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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DEVELOPING YOUR SKILLS
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
steady decline as advertising
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).
Answers may vary.
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 ؍2r: The
formula for the circumference of a circle can be
written as a function of C in terms of r: C1r2 ϭ 2r.
(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
and Trademark Office (USPTO) receives thousands
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
amount of business
(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
steadily until
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
MAINTAINING YOUR SKILLS
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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• 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 .