest?

(d) Graph For what value of x is p largest?

9. A rectangle is inscribed in a circle of radius 2. See the figure.

Let be the point in quadrant I that is a vertex of

the rectangle and is on the circle.

P = 1x, y2

p = p1x2.

A = A1x2.

(a) Express the area A of the rectangle as a function

of x.

(b) Express the perimeter p of the rectangle as a function

of x.

(c) Graph For what value of x is A largest?

(d) Graph For what value of x is p largest?

10. A circle of radius r is inscribed in a square. See the figure.

p = p1x2.

A = A1x2.

(a) Express the area A of the square as a function of the

radius r of the circle.

(b) Express the perimeter p of the square as a function

of r.

11. Geometry A wire 10 meters long is to be cut into two

pieces. One piece will be shaped as a square, and the other

piece will be shaped as a circle. See the figure.

4x

x

10 # 4x

10 m

(a) Express the area A of the rectangle as a function of the

radius r of the semicircle.

(b) Express the perimeter p of the rectangle as a function

of r.

16. Geometry An equilateral triangle is inscribed in a circle of

radius r. See the figure. Express the circumference C of the

circle as a function of the length x of a side of the triangle.

[Hint: First show that ]

17. Geometry An equilateral triangle is inscribed in a circle of

radius r. See the figure in Problem 16. Express the area A

within the circle, but outside the triangle, as a function of the

length x of a side of the triangle.

18. Uniform Motion Two cars leave an intersection at the same

time. One is headed south at a constant speed of 30 miles

per hour, and the other is headed west at a constant speed of

40 miles per hour (see the figure). Build a model that

expresses the distance d between the cars as a function of

the time t.

[Hint: At t = 0, the cars leave the intersection.]

r2 =

x2

3

.

r

x x

x

r

(a) Express the total area A enclosed by the pieces of wire

as a function of the length x of a side of the square.

(b) What is the domain of A?

(c) Graph A = A1x2. For what value of x is A smallest? 262 CHAPTER 3 Functions and Their Graphs

Sphere

r

R

h

19. Uniform Motion Two cars are approaching an intersection.

One is 2 miles south of the intersection and is moving at a

constant speed of 30 miles per hour. At the same time, the

other car is 3 miles east of the intersection and is moving at

a constant speed of 40 miles per hour.

(a) Build a model that expresses the distance d between the

cars as a function of time t.

[Hint: At the cars are 2 miles south and 3 miles

east of the intersection, respectively.]

(b) Use a graphing utility to graph For what value

of t is d smallest?

20. Inscribing a Cylinder in a Sphere Inscribe a right circular

cylinder of height h and radius r in a sphere of fixed radius

1. R. See the illustration. Express the volume V of the cylinder

as a function of h.

[Hint: Note also the right triangle.]

21. Inscribing a Cylinder in a Cone Inscribe a right circular

cylinder of height h and radius r in a cone of fixed radius R

and fixed height H. See the illustration. Express the volume

V of the cylinder as a function of r.

[Hint: Note also the similar triangles.]

22. Installing Cable TV MetroMedia Cable is asked to provide

service to a customer whose house is located 2 miles from

the road along which the cable is buried. The nearest connection

box for the cable is located 5 miles down the road.

See the figure.

V = pr2

h.

V = pr2

h.

d = d1t2.

t = 0,

Cone

r

R

H

h

(a) If the installation cost is $500 permile along the road and

$700 per mile off the road, build a model that expresses

the total cost C of installation as a function of the distance

x (in miles) from the connection box to the point

where the cable installation turns off the road. Give the

domain.

(b) Compute the cost if

(c) Compute the cost if

(d) Graph the function Use TRACE to see how

the cost C varies as x changes from 0 to 5.

(e) What value of x results in the least cost?

23. Time Required to Go from an Island to a Town An island

is 2 miles from the nearest point P on a straight shoreline.

A town is 12 miles down the shore from P. See the

illustration.

C = C1x2.

x = 3 miles.

x = 1 mile.

Box

Stream

House

5 mi

2 mi

x

12 mi

2 mi

x 12 # x

P d2

d1

Island

Town

(a) If a person can row a boat at an average speed of

3 miles per hour and the same person can walk 5 miles

per hour, build a model that expresses the time T that

it takes to go from the island to town as a function of

the distance x from P to where the person lands the

boat.

(b) What is the domain of T?

(c) How long will it take to travel from the island to town if

the person lands the boat 4 miles from P?

(d) How long will it take if the person lands the boat 8 miles

from P?

24. Filling a Conical Tank Water is poured into a container

in the shape of a right circular cone with radius 4 feet and

height 16 feet. See the figure. Express the volume V of the

water in the cone as a function of the height h of the

water.

[Hint: The volume V of a cone of radius r and height h is

V = ]

1

3 pr2

h.

h

16

4

r

25. Constructing an Open Box An open box with a square base

is to be made from a square piece of cardboard 24 inches on

a side by cutting out a square from each corner and turning

up the sides. See the figure. Chapter Review 263

24 in.

24 in.

x x

x x

x

x

x

x

(a) Express the volume V of the box as a function of

the length x of the side of the square cut from each

corner.

(b) What is the volume if a 3-inch square is cut out?

(c) What is the volume if a 10-inch square is cut out?

(d) Graph For what value of x is V largest?

26. Constructing an Open Box An open box with a square

base is required to have a volume of 10 cubic feet.

(a) Express the amount A of material used to make such a

box as a function of the length x of a side of the square

base.

(b) How much material is required for a base 1 foot by

1 foot?

(c) How much material is required for a base 2 feet by

2 feet?

(d) Use a graphing utility to graph For what

value of x is A smallest?

A = A1x2.

V = V1x2.

Library of Functions

CHAPTER REVIEW

x

y

f(x) = b

(0,b)

x

y

3

3

–3

(1, 1)

(–1, –1)

(0, 0)

x

y

4

4

–4

(2, 4)

(0, 0)

(–2, 4)

(–1, 1) (1, 1)

x

y

4

4

#4

(1, 1)

(0, 0)

(#1, #1)

#4

x

y

5

2

#1

(1, 1)

(0, 0)

(4, 2)

( , )

x

y

3

(1, 1)

(#1, #1)

(2, 2 )

(0, 0)

#3

#3

3

3

(#2,# 2 ) 3

1–8

1–2

(# ,# ) 1–8

1–2

Reciprocal function (p. 238) Absolute value function (p. 238) Greatest integer function (p. 238)

f1x2 = ƒxƒ f1x2 = int1x2 f1x2 =

1

x

Cube function (p. 237) Square root function (p. 237) Cube root function (p. 237)

f1x2 = 13 f1x2 = x3 f1x2 = 1x x

Constant function (p. 236)

The graph is a horizontal line with

y-intercept b.

Identity function (p. 237)

The graph is a line with slope 1 and

y-intercept 0.

Square function (p. 237)

The graph is a parabola with intercept

at 10, 02.

f1x2 = b f1x2 = x f1x2 = x2

x

y

2

2

(1, 1)

(#1, #1)

#2

#2 x

y

3

3

#3

(1, 1)

(0, 0)

(#1, 1)

(#2, 2) (2, 2)

x

y

4

2

#2 2 4

#3

Review Exercises

In Problems 1 and 2, determine whether each relation represents a function. For each function, state the domain and range.

1. 51-1, 02, 12, 32, 14, 026 2. 514, -12, 12, 12, 14, 226

In Problems 3–8, find the following for each function:

(a) f122 (b) f1-22 (c) f1-x2 (d) -f1x2 (e) f1x - 22 (f) f12x2

3. f1x2 =

3x

x2 - 1

4. f1x2 =

x2

x + 1 5. f1x2 = 4x2 - 4

6. f1x2 = ƒx2 - 4ƒ 7. f1x2 =

x2 - 4

x2 8. f1x2 =

x3

x2 - 9

In Problems 9–16, find the domain of each function.

9. f1x2 =

x

x2 - 9

10. f1x2 =

3x2

x - 2

11. f1x2 = 22 - x
12. f1x2 = 2x + 2 13. h1x2 =

1x

ƒxƒ

14. g1x2 =

ƒxƒ

x

In Problems 17–22, find and for each pair of functions. State the domain of each of these functions.

f

g

f + g, f - g, f # g,

17. f1x2 = 2 - x; g1x2 = 3x + 1 18. f1x2 = 2x - 1; g1x2 = 2x + 1 19. f1x2 = 3x2 + x + 1; g1x2 = 3x
18. f1x2 = 3x; g1x2 = 1 + x + x2 21. f1x2 =

x + 1

x - 1

; g1x2 =

1

x

22. f1x2 =

1

x - 3

; g1x2 =

3

x

In Problems 23 and 24, find the difference quotient of each function that is, find

23. f1x2 = -2x2 + x + 1 24. f1x2 = 3x2 - 2x + 4In Problems 27 and 28, use the graph of the function to find:

(a) The domain and the range of .

(b) The intervals on which is increasing, decreasing, or constant.

(c) The local minimum values and local maximum values.

(d) The absolute maximum and absolute minimum.

(e) Whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin.

(f) Whether the function is even, odd, or neither.

(g) The intercepts, if any.

27. 28.

In Problems 29–36, determine (algebraically) whether the given function is even, odd, or neither.

f

f

f

In Problems 37–40, use a graphing utility to graph each function over the indicated interval. Approximate any local maximum values

and local minimum values. Determine where the function is increasing and where it is decreasing.

37. 38.
38. 40.

In Problems 41 and 42, find the average rate of change of

(a) From 1 to 2 (b) From 0 to 1 (c) From 2 to 4

41. 42.

In Problems 43–46, find the average rate of change from 2 to 3 for each function Be sure to simplify.

43. f1x2 = 2 - 5x 44. f1x2 = 2x2 + 7 45. f1x2 = 3x - 4x2 46. f1x2 = x2 - 3x + 2

f.

f1x2 = 8x2 - x f1x2 = 2x3 + x

f:

f1x2 = 2x4 - 5x3 + 2x + 1 1-2, 32 f1x2 = -x4 + 3x3 - 4x + 3 1-2, 32

f1x2 = 2x3 - 5x + 1 1-3, 32 f1x2 = -x3 + 3x - 5 1-3, 32

x

y

!5 5

3

!3

(0, 0) (4, 0)

(!4, !2)

(3, !3)

(!1, 1) x

y

!6 6

4

!4

(4, 3)

(!4,!3) (2, !1)

(!2, 1) (3, 0)

(!3, 0)

29. f1x2 = x3 - 4x 30. g1x2 =

4 + x2

1 + x4 31. h1x2 =

1

x4 +

1

x2 + 1 32. F1x2 = 41 - x3

33. G1x2 = 1 - x + x3 34. H1x2 = 1 + x + x2 35. f1x2 =

x

1 + x2 36. g1x2 =

1 + x2

x3

266 CHAPTER 3 Functions and Their Graphs

25. Using the graph of the function shown:

(a) Find the domain and the range of

(b) List the intercepts.

(c) Find

(d) For what value of x does

(e) Solve

(f) Graph

(g) Graph

(h) Graph y = -f1x2.

y = fa

1

2 xb.

y = f1x - 32.

f1x2 7 0.

f1x2 = -3?

f1-22.

f.

f 26. Using the graph of the function g shown:

(a) Find the domain and the range of g.

(b) Find

(c) List the intercepts.

(d) For what value of x does

(e) Solve

(f) Graph

(g) Graph

(h) Graph y = 2g1x2.

y = g1x2 + 1.

y = g1x - 22.

g1x2 7 0.

g1x2 = -3?

g1-12.

(3, 3)

(!2, !1)

(!4, !3)

x

y

!5 5

4

!4

(0, 0) x

y

!5 5

3

!3

(4, 0)

(0, 0)

(3, !3)

(!5, 1) (!1, 1) x

y

x

y

x

y

x

y

In Problems 51–54, sketch the graph of each function. Be sure to label at least three points.

51. 52. 53. 54. f1x2 =

1

x

f1x2 = 13 f1x2 = 1x f1x2 = ƒxƒ x

In Problems 55–66, graph each function using the techniques of shifting, compressing or stretching, and reflections. Identify any

intercepts on the graph. State the domain and, based on the graph, find the range.

55. 56. 57. 58.
56. 60. 61. 62.
57. h1x2 = 1x - 122 + 2 64. h1x2 = 1x + 222 - 3 65. g1x2 = 31x - 123 + 1 66. g1x2 = -21x + 223 - 8

h1x2 = 2x - 1 h1x2 = 1x - 1 f1x2 = 21 - x f1x2 = -2x + 3

g1x2 =

1

2 F1x2 = ƒxƒ - 4 f1x2 = ƒxƒ + 4 g1x2 = -2ƒxƒ ƒxƒ

Chapter Review 267

In Problems 47–50, is the graph shown the graph of a function?

47. 48. 49. 50.

In Problems 67–70,

(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function.

(d) Based on the graph, find the range. (e) Is f continuous on its domain?

67. 68.
68. 70. f1x2 = b x2 if -2 … x … 2

2x - 1 if x 7 2

f1x2 = c

x if -4 … x 6 0

1 if x = 0

3x if x 7 0

f1x2 = b x - 1 if -3 6 x 6 0

3x - 1 if x Ú 0

f1x2 = b 3x if -2 6 x … 1

x + 1 if x 7 1

71. A function is defined by

If f112 = 4, findA.

f1x2 =

Ax + 5

6x - 2

f 72. A function g is defined by

If g1-12 = 0, findA.

g1x2 =

A

x +

8

x2

73. Page Design A page with dimensions of by

11 inches has a border of uniform width x surrounding the

printed matter of the page, as shown in the figure.

(a) Develop a model that expresses the area A of the

printed part of the page as a function of the width x of

the border.

(b) Give the domain and the range of A.

(c) Find the area of the printed page for borders of widths

1 inch, 1.2 inches, and 1.5 inches.

(d) Graph the function A = A1x2.

8

1

2 inches

11 in.

8 in. 1–2

x x

x

x

The most important Beatle album to come out in 1968

was simply entitled The Beatles. It has become known

as the “White Album” because its cover is completely

white and devoid of any front or graphics except on

the spine and a number on the front cover

representing the order of production. Having launched

an explosion of garish, elaborate album art with Sgt.

Pepper, the Beatles now went to the opposite extreme

with the ultimate in plain simplicity.

The White Album was a double album (previously rare

in pop music except for special collections) and

contained thirty songs. Beatle fans consider it either

their heroes’ best or worst album! The controversy

arises from the extreme eclecticism of the music: there

is a bewildering variety of styles on this album.

Although the reason for for this eclecticism was not

apparent at the time, it has since become obvious. The

White Album was not so much the work of one group

but four individuals each of whom was heading in a

different direction.

74. Constructing a Closed Box A closed box with a square

base is required to have a volume of 10 cubic feet.

(a) Build a model that expresses the amount A of material

used to make such a box as a function of the length x of

a side of the square base.

(b) How much material is required for a base 1 foot by

1 foot?

(c) How much material is required for a base 2 feet by

2 feet?

(d) Graph For what value of x is A smallest?

75. A rectangle has one vertex in quadrant I on the graph of

another at the origin, one on the positive

x-axis, and one on the positive y-axis.

(a) Express the area A of the rectangle as a function of x.

(b) Find the largest area A that can be enclosed by the

rectangle.

y = 10 - x2,

A = A1x2. 1. Determine whether each relation represents a function. For

each function, state the domain and the range.

(a)

(b)

(c)

511, 32, 14, -22, 1-3, 52, 11, 726

512, 52, 14, 62, 16, 72, 18, 826

(d)

CHAPTER TEST

The Chapter Test Prep Videos are step-by-step test solutions available in the

Video Resources DVD, in , or on this text’s Channel. Flip

back to the Student Resources page to see the exact web address for this

text’s YouTube channel.

y

6

4

2

!2

!4

x

!4 !2 2 4

y

6

4

2

!2

x

!4 !2 2 4

y

4

!2

!4

x

!4 4

(0, 2)

(1, 3)

(2, 0)

(!5, !3)

(!2, 0)

(3, !3)

(5, !2)

In Problems 2–4, find the domain of each function and evaluate

each function at

2. 3.

4.

5. Using the graph of the function f:

h1x2 =

x - 4

x2 + 5x - 36

g1x2 =

x + 2

ƒx + 2ƒ

f1x2 = 24 - 5x

x = -1.

(a) Find the domain and the range of

(b) List the intercepts.

(c) Find

(d) For what value(s) of x does

(e) Solve f1x2 6 0.

f1x2 = -3?

f112.

f.

6. Use a graphing utility to graph the function

on the interval

Approximate any local maximum values and local minimum

values rounded to two decimal places. Determine where the

function is increasing and where it is decreasing.

7. Consider the function

(a) Graph the function.

(b) List the intercepts.

(c) Find

(d) Find

8. For the function find the average

rate of change of f from 3 to 4.

9. For the functions and find

the following and simplify:

(a)

(b)

(c)

10. Graph each function using the techniques of shifting, compressing

or stretching, and reflections. Start with the graph

of the basic function and show all stages.

(a)

(b)

11. The variable interest rate on a student loan changes each

July 1 based on the bank prime loan rate. For the years

1992–2007, this rate can be approximated by the model

where x is the number

of years since 1992 and r is the interest rate as a percent.

(a) Use a graphing utility to estimate the highest rate

during this time period.Duringwhich yearwas the interest

rate the highest?

(b) Use the model to estimate the rate in 2010. Does this

value seem reasonable?

Source: U.S. Federal Reserve

12. A community skating rink is in the shape of a rectangle with

semicircles attached at the ends.The length of the rectangle

is 20 feet less than twice the width.The thickness of the ice is

0.75 inch.

(a) Build a model that expresses the ice volume, V, as a

function of the width, x.

(b) How much ice is in the rink if the width is 90 feet?

r1x2 = -0.115x2 + 1.183x + 5.623,

g1x2 = ƒx + 4ƒ + 2

h1x2 = -21x + 123 + 3

f1x + h2 - f1x2

f # g

f - g

f1x2 = 2x2 + 1 g1x2 = 3x - 2,

f1x2 = 3x2 - 2x + 4,

g122.

g1-52.

g1x2 = b 2x + 1 if x 6 -1

x - 4 if x Ú -1

f1x2 = -x4 + 2x3 + 4x2 - 2 1-5, 52.

268Chapter Projects 269

In Problems 1–6, find the real solutions of each equation. In Problems 11–14, graph each equation.

11. 12.
12. 14.
13. For the equation , find the intercepts and

check for symmetry.

16. Find the slope–intercept form of the equation of the line

containing the points and .

In Problems 17–19, graph each function.

17.

18.

19. f1x2 = e

2 - x if x … 2

ƒxƒ if x 7 2

f1x2 =

1

x

f1x2 = 1x + 222 - 3

1-2, 42 16, 82

3x2 - 4y = 12

x2 + 1y - 322 = 16 y = 2x

3x - 2y = 12 x = y2

CUMULATIVE REVIEW

1. 3x - 8 = 10 2. 3x2 - x = 0
2. x2 - 8x - 9 = 0 4. 6x2 - 5x + 1 = 0
3. ƒ2x + 3ƒ = 4 6. 22x + 3 = 2

In Problems 7–9, solve each inequality. Graph the solution set.

7. 2 - 3x 7 6 8. ƒ2x - 5ƒ 6 3 9. ƒ4x + 1ƒ Ú 7
8. (a) Find the distance from to

(b) What is the midpoint of the line segment from to ?

(c) What is the slope of the line containing the points

and P2?

P1

P1 P2

P1 = 1-2, -32 P2 = 13, -52.

CHAPTER PROJECTS

3. Suppose you expect to use 500 anytime minutes with

unlimited texting and an unlimited data plan.What would

be the monthly cost of each plan you are considering?

4. Suppose you expect to use 500 anytime minutes with

unlimited texting and 20 MB of data.What would be the

monthly cost of each plan you are considering?

5. Build a model that describes the monthly cost C as a

function of the number of anytime minutes used massuming

unlimited texting and 20 MB of data each month for each

plan you are considering.

6. Graph each function from Problem 5.
7. Based on your particular usage,which plan is best for you?
8. Now, develop an Excel spreadsheet to analyze the various

plans you are considering. Suppose you want a plan that

offers 700 anytime minutes with additional minutes

costing $0.40 per minute that costs $39.99 per month. In

addition, you want unlimited texting, which costs an

additional $20 per month, and a data plan that offers up

to 25 MB of data each month, with each additional MB

costing $0.20. Because cellular telephone plans cost

structure is based on piecewise-defined functions, we

need “if-then” statements within Excel to analyze the

cost of the plan. Use the Excel spreadsheet below as a

guide in developing your worksheet. Enter into your

spreadsheet a variety of possible minutes and data used

to help arrive at a decision regarding which plan is best

for you.

9. Write a paragraph supporting the choice in plans that

best meets your needs.

10. How are “if/then” loops similar to a piecewise-defined

function?

Internet-based Project

1. Choosing a Cellular Telephone Plan Collect information

from your family, friends, or consumer agencies such as

Consumer Reports. Then decide on a cellular telephone

provider, choosing the company that you feel offers the best

service. Once you have selected a service provider, research

the various types of individual plans offered by the company

by visiting the provider’s website.

1. Suppose you expect to use 400 anytime minutes without a

texting or data plan.What would be the monthly cost of

each plan you are considering?

2. Suppose you expect to use 600 anytime minutes with

unlimited texting, but no data plan. What would be the

monthly cost of each plan you are considering? 270 CHAPTER 3 Functions and Their Graphs

The following projects are available on the Instructor’s Resource Center (IRC):

1. Project at Motorola: Wireless Internet Service Use functions and their graphs to analyze the total cost of various wireless

Internet service plans.

III. Cost of Cable When government regulations and customer preference influence the path of a new cable line, the Pythagorean

Theorem can be used to assess the cost of installation.

1. Oil Spill Functions are used to analyze the size and spread of an oil spill from a leaking tanker.

Citation: Excel © 2010 Microsoft Corporation. Used with permission from Microsoft.

Monthly Fee