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E. Linear Regression and the Line of Best Fit

# E. Linear Regression and the Line of Best Fit

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the desired option and press . Note the data in

L1 ranges from 76 to 108, while the data in L2

ranges from 221 to 232. This means an appropriate viewing window might be [70, 120] for the

x-values, and [210, 240] for the y-values. Press

the

key and set up the window accordingly.

After you’re finished, pressing the GRAPH key

should produce the graph shown in Figure 1.99.

Figure 1.99

ENTER

WINDOW

240

70

120

Step 4: Calculate the Regression Equation

210

To have the calculator compute the regression

equation, press the STAT and

keys to move

the cursor over to the CALC options (see Figure 1.100). Since it appears the data is best modeled by a linear equation, we choose option

4:LinReg(ax ؉ b). Pressing the number 4

places this option on the home screen, and

pressing

computes the values of a and b (the

calculator automatically uses the values in L1

and L2 unless instructed otherwise). Rounded to

hundredths, the linear regression model is

y ϭ Ϫ0.33x ϩ 257.06 (Figure 1.101).

Figure 1.100

ENTER

Figure 1.101

Step 5: Display and Use the Results

Although graphing calculators have the ability

to paste the regression equation directly into

Y1 on the Y= screen, for now we’ll enter

Y1 ϭ Ϫ0.33x ϩ 257.06 by hand. Afterward,

pressing the GRAPH key will plot the data points (if

Plot1 is still active) and graph the line. Your

display screen should now look like the one in

Figure 1.102. The regression line is the best estimator for the set of data as a whole, but there will

still be some difference between the values it

generates and the values from the set of raw data

(the output in Figure 1.102 shows the estimated

time for the 2000 Olympics was about 224 sec,

when actually it was the year Ian Thorpe of

Australia set a world record of 221 sec).

EXAMPLE 6

Figure 1.102

240

70

210

Using Regression to Model Employee Performance

Riverside Electronics reviews employee performance

semiannually, and awards increases in their hourly rate

of pay based on the review. The table shows Thomas’

hourly wage for the last 4 yr (eight reviews). Find the

regression equation for the data and use it to project his

hourly wage for the year 2011, after his fourteenth

review.

Solution

120

Following the prescribed sequence produces the

equation y ϭ 0.48x ϩ 9.09. For x ϭ 14 we obtain

y ϭ 0.481142 ϩ 9.09 or a wage of \$15.81. According to this

model, Thomas will be earning \$15.81 per hour in 2011.

Review (x)

Wage ( y)

(2004) 1

\$9.58

2

\$9.75

(2005) 3

\$10.54

4

\$11.41

(2006) 5

\$11.60

6

\$11.91

(2007) 7

\$12.11

8

\$13.02

Now try Exercises 27 through 34

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171

With each linear regression, the calculator can be set to compute a correlation coefficient that is a measure of how well the equation fits the data (see Subsection C). To

0

display this “r-value” use 2nd

(CATALOG) and activate DiagnosticOn. Figure 1.103 shows a scatterplot with perfect negative correlation 1r ϭ Ϫ12 and notice all

data points are on the line. Figure 1.104 shows a strong positive correlation 1r Ϸ 0.982

for the data from Example 6. See Exercise 35.

Figure 1.103

Figure 1.104

E. You’ve just seen how

we can use a linear regression

to find the line of best fit

1.6 EXERCISES

CONCEPTS AND VOCABULARY

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

1. When the ordered pairs from a set of data are

plotted on a coordinate grid, the result is called a

.

2. If the data points seem to form a curved pattern or

if no pattern is apparent, the data is said to have a

association.

3. If the data points seems to cluster along an

imaginary line, the data is said to have a

association.

4. If the pattern of data points seems to increase as

they are viewed left to right, the data is said to have

a

association.

5. Compare/Contrast: One scatterplot is linear, with a

weak and positive association. Another is linear,

with a strong and negative association. Give a

written description of each scatterplot.

6. Discuss/Explain how this is possible: Working

from the same scatterplot, Demetrius obtained the

equation y ϭ Ϫ0.64x ϩ 44 for his equation model,

while Jessie got the equation y ϭ Ϫ0.59x ϩ 42.

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7. For mail with a high priority,

x

“Express Mail” offers next day

1981

delivery by 12:00 noon to most

1985

destinations, 365 days of the

1988

year. The service was first

offered by the U.S. Postal

1991

Service in the early 1980s and

1995

has been growing in use ever

1999

since. The cost of the service

2002

(in cents) for selected years is

shown in the table. (a) Draw a

2010

scatterplot of the data, then

(b) decide if the association is

positive, negative, or cannot be determined.

y

935

1075

1200

1395

1500

1575

1785

1830

Source: 2004 Statistical Abstract of the United States; USPS.com

8. After the Surgeon General’s

x

y

first warning in 1964,

1965

42.4

cigarette consumption began a

1974

37.1

was banned from television

1979

33.5

and radio, and public

1985

29.9

awareness of the dangers of

1990

25.3

cigarette smoking grew. The

1995

24.6

percentage of the U.S. adult

population who considered

2000

23.1

themselves smokers is shown

2002

22.4

in the table for selected years.

2005

16.9

(a) Draw a scatterplot of the

data, then (b) decide if the

association is positive, negative, or cannot be

determined.

Source: 1998 Wall Street Journal Almanac and 2009 Statistical Abstract of the

United States, Table 1299

9. Since the 1970s women have

x

y

made tremendous gains in the

1972

32

political arena, with more and

1978

46

more female candidates running

1984

65

for, and winning seats in the

U.S. Senate and U.S. Congress.

1992

106

The number of women

1998

121

candidates for the U.S. Congress

2004

141

is shown in the table for selected

years. (a) Draw a scatterplot of the data, (b) decide if

the association is linear or nonlinear and (c) if the

association is positive, negative, or cannot be

determined.

Source: Center for American Women and Politics at

www.cawp.rutgers.edu/Facts3.html

10. The number of shares traded on the New York

Stock Exchange experienced dramatic change in

the 1990s as more and more individual investors

gained access to the stock market via the Internet

and online brokerage houses.

The volume is shown in the

table for 2002, and the odd

numbered years from 1991 to

2001 (in billions of shares).

(a) Draw a scatterplot of the

data, (b) decide if the

association is linear or

nonlinear, and (c) if the

association is positive,

negative, or cannot be determined.

x

y

1991

46

1993

67

1995

88

1997

134

1999

206

2001

311

2002

369

Source: 2000 and 2004 Statistical Abstract of the United States, Table 1202

The data sets in Exercises 11 and 12 are known

to be linear.

11. The total value of the goods

x

and services produced by a

(1970 S 0)

y

nation is called its gross

0

5.1

domestic product or GDP. The

5

7.6

GDP per capita is the ratio of

10

12.3

the GDP for a given year to the

15

17.7

population that year, and is

one of many indicators of

20

23.3

economic health. The GDP per

25

27.7

capita (in \$1000s) for the

30

35.0

United States is shown in the

33

37.8

table for selected years.

(a) Draw a scatterplot using

scales that appropriately fit the data, then sketch an

estimated line of best fit, (b) decide if the

association is positive or negative, then (c) decide

whether the correlation is weak or strong.

Source: 2004 Statistical Abstract of the United States, Tables 2 and 641

12. Real estate brokers carefully

Price

Sales

track sales of new homes

130’s

126

looking for trends in location,

150’s

95

price, size, and other factors.

170’s

103

The table relates the average

selling price within a price

190’s

75

range (homes in the \$120,000

210’s

44

to \$140,000 range are

230’s

59

represented by the \$130,000

250’s

21

figure), to the number of new

homes sold by Homestead Realty in 2004.

(a) Draw a scatterplot using scales that

appropriately fit the data, then sketch an estimated

line of best fit, (b) decide if the association is

positive or negative, then (c) decide whether the

correlation is weak or strong.

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For the scatterplots given: (a) Arrange them in order

from the weakest to the strongest correlation, (b) sketch

a line that seems to approximate the data, (c) state

whether the association is positive, negative, or cannot

be determined, and (d) choose two points on (or near)

the line and use them to approximate its slope (rounded

to one decimal place).

13. A.

y

60

55

50

45

40

35

30

25

B.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

C.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

D.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

14. A.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

B.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

C.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

For the scatterplots given, (a) determine whether a linear

or nonlinear model would seem more appropriate.

(b) Determine if the association is positive or negative.

(c) Classify the correlation as weak or strong. (d) If

linear, sketch a line that seems to approximate the data

and choose two points on the line and use them to

approximate its slope.

15.

y

60

55

50

45

40

35

30

25

16.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

17.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

0 1 2 3 4 5 6 7 8 9 x

18.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

20.

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

21. In most areas of the country,

x

y

law enforcement has become

(1990→0) (1000s)

a major concern. The number

3

68.8

of law enforcement officers

6

74.5

employed by the federal

8

83.1

government and having the

authority to carry firearms

10

88.5

and make arrests is shown in

14

93.4

the table for selected years.

(a) Draw a scatterplot using scales that

appropriately fit the data and sketch an estimated

line of best fit and (b) decide if the association is

positive or negative. (c) Choose two points on or

near the estimated line of best fit, and use them to

find a function model and predict the number of

federal law enforcement officers in 1995 and the

projected number for 2011. Answers may vary.

Source: U.S. Bureau of Justice, Statistics at www.ojp.usdoj.gov/bjs/fedle.htm

0 1 2 3 4 5 6 7 8 9 x

D.

19.

173

y

60

55

50

45

40

35

30

25

0 1 2 3 4 5 6 7 8 9 x

22. Due to atmospheric pressure,

x

y

the temperature at which

Ϫ1000

213.8

water will boil varies

0

212.0

predictably with the altitude.

Using special equipment

1000

210.2

designed to duplicate

2000

208.4

atmospheric pressure, a lab

3000

206.5

experiment is set up to study

4000

204.7

this relationship for altitudes

5000

202.9

up to 8000 ft. The set of data

6000

201.0

collected is shown in the table,

with the boiling temperature y

7000

199.2

in degrees Fahrenheit,

8000

197.4

depending on the altitude x in

feet. (a) Draw a scatterplot using scales that

appropriately fit the data and sketch an estimated

line of best fit, (b) decide if the association is

positive or negative. (c) Choose two points on or

near the estimated line of best fit, and use them to

find a function model and predict the boiling point

of water on the summit of Mt. Hood in Washington

State (11,239 ft height), and along the shore of the

Dead Sea (approximately 1312 ft below sea level).

23. For the data given in Exercise 11 (Gross Domestic

Product per Capita), choose two points on or near

the line you sketched and use them to find a function

model for the data. Based on this model, what is the

projected GDP per capita for the year 2010?

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24. For the data given in Exercise 12 (Sales by Real

Estate Brokers), choose two points on or near the

line you sketched and use them to find a function

model for the data. Based on this model, how many

sales can be expected for homes costing \$275,000?

\$300,000?

WORKING WITH FORMULAS

25. Circumference of a Circle: C ‫ ؍‬2␲r: The

formula for the circumference of a circle can be

written as a function of C in terms of r: C1r2 ϭ 2␲r.

(a) Set up a table of values for r ϭ 1 through 6 and

draw a scatterplot of the data. (b) Is the association

positive or negative? Why? (c) What can you say

about the strength of the correlation? (d) Sketch a

line that “approximates” the data. What can you say

about the slope of this line?

26. Volume of a Cylinder: V ‫␲ ؍‬r2h: As part of a

project, students cut a long piece of PVC pipe with

a diameter of 10 cm into sections that are 5, 10,

15, 20, and 25 cm long. The bottom of each is then

made watertight and each section is filled to the

1–90

CHAPTER 1 Relations, Functions, and Graphs

brim with water. The volume

Height Volume

is then measured using a flask

(cm)

(cm3)

marked in cm3 and the results

5

380

collected into the table

shown. (a) Draw a scatterplot

10

800

of the data. (b) Is the

15

1190

association positive or

20

1550

negative? Why? (c) What can

25

1955

you say about the strength of

the correlation? (d) Would the

correlation here be stronger or weaker than the

correlation in Exercise 25? Why? (e) Run a linear

regression to verify your response.

APPLICATIONS

Use the regression capabilities of a graphing calculator to complete Exercises 27 through 34.

27. Height versus wingspan: Leonardo da Vinci’s

famous diagram is an illustration of how the human

body comes in predictable proportions. One such

comparison is a person’s height to their wingspan

(the maximum distance from the outstretched

tip of one middle finger to the other). Careful

measurements were taken on eight students and the

set of data is shown here. Using the data, (a) draw

the scatterplot; (b) determine whether the

association is linear or nonlinear; (c) determine

whether the association is positive or negative; and

(d) find the regression equation and use it to predict

the wingspan of a student with a height of 65 in.

Height (x)

Wingspan ( y)

28. Patent applications: Every year the U.S. Patent

of applications from scientists and inventors. The

table given shows the number of applications

received for the odd years from 1993 to 2003

(1990 S 0). Use the data to (a) draw the scatterplot;

(b) determine whether the association is linear or

nonlinear; (c) determine whether the association is

positive or negative; and (d) find the regression

equation and use it to predict the number of

applications that will be received in 2011.

Source: United States Patent and Trademark Office at www.uspto.gov/web

Year

(1990 S 0)

Applications

(1000s)

61

60.5

3

188.0

61.5

62.5

5

236.7

54.5

54.5

7

237.0

73

71.5

9

278.3

67.5

66

11

344.7

51

50.75

13

355.4

57.5

54

52

51.5

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Section 1.6 Linear Function Models and Real Data

29. Patents issued: An

Year

Patents

increase in the

(1990 S 0)

(1000s)

number of patent

3

107.3

applications (see

Exercise 28), typically

5

114.2

brings an increase in

7

122.9

the number of patents

9

159.2

issued, though many

11

187.8

applications are

denied due to

13

189.6

improper filing, lack

of scientific support, and other reasons. The table

given shows the number of patents issued for the

odd years from 1993 to 2003 (1999 S 0). Use the

data to (a) draw the scatterplot; (b) determine

whether the association is linear or nonlinear;

(c) determine whether the association is positive or

negative; and (d) find the regression equation and

use it to predict the number of applications that

will be approved in 2011. Which is increasing

faster, the number of patent applications or the

number of patents issued? How can you tell

for sure?

Source: United States Patent and Trademark Office at www.uspto.gov/web

30. High jump records: In the

Year

Height

sport of track and field, the

(1900S 0)

in.

high jumper is an unusual

0

75

athlete. They seem to defy

gravity as they launch their

12

76

bodies over the high bar.

24

78

The winning height at the

36

80

summer Olympics (to the

56

84

nearest unit) has steadily

increased over time, as

68

88

shown in the table for

80

93

selected years. Using the

88

94

data, (a) draw the

92

92

scatterplot, (b) determine

whether the association

96

94

is linear or nonlinear,

100

93

(c) determine whether the

104

association is positive or

108

negative, and (d) find the

regression equation using

t ϭ 0 corresponding to 1900 and predict the

winning height for the 2004 and 2008 Olympics.

How close did the model come to the actual

heights?

Source: athens2004.com

31. Females/males in the workforce: Over the last

4 decades, the percentage of the female population

in the workforce has been increasing at a fairly

steady rate. At the same time, the percentage of the

male population in the workforce has been declining.

The set of data is shown in the tables. Using the data,

(a) draw scatterplots for both data sets, (b) determine

whether the associations are linear or nonlinear,

(c) determine whether the associations are positive or

negative, and (d) determine if the percentage of

females in the workforce is increasing faster than the

percentage of males is decreasing. Discuss/Explain

how you can tell for sure.

Source: 1998 Wall Street Journal Almanac, p. 316

Exercise 31 (women)

Exercise 31 (men)

Year (x)

(1950 S 0)

Percent

Year (x)

(1950 S 0)

Percent

5

36

5

85

10

38

10

83

15

39

15

81

20

43

20

80

25

46

25

78

30

52

30

77

35

55

35

76

40

58

40

76

45

59

45

75

50

60

50

73

32. Height versus male shoe

Height

Shoe Size

size: While it seems

66

8

reasonable that taller

people should have larger

69

10

feet, there is actually a

72

9

wide variation in the

75

14

relationship between

74

12

height and shoe size. The

data in the table show the

73

10.5

height (in inches)

71

10

compared to the shoe size

69.5

11.5

worn for a random sample

66.5

8.5

of 12 male chemistry

students. Using the data,

73

11

(a) draw the scatterplot,

75

14

(b) determine whether the

65.5

9

association is linear or

nonlinear, (c) determine

whether the association is positive or negative, and

(d) find the regression equation and use it to predict

the shoe size of a man 80 in. tall and another that is

60 in. tall. Note that the heights of these two men

fall outside of the range of our data set (see

comment after Example 5 on page 168).

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33. Plastic money: The total

x

(1990 S 0)

y

transacted using credit

1

481

cards has been changing

2

539

rapidly over the last 15 to

20 years. The total

4

731

volume (in billions of

7

1080

dollars) is shown in the

8

1157

table for selected years.

9

1291

(a) Use a graphing

calculator to draw a

10

1458

scatterplot of the data

12

1638

and decide whether the

association is linear or nonlinear. (b) Calculate a

regression equation with x ϭ 0 corresponding to

1990 and display the scatterplot and graph on the

same screen. (c) According to the equation model,

how many billions of dollars were transacted in

2003? How much will be transacted in the

year 2011?

Source: Statistical Abstract of the United States, various years

34. Sales of hybrid

Year

Hybrid Sales

cars: Since their

(2000 S 0)

(in thousands)

mass introduction

2

35

near the turn of

the century, the

3

48

sales of hybrid

4

88

cars in the United

5

200

States grew

6

250

late 2007, when

7

352

the price of

8

313

gasoline began

9

292

showing signs of

weakening and eventually dipped below \$3.00/gal.

Estimates for the annual sales of hybrid cars are

given in the table for the years 2002 through 2009

12000 S 02 . (a) Use a graphing calculator to draw a

scatterplot of the data and decide if the association

is linear or nonlinear. (b) If linear, calculate a

regression model for the data and display the

scatterplot and data on the same screen.

(c) Assuming that sales of hybrid cars recover,

how many hybrids does the model project will be

sold in the year 2012?

Source: http://www.hybridcar.com

EXTENDING THE CONCEPT

35. It can be very misleading to

x

y

rely on the correlation

50

67

coefficient alone when

100

125

selecting a regression model.

150

145

To illustrate, (a) run a linear

regression on the data set

200

275

given (without doing a

250

370

scatterplot), and note the

300

550

strength of the correlation

350

600

(the correlation coefficient).

(b) Now run a quadratic

regression ( STAT CALC 5:QuadReg) and note the

strength of the correlation. (c) What do you

notice? What factors other than the correlation

coefficient must be taken into account when

choosing a form of regression?

36. In his book Gulliver’s Travels, Jonathan Swift

describes how the Lilliputians were able to

measure Gulliver for new clothes, even though he

was a giant compared to them. According to the

text, “Then they measured my right thumb, and

desired no more . . . for by mathematical

computation, once around the thumb is twice

around the wrist, and so on to the neck and waist.”

Is it true that once around the neck is twice around

the waist? Find at least 10 willing subjects and take

measurements of their necks and waists in

millimeters. Arrange the data in ordered pair form

(circumference of neck, circumference of waist).

Draw the scatterplot for this data. Does the

association appear to be linear? Find the equation

of the best fit line for this set of data. What is the

slope of this line? Is the slope near m ϭ 2?

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Making Connections

37. (1.3) Is the graph shown here, the graph of a

function? Discuss why or why not.

38. (R.2/R.3) Determine

the area of the figure

shown

1A ϭ LW, A ϭ ␲r2 2.

18 cm

24 cm

39. (1.5) Solve for r: A ϭ P ϩ Prt

40. (R.3) Solve for w (if possible):

Ϫ216w2 ϩ 52 Ϫ 1 ϭ 7w Ϫ 413w2 ϩ 12

MAKING CONNECTIONS

Making Connections: Graphically, Symbolically, Numerically, and Verbally

Eight graphs (a) through (h) are given. Match the characteristics shown in 1 through 16 to one of the eight graphs.

y

(a)

Ϫ5

y

(b)

5

Ϫ5

5 x

y

5 x

Ϫ5

5 x

Ϫ5

1

1. ____ y ϭ x ϩ 1

3

2. ____ y ϭ Ϫx ϩ 1

3. ____ m 7 0, b 6 0

4. ____ x ϭ Ϫ1

Ϫ5

5 x

5 x

Ϫ5

y

(g)

5

Ϫ5

5

Ϫ5

y

(f)

5

Ϫ5

Ϫ5

5 x

y

(d)

5

Ϫ5

Ϫ5

(e)

y

(c)

5

Ϫ5

y

(h)

5

5 x

5

Ϫ5

Ϫ5

5 x

Ϫ5

9. ____ f 1Ϫ32 ϭ 4, f 112 ϭ 0

10. ____ f 1Ϫ42 ϭ 3, f 142 ϭ 3

11. ____ f 1x2 Ն 0 for x ʦ 3Ϫ3, q2

12. ____ x ϭ 3

5. ____ y ϭ Ϫ2

13. ____ f 1x2 Յ 0 for x ʦ 31, q 2

6. ____ m 6 0, b 6 0

14. ____ m is zero

7. ____ m ϭ Ϫ2

15. ____ function is increasing, y-intercept is negative

8. ____ m ϭ

2

3

16. ____ function is decreasing, y-intercept is negative

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1–94

SUMMARY AND CONCEPT REVIEW

SECTION 1.1

Rectangular Coordinates; Graphing Circles and Other Relations

KEY CONCEPTS

• A relation is a collection of ordered pairs (x, y) and can be stated as a set or in equation form.

• As a set of ordered pairs, we say the relation is pointwise-defined. The domain of the relation is the set of all first

coordinates, and the range is the set of all corresponding second coordinates.

• A relation can be expressed in mapping notation x S y, indicating an element from the domain is mapped to

(corresponds to or is associated with) an element from the range.

• The graph of a relation in equation form is the set of all ordered pairs (x, y) that satisfy the equation. We plot a

sufficient number of points and connect them with a straight line or smooth curve, depending on the pattern

formed.

• The x- and y-variables of linear equations and their graphs have implied exponents of 1.

• With a relation entered on the Y= screen, a graphing calculator can provide a table of ordered pairs and the

related graph.

x1 ϩ x2 y1 ϩ y2

,

b.

• The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is a

2

2

• The distance between the points (x1, y1) and (x2, y2) is d ϭ 21x2 Ϫ x1 2 2 ϩ 1y2 Ϫ y1 2 2.

• The equation of a circle centered at (h, k) with radius r is 1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2.

EXERCISES

1. Represent the relation in mapping notation, then state the domain and range.

51Ϫ7, 32, 1Ϫ4, Ϫ22, 15, 12, 1Ϫ7, 02, 13, Ϫ22, 10, 82 6

2. Graph the relation y ϭ 225 Ϫ x2 by completing the table, then state the domain and range of the relation.

x

Ϫ5

Ϫ4

Ϫ2

0

2

4

5

y

3. Use a graphing calculator to graph the relation 5x ϩ 3y ϭ Ϫ15. Then use the TABLE feature to determine the

value of y when x ϭ 0, and the value(s) of x when y ϭ 0, and write the results in ordered pair form.

Mr. Northeast and Mr. Southwest live in Coordinate County and are good friends. Mr. Northeast lives at 19 East and

25 North or (19, 25), while Mr. Southwest lives at 14 West and 31 South or (Ϫ14, Ϫ31). If the streets in Coordinate

County are laid out in one mile squares,

4. Use the distance formula to find how far apart they live.

5. If they agree to meet halfway between their homes, what are the coordinates of their meeting place?

6. Sketch the graph of x2 ϩ y2 ϭ 16.

7. Sketch the graph of x2 ϩ y2 ϩ 6x ϩ 4y ϩ 9 ϭ 0.

8. Find an equation of the circle whose diameter has the endpoints (Ϫ3, 0) and (0, 4).

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Summary and Concept Review

179

Linear Equations and Rates of Change

SECTION 1.2

KEY CONCEPTS

• A linear equation can be written in the form ax ϩ by ϭ c, where a and b are not simultaneously equal to 0.

y2 Ϫ y1

• The slope of the line through (x1, y1) and (x2, y2) is m ϭ x Ϫ x , where x1 x2.

2

1

¢y

vertical change

change in y

rise

ϭ

ϭ

.

• Other designations for slope are m ϭ run

change in x

Âx

horizontal change

Lines with positive slope 1m 7 02 rise from left to right; lines with negative slope (m 6 0) fall from left to right.

• The equation of a horizontal line is y ϭ k; the slope is m ϭ 0.

• The equation of a vertical line is x ϭ h; the slope is undefined.

• Lines can be graphed using the intercept method. First determine (x, 0) (substitute 0 for y and solve for x), then

(0, y) (substitute 0 for x and solve for y). Then draw a straight line through these points.

Parallel

lines have equal slopes (m1 ϭ m2); perpendicular lines have slopes that are negative reciprocals

1

(m1 or m1 # m2 1).

m2

EXERCISES

Ây

rise

9. Plot the points and determine the slope, then use the ratio

ϭ

to find an additional point on the line:

run

¢x

a. 1Ϫ4, 32 and 15, Ϫ22 and b. (3, 4) and 1Ϫ6, 12.

10. Use the slope formula to determine if lines L1 and L2 are parallel, perpendicular, or neither:

a. L1: 1Ϫ2, 02 and (0, 6); L2: (1, 8) and (0, 5)

b. L1: (1, 10) and 1Ϫ1, 72 : L2: 1Ϫ2, Ϫ12 and 11, Ϫ32

11. Graph each equation by plotting points: (a) y ϭ 3x Ϫ 2

(b) y ϭ Ϫ32x ϩ 1.

12. Find the intercepts for each line and sketch the graph: (a) 2x ϩ 3y ϭ 6

(b) y ϭ 43x Ϫ 2.

13. Identify each line as either horizontal, vertical, or neither, and graph each line.

a. x ϭ 5

b. y ϭ Ϫ4

c. 2y ϩ x ϭ 5

14. Determine if the triangle with the vertices given is a right triangle: 1Ϫ5, Ϫ42, (7, 2), (0, 16).

15. Find the slope and y-intercept of the line shown and discuss the slope ratio in this context.

Hawk population (100s)

10

y

8

6

4

2

2

4

6

x

8

Rodent population

(1000s)

SECTION 1.3

Functions, Function Notation, and the Graph of a Function

KEY CONCEPTS

• A function is a relation, rule, or equation that pairs each element from the domain with exactly one element of the

range.

• The vertical line test says that if every vertical line crosses the graph of a relation in at most one point, the relation

is a function.

• The domain and range can be stated using set notation, graphed on a number line, or expressed using interval

notation.

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CHAPTER 1 Relations, Functions, and Graphs

• On a graph, vertical boundary lines can be used to identify the domain, or the set of “allowable inputs” for a

function.

On a graph, horizontal boundary lines can be used to identify the range, or the set of y-values (outputs) generated

by the function.

When a function is stated as an equation, the implied domain is the set of x-values that yield real number outputs.

x-values that cause a denominator of zero or that cause the radicand of a square root expression to be negative

must be excluded from the domain.

The phrase “y is a function of x,” is written as y ϭ f 1x2 . This notation enables us to summarize the three most

important aspects of a function with a single expression (input, sequence of operations, output).

EXERCISES

16. State the implied domain of each function:

a. f 1x2 ϭ 24x ϩ 5

b. g1x2 ϭ

17. Determine h1Ϫ22, h1Ϫ23 2 , and h(3a) for h1x2 ϭ 2x2 Ϫ 3x.

xϪ4

x ϪxϪ6

2

18. Determine if the mapping given represents a function. If not, explain how the definition of a function is violated.

Mythological

deities

Primary

concern

Apollo

Jupiter

Ares

Neptune

Mercury

Venus

Ceres

Mars

messenger

war

craftsman

love and beauty

music and healing

oceans

all things

agriculture

19. For the graph of each function shown, (a) state the domain and range, (b) find the value of f (2), and (c) determine

the value(s) of x for which f 1x2 ϭ 1.

I.

II.

y

5

Ϫ5

5 x

Ϫ5

SECTION 1.4

III.

y

5

Ϫ5

5 x

Ϫ5

y

5

Ϫ5

5 x

Ϫ5

Linear Functions, Special Forms, and More on Rates of Change

KEY CONCEPTS

• The equation of a nonvertical line in slope-intercept form is y ϭ mx ϩ b or f 1x2 ϭ mx ϩ b. The slope of the line

is m and the y-intercept is (0, b).

Ây

to

To graph a line given its equation in slope-intercept form, plot the y-intercept, then use the slope ratio m ϭ

¢x

find a second point, and draw a line through these points.

• If the slope m and a point (x1, y1) on the line are known, the equation of the line can be written in point-slope

form: y Ϫ y1 ϭ m1x Ϫ x1 2 .

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