Practice Makes Perfect Algebra II Review and Workbook, Second Edition

By Christopher Monahan

The winning formula for success in algebra is practice, practice, practice! This book will help you increase your grasp of advanced algebra concepts. Numerous lessons will teach you such essential skills as transforming functions, completing the square, working with matrices, and determining probability. These lessons are accompanied by a variety of exercises to practice what you’ve learned, along with a complete answer key to check your work. Throughout this book you will learn terms to further your understanding of algebra, and you will expand your knowledge of the subject through dozens of sample problems and their solutions. With the lessons in this book, you will find it easier than ever to grasp concepts in advanced algebra. And with hundreds of exercises for practice, you will gain confidence using your new algebra skills in your classwork and on exams. You’ll be on your way to mastering these topics and more: •Functions •Exponential and logarithmic equations •Arithmetic of complex numbers •The factor theorem •Polynomial and rational equations •Regression equations •Inferential statistics

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Dec 27, 2017
• ISBN: 9781260116038

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To Diane—thank you for all your support.

Contents

Introduction

1   Functions: An introduction

Relations and inverses

Functions

Function notation

Arithmetic of functions

Transformation of functions

Inverse of a function

Graphical representation of functions

2   Linear equations and inequalities

Simple linear equations

Linear inequalities

System of linear equations—graphical

System of linear equations—substitution

System of linear equations—elimination

System of linear equations—three variables

System of linear equations—matrix

System of linear equations—application

System of linear inequalities

Absolute value equations

Absolute value inequalities

3   Quadratic relationships

The parabola

Special factoring formulas

Trial and error

Completing the square

Quadratic formula

Applications

Square root function

Circles

Ellipses

Hyperbolas

Systems of equations

4   Complex numbers

Powers of i

Arithmetic of complex numbers

The discriminant and nature of the roots of a quadratic equation

Sum and product of roots of a quadratic equation

5   Polynomial functions

Even and odd functions

Inverse functions

End behavior

Factor theorem

6   Rational and irrational functions

Rational functions

Multiplying and dividing rational expressions

Adding and subtracting rational expressions

Solving rational equations

Irrational functions

Simplifying irrational expressions

Solving irrational equations

7   Exponential and logarithmic functions

Properties of exponents

Exponential and logarithmic functions

Properties of logarithms

Solving exponential and logarithmic equations

8   Sequences and series

Summation notation

Recursion

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

9   Introduction to probability

Fundamental theorem of counting

Permutations

Combinations

Binomial expansions

Conditional probability

Binomial probability/Bernoulli trial

10   Introduction to statistics

Measures of central tendency

Measures of dispersion

Normal distribution

Regressions

11   Inferential statistics

Basics

Central limit theorem and standard error

Standardized (z) scores

The basics of inferential statistics

Confidence intervals

Tests of hypotheses

Simulation

12   Trigonometry: Right triangles and radian measure

Right triangle trigonometry

Special right triangles

The unit circle: First quadrant

The unit circle—beyond the first quadrant

Radian measure

Basic trigonometric identities

Area of a triangle

Law of sines

Ambiguous case

Law of cosines

13   Graphs of trigonometric functions

Amplitude and period

Graphing trigonometric functions

Inverse trigonometric functions

Solving trigonometric equations

Answer key

Introduction

Algebra II uses the skills and concepts learned in Algebra I as well as many of the concepts learned in Geometry. You will learn many new concepts in Algebra II, the most important of which is the idea of a function. Functions are a fundamental building block for the development of higher mathematics. Questions about domain and range will appear in almost every chapter of the course. Using transformations to extend a basic function into a family of functions will help you develop a better understanding of functions and allow you to develop a mental image of the function before you look at the graph on your graphing calculator or computer. Mathematics is a tool used in nearly every aspect of the world of work. Examples for many of these applications are included in Practice Makes Perfect Algebra II Review and Workbook.

Whereas you can read a piece of literature or a document for a social studies class and then quietly contemplate the meaning of what you read, mathematics requires a more active approach. You should read the text and the examples provided for you. You should also do the guided exercises after you have finished reading a section to ensure you understand the steps involved in solving the problem. Once you have done this, then you should do the exercises at the end of the section. Check your answers with the answer key in the back of the book to verify you have done the problem correctly. You will find many problems in this book. Some require very little writing, and some require the use of technology such as a graphing calculator or computer software. Most of the exercises can be done with paper and pencil. Take the time to do all of the exercises. You will learn a great deal from the time and effort you put forth.

Algebra II Review and Workbook is written so that you can practice a few concepts at a time. This does not meant that the examples and exercises will not apply what you learned in a previous lesson or from other courses, but that the crux of the problem is to help you better understand the concept from that particular section. If you find that a prior concept is slowing you down, take the time to go back to that section to get a better understanding of what you are missing.

   Functions: An introduction

Mathematics is known for its ability to convey a great deal of information with the use of the minimum number of symbols. While this may be initially confusing (if not frustrating) for the learner, the notation of mathematics is a universal language. In this chapter, you will learn about function notation.

Relations and inverses

One of the major concepts used in mathematics is relations. A relation is any set of ordered pairs. The set of all first elements (the input values) is called the domain, while the set of second elements (the output values) is called the range. Relations are traditionally named with a capital letter. For example, given the relation

A = {(2, 3), (−1, 5), (4, −3), (2, 0), (−9, 1)}

the domain of A (written DA) is {−9, −1, 2, 4}. The domain was written in increasing order for the convenience of reading, but this is not required. The element 2, which appears as the input for two different ordered pairs, needs to be written only one time in the domain. The range of A (written RA) is {−3, 0, 1, 3, 5}.

The inverse of a relation is found by interchanging the input and output values. For example, the inverse of A (written A−1) is

A−1 = {(3, 2), (5, −1), (−3, 4), (0, 2), (1, −9)}

Do you see that the domain of the inverse of A is the same set as the range of A, and that the range of the inverse of A is the same as the domain of A? This is very important.

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Given the relationships:

1.   Find the domain of A.

2.   Find the range of A.

3.   Find the domain of B.

4.   Find the range of B.

5.   Find the domain of C.

6.   Find the range of C.

7.   Find A−1.

8.   Find B−1.

9.   Find C−1.

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Functions

Functions are a special case of a relation. By definition, a function is a relation in which each element of the domain (the input value) has a unique element in the range (the output value). In other words, for each input value there can be only one output value. Looking at the relations for A and A−1 in the previous section, you can see that A is not a function because the input value of 2 is associated with the output values 3 and 0. The relation A−1 is a function because each input value is paired with a unique output value. (Don’t be confused that the number 2 is used as an output value for two different input values. The definition of a function does not place any stipulations on the output values.)

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Given the relationships:

1.   Which of the relations A, B, and C are functions?

2.   Which of the relations A−1, B−1, and C−1 are functions?

3.   A relation is defined by the sets {(students in your math class), (telephone numbers at which they can be reached)}. That is, the input is the set of students in your math class and the output is the set of telephone numbers at which they can be reached. Must this relationship be a function? Explain.

4.   Is the inverse of the relation in question 3 a function? Explain.

5.   A relation is defined by