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A Graphical Approach to
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Precalculus with Limits
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Hornsby Lial Rockswold
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ISBN 978-1-29204-080-6 5
e
9 781292 040806
Pearson New International Edition
A Graphical Approach to
Precalculus with Limits
Hornsby Lial Rockswold
Fifth Edition
International_PCL_TP.indd 1 7/29/13 11:23 AM
ISBN 10: 1-292-04080-7
ISBN 13: 978-1-292-04080-6
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ISBN 10: 1-292-04080-7
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-04080-6
ISBN 13: 978-1-269-37450-7
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A catalogue record for this book is available from the British Library
Printed in the United States of America
Copyright_Pg_7_24.indd 1 7/29/13 11:28 AM
123345667888551932396271959167014931
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Chapter 1 Linear Functions, Equations, and Inequalities
John Hornsby/Margaret Lial/Gary K. Rockswold 1
Chapter 2 Analysis of Graphs of Functions
John Hornsby/Margaret Lial/Gary K. Rockswold 81
Chapter 3 Polynomial Functions
John Hornsby/Margaret Lial/Gary K. Rockswold 159
Chapter 4 Rational, Power, and Root Functions
John Hornsby/Margaret Lial/Gary K. Rockswold 255
Chapter 5 Inverse, Exponential, and Logarithmic Functions
John Hornsby/Margaret Lial/Gary K. Rockswold 319
Chapter 6 Analytic Geometry
John Hornsby/Margaret Lial/Gary K. Rockswold 391
Chapter 7 Systems of Equations and Inequalities; Matrices
John Hornsby/Margaret Lial/Gary K. Rockswold 436
Chapter 8 The Unit Circle and the Functions of Trigonometry
John Hornsby/Margaret Lial/Gary K. Rockswold 527
Chapter 9 Trigonometric Identities and Equations
John Hornsby/Margaret Lial/Gary K. Rockswold 630
Chapter 10 Applications of Trigonometry and Vectors
John Hornsby/Margaret Lial/Gary K. Rockswold 691
Chapter 11 Further Topics in Algebra
John Hornsby/Margaret Lial/Gary K. Rockswold 764
Chapter 12 Limits, Derivatives, and Definite Integrals
John Hornsby/Margaret Lial/Gary K. Rockswold 829
Chapter R Reference: Basic Algebraic Concepts
John Hornsby/Margaret Lial/Gary K. Rockswold 873
I
999022723
Appendices
John Hornsby/Margaret Lial/Gary K. Rockswold 907
Photo Credits
John Hornsby/Margaret Lial/Gary K. Rockswold 922
Index 923
II
1
Linear Functions,
Equations, and Inequalities
DURING RECENTYEARS, temperatures have risen over much of the Arctic and
Antarctic regions. A sustained warming of 4°F to 11°F above average twentieth-
century temperatures could cause the Greenland ice sheet to melt. Sea levels
may rise as much as 25 feet. (See Exercise 57 in Section 1.6.)
This melting phenomenon is also occurring in the Antarctic, where the
Wilkins Ice Shelf is melting at an alarming rate. During the past 50 years,
the rate of warming at Antarctica has been about 0.9°F per decade. If the
entire Antarctic ice cap were to melt, sea levels could rise over 200 feet.
To bring attention to global warming, British adventurer and endurance
swimmer Lewis Gordon Pugh dove into a region of the Antarctic that used
to be covered by ice.
What will the future bring? Linear functions can help us make predic-
tions. In this chapter, we learn how to model many different phenomena with
linear functions and equations.
Source: National Snow and Ice Data Center, March 25, 2008.
CHAPTER OUTLINE
1.1 Real Numbers and the Rectangular Coordinate System
1.2 Introduction to Relations and Functions
1.3 Linear Functions
1.4 Equations of Lines and Linear Models
1.5 Linear Equations and Inequalities
1.6 Applications of Linear Functions
From Chapter 1 of A Graphical Approach to Precalculus with Limits: A Unit Circle Approach, Fifth Edition. John Hornsby,
Margaret Lial, Gark K. Rockswold. Copyright © 2011 by Pearson Education, Inc. All rights reserved.
(cid:20)
2 CHAPTER 1 Linear Functions, Equations, and Inequalities
Real Numbers and the Rectangular
1.1
Coordinate System
Sets of Real Numbers • The Rectangular Coordinate System • Viewing Windows • Approximations of Real Numbers
• Distance and Midpoint Formulas
Sets of Real Numbers
There are several important sets of numbers that are used in mathematics. Some of
these sets are listed in the following table.
Sets of Numbers
Set Description Examples
Natural Numbers 51, 2, 3, 4, Á6 1, 45, 127, 103
Whole Numbers 50, 1, 2, 3, 4, Á6 0, 86, 345, 23
Integers 5Á, -2, -1, 0, 1, 2, Á6 0, -5, -102, 99
Rational Numbers Epq|p and q are integers, q Z 0F 0, -56, -2,272, 0.5
Irrational Numbers 5x|x is not rational6 22,p, -237
Real Numbers 5x|x is a decimal number6 -26,p,2, 245, 0.41
3
Whole numbersinclude the natural numbers; integersinclude the whole num-
bers and the natural numbers. The result of dividing two integers (with a nonzero divi-
sor) is a rational number,orfraction. Rational numbers include the natural numbers,
Origin whole numbers, and integers. For example, the integer -3is a rational number because
-3
it can be written as .Every rational number can be written as a repeating or termi-
1
nating decimal. For example, 0.6 = 0.66666Árepresents the rational number 2.
–5–4 –3–2 –1 0 1 2 3 4 5 3
Numbers that can be written as decimal numbers are real numbers.Real numbers
Graph of {–3, –1, 0, 1, 3, 5}
include rational numbers and can be shown pictorially—that is, graphed—on a
FIGURE 1
number line. See FIGURE 1.The point corresponding to 0 is called the origin. Every
real number corresponds to one and only one point on the number line, and each point
corresponds to one and only one real number. This correspondence is called a
coordinate system.The number associated with a given point is called the coordinate
–5–4–3–2–1 0 1 2 3 4 5 of the point. The set of all real numbers is graphed in FIGURE 2.
Graph of the set of real numbers Some real numbers cannot be represented by quotients of integers or by repeating
FIGURE 2 or terminating decimals. These numbers are called irrational numbers.Examples of
irrational numbers include 23, 25, 23 10, and 25 20, but not 21, 24, 29, Á,
which equal 1, 2, 3, Á,and hence are rational numbers. If ais a natural number but
–23 √2 √5 (cid:2) 2ais not a natural number, then 2ais an irrational number. Another irrational num-
ber is p, which is approximately equal to 3.14159. In FIGURE 3the numbers in the set
–1Graph0 of – 1 2 , 0, 2√2,√5,3(cid:2),4 4 E-23, 0, 22, 25,p, 4F are located on a number line. Note that 22is approximately
3 equal to 1.41, so it is located between 1 and 2, slightly closer to 1.
FIGURE 3
The Rectangular Coordinate System
If we place two number lines at right angles, intersecting at their origins, we obtain a
two-dimensional rectangular coordinate system. The number lines intersect at the
(cid:21)
1.1 Real Numbers and the Rectangular Coordinate System 3
y-axis originof the system, designated 0. The horizontal line is called the x-axis,and the ver-
tical line is called the y-axis. On the x-axis, positive numbers are located to the right
Quadrant Quadrant of the origin, and negative numbers are located to the left. On the y-axis, positive num-
II I
bers are located above the origin, negative numbers below.
P(a, b) b
This rectangular coordinate system is also called the Cartesian coordinate system,
x-axis
a 0 Origin named after René Descartes (1596–1650). The plane into which the coordinate system
is introduced is the coordinate plane, or xy-plane.The x-axis and y-axis divide the
Quadrant Quadrant
III IV plane into four regions, or quadrants,as shown in FIGURE 4.The points on the x-axis
or y-axis belong to no quadrant.
Rectangular coordinate system
Each point P in the xy-plane corresponds to a unique ordered pair (a,b) of real
FIGURE 4
numbers. We call a the x-coordinate and b the y-coordinate of point P. The point P
y corresponding to the ordered pair (a,b)is often written as P(a,b), as in FIGURE 4,and
referred to as “the point (a,b).” FIGURE 5illustrates how to plot the point A(3, 4). Addi-
B(–5, 6) A(3, 4) tional points are labeled B–E. The coordinates of the origin are (0, 0).
4
x
E(–3, 0)0 3
Viewing Windows
D(4, –3)
C(–2, –4) The rectangular (Cartesian) coordinate system extends indefinitely in all directions.
We can show only a portion of such a system in a text figure. Similar limitations occur
Plotting points in the xy-plane
with the viewing “window” on a calculator screen. FIGURE 6shows a calculator screen
FIGURE 5 that has been set to have a minimum x-value of -10,a maximum x-value of 10, a min-
imumy-value of -10,and a maximum y-value of 10. The tick marks on the axes have
10
been set to be 1 unit apart. Thus, there are 10 tick marks on the positive x-axis. This
window is called the standard viewing window.
To convey information about a viewing window, we use the following abbreviations:
–10 10
Xmin: minimum value of x Ymin: minimum value of y
Xmax: maximum value of x Ymax: maximum value of y
–10 Xscl: scale (distance between Yscl: scale (distance between
Standard viewing window
tick marks) on the x-axis tick marks) on the y-axis.
FIGURE 6
To further condense this information, we use the following symbolism, which gives
viewing information for the window in FIGURE 6.
GCMTECHNOLOGY NOTE
Xmax Ymin
You should consult the graphing Xmin Ymax
calculator manual that accompanies 3-10, 104 by 3-10, 104.
this text or your owner’s manual to Xscl = 1 Yscl = 1
see how to set the viewing window
on your screen. Remember that FIGURE 7 shows several other viewing windows. Notice that FIGURES 7(b) and 7(c)
different settings will result in look exactly alike, and unless we are told what the settings are, we have no way of dis-
different views of graphs. tinguishing between them. What are Xscl and Yscl in each figure?
3.1 10 100
–4.7 4.7 –10 10 –100 100
–3.1 –10 –100
[–4.7, 4.7] by [–3.1, 3.1] [–10, 10] by [–10, 10] [–100, 100] by [–100, 100]
(a) (b) (c)
FIGURE 7
(cid:22)
4 CHAPTER 1 Linear Functions, Equations, and Inequalities
WHAT WENT WRONG?
A student learning how to use a graphing calculator could not understand why the axes
on the graph were so “thick,” as seen in FIGURE A,while those on a friend’s calculator
were not, as seen in FIGURE B.
40 40
–60 60 –60 60
–40 –40
Xscl = 1, Yscl = 1 Xscl = 10, Yscl = 10
FIGURE A FIGURE B
What Went Wrong? How can the student correct the problem in FIGURE Aso that the
axes look like those in FIGURE B?
Approximations of Real Numbers
Although calculators have the capability to express numbers like 22, 23 5,and pto
many decimal places, we often ask that answers be rounded. The following table
reviews rounding numbers to the nearest tenth, hundredth, or thousandth.
TI-84 Plus
Rounding Numbers
FIGURE 8
Number Nearest Tenth Nearest Hundredth Nearest Thousandth
1.3782 1.4 1.38 1.378
201.6666 201.7 201.67 201.667
0.0819 0.1 0.08 0.082
FIGURE 9 InFIGURE 8,the TI-84 Plus graphing calculator is set to round values to the near-
est hundredth (two decimal places). In FIGURE 9,the numbers from the preceding table
are rounded to the nearest hundredth.
The symbol L indicates that two expressions are approximately equal.For exam-
ple, p L 3.14,but p Z 3.14,since p = 3.141592654.Á When using pin calcula-
tions, be sure to use the built-in key for prather than 3.14. See FIGURE 10.
FIGURE 10
Answer to What Went Wrong?
Since Xscl = 1and Yscl = 1in FIGURE A,there are 120 tick marks along the x-axis and 80 tick marks along the
y-axis. The resolution of the graphing calculator screen is not high enough to show all these tick marks, so the
axes appear as heavy black lines instead. The values for Xscl and Yscl need to be larger, as in FIGURE B.
(cid:23)
1.1 Real Numbers and the Rectangular Coordinate System 5
GCM EXAMPLE 1 Finding Roots on a Calculator
Approximate each root to the nearest thousandth. (Note: You can use the fact that
2n a = a1/nto find roots.)
(a) 223 (b) 23 87 (c) 24 12
Solution
(a) The screen in FIGURE 11(a) shows that an approximation for 223, to the nearest
thousandth, is 4.796. It is displayed twice, once for 223and once for 231/2.
(b) To the nearest thousandth, 23 87 L 4.431.See FIGURE 11(b).
(c) FIGURE 11(c)indicates 24 12 L 1.861in three different ways.
TECHNOLOGY NOTE In all the screens, note the inclusion of parentheses.
Many graphing calculators have
built-in keys for calculating square
roots and menus for calculating
other types of roots.
(a) (b) (c)
FIGURE 11
EXAMPLE 2 Approximating Expressions with a Calculator
Approximate each expression to the nearest hundredth.
3.8 - 1.4
(a) (b) 3p4 - 92 (c) 2(4 - 1)2 + (-3 - 2)2
5.4 + 3.5
Solution
TECHNOLOGY NOTE (a) SeeFIGURE 12(a).To the nearest hundredth,
Some graphing calculators display 3.8 - 1.4
L 0.27.
leading zeros in decimal numbers, 5.4 + 3.5
whereas others do not. For
example, 1might be displayed as (b) Many calculators also have a special key to calculate the square of a number. To
either 0.254 or .25. In this text, the nearest hundredth, 3p4 - 92 L 211.23.See FIGURE 12(b).
graphing calculator screens do not (c) From FIGURE 12(c), 2(4 - 1)2 + (-3 - 2)2 L 5.83.
usually show leading zeros. See
FIGURES 9and12(a).
Do not confuse the negation
and subtraction symbols.
Insert parentheses
around both the
numerator and the
denominator.
(a) (b) (c)
Use parentheses
FIGURE 12
carefully.
(cid:24)
