Algebra and Trigonometry

1008 Pages·2013·65.573 MB·English

by  Beecher, Judith A;Penna, Judith A;Bittinger, Marvin L

Pearson New International Edition

Algebra and Trigonometry
Judith A. Beecher Judith A. Penna
Marvin L. Bittinger
Fourth Edition
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ISBN 10: 1-292-04020-3
ISBN 13: 978-1-292-04020-2
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Basic Concepts of Algebra
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 1
1. Graphs, Functions, and Models
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 61
2. More on Functions
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 165
3. Quadratic Functions and Equations; Inequalities
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 249
4. Polynomial Functions and Rational Functions
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 315
5. Exponential Functions and Logarithmic Functions
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 417
6. The Trigonometric Functions
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 515
7. Trigonometric Identities, Inverse Functions, and Equations
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 631
8. Applications of Trigonometry
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 705
9. Systems of Equations and Matrices
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 799
10. Analytic Geometry Topics
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 899
Useful Mathematical References
Judith A. Beecher/Judith A. Penna/Marvin L. Bittinger 989
Index 995
I
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Basic Concepts
of Algebra
Phovoir/Jupiter Images Yellowdog Productions/Jupiter Images
1 The Real-Number
System
2 Integer Exponents,
Scientific Notation,
and Order of
Operations
3 Addition, Subtraction,
and Multiplication
ofPolynomials
4 Factoring
5 The Basics of
e Equation Solving
p
u
Application Lo 6 Rational Expressions
y
M
e/es 7 Radical Notation and
ug
rdma Rational Exponents
It is estimated that there were b Pack I Study Guide
51.2 billion pieces oftrash on 76 million mi of BoSto Review Exercises
U.S.roadways in a recent year (Source: Keep America Beautiful).
Test
On average,how many pieces oftrash were on each mile ofroadway?
This problem appears as Exercise 79 in Section 2.
From Chapter R ofAlgebra and Trigonometry,Fourth Edition,Judith A.Beecher,Judith A.Penna and Marvin L.Bittinger.
Copyright © 2012 by Pearson Education,Inc.All rights reserved.
1
Basic Concepts of Algebra
The Real-Number System
3
Identify various kinds of real numbers.
1
3
Use interval notation to write a set of numbers.
3
Identify the properties of real numbers.
3
Find the absolute value of a real number.
3
Real Numbers
In applications of algebraic concepts, we use real numbers to represent
quantities such as distance,time,speed,area,profit,loss,and temperature.
Some frequently used sets of real numbers and the relationships among
them are shown below.
Natural numbers
(positive integers):
1, 2, 3, ...
Whole numbers:
0, 1, 2, 3, ...
Integers:
..., (cid:2)3, (cid:2)2, (cid:2)1, 0, Zero: 0
1, 2, 3, ...
Rational Negative integers:
numbers (cid:2)1, (cid:2)2, (cid:2)3, ...
Rational numbers
Real that are not integers:
numbers (cid:2)2, (cid:2)(cid:2)4, (cid:2)19, (cid:2)(cid:2)7, 8.3,
3 5 (cid:2)5 8
0.56, ...
Irrational numbers:
π,(cid:2)√53,√427,
(cid:2)4.030030003..., ...
Numbers that can be expressed in the form p q,where pand qare inte-
>
gers and q Z 0,are rational numbers.Decimal notation for rational num-
bers either terminates(ends) or repeats.Each ofthe following is a rational
number.
0
a) 0 0 (cid:3) for any nonzero integer a
a
(cid:2)7 7
b) -7 (cid:2)7 (cid:3) ,or
1 (cid:2)1
1
c) = 0.25 Terminating decimal
4
5
d) - = -0.454545Á = -0.45 Repeating decimal
11
5
e) = 0.8333Á = 0.83 Repeating decimal
6
2
Basic Concepts of Algebra
The real numbers that are not rational are irrational numbers.Decimal
notation for irrational numbers neither terminates nor repeats.Each ofthe
following is an irrational number.
a) p = 3.1415926535Á There is no repeating block ofdigits.
22and 3.14 are rational approximationsofthe irrational number p.
A7 B
b) 22 = 1.414213562Á There is no repeating block ofdigits.
c) -6.12122122212222Á Although there is a pattern,there is no
repeating block ofdigits.
The set ofall rational numbers combined with the set ofall irrational
numbers gives us the set of real numbers.The real numbers are modeled
using a number line,as shown below.
Each point on the line represents a real number,and every real number
is represented by a point on the line.
(cid:2)2.9 (cid:2)E (cid:2)3 p *
(cid:2)5 (cid:2)4 (cid:2)3 (cid:2)2 (cid:2)1 0 1 2 3 4 5
The order ofthe real numbers can be determined from the number line.
Ifa number ais to the left ofa number b,then ais less thanb a 6 b .Sim-
1 2
ilarly,ais greater thanb a 7 b ifais to the right ofbon the number line.
1 2
For example,we see from the number line above that -2.9 6 -3,because
5
-2.9is to the left of -3.Also,17 7 13,because 17is to the right of 13.
5 4 4
The statement a … b,read “ais less than or equal to b,”is true ifeither
a 6 bis true or a = bis true.
The symbol(cid:5)is used to indicate that a member,or element,belongs to
a set.Thus ifwe let (cid:3)represent the set ofrational numbers,we can see from
the diagram on the previous page that 0.56(cid:5)(cid:3).We can also write 12 (cid:4)(cid:3)to
indicate that 12is notan element ofthe set ofrational numbers.
When allthe elements of one set are elements of a second set,we say
that the first set is a subsetofthe second set.The symbol8is used to denote
this.For instance,if we let (cid:2)represent the set of real numbers,we can see
from the diagram that (cid:3)8(cid:2)(read “(cid:3)is a subset of(cid:2)”).
3
Interval Notation
Sets of real numbers can be expressed using interval notation.For exam-
ple,for real numbers aand bsuch that a 6 b,the open interval a, b is
1 2
the set ofreal numbers between,but not including,aand b.That is,
( ) a, b = {x|a 6 x 6 b}.
1 2
a (a, b) b
The points aand bare endpointsofthe interval.The parentheses indicate
that the endpoints are not included in the interval.
Some intervals extend without bound in one or both directions.The
q
interval a, ,for example,begins at aand extends to the right without
3 2
bound,that is,
[
a, q = {x|x Ú a}.
a [a, ∞) 3 2
The bracket indicates that ais included in the interval.
3
Basic Concepts of Algebra
The various types ofintervals are listed below.
Intervals: Types, Notation, and Graphs
INTERVAL SET
TYPE NOTATION NOTATION GRAPH
Open a, b x|a 6 x 6 b ( )
1 2 5 6
a b
Closed a,b x|a … x … b [ ]
3 4 5 6
a b
Half-open a, b x|a … x 6 b [ )
3 2 5 6
a b
Half-open a, b x|a 6 x … b ( ]
1 4 5 6
a b
Open a, q x|x 7 a (
1 2 5 6
a
Half-open a, q x|x Ú a [
3 2 5 6
a
Open -q, b x|x 6 b )
1 2 5 6
b
Half-open -q, b x|x … b ]
1 4 5 6
b
The interval -q, q , graphed below, names the set of all real
1 2
numbers,(cid:2).
EXAMPLE 1 Write interval notation for each set and graph the set.
a) x|-4 6 x 6 5 b) x|x Ú 1.7
5 6 5 6
c) x|-5 6 x … -2 d) x|x 6 15
5 6 5 6
Solution
a) x|-4 6 x 6 5 = -4, 5 ;
5 6 1 2
(cid:2)5(cid:2)4(cid:2)3(cid:2)2(cid:2)1 0 1 2 3 4 5
b) x|x Ú 1.7 = 1.7, q ;
5 6 3 2
(cid:2)5(cid:2)4(cid:2)3(cid:2)2(cid:2)1 0 1 2 3 4 5
c) x|-5 6 x … -2 = -5, -2 ;
5 6 1 4
(cid:2)5(cid:2)4(cid:2)3(cid:2)2(cid:2)1 0 1 2 3 4 5
d) x|x 6 25 = - q, 25 ;
E F A B
(cid:2)5(cid:2)4(cid:2)3(cid:2)2(cid:2)1 0 1 2 3 4 5
k
Now Try Exercises 13 and 15.
4
Basic Concepts of Algebra
3
Properties of the Real Numbers
The following properties can be used to manipulate algebraic expressions as
well as real numbers.
Properties of the Real Numbers
For any real numbers a,b,and c:
a + b = b + aand Commutative properties of
ab = ba addition and multiplication
a + b + c = a + b + cand Associative properties of
1 2 1 2
a bc = ab c addition and multiplication
1 2 1 2
a + 0 = 0 + a = a Additive identity property
-a + a = a + -a = 0 Additive inverse property
# # 1 2
a 1 = 1 a = a Multiplicative identity property
#1 1#
a = a = 1 a Z 0 Multiplicative inverse property
a a 1 2
a b + c = ab + ac Distributive property
1 2
Note that the distributive property is also true for subtraction since
a b - c = a b + -c = ab + a -c = ab - ac.
1 2 3 1 24 1 2
EXAMPLE 2 State the property being illustrated in each sentence.
# #
a) 8 5 = 5 8 b) 5#+ m +# n = 5 + m + n
c) 14 + -14 = 0 d) 6 1 1= 1 6 =2 6 1 2
1 2
e) 2 a - b = 2a - 2b
1 2
Solution
SENTENCE PROPERTY
# #
a) 8 5 = 5 8 Commutative property of
multiplication:ab = ba
b) 5 + m + n = 5 + m + n Associative property ofaddition:
1 2 1 2
a + b + c = a + b + c
1 2 1 2
c) 14 + -14 = 0 Additive inverse property:
1 2
# # a + -a = 0
1 2
d) 6 1 = 1 6 = 6 M#ultiplica#tive identity property:
a 1 = 1 a = a
e) 2 a - b = 2a - 2b Distributive property:
1 2
a b + c = ab + ac
1 2
k
Now Try Exercises 49 and 55.
5