Jay Abramson, OpenStax College – Algebra and Trigonometry (2015)

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Автор: Jay Abramson, OpenStax College
Название книги: Algebra and Trigonometry (2015)
Формат: PDF
Жанр: Математика
Страницы: 1566
Качество: Изначально компьютерное, E-book

OpenStax College
OpenStax College is a non-profit organization committed to improving student access to quality learning materials. Our free textbooks
are developed and peer-reviewed by educators to ensure they are readable, accurate, and meet the scope and sequence requirements
of modern college courses. Through our partnerships with companies and foundations committed to reducing costs for students,
OpenStax College is working to improve access to higher education for all.

Chapter 1: Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Real Numbers: Algebra Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Exponents and Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3 Radicals and Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.4 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1.5 Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.6 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 2: Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.1 The Rectangular Coordinate Systems and Graphs . . . . . . . . . . . . . . . . . . . . . . 108
2.2 Linear Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.3 Models and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.5 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
2.6 Other Types of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
2.7 Linear Inequalities and Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . 204
Chapter 3: Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.1 Functions and Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.2 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
3.3 Rates of Change and Behavior of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
3.4 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
3.5 Transformation of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
3.6 Absolute Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
3.7 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Chapter 4: Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
4.1 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
4.2 Modeling with Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
4.3 Fitting Linear Models to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Chapter 5: Polynomial and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
5.1 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
5.2 Power Functions and Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 499
5.3 Graphs of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
5.4 Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
5.5 Zeros of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
5.6 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
5.7 Inverses and Radical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
5.8 Modeling Using Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
Chapter 6: Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 643
6.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
6.2 Graphs of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
6.3 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
6.4 Graphs of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
6.5 Logarithmic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
6.6 Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
6.7 Exponential and Logarithmic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
6.8 Fitting Exponential Models to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774
Chapter 7: The Unit Circle: Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . 809
7.1 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810
7.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835
7.3 Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850
7.4 The Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
Chapter 8: Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
8.1 Graphs of the Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 902
8.2 Graphs of the Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 925
8.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
Chapter 9: Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . . . . . . 975
9.1 Solving Trigonometric Equations with Identities . . . . . . . . . . . . . . . . . . . . . . . . 976
9.2 Sum and Difference Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989
9.3 Double-Angle, Half-Angle, and Reduction Formulas . . . . . . . . . . . . . . . . . . . . . 1007
9.4 Sum-to-Product and Product-to-Sum Formulas . . . . . . . . . . . . . . . . . . . . . . . 1022
9.5 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031
Chapter 10: Further Applications of Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 1059
10.1 Non-right Triangles: Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060
10.2 Non-right Triangles: Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
10.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095
10.4 Polar Coordinates: Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111
10.5 Polar Form of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134
10.6 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148
10.7 Parametric Equations: Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164
10.8 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178
Chapter 11: Systems of Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1211
11.1 Systems of Linear Equations: Two Variables . . . . . . . . . . . . . . . . . . . . . . . . 1212
11.2 Systems of Linear Equations: Three Variables . . . . . . . . . . . . . . . . . . . . . . . 1233
11.3 Systems of Nonlinear Equations and Inequalities: Two Variables . . . . . . . . . . . . . 1246
11.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260
11.5 Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272
11.6 Solving Systems with Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . 1286
11.7 Solving Systems with Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301
11.8 Solving Systems with Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317
Chapter 12: Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341
12.1 The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342
12.2 The Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364
12.3 The Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386
12.4 Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404
12.5 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422
Chapter 13: Sequences, Probability, and Counting Theory . . . . . . . . . . . . . . . . . . . . 1445
13.1 Sequences and Their Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446
13.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463
13.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476
13.4 Series and Their Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487
13.5 Counting Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502
13.6 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514
13.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522
A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555

Welcome to Algebra and Trigonometry, an OpenStax College resource. This textbook has been created with several goals in
mind: accessibility, customization, and student engagement—all while encouraging students toward high levels of academic
scholarship. Instructors and students alike will find that this textbook offers a strong foundation in algebra and trigonometry
in an accessible format.
About OpenStax College
OpenStax College is a non-profit organization committed to improving student access to quality learning materials. Our
free textbooks go through a rigorous editorial publishing process. Our texts are developed and peer-reviewed by educators
to ensure they are readable, accurate, and meet the scope and sequence requirements of today’s college courses. Unlike
traditional textbooks, OpenStax College resources live online and are owned by the community of educators using them.
Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College is
working to improve access to higher education for all. OpenStax College is an initiative of Rice University and is made
possible through the generous support of several philanthropic foundations. OpenStax College textbooks are used at many
colleges and universities around the world. Please go to https://openstaxcollege.org/pages/adoptions to see our rapidly
expanding number of adoptions.
About OpenStax College’s Resources
OpenStax College resources provide quality academic instruction. Three key features set our materials apart from others:
they can be customized by instructors for each class, they are a “living” resource that grows online through contributions
from educators, and they are available free or for minimal cost.
Customization
OpenStax College learning resources are designed to be customized for each course. Our textbooks provide a solid
foundation on which instructors can build, and our resources are conceived and written with flexibility in mind. Instructors
can select the sections most relevant to their curricula and create a textbook that speaks directly to the needs of their classes
and student body. Teachers are encouraged to expand on existing examples by adding unique context via geographically
localized applications and topical connections.
Algebra and Trigonometry can be easily customized using our online platform (http://cnx.org/content/col11758/latest/).
Simply select the content most relevant to your current semester and create a textbook that speaks directly to the needs of
your class. Algebra and Trigonometry is organized as a collection of sections that can be rearranged, modified, and enhanced
through localized examples or to incorporate a specific theme to your course. This customization feature will ensure that
your textbook truly reflects the goals of your course.
Curation
To broaden access and encourage community curation, Algebra and Trigonometry is “open source” licensed under a
Creative Commons Attribution (CC-BY) license. The mathematics community is invited to submit feedback to enhance
and strengthen the material and keep it current and relevant for today’s students. Submit your suggestions to
info@openstaxcollege.org, and check in on edition status, alternate versions, errata, and news on the StaxDash at
http://openstaxcollege.org.
Cost
Our textbooks are available for free online, and in low-cost print and e-book editions.
About Algebra and Trigonometry
Written and reviewed by a team of highly experienced instructors, Algebra and Trigonometry provides a comprehensive
and multi-layered exploration of algebraic principles. The text is suitable for a typical introductory algebra course, and was
developed to be used flexibly. While the breadth of topics may go beyond what an instructor would cover, the modular
approach and the richness of content ensures that the book meets the needs of a variety of programs.
Algebra and Trigonometry guides and supports students with differing levels of preparation and experience with
mathematics. Ideas are presented as clearly as possible, and progress to more complex understandings with considerable reinforcement along the way. A wealth of examples – usually several dozen per chapter – offer detailed, conceptual
explanations, in order to build in students a strong, cumulative foundation in the material before asking them to apply what
they’ve learned.
Coverage and Scope
In determining the concepts, skills, and topics to cover, we engaged dozens of highly experienced instructors with a range of
student audiences. The resulting scope and sequence proceeds logically while allowing for a significant amount of flexibility
in instruction.
Chapters 1 and 2 provide both a review and foundation for study of Functions that begins in Chapter 3. The authors
recognize that while some institutions may find this material a prerequisite, other institutions have told us that they have a
cohort that need the prerequisite skills built into the course.
Chapter 1: Prerequisites
Chapter 2: Equations and Inequalities
Chapters 3-6: The Algebraic Functions
Chapter 3: Functions
Chapter 4: Linear Functions
Chapter 5: Polynomial and Rational Functions
Chapter 6: Exponential and Logarithm Functions
Chapters 7-10: A Study of Trigonometry
Chapter 7: The Unit Circle: Sine and Cosine Functions
Chapter 8: Periodic Functions
Chapter 9: Trigonometric Identities and Equations
Chapter 10: Further Applications of Trigonometry
Chapters 11-13: Further Study in Algebra and Trigonometry
Chapter 11: Systems of Equations and Inequalities
Chapter 12: Analytic Geometry
Chapter 13: Sequences, Probability, and Counting Theory
All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.
Development Overview
Openstax Algebra and Trigonometry is the product of a collaborative effort by a group of dedicated authors, editors, and
instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and voice.
Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision for
the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading numerous drafts,
and assimilating field reviews into actionable revision plans for our authors and editors.
The collective experience of our author team allowed us to pinpoint the subtopics, exceptions, and individual connections
that give students the most trouble. And so the textbook is replete with well-designed features and highlights, which help
students overcome these barriers. As the students read and practice, they are coached in methods of thinking through
problems and internalizing mathematical processes.
For example, narrative text is often followed with the “How To” feature, which summarizes the presentation into a series
of distinct steps. This approach addresses varying learning styles, and models for students an important learning skill for
future studies. Furthermore, the extensive graphical representations immediately connect concepts with visuals.

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Jay Abramson, OpenStax College - Algebra and Trigonometry (2015)

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