Algebra ii all-in-one for dummies 1st edition mary jane sterling 2024 scribd download
Algebra
For Dummies 1st Edition Mary Jane Sterling
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Library of Congress Control Number: 2022940643
ISBN 978-1-119-89626-5 (pbk); ISBN 978-1-119-89627-2 (ebk); ISBN 978-1-119-89628-9 (ebk)
Contents at a Glance
Part 1: Getting to First Base with the Basics
CHAPTER 1: Beginning at the Beginning of Algebra
CHAPTER 2: Taking on Linear Equations and Inequalities
CHAPTER 3: Handling Quadratic and Other Polynomial Equations
CHAPTER 4: Controlling Quadratic and Rational Inequalities
CHAPTER 5: Soothing the Rational, the Radical, and the Negative
CHAPTER 6: Giving Graphing a Gander
Part 2: Figuring on Functions
CHAPTER 7: Formulating Functions
CHAPTER 8: Specializing in Quadratic Functions
CHAPTER 9: Plugging In Polynomials
CHAPTER 10: Acting Rationally with Functions
CHAPTER 11: Exploring Exponential and Logarithmic Functions
CHAPTER 12: Transforming and Critiquing Functions
Part 3: Using Conics and Systems of Equations
CHAPTER 13: Slicing the Way You Like It: Conic Sections
CHAPTER 14: Solving Systems of Linear Equations
Part 4: Making
Part 5: Applying Known Formulas
Solving Quadratics by Completing the Square
Composing Functions and Applying the Difference Quotient
Performing compositions
Simplifying the difference quotient
Dealing with Inverse Functions
Determining if functions are inverses
Solving for the inverse of a function
Practice Questions Answers and Explanations
Whaddya Know? Chapter 7 Quiz
CHAPTER 8: Specializing in Quadratic Functions
Setting the Standard to Create a Parabola
Starting with “a” in the standard form
Following up with “b” and “c”
Recognizing the Intercepts and Vertex
Finding the one and only y-intercept
Finding the x-intercepts
Going to the extreme: Finding the vertex
Making Symmetry Work with an Axis
Graphing Parabolas
Determining Positive and Negative Intervals
Graphing Polynomials
Practice Questions Answers and Explanations
Whaddya Know? Chapter 9 Quiz
Answers to Chapter 9 Quiz
CHAPTER 10: Acting Rationally with Functions
Exploring the Domain and Intercepts of Rational Functions
Introducing intercepts
Adding Asymptotes to the Rational Pot
Determining the equations of vertical asymptotes
Heading after horizontal asymptotes
Graphing vertical and horizontal asymptotes
Crunching the numbers and graphing oblique asymptotes
Removing Discontinuities
Removal by factoring
Evaluating
Showing
Going the Limit: Limits at a Number and Infinity
Evaluating limits at discontinuities
Determining an existent limit without tables
Noting the Ups,
Determining increasing and decreasing intervals
Charting the highs and
Whaddya Know? Chapter 12
and Explanations
Looking at the Standard Linear-Systems Form
Solving Systems of Two Linear Equations by Using Elimination
Getting
the point with elimination
Recognizing solutions indicating parallel or coexisting lines
Making Substitution the Choice
Variable substitution made easy
Identifying parallel and coexisting lines
Graphing Solutions of
Pinpointing the intersection
systems
PART 4: MAKING LISTS AND CHECKING FOR IMAGINARY NUMBERS
CHAPTER 17: Getting More Complex with Imaginary Numbers
Simplifying Powers of i
Understanding the Complexity of Complex Numbers
Operating on complex numbers
Multiplying by the conjugate to perform division
Simplifying radicals
Solving Quadratic Equations with Complex Solutions
Working Polynomials with Complex Solutions
Identifying conjugate pairs
Interpreting complex zeros
Practice Questions Answers and Explanations
Whaddya Know? Chapter 17 Quiz
Answers to Chapter 17 Quiz
CHAPTER 18: Making Moves with Matrices
Describing the Different Types of Matrices
Making a Series of Moves
Introducing summation notation
Summing arithmetically
Summing geometrically
Highlighting Special Formulas
Practice Questions Answers and Explanations
Whaddya Know? Chapter 19 Quiz
Answers to Chapter 19 Quiz
CHAPTER 20: Everything You Wanted to Know about Sets and Counting
Revealing the Set Rules
Listing elements with a roster
Building sets from scratch
Going for all (universal set) or nothing (empty set)
Subbing in with subsets
Operating
Celebrating
Looking both ways for set intersections
Feeling complementary about sets
Counting the elements in sets
Applying the Venn diagram
Adding a set to a Venn diagram
Focusing on Factorials
How Do I Love Thee? Let Me Count Up the Ways
PART 5: APPLYING KNOWN FORMULAS
Getting All Geometric
Telling the plain truth with plane geometry
Keeping it solid
Solving a Formula for a Variable
Practice Questions Answers and Explanations
Whaddya Know? Chapter 21 Quiz
Answers to Chapter 21 Quiz
CHAPTER 22: Taking on Applications
Making Mixtures Magically Mathematical
Keeping money matters solvent
Working
Whaddya Know? Chapter 22 Quiz
Introduction
Here you are: contemplating the study of Algebra II. You just couldn’t get enough of Algebra I? Good for you! You’re moving along and preparing yourself for more of the fun and mystery and challenge of higher mathematics.
Algebra is really the basis of most courses that you take in high school and college. You can’t do anything in calculus without a good algebra background. And there’s a lot of algebra in geometry. You even need algebra in computer science! Algebra was created, modified, and continues to be tweaked so that ideas and procedures can be shared by everyone. With all people speaking the same “language,” there are fewer misinterpretations.
What you find in this book is a glimpse into the way I teach: uncovering mysteries, working in historical perspectives, providing information, and introducing the topic of Algebra II with good-natured humor. Over the years, I’ve tried many approaches to teaching algebra, and I hope that with this book I’m helping you cope with and incorporate other teaching methods.
About This Book
Because you’re interested in this book, you probably fall into one of four categories:
» You’re fresh off Algebra I and feel eager to start on this new venture.
» You’ve been away from algebra for a while, but math has always been your main interest, so you don’t want to start too far back.
» You’re a parent of a student embarking on or having some trouble with an Algebra II class and you want to help.
» You’re just naturally curious about science and mathematics and you want to get to the good stuff that’s in Algebra II.
Whichever category you represent (and I may have missed one or two), you’ll find what you need in this book. You can find some advanced algebraic topics, but I also cover the necessary basics, too. You can also find plenty of connections — the ways different algebraic topics connect with each other and the ways that algebra connects with other areas of mathematics.
After all, many other math areas drive Algebra II. Algebra is the passport to studying calculus, trigonometry, number theory, geometry, all sorts of good mathematics, and much of science. Algebra is basic, and the algebra you find here will help you grow your skills and knowledge so you can do well in math courses and possibly pursue other math topics.
Each new topic provides:
» Example problems with answers and solutions
» Practice problems with answers and solutions
Each chapter provides:
» An end-of-chapter quiz with problems representing the topics covered
» Solutions to those quiz questions
Online quizzes are also available for even more practice and confidence-building.
Foolish Assumptions
You are reading this book to learn more about algebra, so I’m assuming that you already have some of the other basic math skills: familiarity with fractions and their operations, comfort with handling decimals and the operations involved, some experience with integers (signed whole numbers) and how they operate, and some graphing knowledge — how to place points on a graphing plane. If you don’t have as much knowledge as you’d like of some items mentioned, you may want to refer to some resources such as Algebra I All In One For Dummies, Basic Math & Pre-Algebra For Dummies, or Pre-Algebra Essentials For Dummies (John Wiley & Sons, Inc.).
My second assumption is that you’re as excited about mathematics as I am. Oh, okay, you don’t have to be that excited. But you’re interested and eager and anxious to increase your mathematical abilities. That’s the main thing you need.
Read on. Work through the book at your own pace and in the order that works for you.
Icons Used in This Book
In this book, I use these five icons to signal what’s most important along the way:
Each example is an algebra question based on the discussion and explanation, followed by a step-by-step solution. Work through these examples, and then refer to them to help you solve the practice test problems at the end of the chapter.
This icon points out important information that you need to focus on. Make sure that you understand this information fully before moving on. You can skim through these icons when reading a chapter to make sure that you remember the highlights.
Tips are hints that can help speed you along when answering a question. See whether you find them useful when working on practice problems.
This icon flags common mistakes that students make if they’re not careful. Take note and proceed with caution!
When you see this icon, it’s time to put on your thinking cap and work out a few practice problems on your own. The answers and detailed solutions are available so you can feel confident about your progress.
Beyond the Book
In addition to what you’re reading right now, this book comes with a Cheat Sheet that provides quick access to some formulas, rules, and processes that are frequently used. To get this Cheat Sheet, simply go to www.dummies.com and type Algebra II All in One For Dummies Cheat Sheet in the Search box.
You’ll also have access to online quizzes related to each chapter. These quizzes provide a whole new set of problems for practice and confidence-building. To access the quizzes, follow these simple steps:
1. Register your book or ebook at Dummies.com to get your PIN. Go to www.dummies. com/go/getaccess.
2. Select your product from the drop-down list on that page.
3. Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.
If you do not receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.
Now you’re ready to go! You can come back to the practice material as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Your registration is good for one year from the day you activate your PIN.
Where to Go from Here
This book is organized so that you can safely move from whichever chapter you choose to start with and in whatever order you like. You can strengthen skills you feel less confident in or work on those that need some attention.
If you haven’t worked on any algebra recently, I’d recommend that you start out with Chapter 1 and some other chapters in the first unit. It’s important to know the vocabulary and basic notation so you understand what is being presented in later chapters.
I’m so pleased that you’re willing, able, and ready to begin an investigation of Algebra II. If you’re so pumped up that you want to tackle the material cover to cover, great! But you don’t have to read the material from page 1 to page 2 and so on. You can go straight to the topic or topics you want or need, and refer to earlier material if necessary. You can also jump ahead if so inclined. I include clear cross-references in chapters that point you to the chapter or section where you can find a particular topic — especially if it’s something you need for the material you’re looking at or if it extends or furthers the discussion at hand.
You can use the table of contents at the beginning of the book and the index in the back to navigate your way to the topic that you need to brush up on. Regardless of your motivation or what technique you use to jump into the book, you won’t get lost because you can go in any direction from there.
Enjoy!
1 Getting to First Base with the Basics
In This Unit. . .
CHAPTER 1: Beginning at the Beginning of Algebra
Following the Order of Operations and Other Properties
Specializing in Products and “FOIL”
Expounding on Exponential Rules
Taking On Special Operators
Simplifying Radical Expressions
Factoring in Some Factoring Techniques
Practice Questions Answers and Explanations
Whaddya Know? Chapter 1 Quiz
Answers to Chapter 1 Quiz
CHAPTER 2: Taking on Linear Equations and Inequalities
Variables on the Side: Solving Linear Equations
Making Fractional Terms More Manageable
Solving for a Variable
Making Linear Inequalities More Equitable
Compounding the Situation with Compound Statements
Dealing with Linear Absolute Value Equations and Inequalities
Practice Questions Answers and Explanations
Whaddya Know? Chapter 2 Quiz
Answers to Chapter 2 Quiz
CHAPTER 3: Handling Quadratic and Other Polynomial Equations
Implementing the Square Root Rule
Successfully Factoring for Solutions
Resorting to the Quadratic Formula
Solving Quadratics by Completing the Square
Tackling Higher-Powered Polynomials
Using the Rational Root Theorem and Synthetic Division
Practice Questions Answers and Explanations
Whaddya Know? Chapter 3 Quiz
Answers to Chapter 3 Quiz
CHAPTER 4: Controlling Quadratic and Rational Inequalities
Checking Out Quadratic Inequalities
Compounding the Situation with Compound Inequalities
Solving Inequality Word Problems
Practice Questions Answers and Explanations
Whaddya Know? Chapter 4 Quiz
Answers to Chapter 4 Quiz
CHAPTER 5: Soothing the Rational, the Radical, and the Negative
» Raising your exponential power and taming radicals
» Looking at special products and factoring
Chapter 1
Beginning at the Beginning of Algebra
Algebra is a branch of mathematics that people study before they move on to other areas or branches of mathematics and science. For example, you use the processes and mechanics of algebra in calculus to complete the study of change; you use algebra in probability and statistics to study averages and expectations; and you use algebra in chemistry to solve equations involving chemicals.
Any study of science or mathematics involves rules and patterns. You approach the subject with the rules and patterns you already know, and you build on them with further study. And any discussion of algebra presumes that you’re using the correct notation and terminology.
Going into a bit more detail, the basics of algebra include rules for dealing with equations, rules for using and combining terms with exponents, patterns to use when factoring expressions, and a general order for combining all of the above. In this chapter, I present these basics so you can further your study of algebra and feel confident in your algebraic ability. Refer to these rules whenever needed as you investigate the many advanced topics in algebra.
Following the Order of Operations and Other Properties
Mathematicians developed the rules and properties you use in algebra so that every student, researcher, curious scholar, and bored geek working on the same problem would get the same answer — regardless of the time or place. You don’t want the rules changing on you every day; you want consistency and security, which you get from the strong algebra rules and properties presented in this section.
Ordering your operations
When mathematicians switched from words to symbols to describe mathematical processes, their goal was to make dealing with problems as simple as possible; however, at the same time, they wanted everyone to know what was meant by an expression and for everyone to get the same answer to the same problem. Along with the special notation came a special set of rules on how to handle more than one operation in an expression. For instance, if you do the problem 4 35 6237 14 2 2 , you have to decide when to add, subtract, multiply, divide, take the root, and deal with the exponent.
The order of operations dictates that you follow this sequence:
1. Perform all operations inside/on grouping symbols (parentheses, brackets, braces, fraction lines, and so on).
2. Raise to powers or find roots.
3. Multiply and divide, moving from left to right.
4. Add and subtract, moving from left to right.
Q. Simplify: 4 35 6237 14 2 2
A. 4 9304 76. Follow the order of operations. The radical acts like a grouping symbol, so you subtract what’s in the radical first: 4 35 616 14 2 2 .
Raise the power and find the root: 4 95 64 14 2 .
Multiply and divide, working from left to right: 4 9304 7.
Add and subtract, moving from left to right: 4 9304 76.
Q. Simplify: 33 42 2 22 2 () ()() . A. 1 2 2 or .
Follow the order of operations: The fraction line serves as a grouping symbol, so you simplify the numerator and denominator separately before dividing. Also, the radical is a grouping symbol, so the value under the radical is determined before finding the square root. Under the radical, first square the 3, then multiply, and finally subtract. You can then take the root.
Now multiply the two numbers in the denominator: 35 22 35 4 .
Apply the two operations, and perform those operations before dividing: 35 4 2 4 1 2 or 35 4 8 4 2.
1 Beginning at the Beginning of Algebra 9
Examining various properties
Mathematics has a wonderful structure that you can depend on in every situation. Most importantly, it uses various properties, such as associative, commutative, distributive, and so on. These properties allow you to adjust mathematical statements and expressions without changing their values. You make them more usable without changing the outcome.
Property Description
Commutative (of addition)
Commutative (of multiplication)
Associative (of addition)
Math Statement
You can change the order of the values in an addition statement without changing the final result. a bb a
You can change the order of the values in a multiplication statement without changing the final result. a bb a
You can change the grouping of operations in an addition statement without changing the final result. a bc ab c () ()
Associative (of multiplication) You can change the grouping of operations in a multiplication statement without changing the final result. a bc ab c
Distributive (multiplication over addition)
Distributive (multiplication over subtraction)
You can multiply each term in an expression within parentheses by the multiplier outside the parentheses and not change the value of the expression. It takes one operation, multiplication, and spreads it out over terms that you add to one another. a bc ab ac
You can multiply each term in an expression within parentheses by the multiplier outside the parentheses and not change the value of the expression. It takes one operation, multiplication, and spreads it out over terms that you subtract from one another. a bc ab ac
Identity (of addition)
Identity (of multiplication)
Multiplication property of zero
The additive identity is zero. Adding zero to a number doesn’t change that number; it keeps its identity. a aa 00
The multiplicative identity is 1. Multiplying a number by 1 doesn’t change that number; it keeps its identity. a aa 11
The only way the product of two or more values can be zero is for at least one of the values to actually be zero. a bc de f ab cd ef 0 0 ,, ,, or
Additive inverse A number and its additive inverse add up to zero. a a 0
Multiplicative inverse A number and its multiplicative inverse have a product of 1. The only exception is that zero does not have a multiplicative inverse. a a 1 1, a 0
Q. Use the commutative and associative properties of addition to simplify the expression: 16 11 5 aa .
A. 1121a . First, reverse the order of the terms in the parentheses using the commutative property: 11 16 5 aa .
Next, reassociate the terms: 11 16 5 aa .
Finally, add the two terms in the parentheses: 11 21a .
Q. Use the distributive property to simplify the expression: 12 1 2 2 3 3 4 .
A. 5.
Q. Use the multiplicative identity to find a common denominator and add the terms: 3 8 5 6 .
A. 1 5 24 . Multiply the first fraction by 1 in the form of the fraction 3 3 , and the second fraction by 4 4 : 3
Q. Use an additive inverse to solve the equation 3 112 xy for y.
A. y = 112 – 3x. Add 3 x to each side of the equation: 33 112 3 112 3 xx yx yx
Q. Apply the multiplication property of zero to solve the equation: xx x ()() 54 0
A. –5, 4, 0. At least one of the factors has to equal 0. So, if x 0, then the product is 0. Or, if x 5, then the second factor is 0 and the product is 0. Likewise if x 4 , then the third factor is 0 and the product is 0.
Q. Use the additive inverse to add and subtract a number that makes the expression xx 2 10 factorable as the square of a binomial.
A. x 525 2 . Adding and subtracting 25 will do the trick:
1 Beginning at the Beginning of Algebra 11
3 Use the associative and commutative properties of multiplication to simplify 4 11 15 11 8 .
4 Use the distributive property to simplify 10 3 5 xx
5 Use a multiplicative inverse to solve the equation 4 100 x for the value of x.
THE BIRTH OF NEGATIVE NUMBERS
In the early days of algebra, negative numbers weren’t an accepted entity. Mathematicians had a hard time explaining exactly what the numbers illustrated; it was too tough to come up with concrete examples. One of the first mathematicians to accept negative numbers was Fibonacci, an Italian mathematician. When he was working on a financial problem, he saw that he needed what amounted to a negative number to finish the problem. He described it as a loss and proclaimed, “I have shown this to be insoluble unless it is conceded that the man had a debt.”
