package depends on its weight. The probability that you and another person in a room share the ...

138 Pages·2010·44.75 MB·English

m m HE COST OF MAILING A
package depends on its
weight. The probability that
you and another person in a
room share the same birthday depends
on the number of people in the room. In
E- _-^ - both these situations, the relationship
between variables can be described by a
function. understanding this concept will
give you a new perspective on many
ordinary situations.
'TIST HE SEASON AND YOU'VE
waited until the last minute to
mail your holiday gifts.Your only
option is overnight express mail.
You realize that the cost of mailing
a gift depends on its weight. but the
mailing costs seem somewhat odd.
Your packages that weigh 1.1 pounds,
1.5 pounds, and 2 pounds cost $15.75
each to send overnight. Packages that
weigh 2.01 pounds and 3 pounds cost
you $18.50 each. Finally, your heaviest
gift is barely over 3 pounds and its
mailing cost is $21.25. What sort of
system is this in which costs increase
by $2.75, stepping from $15.75 to
$18.50 and from $18.50 to $21.25?
Graphs that ascend in steps
are discussed on page 162 in
Section 1.3.
128 Chapter 1 Functions and Graphs
'lot points in the
rectangular coordinate
system.
Graph equations in the
rectangular coordinate
system.
Interpret information
about a graphing utility's
viewing rectangle or table.
Use a graph to determine
intercepts.
Interpret information given The beginning of the .. . ..)teenth century was a time of innovative ideas and enor-
mous intellectual progress in Europe. English theatergoers enjoyed a succession of
by graphs.
exciting new plays by Shakespeare. William Harvey proposed the radical notion that
the heart was a pump for blood rather than the center of emotion. Galileo. with his
new-fangled invention called the telescope, supported the theory of Polish
astronomer Copernicus that the sun, not the Earth, was the center of the solar
system. Monteverdi was writing the world's first grand operas. French mathemati-
cians Pascal and Fermat invented a new field of mathematics called probability theory.
Into this arena of intellectual electricity stepped French aristocrat Rent5 Descartes
(1596-1650). Descartes (pronounced "day cart"), propelled by the creativity surround-
ing him, developed a new branch of mathematics that brought together algebra and
geometry in a unified way-a way that visualized numbers as points on a graph, equa-
tions as geometric figures, and geometric figures as equatiomThis new branch of math-
ematics, called analytic geometry, established Descartes as one of the founders of
modem thought and among the most original mathematicians and philosophers of any
age. We begin this section by looking at Descartes's deceptively simple idea, called the
rectangular coordinate system or (in his honor) the Cartesian coordinate system.
0 Plot points in the rectangular
coordinate system. Point~andO rdered Pairs'
. , .
Descartes used two number lines that intersect at right angles at their zero points, as
shown in Figure 1.1. The horizontal number line is thex-axis.The vertical number line
is the y-axis. The point of intersection of these axes is their zero points, called the
origin. Positive numbers are shown to the right and above the origin. Negative num-
bers are shown to the left and below the origin. The axes divide the plane into four
quarters,called quadrants.The points located on the axes+ are not in any quadrant.
Each point in the rectangular coordinate system corresponds to an ordered
pair of real numbers, (x, y). Examples of such pairs are (-5,3) and (3, -5). The
first number in each pair, called the x-coordinate, denotes the distance and direction
from the origin along the x-axis. The second num-
ber in each pair, called the y-coordinate, denotes
vertical distance and direction along a line parallel
to the y-axis or along the y-axis itself.
(-5.3)
Figure 1.2 shows how we plot, or locate, the
points corresponding to the ordered pairs (-5.3)
aton dth (e3 l,e -f5t )a.l oWnge thpelo.rt- a(x-5is..3T)h ebny wgeo ignog 35 uunniittss ufpro pma r0- -5-4-3-2-h - ..
The phrase ordered pc allel to the y-axis. We plot (3. -5) by going 3 units
because order is important. The from 0 to the right along the x-axis and 5 units down
order in which coordinates parallel to the y-axis. The phrase "the points corre-
appear makes a difference in a spending to the ordered pairs (-5,3) and (3, -5)" -5 (3.-5)'
point's location.This is
is often abbreviated as "the points (-5.3) : t t o ~ fpl,,,ti~ine (-5.3) and
illustrated in Figure 1.2.
and (3. -5):' (3. -5)
.
129
- Section 1.1 Graphs and Graphing Utilities
Plotting Points in the Rectangular Coordinate System
he points: A(-3.5). B(2. -4), C(5.0 ),D (-5. -3). E(0,4) and F(O.0).
Solution See Figure 1.3. We move from the A1-3, 51 b
origin and plot the points in the Collowi~
Â
way:
A(-3.5): 3 units left, 5 units up
B(2, -4): 2 units right, 4 units down
C(5.0): 5 units right,O units up or down
-2
D(-5, -3): 5 units left. 3 units down
.
E(O, 4): 0 units right or left. 4 units up ,,
01-5, -31 -5
F(O.O): 0 units right or left, 0 units up '712. -4)
or down Plotting points
Reminder: Answers to all Check Paint
axnreisei are given in the mwar sootion.
Cheok your answer befon eontindng your Plot the points: A[- 2.4), B(4, -2). C(-3. O), and D(O, -3).
reading to verify that you andantand
tha concept.
Q Graph equations in the Graphs of Equations
rectangular coordinate
A relationship between two quantities can be expressed as an equation in two
system
variables, such as
A solution of an equation in two variables,.a and y, is an ordered pair of real num-
bers with the following property: When the r-coordinate is substituted for x and the
y-coordinate is substituted for y in the equation, we obtain a true statement. For
example,consider the equation y = 4 - x2 and the ordered pair (3, -5). When 3 is
substituted for A and -5 is substituted for y, we obtain the statement -5 = 4 - 3',
or -5 = 4 9, or -5 = -5 Because this statement is true, the ordered pair
-
(3,- 5) is a solution of the equation y = 4 - x2. We also say that (3. -5) satisfies
the equation.
We can generate as many ordered-pair solutions as desired to y = 4 - x2 by
substituting numbers lor x and then finding the corresponding values for y. For
example. suppose we let x = 3:
Slid with A tompute y.
The graph of an equation in two variables is the set of all points who'i e
coordinates satisfy the equation. One method for graphing such equations is the
point-plotting method. First, we find several ordered pairs that are solutions of
the equation. Next, we plot these ordered pairs as points in the rectangular coordi-
nate system. Finally, we connect the points with a smooth curve or line. This often
gives us a picture of all ordered pairs that satisfy the equation.
1
130 Chapter 1 Functions and Graphs
' 1
I1
Graphing an Equation Using
r the Point-Plotting Method
4 - .r2. Select integers for x, starting with -3 and ending with 3.
L . . For each value of x, we find the corresponding value for y.
Now we plot the seven points and join them with a
smooth curve, as shown in Figure 1.4. The graph of
y = 4 - .r2 is a curve where the part of the graph to
the right of the y-axis is a reflection of the part to the
left of it and vice versa. The arrows on the left and
x
the right of the curve indicate that it extends indefi-
nitely in both directions.
-2
. .
uf2
Graph y = 4 - x. Select integers for r. starting with -3 and ending with 3.
Graphing an Equation Using
the Point-Plotting Method
Graph y = 1x1. Select integers for x, starting with -3 and ending with 3.
Solution For each value of ,i,w e find the corresponding value for y.
We plot the points and connect
them, resulting in the graph
shown in Figure 1.5. The graph is
V-shaped and centered a1 the ori-
gin. For every point (.r, y) on the
graph, the point (-.v. y) is also
on the graph.This shows that the
absolute value of a positive num-
Figure IS The graph of ber is the same as the absolute
? = 1x1 value of its opposite.
586
+
Graph y = 1.t 11. Select integers tor,r.starting with -4 andending with 2.
131
Section 1.1 Graphs and Graphing Utilities
Interpret information about a Graphing Equations and Creating Tables
graphing utility's viewing Using a Graphing Utility
rectangle or table
Graphing calculators or graphing software packages for computers are referred to as
graphing utilities or graphers. A graphing utility is a powerful tool that quickly gen-
erates the graph of an equation in two variables. Figures 1.6(a) and 1.6(b) show two
such eraphs for the equations in Examples 2 and 3.
The graph of
The graph of
What differences do you notice between these graphs and the graphs that we
drew by hand? They do seem a bit "jittery." Arrows do not appear on the left and right
ends of the graphs. Furthermore, numbers are not given along the axes. For both graphs
in Figure 1.6, the.\-axis extends from -10 lo 10 and the y-axis also extends from -10 to
10.The distance represented by each consecutive tick mark is one unit We say that the
viewing rectangle,or the viewing window, is [-lo, 10,ll by [-lo, 10,1].
u if you are not using a
I graphing utility in the course,
read this part of the section.
Knowing about viewing rectan-
The minimm The maximum Diitanca batwefin The mintmum The maximum Distance b~lwnen
gles will enable you to under-
x-valua along r-valm along conieeutive tick y-vahe along y-value along eonieoulive tick
stand thegraphs that we display thex-axis is thex-axis is marks on !he !he y-axis is the y-axis is marks on lhn
in the technology boxes through- -10. 10. x-axis is DMu nit. -10. 10. y-axis is one unit.
the bonk.
-"+
To graph an equation in x and y using a graphing utility, enter the equation and
specify the size of the viewing rectangle. The size of the viewing rectangle sets minimum
and maximum values for both the x- and y-axesEnter these valnes,as well as the values
between consecutive tickmarks.on the respective axes.The [-lo, 10,1] by [-lo, 10,1]
viewing rectangle used in Figure 1.6 is called the standard viewing rectangle.
Understanding the Viewing Rectangle
What is the meaning of a [-2,3,0.5] by [-lo, 20,5] viewing rectangle?
Solution We begin with [-2,3,0.5]. which describes the x-axis. The minimum
x-value is -2 and the maximum x-value is 3.The distance between consecutive tick
marks is 0.5.
Next, consider [-lo, 20, 51, which describes the y-axis. The minimum y-value is
-10 and the maximumy-value is 20 The distance between consecutive tick marks is 5.
Figure 1.7 illustrates a [-2,3,0.5] by [-lo, 20,5] viewing rectangle.To make
things clearer, we've placed numbers by each tick mark. These numbers do not
appear on the axes when you use a graphing utility to graph an equation.
A [-2,3.0.5] by
. ] viewing rectangle
-1
What is the meaning of a [-loo, 100,501 by [-loo. 10C. viewing
rectangle? Create a figure like the one in Figure 1.7 that illustrates this
viewing rectangle.
On most graphing utilities, the display screen is two-thirds as high as it is wide. By
using a square setting, you can equally space the x and y tick marks. (This does not
occur in the standard viewing rectangle.) Graphing utilities can also zoom in and zoom
out. When you zoom in. you see a smaller portion of the graph. but you do so in greater
detail. When you zoom out, you see a larger portion of the graph.Thus, zooming out
132 Chapter 1 Functions and Graphs
may help you to develop a better understanding of the overall character of the graph.
With practice, you will become more comfortable with gmphing equations in two vari-
ables using your graphing utility. You will also develop a better sense of the size of the
viewing rectangle that will reveal needed information about a particular graph.
Graphing utilities can also be used to create tables showing solutions of equa-
tions in two variables. Use the Table Setup function to choose the starting value of x
and to input the increment, or change, between the consecutive .r-values. The
corresponding y-values are calculated based on the equation(s) in two variables in the
screen. In Figure 1.8,w e used a TI-83 Plus to create a table for y = 4 - x2 and
1.~1,
y = the equations in Examples 2 and 3.
We entered two aqatloni:
-
Wt antirid -3 far tho stirtint
H,Ic +E s.-.+d --711 -value and I as the increment
I
butwacn x-voluii.
.
ctlunii, risptlittl~.A now hyip irail
~rolliiÃth ra.~h tha tibia to fiod other
1
,: -I$ I?= I $ x-values and oorreiponding y-values.
q,,@
T8 Creating a table for
\x\
y, = 4 - .r' and )$ =
0 Intercepts
A a graph to determine
intercepts. An x-intercept of a graph is thex-coordinate of a point where the graph intersects the
x-axis. For example, look at the graph of y = 4 x2 in Figure 1.9. The graph cross-
J' -
4 es the x-axis at (-2.0) and (2.0). Thus, the x-intercepts are -2 and 2.
y-intercept: 4
The y-coordinate corresponding to an x-intercept is always zero.
A y-intercept of a graph is the y-coordinate of a point where the graph inter-
sects the y-axis.The graph of != 4 - .v2 in Figure 1.9 shows that the graph crosses
the y-axis at (0,4). Thus, the y-intercept is 4. The x-coordinate corresponding to a
y-intercept is always zero.
, Study Tip
Mathematicians tend to use two ways to describe intercepts. id you notice that we are
using single numbers'? If a is an x-intercept of a graph, then the graph passes through the
point (a,0 ). If b is a y-intercept of a graph, then the graph passes through the point (0, b}.
*,*-, 3. Intercepts of y = 4 - .? Some hooks state that Lhe x-intercept is the point (a, 0) and the s-intercept is at a on
thex-axis Similarly, the !-intercept is the point (0, b) and the y-intercept is at b on the y-axis.
In these descriptions, the intercepts are the actual points where the graph intersects the axes.
Although we'll describe intercepts as single numbers, we'll immediately state the point
on the x- or }'-axis that the graph passes through. Here's the important thing to keep in mind:
,v-intercep1:The corresponding value of y is 0.
yin1ercept:The corresponding value of A- is 0.
-
~
Identifying Intercepts
Ident ex- a y-intercepts.
a. b.
t'
Section 1.1 Graphs and Graphing Utilities 133
Solution
a. The graph crosses the x-axis at (-1,O). Thus, the x-intercept is -1.The graph
crosses the y-aris at (0,2). Thus, the y-intercept is 2.
b. The graph crosses the x-axis at (3, O), so the x-intercept is 3.This vertical line
does not cross the y-axis.Thus, there is no y-intercept.
c. This graph crosses the x- and y-axes at the same po-int, the o rigin. Because the
graph crosses both axes at (0,0), the x-intercept is 0 and A- ., :-*-----* ;* 1.
Identify the .1- and y-intercepts.
c. V
x x
Figure 1.10 illustrates that a graph may have no intercepts or several intercepts.
No x-intercept One -(Â¥-intercep No intercepts One T-mlercepi The samex-mtercepi
figure 1.10 One y-intercept No y-intercept Threey-intercepts andy-intercept
Interpret information given Interpreting Information Given by Graphs
by graphs.
Line graphs are often used to illustrate trends over time. Some measure of time, such
as months or years,frequently appears on the horizontal axis.Amounts are generally
listed on the vertical axis. Points are drawn to represent the given information. The
graph is formed by connecting the points with line segments.
Figure 1.1 1 is an example of a typical line graph.The graph shows the average age
at which women in the United States married for the first time from 1890 through2003.
The years are listed on the horizontal axis and the ages are listed on the vertical axis.
The symbol+on the vertical axis shows that there is a break in values between 0 and 20.
Thus, the fist tick mark on the vertical axis represents an average age of 20.
Women%A verage Age of First Marriage
r
26
,
t
I I I I I I I 1 I I 1 1
1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Year
Lwws Bureau 2003
134 Chapter 1 Functions and Graphs
Women's Average Age A line graph displays information in the first quadrant of a rectangular
of Firsi Marriage coordinate system. By identifying points on line graphs and their coordinates, you
can interpret specific information given by the graph.
For example, the red lines in Figure 1.11 show how to find the average age at
which women married for the first time in 1980.
Step 1 Locate 1980 on the horizontal axis.
Step 2 Locate the point on the line graph above 1980.
Step 3 Read across to the corresponding age on the vertical axis.
The age appears to be exactly at 22.Thus. in 1980, women in the United States mar-
ried for the first time at an average age of 22.
FigUrti 1.11 (partly repeated) iterpreting Information Given by a Graph
line graph Figure 1.12 Percentage of US. Federal Prisoners
snows the percentage of federal Sentenced for Drug Offenses
prisoners in the United States
sentenced for drug offenses
from 1970 through 2003.
a. For the period shown,
estimate the maximum
Table 1.1 Number of U.S. percentage of federal 3 l
30%
un+ial Prisoners prisoners sentenced for
drug offenses. When did
Federal this occur? 0-
10%
Table 1.1 shows the
number, in thousands, of
1 federal prisoners in the year
89,538
United States for four W W
2000 133.9;
Selected years, ptimaÂt¥eSà §'w:Pm Schmalltger,CrimÈlçiJt~tTlm tor. 7th Edition,
159,Tra Prentice Hall. 2003.
J the number of federal
r
Source: Bureau ofJ astii- _ itics prisoners sentenced for drug offenses for H19hwt point
the year in part (a). 70% on graph
Solution
L
a. The maximum percentage of federal pris-
40%
oners sentenced for drug offenses can be
found by locating the highest point on the 30%
graph. This point lies above 1995 on the Paint Un
horizontal axis. Read across to the corre- lhlà 1115.
spending percent on the vertical axis. The 10%
number falls below 60% by
approximately one unit. It appears that I980 1985 1990 1995 2000 2003
59% is a reasonable estimate. Thus, the Year (1 980-2003)
maximum percentage of federal prisoners sentenced for drug offenses is
approximately 59%.This occurred in 1995.
b. Table 1.1 shows that there were 89,538 federal prisoners in 1995.The number
sentenced for drug offenses can be determined as follows:
Number sentenced
= 0.59 X 89,538.
for drug offenses
To estimate the number sentenced for drug offenses, we round the percent to
60% and the total federal prison population to 90,000.
Number sentenced for drug offenses s= 60% of 90,000
= 0.6 X 90,000
= 54,000
-
Section 1.1 Graphs and Graphing Utilities 135
In 1995, approximately 54.000 federal prisoners were sentenced for drug
offenses.
Use Figure 1.12 and Table 1.1 to estimate the number of federal prisoners
"entenced for drug offenses in 2003.
SET 1.
5. Which equalio~ Is to Y->i n the lab1
a. y2=x+8 b. y,=.r-2
n Exercises 1-12, plot the given point in a rectangular
coordinate system. c.yi=ZÑ. d. y, = 1 - 2.r
34. Which equation corresponds to Yl in the table?
1. (1,4) 2. (2.5) 3. (-2.3)
a. v\ = -3x b. yl = .r2
c. yl = -xL d. y, = 2 - .r
35. Does the graph of Y2 pass through the origin?
36. Does the graph ofYl pass through the origin?
Graph each equation in Exercises 13-28. Let x = -3, -2, -1,0, 37. At which point does the graph of Y2 cross the.r-axis?
I,2 , and 3. 38. At which point does the graph of Y2 cross the y-axis?
+
13. y = .r2 - 2 14. y = ,r2 2 15. y = x - 2 39. At which points do the graphs of Y, and Y; intersect?
16. y = .r + 2 17. y = 2.1 + 1 18. y = 2.t - 4
19. y = -\x 20. y = -,.rI + 2 21. y = 21x1 40. For which values ofx is Yl = Y,?
1.~1 +
22. y = -21x1 23. y = 1 24. y = 1.rI - 1
In Exercises 41-46, use the graph to a. determine the
25. y = 9 - x2 26. y = -x2 27. y = ,r3
x-inrercepts, if any; b. determine the y-intercepts, if any. For each
28. y = x3 - I graph, tick marks along the axes represent one unit each.
In Exercises 29-32, match the viewing rectangle with the correct 41. s 42.
figure. Then label the tick marks in the figure to illustrate this
viewing rectangle.
29. [-5.5, 11 by [-5,5, 11 30. [-10,10,2] by [-4.4.21
31. [-20.80,10] by [-30,70. 101
32. [-40,40,20; by [-1000,1000.100]
The table of values was generated by a graphing utility with a
TABLEf eature. Use the table to solve Exercises 33-40.
1
136 Chapter 1 Functions and Graphs
Practice Plus where A is the person's age. Use these mathematical models to
solve Exercises 61-62.
In Exercises 47-50, write each English sentence as an
equation in two variables. Then graph the equation. f
1
- -
47. The y-value is four more than twice the A'-value. .WS- 170
+. 0
1; dl 48. .TKh-ev aylu-ev.a lue is the difference between four and twice the Qs .Ea 160-
1
49. The y-value is three decreased by the square of the .K-value.
50. The y-value is two more than the square of the x-value.
11
In Exercises 51-54, graph each equation. 30Â .aBa2 110 -
51. y = 5 (Let .1 = -3, -2, -1,0, 1.2, and 3.)
52. y = -1 (Let x = -3, -2, -1,O. 1.2. and 3.)
1 1 1 1 1 61. What is the desirable heart rate during exercise for a 40-year-
(Let .r = -2 -1 -- -- - - 1, and 2.) old man^ Identify your computation as an appropriate point
I ' ' 2' 3'3'2'
.I1 on the blue graph.
:ation Exercises 62. What is the desirable heart rate during exercise for a &-year-
old woman? Identify your computation as an appropriate
 ¥ -
We live in an era of democratic aspiration. The number point on the red graph.
of democracies worldwide is on the rise. The line graph -
Autism is a neuroloeical disorder that impedes lan- mee and derails
shows the number of democracies worldwide, in four-yearperi- social and emotiona"l development. New findings suggest that the
oh,f rom 1973 through 2001, including2002. Use the graph to condition is not a sudden calamity that strikes children at the age of
solve Exewes 55-69.
2 or3, but a developmentalproblem linked to abnormally rapid
I brain growth during infancy The graphs show that the heads of
Tnracking Democracy: Number
severely autistic children start out smaller than average and then go
of Democracies Worldwide
!- through a period of explosive growth Exercises 63-64 involve
130 mathematical models for the data shown by the graphs.
Developmental Differences between
Hea"l thy Children and Severe Autisfics
Age (months)
Source.The Freedom House
Source The Jozmnd of dreAmerican MedicalAssocialio!t
63. The data for one of the two groups shown by the graphs can
55. Find an estimate for the number of democracies in 1989.
be modeled by
56. How many more democracies were there in 2002 than in 1973? y = 2.9& + 36,
57. In which four-year period did the number of democracies
where y is the head circumference, in centimeters, at age x
increase at the greatest rate?
months, 0 5 A' 5 14.
58. In which four-year period did the number of democracies
a. According to the model, what is the head circumference
increase at the slowest rate?
at birth?
59. In which year were there 49 democracies?
b. According to the model. what is the head circumfereuce
60. In which year were there 110 democracies? at 9 mouths?
I
Medical researchers have found that the desirable heart rate. R, in c According to the model, what is the head circumference
beats per minute, for beneficial exercise is approximated by the at 14 months? Use a calculator and round to the nearest
mathematical models tenth of a centimeter.
d. Use the values that you obtained m parts (a) through (c)
and the graphs shown above to determine whether the
given model describes healthy children or severe autistics.