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1 Numbers, Data, and Problem Solving

# 1 Numbers, Data, and Problem Solving

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1.1 Numbers, Data, and Problem Solving

EXAMPLE 1

3

Classifying numbers

Classify each real number as one or more of the following: natural number, integer, rational

number, or irrational number.

5, -1.2,

13

, - 27, -12, 216

7

SOLUTION

5: natural number, integer, and rational number

-1.2: rational number

13

: rational number

7

- 27: irrational number

-12: integer and rational number

216 = 4: natural number, integer, and rational number

Now Try Exercise 7

Order of Operations

6Ϫ3‫ء‬2

–52

Figure 1.1

0

–25

Does 6 - 3 # 2 equal 0 or 6? Does - 52 equal 25 or - 25? Figure 1.1 correctly shows that

6 - 3 # 2 = 0 and that - 52 = - 25. Because multiplication is performed before subtraction, 6 - 3 # 2 = 0. Similarly, because exponents are evaluated before performing negation, -52 = - 25. It is essential that algebraic expressions be evaluated consistently, so the

following rules have been established.

Order of Operations

Using the following order of operations, first perform all calculations within parentheses, square roots, and absolute value bars and above and below fraction bars. Then use

the same order of operations to perform any remaining calculations.

1. Evaluate all exponents. Then do any negation after evaluating exponents.

2. Do all multiplication and division from left to right.

3. Do all addition and subtraction from left to right.

EXAMPLE 2

Evaluating arithmetic expressions

Evaluate each expression by hand.

10 - 6

- 4 - ƒ7 - 2ƒ

(a) 3(1 - 5)2 - 42

(b)

5 - 3

SOLUTION

(a) 3(1 - 5)2 - 42 = 3( -4)2 - 42

NOTE

(b)

10 - 6

4

- 4 - ƒ7 - 2ƒ = - 4 - ƒ5ƒ

5 - 3

2

= 3(16) - 16

= 2 - 4 - 5

= 48 - 16

= -2 - 5

= 32

= -7

(-4)2 = ( -4)( -4) = 16 and -42 = - (4)(4) = - 16.

Now Try Exercises 19 and 21 ᭣

4

CHAPTER 1 Introduction to Functions and Graphs

Scientific Notation

Numbers that are large or small in absolute value are often expressed in scientific notation. Table 1.1 lists examples of numbers in standard (decimal) form and in scientific

notation.

Table 1.1

Standard Form

Scientific Notation

93,000,000 mi

9.3 * 10 mi

Distance to the sun

13,517

1.3517 * 10

9,000,000,000

9 * 10

Estimated world population in 2050

7

4

9

-6

0.00000538 sec

5.38 * 10

0.000005 cm

5 * 10-6 cm

sec

Application

Time for light to travel 1 mile

Size of a typical virus

To write 0.00000538 in scientific notation, start by moving the decimal point to the

right of the first nonzero digit, 5, to obtain 5.38. Since the decimal point was moved six

places to the right, the exponent of 10 is -6. Thus, 0.00000538 = 5.38 * 10-6. When the

decimal point is moved to the left, the exponent of 10 is positive, rather than negative. Here

is a formal definition of scientific notation.

Calculator Help

To display numbers in scientific

notation, see Appendix A

(page AP-2).

Scientific Notation

A real number r is in scientific notation when r is written as c * 10n, where

1 … ƒ c ƒ 6 10 and n is an integer.

An Application Nanotechnology involves extremely small electrical circuits. Someday

this technology may use the movement of the human body to power tiny devices such as

pacemakers. The next example demonstrates how scientific notation appears in the

description of this new technology.

EXAMPLE 3

Analyzing the energy produced by your body

Nanotechnology is a technology of the very small: on the order of one billionth of a meter.

Researchers are looking to power tiny devices with energy generated by the human body.

(Source: Z. Wang, “Self-Powered Nanotech,” Scientific American, January 2008.)

(a) Write one billionth in scientific notation.

(b) While typing, a person’s fingers generate about 2.2 * 10-3 watt of electrical energy.

Write this number in standard (decimal) form.

SOLUTION

1

(a) One billionth can be written as 1,000,000,000

= 101 9 = 1 * 10-9.

(b) Move the decimal point in 2.2 three places to the left: 2.2 * 10-3 = 0.0022.

Now Try Exercise 83 ᭣

The next two examples illustrate how to evaluate expressions involving scientific

notation.

1.1 Numbers, Data, and Problem Solving

EXAMPLE 4

5

Evaluating expressions by hand

Evaluate each expression. Write your result in scientific notation and standard form.

4.6 * 10-1

(a) (3 * 103)(2 * 104) (b) (5 * 10-3)(6 * 105) (c)

2 * 102

SOLUTION

(3 * 103)(2 * 104) =

=

=

=

(b) (5 * 10-3)(6 * 105) =

=

=

=

4.6 * 10-1

(c)

=

2 * 102

=

=

=

(a)

Algebra Review

To review exponents, see Chapter R

(page R-7).

3 * 2 * 103 * 104

6 * 103 + 4

6 * 107

60,000,000

5 * 6 * 10-3 * 105

30 * 102

3 * 103

3000

4.6

10-1

*

2

102

2.3 * 10-1 - 2

2.3 * 10-3

0.0023

Commutative property

Scientific notation

Standard form

Commutative property

Scientific notation

Standard form

Multiplication of fractions

Subtract exponents.

Scientific notation

Standard form

Now Try Exercises 53, 55, and 57 ᭣

Calculators Calculators often use E to express powers of 10. For example, 4.2 * 10-3

might be displayed as 4.2E–3. On some calculators, numbers can be entered in scientific

notation with the EE key, which you can find by pressing 2nd , .

EXAMPLE 5

Computing in scientific notation with a calculator

6 * 103

103 + 450 3

2

(a) a

(b) 24500pa

b

6 b(1.2 * 10 )

4 * 10

0.233

SOLUTION

(a) The given expression is entered in two ways in Figure 1.2. Note that in both cases

a

6 * 103

b(1.2 * 102) = 0.18 = 1.8 * 10-1.

4 * 106

(b) Be sure to insert parentheses around 4500p and around the numerator, 103 + 450, in

the ratio. From Figure 1.3 we can see that the result is approximately 1.59 * 1012.

Calculator Help

To enter numbers in scientific

notation, see Appendix A

(page AP-2).

(6‫ء‬10^3)/(4‫ء‬10^6

)‫(ء‬1.2‫ء‬10^2)

.18

( 6 E3 ) / ( 4 E6 ) ‫ ( ء‬1 . 2

E2 )

.18

Figure 1.2

√ (4500␲) ‫(( ء‬103ϩ4

50) / .233)^3

1.58960355 E 12

Figure 1.3

Now Try Exercises 61 and 63 ᭣

6

CHAPTER 1 Introduction to Functions and Graphs

EXAMPLE 6

Computing with a calculator

Use a calculator to evaluate each expression. Round answers to the nearest thousandth.

1 + 22

(a) 2

3 131 (b) p3 + 1.22 (c)

(d) ƒ 23 - 6 ƒ

3.7 + 9.8

Algebra Review

To review cube roots, see Chapter R

(page R-40).

SOLUTION

(a) On some calculators the cube root can be found by using the MATH menu. If your calculator does not have a cube root key, enter 131^(1/3). From the first two lines in

Figure 1.4, we see that 2

3 131 L 5.079.

(b) Do not use 3.14 for the value of p. Instead, use the built-in key to obtain a more accurate value of p. From the bottom two lines in Figure 1.4, p3 + 1.22 L 32.446.

(c) When evaluating this expression be sure to include parentheses around the numerator

and around the denominator. Most calculators have a special square root key that can

1 + 22

be used to evaluate 22. From the first three lines in Figure 1.5, 3.7

+ 9.8 L 0.179.

(d) The absolute value can be found on some calculators by using the MATH NUM

menus. From the bottom two lines in Figure 1.5, ƒ 23 - 6 ƒ L 4.268.

Calculator Help

3 √ (131)

5.078753078

␲^3ϩ1.2 2

32.44627668

To enter expressions such as

3

2131 , 22, p, and ƒ 23 - 6 ƒ ,

see Appendix A (page AP-2).

Figure 1.4

(1ϩ √ (2)) / (3.7ϩ9.

8)

.1 7 8 8 3 0 6 3 4 2

abs(√ (3)Ϫ6)

4.267949192

Figure 1.5

Now Try Exercises 67, 69, 71, and 73 ᭣

Problem Solving

Many problem-solving strategies are used in algebra. However, in this subsection we focus

on two important strategies that are used frequently: making a sketch and applying one or

more formulas. These strategies are illustrated in the next three examples.

EXAMPLE 7

Finding the speed of Earth

Earth travels around the sun in an approximately circular orbit with an average radius of

93 million miles. If Earth takes 1 year, or about 365 days, to complete one orbit, estimate

the orbital speed of Earth in miles per hour.

SOLUTION

D

Getting Started Speed S equals distance D divided by time T, S = T . We need to find the

Geometry Review

To find the circumference of a circle,

see Chapter R (page R-2).

number of miles Earth travels in 1 year and then divide it by the number of hours in

1 year. ᭤

Distance Traveled A sketch of Earth orbiting the sun is shown in Figure 1.6. In 1 year

Earth travels the circumference of a circle with a radius of 93 million miles. The

circumference of a circle is 2pr, where r is the radius, so the distance D is

D = 2pr = 2p(93,000,000) L 584,300,000 miles.

1.1 Numbers, Data, and Problem Solving

Hours in 1 Year

7

The number of hours H in 1 year, or 365 days, equals

H = 365 * 24 = 8760 hours.

93,000,000 mi

Earth

Speed of Earth

S =

D

H

=

584,300,000

8760

L 66,700 miles per hour.

Now Try Exercise 85 ᭣

Sun

(Not to scale)

Figure 1.6 Earth’s Orbit

Many times in geometry we evaluate formulas to determine quantities, such as

perimeter, area, and volume. In the next example we use a formula to determine the number of fluid ounces in a soda can.

EXAMPLE 8

Finding the volume of a soda can

The volume V of the cylindrical soda can in Figure 1.7 is given by V = pr2h, where r is

its radius and h is its height.

(a) If r = 1.4 inches and h = 5 inches, find the volume of the can in cubic inches.

(b) Could this can hold 16 fluid ounces? (Hint: 1 cubic inch equals 0.55 fluid ounce.)

r

SOLUTION

(a) V = pr 2h = p(1.4)2(5) = 9.8p L 30.8 cubic inches.

(b) To find the number of fluid ounces, multiply the number of cubic inches by 0.55.

30.8 * 0.55 = 16.94

h

Now Try Exercise 93 ᭣

Yes, the can could hold 16 fluid ounces.

Figure 1.7 A Soda Can

Measuring the thickness of a very thin layer of material can be difficult to do directly.

For example, it would be difficult to measure the thickness of a sheet of aluminum foil or

a coat of paint with a ruler. However, it can be done indirectly using the following formula.

Thickness =

Volume

Area

That is, the thickness of a thin layer equals the volume of the substance divided by the area

that it covers. For example, if a volume of 1 cubic inch of paint is spread over an area of

1

100 square inches, then the thickness of the paint equals 100

inch. This formula is illustrated

in the next example.

EXAMPLE 9

Calculating the thickness of aluminum foil

A rectangular sheet of aluminum foil is 15 centimeters by 35 centimeters and weighs

5.4 grams. If 1 cubic centimeter of aluminum weighs 2.7 grams, find the thickness of the

aluminum foil. (Source: U. Haber-Schaim, Introductory Physical Science.)

SOLUTION

Getting Started Start by making a sketch of a rectangular sheet of aluminum, as shown in

Figure 1.8. To complete this problem we need to find the volume V of the aluminum foil

and its area A. Then we can determine the thickness T by using the formula T = VA. ᭤

15 cm

T

35 cm

(Not to scale)

Figure 1.8 Aluminum Foil

8

CHAPTER 1 Introduction to Functions and Graphs

NOTE

For the rectangular box shape shown in Figure 1.8 on the previous page,

Volume = Length * Width * Thickness.

('')''*

Area

It follows that Thickness =

Geometry Review

To find the area of a rectangle, see

Chapter R (page R-1). To find the

volume of a box, see Chapter R

(page R-3).

Volume

Area .

Volume Because the aluminum foil weighs 5.4 grams and each 2.7 grams equals 1 cubic

centimeter, the volume of the aluminum foil is

5.4

= 2 cubic centimeters.

2.7

Area

Divide weight by density.

The aluminum foil is rectangular with an area of 15 * 35 = 525 square centimeters.

Thickness The thickness of 2 cubic centimeters of aluminum foil with an area of 525

square centimeters is

Thickness =

Volume

2

=

L 0.0038 centimeter.

Area

525

Now Try Exercise 89 ᭣

1.1 Putting

It All

Together

Concept

Numbers play a central role in our society. Without numbers, data could be

described qualitatively but not quantitatively. For example, we could say that

the day seems hot but would not be able to give an actual number for the temperature. Problem-solving strategies are used in almost every facet of our lives,

providing the procedures needed to systematically complete tasks and perform computations.

The following table summarizes some of the concepts in this section.

Examples

Natural numbers

Sometimes referred to as the counting numbers

1, 2, 3, 4, 5, . . .

Integers

Include the natural numbers, their opposites, and 0

. . . , -2, -1, 0, 1, 2, . . .

Rational numbers

Include integers; all fractions q, where p and q are integers

with q Z 0; all repeating and all terminating decimals

p

1

128

, -3,

, -0.335, 0,

2

6

1

1

0.25 = , 0.33 =

4

3

Irrational numbers

Can be written as nonrepeating, nonterminating decimals;

cannot be a rational number; if a square root of a positive integer is not an integer, it is an irrational number.

p, 22, - 25, 2

3 7, p4

1.1 Numbers, Data, and Problem Solving

Concept

Examples

Real numbers

Any number that can be expressed in

standard (decimal) form

Include the rational numbers and

irrational numbers

4

p, 27, - , 0, - 10, 1.237

7

2

0.6 = , 1000, 215, - 25

3

Order of

operations

Using the following order of operations,

first perform all calculations within parentheses, square roots, and absolute value

bars and above and below fraction bars.

Then perform any remaining calculations.

1. Evaluate all exponents. Then do any

negation after evaluating exponents.

2. Do all multiplication and division

from left to right.

3. Do all addition and subtraction from

left to right.

-42 - 12 , 2 - 2 =

=

=

=

2 + 42

=

3 - 3#5

Scientific notation

A number in the form c * 10n, where

1 … ƒ c ƒ 6 10 and n is an integer

3.12 * 104 = 31,200

- 1.4521 * 10-2 = - 0.014521

5 * 109 = 5,000,000,000

1.5987 * 10-6 = 0.0000015987

Used to represent numbers that are large

or small in absolute value

1.1

9

- 16 - 12 , 2 - 2

- 16 - 6 - 2

- 22 - 2

- 24

2 + 16

3 - 15

18

=

-12

3

= 2

Exercises

Classifying Numbers

Exercises 1–6: Classify the number as one or more of the following: natural number, integer, rational number, or real number.

1. 21

24 (Fraction of people in the United States completing at

least 4 years of high school)

2. 20,082 (Average cost in dollars of tuition and fees at a

private college in 2004)

3. 7.5 (Average number of gallons of water used each minute

while taking a shower)

4. 25.8 (Nielsen rating of the TV show Grey’s Anatomy the

week of February 12–18, 2007)

5. 9022 (Distance in feet from home plate to second base

on a baseball field)

6. -71 (Wind chill when the temperature is -30°F and the

wind speed is 40 mph)

Exercises 7–10: Classify each number as one or more of the

following: natural number, integer, rational number, or irrational number.

7. p, -3, 29, 29, 1.3, - 22

8. 31, - 58, 27, 0.45, 0, 5.6 * 103

9. 213, 13, 5.1 * 10-6, -2.33, 0.7, - 24

5.7 2

10. -103, 21

25 , 2100, - 10 , 9 , -1.457, 23

Exercises 11–16: For the measured quantity, state the set of

numbers that most appropriately describes it. Choose from

the natural numbers, integers, and rational numbers. Explain

11. Shoe sizes

12. Populations of states

13. Gallons of gasoline

14. Speed limits

10

CHAPTER 1 Introduction to Functions and Graphs

15. Temperatures in a winter weather forecast in Montana

49. 0.045 * 105

50. -5.4 * 10-5

16. Numbers of compact disc sales

51. 67 * 103

52. 0.0032 * 10-1

Order of Operations

Exercises 53–60: Evaluate the expression by hand. Write

your result in scientific notation and standard form.

53. (4 * 103)(2 * 105)

54. (3 * 101)(3 * 104)

Exercises 17–28: Evaluate by hand.

17. ƒ 5 - 8 # 7 ƒ

18. -2(16 - 3 # 5) , 2

19. - 62 - 3(2 - 4)4

20. (4 - 5)2 - 32 - 329

55. (5 * 102)(7 * 10-4)

8 - 4

21. 29 - 5 4 - 2

6 - 42 , 23

22.

3 - 4

57.

6.3 * 10-2

3 * 101

58.

8.2 * 102

2 * 10-2

23. 213 - 12

13 - 29 + 16

24.

ƒ 5 - 7 ƒ2

59.

4 * 10-3

8 * 10-1

60.

2.4 * 10-5

4.8 * 10-7

2

25.

2

4 + 9

-32 # 3

2 + 3

5

27. - 52 - 20 , 4 - 2

26. 10 , 2 ,

5 + 10

5

28. 5 - (-4)3 - (4)3

56. (8 * 10-3)(7 * 101)

Exercises 61–66: Use a calculator to approximate the expression. Write your result in scientific notation.

8.947 * 107

(4.5 * 108)

61.

0.00095

Scientific Notation

62. (9.87 * 106)(34 * 1011)

Exercises 29–40: Write the number in scientific notation.

29. 184,800 (New lung cancer cases reported in 2005)

63. a

30. 29,285,000 (People worldwide living with HIV)

3 (2.5 * 10-8) + 10-7

64. 2

31. 0.04361 (Proportion of U.S. deaths attributed to accidents in 2004)

65. (8.5 * 10-5)(- 9.5 * 107)2

32. 0.62 (Number of miles in 1 kilometer)

33. 2450

34. 105.6

35. 0.56

36. -0.00456

37. - 0.0087

38. 1,250,000

39. 206.8

40. 0.00007

101 + 23 2

b + 23.4 * 10-2

0.42

66. 2p(4.56 * 104) + (3.1 * 10-2)

Exercises 67–76: Use a calculator to evaluate the expression. Round your result to the nearest thousandth.

3 192

67. 2

68. 2(32 + p3)

69. ƒ p - 3.2 ƒ

70.

1.72 - 5.98

35.6 + 1.02

Exercises 41–52: Write the number in standard form.

41. 1 * 10-6 (Wavelength in meters of visible light)

71.

0.3 + 1.5

5.5 - 1.2

72. 3.2(1.1)2 - 4(1.1) + 2

42. 9.11 * 10-31 (Weight in kilograms of an electron)

73.

1.53

12 + p - 5

74. 4.32 -

43. 2 * 108 (Years required for the sun to orbit our galaxy)

44. 9 * 10 (Federal debt in dollars in 2007)

12

75. 15 +

4 + 23

7

76.

5

17

5 + 25

2

45. 1.567 * 102

46. -5.68 * 10-1

Applications

47. 5 * 105

48. 3.5 * 103

Exercises 77–80: Percent Change If an amount changes

from A to a new amount B, then the percent change is

B - A

* 100.

A

1.1 Numbers, Data, and Problem Solving

Calculate the percent change for the given A and B. Round your

answer to the nearest tenth of a percent when appropriate.

77. A = \$8, B = \$13

78. A = \$0.90, B = \$13.47

79. A = 1.4, B = 0.85

80. A = 1256, B = 1195

81. Percent Change Suppose that tuition is initially \$100

per credit and increases by 6% from the first year to the

second year. What is the cost of tuition the second year?

Now suppose that tuition decreases by 6% from the second to the third year. Is tuition equal to \$100 per credit

the third year? Explain.

82. Tuition Increases From 1976 to 2004, average annual

tuition and fees at public colleges and universities

increased from \$433 to \$5132. Calculate the percent

change over this time period.

11

87. Federal Debt The amount of federal debt changed dramatically during the 30 years from 1970 to 2000.

(Sources: Department of the Treasury, Bureau of the Census.)

(a) In 1970 the population of the United States was

203,000,000 and the federal debt was \$370 billion.

Find the debt per person.

(b) In 2000 the population of the United States was

approximately 281,000,000 and the federal debt was

\$5.54 trillion. Find the debt per person.

88. Discharge of Water The Amazon River discharges

water into the Atlantic Ocean at an average rate of

4,200,000 cubic feet per second, the highest rate of any

river in the world. Is this more or less than 1 cubic mile

of water per day? Explain your calculations. (Source:

The Guinness Book of Records 1993.)

83. Nanotechnology (Refer to Example 3.) During inhalation, the typical body generates 0.14 watt of electrical

power, which could be used to power tiny electrical circuits. Write this number in scientific notation. (Source:

Scientific American, January 2008.)

84. Movement of the Pacific Plate The Pacific plate (the

floor of the Pacific Ocean) near Hawaii is moving at

speed at which a fingernail grows. Use scientific notation to determine how many kilometers the Pacific plate

travels in one million years.

85. Orbital Speed (Refer to Example 7.) The planet Mars

travels around the sun in a nearly circular orbit with a

radius of 141 million miles. If it takes 1.88 years for

Mars to complete one orbit, estimate the orbital speed of

Mars in miles per hour.

Sun

Mars

141,000,000 mi

89. Thickness of an Oil Film (Refer to Example 9.) A drop

of oil measuring 0.12 cubic centimeter is spilled onto a

lake. The oil spreads out in a circular shape having a

diameter of 23 centimeters. Approximate the thickness

of the oil film.

90. Thickness of Gold Foil (Refer to Example 9.) A flat,

rectangular sheet of gold foil measures 20 centimeters

by 30 centimeters and has a mass of 23.16 grams. If

1 cubic centimeter of gold has a mass of 19.3 grams,

find the thickness of the gold foil. (Source: U. HaberSchaim, Introductory Physical Science.)

(Not to scale)

86. Size of the Milky Way The speed of light is about

186,000 miles per second. The Milky Way galaxy has an

approximate diameter of 6 * 1017 miles. Estimate, to

the nearest thousand, the number of years it takes for

light to travel across the Milky Way. (Source: C. Ronan,

The Natural History of the Universe.)

91. Analyzing Debt A 1-inch-high stack of \$100 bills contains about 250 bills. In 2000 the federal debt was

approximately 5.54 trillion dollars.

(a) If the entire federal debt were converted into a stack

of \$100 bills, how many feet high would it be?

(b) The distance between Los Angeles and New York is

approximately 2500 miles. Could this stack of \$100

bills reach between these two cities?

12

CHAPTER 1 Introduction to Functions and Graphs

92. Volume of a Cone The volume V of a cone is given by

V = 13pr2h, where r is its radius and h is its height. Find

V when r = 4 inches and h = 1 foot. Round your

95. Thickness of Cement (Refer to Example 9.) A 100foot-long sidewalk is 5 feet wide. If 125 cubic feet of

cement are evenly poured to form the sidewalk, find the

thickness of the sidewalk.

96. Depth of a Lake (Refer to Example 9.) A lake covers

2.5 * 107 square feet and contains 7.5 * 108 cubic feet

of water. Find the average depth of the lake.

r

h

97. Describe some basic sets of numbers that are used in

mathematics.

93. Size of a Soda Can (Refer to Example 8.) The volume

V of a cylindrical soda can is given by V = pr2h, where

r is its radius and h is its height.

(a) If r = 1.3 inches and h = 4.4 inches, find the volume of the can in cubic inches.

(b) Could this can hold 12 fluid ounces? (Hint: 1 cubic

inch equals about 0.55 fluid ounce.)

94. Volume of a Sphere The volume of a sphere is given by

V = 43pr3, where r is the radius of the sphere. Calculate

the volume if the radius is 3 feet. Approximate your

98. Suppose that a positive number a is written in scientific

notation as a = b * 10n, where n is an integer and

1 … b … 10. Explain what n indicates about the size of a.

EXTENDED AND DISCOVERY EXERCISE

1. If you have access to a scale that weighs in grams, find

the thickness of regular and heavy-duty aluminum foil.

Is heavy-duty foil worth the price difference? (Hint:

Each 2.7 grams of aluminum equals 1 cubic centimeter.)

1.2 Visualizing and Graphing Data

• Analyze one-variable data

• Find the domain and range

of a relation

• Graph in the xy-plane

• Calculate distance

• Find the midpoint

• Learn the standard

equation of a circle

• Learn to graph equations

with a calculator (optional)

Introduction

Technology is giving us access to huge amounts of data. For example, space telescopes,

such as the Hubble telescope, are providing a wealth of information about the universe.

The challenge is to convert the data into meaningful information that can be used to solve

important problems. Before conclusions can be drawn, data must be analyzed. A powerful

tool in this step is visualization, as pictures and graphs are often easier to understand than

words. This section discusses how different types of data can be visualized by using various mathematical techniques.

One-Variable Data

Data often occur in the form of a list. A list of test scores without names is an example;

the only variable is the score. Data of this type are referred to as one-variable data. If the

values in a list are unique, they can be represented visually on a number line.

Means and medians can be found for one-variable data sets. To calculate the mean (or

average) of a set of n numbers, we add the n numbers and then divide the sum by n. The

median is equal to the value that is located in the middle of a sorted list. If there is an odd

1.2 Visualizing and Graphing Data

13

number of data items, the median is the middle data item. If there is an even number of

data items, the median is the average of the two middle items.

EXAMPLE 1

Analyzing a list of data

Table 1.2 lists the monthly average temperatures in degrees Fahrenheit at Mould Bay,

Table 1.2

Monthly Average Temperatures at Mould Bay, Canada

Temperature (°F)

- 27

- 31

- 26

-9

12

32

39

36

21

1

-17

-24

Source: A. Miller and J. Thompson, Elements of Meteorology.

(a)

(b)

(c)

(d)

Plot these temperatures on a number line.

Find the maximum and minimum temperatures.

Determine the mean of these 12 temperatures.

Find the median and interpret the result.

SOLUTION

(a) In Figure 1.9 the numbers in Table 1.2 are plotted on a number line.

–40

–30

–20

–10

0

10

20

30

40

Figure 1.9 Monthly Average Temperatures

CLASS DISCUSSION

In Example 1(c), the mean of the

temperatures is approximately

0.6°F. Interpret this temperature.

(b) The maximum temperature of 39°F is plotted farthest to the right in Figure 1.9.

Similarly, the minimum temperature of - 31°F is plotted farthest to the left.

(c) The sum of the 12 temperatures in Table 1.2 equals 7. The mean, or average, of these

7

temperatures is 12

L 0.6°F.

(d) Because there is an even number of data items, the median is the average of the middle two values. From the number line we see that the middle two values are -9°F and

1°F. Thus the median is - 9 2+ 1 = - 4°F. This result means that half the months have

an average temperature that is greater than -4°F and half the months have an average

Now Try Exercises 1 and 5 ᭣

temperature that is below -4°F.

Two-Variable Data

Sometimes a relationship exists between two lists of data. Table 1.3 lists the monthly average precipitation in inches for Portland, Oregon. In this table, 1 corresponds to January, 2

to February, and so on, until 12 represents December. We show the relationship between a

month and its average precipitation by combining the two lists so that corresponding

months and precipitations are visually paired.

Table 1.3 Average Precipitation for Portland, Oregon

Month

Precipitation (inches)

1

2

3

4

5

6

7

8

9

10

11

12

6.2

3.9

3.6

2.3

2.0

1.5

0.5

1.1

1.6

3.1

5.2

6.4

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