# Algebra 2 Companion Book, Volume 1

This companion book to Thinkwell's Algebra 2 course includes: • Illustrated lesson notes • Lesson worksheets • Practice exercises • Review worksheets • Suggested pacing guide • Formula and resource worksheets Volume 1 covers: Ch 1 - Foundations for Functions Ch 2 - Linear Functions Ch 3 - Linear Systems Ch 4 - Matrices Ch 5 - Quadratic Functions

A companion book for Thinkwell's Algebra 2 online video course. Volume #1 covers Chapters 1 ‒ 5.

No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without prior written permission of Thinkwell Corporation unless such copying is expressly permitted by federal copyright law. Address inquiries to: Permissions, Thinkwell Corp., 505 E. Huntland Drive, Suite 150, Austin, TX 78752.

Welcome to Algebra 2 !

Before we get too far into the fun, I wanted to personally introduce myself: I'm your virtual instructor, Professor Burger. Welcome to Thinkwell's wonderful world of Algebra 2!

Together, you and I will build on the concepts we developed in Algebra 1 and explore new, more challenging ideas that are central to Algebra 2. Through my lessons I hope to help you to MAKE MEANINGFUL CONNECTIONS with the math ideas in Algebra 2 — there ’ s no need to memorize if you focus on deep understanding. I invite (urge) you to take the time to truly think through the math we ’ ll explore together and to mindfully practice the skills behind the ideas every day. If you do, you ’ ll not only succeed in Algebra 2, but you ’ ll also be on solid ground for all the other math in your future! About This Book This is a companion book to Thinkwell's Algebra 2 online video course. Use this book as a complement to the online materials. I always say that to learn math you must DO MATH! I encourage you to put pencil to paper and • take notes in this book • highlight key concepts and earmark ideas you want to remember • doodle, sketch, and visualize the math ideas presented in each topic.

In a nutshell, make this book your own and keep it by your side as you study the concepts in this course. This book is divided into chapters. Within each chapter are a series of Algebra 2 topics. Every topic online contains my Video Lessons along with an electronic version of the Notes, Practice questions and Worksheet questions (although I wish we ’ d call them “ Funsheets ” , but that ’ s another story). How To Use This Book Use this book alongside the online course and TAKE NOTES here while watching my video lessons. Your own notes are a key to your own success — I promise. Since this book summarizes the concepts, vocabulary, and key examples presented in the Video Lessons, it is a great tool to help you navigate the videos — but this companion book is not intended as a shortcut to replace the Video Lessons. To get the most out of this learning experience, I urge you to watch (and think through) all of the online Video Lessons. Maybe even watch some twice! The online Algebra 2 course offers lots of opportunities to practice the skills you ’ ll need for success in Algebra 2. Each topic ’ s Worksheet Practice and Interactive Practice is a collection of questions connected to the content presented in the Video Lessons. I've included those questions here in this book, so you can explore them offline and spend time really thinking through each question. I always say that the best way to learn math is to DO MATH. So, take advantage of all the opportunities to practice what you've learned!

Finally, at the end of each section, I've included a Review Worksheet to give you even more opportunities to review and practice the concepts you learned in the Video Lessons. Put pencil to paper (or pen, if you dare) to answer each question. I recommend you complete these Reviews before taking the course Quizzes and Tests online. LET'S GO! I look forward to our Algebra 2 journey together! Remember to make meaning and focus on deep understanding … and also remember that YOU CAN DO IT! Have fun! If you have any questions, please reach out to my friends at Thinkwell. Email them at support@thinkwell.com. Also, I ’ m on Twitter @ebb663, if you want to say, “ hello ” .

I wish you all the best in your Algebra 2 success!

— Prof. B.

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Tips for Success������������������������������������������������������������������������������������������������������������������������� 1 Suggested Pacing Guide��������������������������������������������������������������������������������������������������������������3 Chapter 1: Foundations for Functions������������������������������������������������������������������������������������������5 1.1 Properties and Operations������������������������������������������������������������������������������������������7 1.1.1 Sets of Numbers ����������������������������������������������������������������������������������������9 1.1.2 Properties of Real Numbers ���������������������������������������������������������������������14 1.1.3 Square Roots ��������������������������������������������������������������������������������������������20 1.1.4 Simplifying Algebraic Expressions ������������������������������������������������������������ 26 1.1.5 Properties of Exponents and Scientific Notation ��������������������������������������31 1.1 Review Worksheet, Part 1 ��������������������������������������������������������������������������37 1.1 Review Worksheet, Part 2 ��������������������������������������������������������������������������39 1.2 Introduction to Functions �����������������������������������������������������������������������������������������41 1.2.1 Relations and Functions ��������������������������������������������������������������������������43 1.2.2 Function Notation �������������������������������������������������������������������������������������48 1.2.3 Exploring Transformations ������������������������������������������������������������������������ 53 1.2.4 Introduction to Parent Functions ��������������������������������������������������������������60 1.2 Review Worksheet ��������������������������������������������������������������������������������������65 Chapter 2: Linear Functions������������������������������������������������������������������������������������������������������69 2.1 Linear Equations and Inequalities ����������������������������������������������������������������������������71 2.1.1 Solving Linear Equations and Inequalities ������������������������������������������������73 2.1.2 Proportional Reasoning ����������������������������������������������������������������������������79 2.1.3 Graphing Linear Functions ������������������������������������������������������������������������85 2.1.4 Writing Linear Functions ��������������������������������������������������������������������������93 2.1.5 Linear Inequalities in Two Variables ������������������������������������������������������� 101 2.1 Review Worksheet, Part 1 ������������������������������������������������������������������������107 2.1 Review Worksheet, Part 2 �������������������������������������������������������������������������110 2.2 Applying Linear Functions���������������������������������������������������������������������������������������113 2.2.1 Transforming Linear Functions �������������������������������������������������������������� 115 2.2.2 Curve Fitting With Linear Models ���������������������������������������������������������� 123 2.2.3 Solving Absolute-Value Equations and Inequalities ������������������������������� 130 2.2.4 Absolute-Value Functions ���������������������������������������������������������������������� 136 2.2 Review Worksheet ������������������������������������������������������������������������������������142 Chapter 3: Linear Systems������������������������������������������������������������������������������������������������������147 3.1 Linear Systems in Two Dimensions����������������������������������������������������������������������� 149 3.1.1 Using Graphs and Tables to Solve Linear Systems ������������������������������� 151 3.1.2 Using Algebraic Methods to Solve Linear Systems �������������������������������� 157 3.1.3 Solving Systems of Linear Inequalities ��������������������������������������������������164

3.1.4 Linear Programming�������������������������������������������������������������������������������������170 3.1 Review Worksheet ������������������������������������������������������������������������������������������176 Chapter 4: Matrices������������������������������������������������������������������������������������������������������������������������181 4.1 Matrix Operations���������������������������������������������������������������������������������������������������������183 4.1.1 Matrices and Data ����������������������������������������������������������������������������������������185 4.1.2 Multiplying Matrices �������������������������������������������������������������������������������������191 4.1.3 Using Matrices to Transform Geometric Figures ������������������������������������������198 4.1 Review Worksheet ������������������������������������������������������������������������������������������204 Chapter 5: Quadratic Functions�����������������������������������������������������������������������������������������������������207 5.1 Quadratic Functions and Complex Numbers ��������������������������������������������������������������209 5.1.1 Using Transformations to Graph Quadratic Functions ��������������������������������� 211 5.1.2 Properties of Quadratic Functions in Standard Form ������������������������������������219 5.1.3 Solving Quadratic Equations by Graphing and Factoring ����������������������������225 5.1.4 Completing the Square ��������������������������������������������������������������������������������233 5.1.5 Complex Numbers and Roots ����������������������������������������������������������������������238 5.1.6 The Quadratic Formula ��������������������������������������������������������������������������������244 5.1 Review Worksheet, Part 1 ������������������������������������������������������������������������������250 5.1 Review Worksheet, Part 2 ������������������������������������������������������������������������������254 5.2 Applying Quadratic Functions �������������������������������������������������������������������������������������257 5.2.1 Solving Quadratic Inequalities ���������������������������������������������������������������������259 5.2.2 Curve Fitting with Quadratic Models ������������������������������������������������������������265 5.2.3 Operations with Complex Numbers �������������������������������������������������������������270 5.2 Review Worksheet ������������������������������������������������������������������������������������������278 Formulas & Symbols����������������������������������������������������������������������������������������������������������������������281

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Check out the suggested pacing guide in this book or download the online Lesson Plan and create a study schedule for yourself. Your schedule will be your plan for Algebra success!

Be an active learner. Before you begin studying, collect the tools you'll need: a pencil, scratch paper, highlighters, or graph paper are great things to have on-hand.

2

As you watch the Video Lessons online, work out the examples along with Prof. Burger on the Lesson Notes here (or on your own paper). Highlight important points in the Lesson Notes, and earmark topics you want to go back to review before a Quiz or Test.

Practice as you go. After each Video example, complete the Worksheet questions for that example. Once you've watched all the video lessons and answered all the Worksheet questions, check your understanding by completing the Practice question set. Go online to check your answers and to see answer feedback with step- by-step explanations. Review to remember.

3

4

Before a Quiz or Test, complete the Review Worksheet and re-do any exercises you need extra practice to master.

Reach out if you need help! Have questions? Need help? Reach out to us at support@thinkwell.com. We're here to help!

5

1

This pacing guide follows a 36-week plan to sequentially progress through Thinkwell's Algebra 2 online course. Since the course is self-paced, feel free to go as quickly or as slowly through the material as you need to – this guide is just a suggestion. The list below corresponds with Thinkwell's Algebra 2 online course scope and sequence. 口 WEEK 1: – 1.1 Properties and Operations 口 WEEK 2: – 1.1 Properties and Operations (Cont.)

3

4

Chapter 1 Foundations for Functions

5

1.1

Properties and Operations

7

1.1.1 Sets of Numbers

Key Objectives • Order and classify real numbers. • Use interval notation to represent a set of numbers. • Translate between methods of set notation. Key Terms • A set is a collection of items called elements .

• Interval notation is a way of writing the set of all real numbers between two endpoints. The symbols [ and ] are used to include an endpoint in an interval, and the symbols ( and ) are used to exclude an endpoint from an interval. • Set-builder notation is a notation for a set that uses a rule to describe the properties of the elements of the set. • Roster notation is a way of representing a set by listing the elements between braces, { }. Example 1 Ordering and Classifying Real Numbers Rational numbers can be expressed as a quotient (or ratio) of two integers, where the denominator is not zero. The decimal form of a rational number either terminates, such as 1/2 = 0.5, or repeats, such as − 1/11 = − 0.090909... = 0.09. Irrational numbers, such as 2 and π , cannot be expressed as a quotient of two integers, and their decimal forms do not terminate or repeat. However, the value of these numbers can be approximated using terminating decimals. Real numbers in their decimal forms can be easily compared, as demonstrated by Prof. Burger in the following example.

9

1.1.1 Sets of Numbers (continued)

Example 2 Interval Notation

There are many ways to represent sets, such as with words, number lines, inequalities, or by using interval notation. Interval notation can be used to represent the set of all real numbers between two endpoints, called an interval. In interval notation, the symbols [ and ] are used to include an endpoint in an interval, and the symbols ( and ) are used to exclude an endpoint from an interval. An interval that extends forever in the positive direction goes to infinity ( ∞ ), and an interval that extends forever in the negative direction goes to negative infinity ( −∞ ). Because ∞ and −∞ are not numbers, they cannot be included in a set of numbers, so parentheses are used to enclose them in an interval.

10

1.1.1 Sets of Numbers (continued)

Example 3 Translating Between Methods of Set Notation

Sets can also be represented using roster notation and set-builder notation. In roster notation, the elements of the set are listed between braces { }. In set-builder notation, a rule (such as an inequality) describes the properties of the elements of the set.

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1.1.1 Sets of Numbers - Worksheet

Example 1: Order the given numbers from least to greatest. Then classify each number by the subsets of the real numbers to which it belongs. (Use “R” for real number, “Q” for rational number, “Z” for integer, “W” for whole number, “N” for natural number, and “irrational” for irrational number.) 1. 3 2, 7, 5.125, 4 , 4.6 , 6.897, 4, , 6 3. − − π 5, , 3,1.3, 1

1 3

3 5

1 8

2. − − 100 4

2

Example 2: Use interval notation to represent each set of numbers. 4. − 10 < x ≤ 10 5.

–15 –10 –5 0 5

6. 1 ≤ x < 20 or x > 30

Example 3: Rewrite each set in the indicated notation. 7. { x | x = 1}; words

8.

–6–4–2 0 2 4 6 set-builder notation

9. {0, 5, 10, 15, 20, ...}; words

10. integers from − 5 to 5; roster notation

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1.1.1 Sets of Numbers - Practice

1.  Order the list of numbers from least to greatest. 0.5, 5, 0, , and 3.1

2.  Classify each number by the subsets of the real numbers to which it belongs. − π 3, , 5, , and 0

9 4

8 3

4.  Write the set of numbers using interval notation.

3.  Write the set of numbers using interval notation. − 6 < x ≤ 6

5.  Write the set in words. { x | x ≥ 0}

6.  Write the set “all values of x greater than or equal to − 1 and less than 2” in set builder notation.

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1.1.2 Properties of Real Numbers

Key Objectives • Identify and use properties of real numbers. • Use mental math in real world applications. • Classify statements as sometimes, never, or always true. Key Terms • An additive inverse is the opposite of a number. Two numbers are additive inverses if their sum is zero. • The multiplicative inverse of a nonzero number is the number's reciprocal. • The Associative Property is the property that states that three or more values being added or multiplied together can be grouped in any order without changing the result. ( a + b ) + c = a + ( b + c ) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) • The Distributive Property is the property that states that the product of a value and a sum (or difference) is equal to that value multiplied by each of the values being added (or subtracted). The four basic math operations are addition, subtraction, multiplication, and division. Recall that the opposite of any number a is − a and the reciprocal of any nonzero number a is 1/ a . Because subtraction is addition of the opposite and division is multiplication by the reciprocal, the properties of real numbers focus on addition and multiplication. Example 1 Finding Inverses The opposite of a number can also be called its additive inverse. The sum of a number and its additive inverse is always 0. The reciprocal of a nonzero number can also be called its multiplicative inverse. The product of a nonzero number and its multiplicative inverse is always 1. a ( b + c ) = ab + ac a ( b − c ) = ab − ac • A commission is a fee paid to a person who negotiates a sale. • A counterexample is an example that shows that a statement is false.

14

1.1.2 Properties of Real Numbers (continued)

Example 2 Identifying Properties of Real Numbers

Properties of real numbers, such as the Associative Property and the Distributive Property, can be applied to numeric expressions without changing the value of the expression. By the Associative Property, the sum or product of three or more real numbers is the same regardless of the way the numbers are grouped. For example, (5 + 2) + 4 = 5 + (2 + 4) = 10. By the Distributive Property, the product of a sum and a number is the same whether the sum is found first or whether each term is multiplied by the number first. For example, 5(6 + 2) = 5(8) = 40 and 5(6 + 2) = 5(6) + 5(2) = 30 + 10 = 40. The Distributive Property can also be applied to the product of a difference and a number.

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1.1.2 Properties of Real Numbers (continued)

Example 3 Consumer Economics Application

Properties of real numbers can be applied to simplify numeric expressions and solve problems mentally. Recall that 10% of a number is equal to that number after the decimal point is moved one place to the left. For example, 10% of 175 is 17.5.

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1.1.2 Properties of Real Numbers (continued) Example 4 Classifying Statements as Sometimes, Always, or Never True A counterexample shows that a statement is false. If a statement has a counterexample, then it is either sometimes true or never true.

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1.1.2 Properties of Real Numbers - Worksheet

Example 1: Find the additive and multiplicative inverse of each number. 1. − 36 2. − 0.05

3. 2 2

1 500

4. 2 5

5. −

6. 0.25

Example 2: Identify the property demonstrated by each equation. 7. ( ) = ⋅ 3 2 5 (3 2) 5 8. x + 7 y = 7 y + x

1 3

1 3

9.

(28)(9)

(9)(28)

=

Example 3: Use mental math to find each value. 10. cost of 3 items at \$2.55 each

1 3 % discount on a \$21.99 item

11. a 33

Example 4: Classify each statement as sometimes, always, or never true. Give examples or properties to support your answer. 12. 20 a + 20 b = 5(4 a + 4 b ) 13. a ÷ b = b ÷ a 14. a + ( bc ) = ( a + b )( a + c )

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1.1.2 Properties of Real Numbers - Practice

1.  Identify the additive and multiplicative inverse of − 17.

2.  Identify the additive and multiplicative inverse of 5/9.

3.  Identify the property demonstrated by the equation. 3 ⋅ 2 = 2 ⋅ 3

4.  Identify the property demonstrated by the equation. 4 + 5 = 5 + 4

5.  Identify the property demonstrated by the equation. + = + (2 2 3)2 (2 2)2 (3)2

6.  Use mental math to find a 15% discount for a \$71.40 bill.

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1.1.3 Square Roots Key Objectives • Estimate square roots. • Simplify square root expressions. • Add, subtract, multiply, and divide square roots. Key Terms

• A principal root is the positive square root of a number, indicated by the radical sign. • Terms that include the square root of the same radicand are like radical terms . • Rationalizing the denominator is a method of rewriting a fraction by multiplying it by another fraction that is equivalent to 1 in order to remove radical terms from the denominator. Example 1 Estimating Square Roots Numbers such as 25 that have integer square roots are called perfect squares. Square roots of integers that are not perfect squares are irrational numbers. The value of the square root of a number that is not a perfect square can be estimated using the perfect squares greater than and less than that number.

Example 2 Simplifying Square-Root Expressions

A square-root expression is in simplest form when the radicand has no perfect-square factors (except 1) and there are no radicals in the denominator. The Product and Quotient Properties of Square Roots can be applied to simplify a square root, as well as to multiply or divide square roots.

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1.1.3 Square Roots (continued)

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1.1.3 Square Roots (continued) Example 3 Rationalizing the Denominator

A fraction that has a square root in the denominator is not simplified. To simplify the fraction, rationalize the denominator by multiplying by another fraction that is equivalent to 1, that produces a perfect-square radicand in the denominator.

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1.1.3 Square Roots (continued) Example 4 Adding and Subtracting Square Roots

Square roots that have the same radicand are called like radical terms. To add or subtract square roots, first simplify each square root and then combine the like radical terms by adding or subtracting their coefficients.

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1.1.3 Square Roots - Worksheet

Example 1: Estimate to the nearest tenth. 1. 75 2. 20

3. − 93

4. 13

Example 2: Simplify each expression. 5. − 300 6.

7. 72 2

8. 80

⋅ 24 6

Example 3: Simplify by rationalizing each denominator. 9. 1 2 10. − 5 6 3

11. 50 12

3 21

12.

Example 4: Add or Subtract. 13. + 6 7 7 7

16. − + 50 6 2

14.

15.

+ 4 5 245

− 5 32 15 2

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1.1.3 Square Roots - Practice

1.  Simplify. 75

2.  Simplify. 81 121

4.  Simplify. 98 2

3.  Simplify. ⋅ 3 27

5.  Subtract.

+ 6 2 50

− 12 3 8 3

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1.1.4 Simplifying Algebraic Expressions Key Objectives • Translate words into algebraic expressions.

• Evaluate algebraic expressions. • Simplify algebraic expressions. • Write, simplify, and evaluate algebraic expressions in real-world applications. Key Terms • A variable is a letter or symbol that represents a quantity that can change. • An algebraic expression contains one or more variables and may contain operation symbols. Example 1 Translating Words into Algebraic Expressions To translate a real-world situation into an algebraic expression, first determine the action being described. Then choose the operation that is indicated by the type of action, such as addition, subtraction, multiplication, or division. Use variables to represent unknown quantities.

Example 2 Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute a number for each variable and simplify using the order of operations.

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1.1.4 Simplifying Algebraic Expressions (continued)

Example 3 Simplifying Expressions

Recall that the terms of an algebraic expression are separated by addition or subtraction symbols. Like terms have the same variables raised to the same exponents. Constant terms are like terms that always have the same value. To simplify an algebraic expression, combine the like terms by adding or subtracting their coefficients. Note that algebraic expressions are equivalent if they contain exactly the same terms when simplified.

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1.1.4 Simplifying Algebraic Expressions (continued) Example 4 Transportation Application

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1.1.4 Simplifying Algebraic Expressions - Worksheet

Example 1: Write an algebraic expression to represent each situation. 1. the cost of c containers of yogurt at \$0.79 each

2. the area of a rectangle with length l meters and width 8 meters

Example 2: Evaluate each expression for the given values of the variables. 3. a 2 + b 2 − 2 ab for a = 5 and b = 8 4. − + xy x y 3 9 2 2

x

= 2 and 4 y

for

=

Example 3: Simplify each expression. 5. − 8 a + 9 − 5 a + a

6. − 2(2 x + y ) − 7 x + 2 y

7. 1 + ( ab − 5 a )5 − b 2

Example 4: 8.  Regan runs and bicycles every day for a total of 60 minutes.

Her body uses 9 Calories per minute during running and 7 Calories per minute during bicycling. a. Write and simplify an expression for the total Calories Regan uses running and bicycling each day. b. How many Calories does she use on a day when she runs for 20 minutes?

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1.1.4 Simplifying Algebraic Expressions - Practice

1.  A chess game begins with 32 pieces on the chess board. Write an algebraic expression that represents the number of pieces on the chess board after n pieces are removed.

2.  Write an algebraic expression that represents the number of hours in y days.

3.  Evaluate the expression 2 qp + 5 p − 3 q for p = 3 and q = 5.

4.  Evaluate the expression uv 2 + 5 uv + u 2 for u = 3 and v = 4.

5.  Simplify.

6.  Susan needs to buy apples and oranges to make fruit salad. She needs 15 fruits in all. Apples cost \$3 per piece, and oranges cost \$2 per piece. Let m represent the number of apples and r represent the number of oranges. Write an expression that represents the amount Susan spent on the fruits. Then determine the amount she spent if she bought 6 apples.

7 a 2 + 3 b + 6 a − 2 a 2

30

1.1.5 Properties of Exponents and Scientific Notation Key Objectives • Write exponential expressions in expanded form. • Simplify expressions involving exponents. • Simplify expressions involving scientific notation. Key Terms

• Scientific notation is a kind of shorthand that can be used to write really large or really small numbers. Numbers expressed in scientific notation are written as the product of a factor and a power of 10. Example 1 Writing Exponential Expressions in Expanded Form An expression of the form a n is an exponential expression where a is the base, n is the exponent, and the quantity a n is called a power. The exponent indicates the number of times that the base is used as a factor. For example, in the exponential expression 5 3 , 5 is the base, 3 is the exponent, and the value of the power is found by multiplying the base 5 by itself 3 times. 5 3 = 5 ⋅ 5 ⋅ 5 = 125 If the base of a power includes more than one factor, then the base is written in parentheses.

31

1.1.5 Properties of Exponents and Scientific Notation (continued) Example 2 Simplifying Expressions with Negative Exponents An exponential expression is simplified when it contains no negative exponents, no grouping symbols, and no like terms. Use the Negative Exponent Property to simplify an exponential expression that contains a negative exponent.

32

1.1.5 Properties of Exponents and Scientific Notation (continued) Example 3 Using Properties of Exponents to Simplify Expressions Like the Negative Exponent Property, the Product of Powers Property, Power of a Quotient Property, and the Power of a Power Property can be used to simplify exponential expressions.

33

1.1.5 Properties of Exponents and Scientific Notation (continued) Example 4 Simplifying Expressions Involving Scientific Notation Scientific notation is a method of writing numbers by using powers of 10. In scientific notation, a numbers takes the form m × 10 n , where 1 ≤ m < 10 and n is an integer. Use the properties of exponents to calculate with numbers expressed in scientific notation.

Example 5 Problem Solving Application

34

1.1.5 Properties of Exponents and Scientific Notation - Worksheet

Example 1: Write each expression in expanded form. 1. 4( a − b ) 2 2. (12 xy ) 4

3

4. −   

   d

1 2

3. − s 3 ( − 2 t ) 5

Example 2: Simplify each expression. 5. − 2

3

   − 3 5

   − 2 3

  

7.   

6. 5 0

8. 10 − 1

Example 3: Simplify each expression. Assume all variables are nonzero. 9. ( − 3 a 2 b 3 ) 2 10. c 3 d 2 ( c − 2 d 4 ) 11. uv u v 5 6 2 2

   2

12.   10

5 2

y



x

2

16. − − − x y x y 1 2 3 5

15. b b (4 ) 2

13. − 2 s − 3 t (7 s − 8 t 5 )

14. − 4 m ( mn 2 ) 3

Example 4: Simplify each expression. Write the answer in scientific notation. 17. (2.2 × 10 5 )(4.5 × 10 11 ) 18. × × − 7.8 10 2.6 10 8 3 Example 5: 20.  Nanotechnology is a branch of engineering that works with devices that are smaller than 100 nanometers. The width of one string on the playable nanoguitar created by scientists at Cornell University in 2003 is 2.0 × 10 − 7 meters. If the width of a human hair is about 80 microns, how many nanoguitar strings would have the same width as a human hair? ( Hint: 1 micron = 10 − 6 meters)

3 4

× × − 16 10 4.0 10

19.

10 microns

35

1.1.5 Properties of Exponents and Scientific Notation - Practice

1.  Evaluate the expression 4 − 3 .

2.  Evaluate the expression (3/7) − 3 .

3.  Simplify 3 m 2 ( − 6 m 3 ). Assume all variables are non-zero.

4. Simplify − 7 p 8 ( − 2 p ). Assume all variables are non-zero.

5.  Simplify. Assume all variables are non-zero.       a b b 2 4 3

6.  Simplify. Write the answer in scientific notation. × × − 8.4 10 2.1 10 6 2

36

1.1 Review Worksheet, Part 1

1.1.1 Sets of Numbers Order the given numbers from least to greatest. Then classify each number by the subsets of the real numbers to which it belongs. 1. − − 2.33, 5.5, 2 5, , 0.75 2. − − − , 2, 2, 2 , 1.25 3. π − − − 9, 2 , 1, 5.12,

7 2

1 2

4 5

3

Use interval notation to represent each set of numbers. 4. x ≠ 5 5. − 15 < x < 0

6.

–6–4–2 0 2 4 6

Rewrite each set in the indicated notation. 7. ( − ∞, 3] or (5, 11]; words

8. positive multiples of 11; roster notation

9.

10. { − 9, − 7, − 5, − 3, − 1}; set-builder notation

; words

–4–3–2–1 0 1 2 3 4

1.1.2 Properties of Real Numbers Find the additive and multiplicative inverse of each number. 11. − 2.5 12. 0.75

13. 2 π

2 3

15. 1 20

14. −

16. 6231

Identify the property demonstrated by each equation. 17. z ( x − y ) = zx − zy 18. 4 abc = 4 acb

19. ( a + 0) + b = a + b

37

1.1 Review Worksheet, Part 1 (continued)

Use mental math to find each value. 20. 9% sales tax on a \$150 purchase

21. cost of 5 items at \$1.96 each

Classify each statement as sometimes, always, or never true. Give examples or properties to support your answer. 22. a − ( b − c ) = a − b + c 23. ≠ ≠ ab a b 1 0 for 0 and 0

  

   =

ab

1.1.3 Square Roots Estimate to the nearest tenth . 24. 60

27. 99

25. − 15

26. 47

Simplify each expression. 28. 162

30. 50 9

1 121

29. −

31. − ⋅ 2 10 8

32. 288 8

34. 2 126 14

33.

35. − 189

⋅ 85 5

Simplify by rationalizing each denominator. 36. 2 3 37. 3 27 2 6

18 6

39. 11

38. −

5 132

Add or subtract. 40. − 4 3 9 3

41.

42.

43.

+ 112 63

− 8 15 2

+ 12 7 27

38

1.1 Review Worksheet, Part 2

1.1.4 Simplifying Algebraic Expressions Write an algebraic expression to represent each situation. 1. the measure of the supplement of an angle whose measure is x °

2. the number of \$0.60 bagels that can be purchased with d dollars

Evaluate each expression for the given values of the variables. 3. 6 c − 3 c 2 + d 3 for c = 5 and d = 3

4. y 2 − 2 xy 2 − x for x = 2 and y = 3

2

5. 3 a 2 b − ab 3 + 5 for a = 5 and b = 2

− s t st 2

2

6.

s

= 5and 3 t

for

=

Simplify each expression. 7. − x − 3 y + 4 x − 9 y + 2

8. − 4( − a + 3 b ) − 3( a − 5 b )

9. 5 − (3 m + 2 n )

10. x (4 + y ) − 2 x ( y + 7)

11.  Enrique is baking muffins and bread. He wants to bake a total of 10 batches. Each batch of muffins bakes for 30 minutes, and each batch of bread bakes for 50 minutes. Let m represent the number of batches of muffins. a. Write an expression for the total time required to bake a combination of muffins and bread if each batch is baked separately.

b. If Enrique makes 2 batches of muffins, how long will it take to bake all 10 batches?

39

1.1 Review Worksheet, Part 2 (continued)

1.1.5 Properties of Exponents and Scientific Notation Write each expression in expanded form. 12. ( m + 2 n ) 3 13. 5 x 3 14. ( − 9 fg ) 3 h 4

15. 2 a ( − b 2 − a ) 2

Simplify each expression. 16. ( − 4) − 2 17.

3

1

   − 5 2

   − 3 4

18. −   

  

19. − 6 0

Simplify each expression. Assume all variables are nonzero. 20. − − − s t s t 100 25 3 5 2 6 21. ( − x 4 y 2 ) 5 22. (16 u 4 v 6 ) − 2

23. 8 a 2 b 5 ( − 2 a 3 b 2 )

Simplify each expression. Write the answer in scientific notation. 24. (3.2 × 10 6 ) (1.7 × 10 − 4 ) 25. × × − 5.1 10 3.4 10 4 5

26. (6.8 × 10 3 ) (9.5 × 10 5 )

27.  A computer with a 5.4 GHz microprocessor can make 5.4 × 10 9 calculations in one second. If a total of 5.02 × 10 11 calculations are required to convert a given MP3 file to audio, how many minutes will the computer take to convert the file? Round your answer to the nearest hundredth.

40

1.2

Introduction to Functions

41

1.2.1 Relations and Functions Key Objectives

• Identify domain and range for relations. • Determine whether relations are functions. • Use the Vertical Line Test to determine whether a relation is a function. Key Terms

• A relation is a pairing of input values with output values; a set of ordered pairs. • The domain is the set of all possible x -values (inputs) of a relation or function. • The range is the set of all possible y -values (outputs) of a relation or function. • A function is a relation in which there is only one ouput ( y -value) for each input ( x -value). • Vertical Line Test If a vertical line intersects the graph of a relation at only one point, then the relation is a function. If the line intersects the graph of a relation at more than one point, then the relation is not a function. Example 1 Identifying Domain and Range A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs ( x , y ), where x is an input value and y is an output value. The set of all input values for a relation is called the domain, and the set of all output values is called the range.

Example 2 Determining Whether a Relation is a Function

A relation in which the first coordinates is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range. Although a single input in a function cannot be mapped to more than one output, two or more different inputs can be mapped to the same output.

43

1.2.1 Relations and Functions (continued) Example 3 Using the Vertical Line Test

Every point on a vertical line has the same x -coordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x -coordinate. Therefore the relation is not a function.

44

1.2.1 Relations and Functions (continued)

45

1.2.1 Relations and Functions - Worksheet

Example 1: Give the domain and range for each relation. 1.

2. Average Movie Ticket Price Year Price 2000 \$5.39 2001 \$5.65 2002 \$5.80 2003 \$6.03

Example 2: Determine whether each relation is a function. 3. Math Test Scores Name Jan Helen Luke Soren Score 90 84 88 84

4. from car models to car colors

Example 3: Use the vertical-line test to determine whether each relation is a function. If not, identify two points a vertical line would pass through. 5. 6. 7.

3

3

2

–3

3

–2

4

–3

3

–2

–3

–3

46

1.2.1 Relations and Functions - Practice

1.  Identify the domain and range for the relation. {(4, 3), (7, 6), (10, 9), (13, 6), (16, 9)}

2.  Identify the relation that is a function. ○ day of the week to the date ○ income to color of their car ○ shoe size to a person’s name ○ professional golfers to their rankings 4.  Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.

3.  Identify the relation that is not a function. ○ time of day to the temperature at that time ○ weight of an apple to the apple’s cost ○ number of days to the height of a tree ○ weight of a person to a person’s height

5.  Use the vertical line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.

6.  Give the domain and range of the relation. Indicate whether the relation is a function. x y 5 − 3 6 − 2

6 0 7 4

47

1.2.2 Function Notation Key Objectives • Evaluate functions. • Write functions using function notation. Key Terms • An independent variable is the input of a function; a variable whose value determines the value of the output, or dependent variable . • Function notation is the notation used to express the equation of a function. If x is the independent variable and y is the dependent variable, then the function can be written using f ( x ) for y , where f names the function. Some sets of ordered pairs can be described by using an equation. When the set of ordered pairs described by an equation satisfies the definition of a function, the equation can be written in function notation. If x is the function’s input and y is its output, then in function notation f ( x ) replaces y , and f names the function. For example, the function described by the equation y = 5 x + 3 is the same as the function described by f ( x ) = 5 x + 3. And both of these functions are the same as the set of ordered pairs ( x , 5 x + 3). Note that the graph of a function is a picture of the function’s ordered pairs. Example 1 Evaluating Functions To evaluate a function, find the output value(s) that corresponds to a given input value.

48

1.2.2 Function Notation (continued) Example 2 Graphing Functions

In the notation f ( x ), f is the name of the function. The output f ( x ) of a function is called the dependent variable because it depends on the input value of the function. The input x is called the independent variable. When a function is graphed, the independent variable is graphed on the horizontal axis (the x -axis) and the dependent variable is graphed on the vertical axis (the y -axis).

49

1.2.2 Function Notation (continued) Example 3 Writing Functions

The algebraic expression used to define a function is called the function rule. For example, the function described by the equation f ( x ) = 5 x + 3 is defined by the function rule 5 x + 3. To write a function rule, first identify the independent and dependent variables.

50

1.2.2 Function Notation - Worksheet

Example 1: For each function, evaluate f (0), f (1.5), and f ( − 4). 1. f ( x ) = 3 x − 4 2. f ( x ) = x 2 + 9

3. f ( x ) = 3 x 2 − x + 2

4.

5.

6.

2

f

4

f

f

4

–3

3

–2

–4

2

–4

4

Example 2: Graph each function. 7. 2 3 4 5

8. g ( x ) = − 3 x + 12

9.

Length of stay (nights)

Daily rate (\$)

1

65 55 45 40

5 10 15

2 5 10 17

Example 3: 10.  A furniture company misprinted a sales ad for a living room set but honors the advertised price. For each customer who purchases the living room set, the company suffers a loss of \$125. Write a function to represent the company’s total loss. What is the value of the function for an input of 50, and what does it represent?

51

1.2.2 Function Notation - Practice

1.  For the function f ( x ) = x 2 − 2, identify the values of f (0), f (1/2), and f ( − 1).

2.  For the function in the graph, identify the values of f (0), f (2), and f ( − 3).

4.  Graph.

3.  Graph.

f ( x ) = − 7 x + 3

f ( x ) = 2 x + 3

6.  A truck rental company charges \$65 plus \$32 per hour to rent a truck. Identify the function that represents the total charge for renting a truck for a certain number of hours. What is the value of the function for an input of 6, and what does it represent?

5.  An interior designer charges \$80 to visit a site, plus \$65 to design each room. Identify a function that represents the total amount he charges for designing a certain number of rooms. What is the value of the function for an input of 5, and what does it represent?

52

1.2.3 Exploring Transformations Key Objectives

• Apply transformations to points and functions. • Interpret transformations of real-world data. Key Terms • A transformation is a change in position or size of a figure.

• An image is a shape that results from a transformation of a figure known as the preimage . • A translation , or slide, is a transformation that moves every point of a figure the same distance and direction. • A reflection is a transformation across a line, called the line of reflection, that results in a mirror image. Each point and its image are the same distance from the line of reflection. • A compression is a transformation that pushes the points of a graph horizontally toward the y -axis or vertically toward the x -axis. • A stretch is a transformation that pulls the points of a graph horizontally away from the y -axis or vertically away from the x -axis. Example 1 Translating Points A translation is a transformation where each point in a figure is moved the same distance in the same direction. When a figure is translated left or right, called a horizontal translation, each point shifts left or right by a number of units and each of the figure’s x -coordinates changes by that number of units. When a figure is translated up or down, called a vertical translation, each point shifts up or down by a number of units and each of the figure’s y -coordinates changes.

53

1.2.3 Exploring Transformations (continued) Example 2 Translating and Reflecting Functions

A reflection is a transformation that flips a figure across a line called the line of reflection. Each reflected point in the image is the same distance from the line of reflection as its corresponding point in the preimage, but on the opposite side of the line of reflection. For a figure reflected across the y -axis, the x -coordinate of each point in the image is the opposite of each corresponding x -coordinate in the preimage. For a figure reflected across the x -axis, the y -coordinate of each point in the image is the opposite of each corresponding y -coordinate in the preimage.

54

1.2.3 Exploring Transformations (continued) Example 3 Stretching and Compressing Functions Horizontal and vertical stretches and compressions are tranformations where the points on a figure are either pulled away from or pushed towards the x - or y -axis. • In a horizontal stretch, each point is pulled away from the y -axis and each x -coordinate changes. • In a vertical stretch, each point is pulled away from the x -axis and each y -coordinate changes. • In a horizontal compression, each point is pushed towards the y -axis and each x -coordinate changes. • In a vertical compression, each point is pushed towards the x -axis and each y -coordinate changes.

55

1.2.3 Exploring Transformations (continued) Example 4 Transformation Application

56

1.2.3 Exploring Transformations - Worksheet

Example 1: Perform the given translation on the point (4, 2) and give the coordinates of the translated point. 1. 5 units left 2. 3 units down 3. 1 unit right, 6 units up

Example 2: Use a table to perform each transformation of y = f ( x ). Use the same coordinate plane as the original function. 4. translation 2 units up

3

f

–2

2

4

6

–3

5. reflection across the y -axis

6. reflection across the x -axis

Example 3: Use a table to perform each transformation of y = f ( x ). Use the same coordinate plane as the original function. 7. horizontal stretch by a factor of 3

4

f

–4

4

–2

9. vertical compression by a factor of 1 3

8. vertical stretch by a factor of 3

57

1.2.3 Exploring Transformations - Worksheet (continued)

Example 4: The graph shows the price for admission by age at a local zoo. Sketch a graph to represent each situation and identify the transformation of the original graph that it represents. 10. Admission is half price on Wednesdays.

8

6

4

2

0

20 30

10

40 50 60

Age (yr)

11. To raise funds for endangered species, the zoo charges \$1.50 extra per ticket.

12. The maximum age for each ticket price is increased by 5 years.

58

1.2.3 Exploring Transformations - Practice

1.  Identify the coordinates of the point (2, − 1), translated 6 units left and 3 units up.

2.  Create a table that shows the reflection of f ( x ) across the y -axis. x f ( x ) − 5 − 3

− 1 2 2 4 6 − 1

4.  Create a table that shows the reflection of f ( x ) across the x -axis. x f ( x ) − 5 − 3

3.  The graph of y = f ( x ) is given below. Use a table to perform the reflection of y = f ( x ) across the y -axis. Graph the reflection.

− 1 2 2 4 6 − 1

5.  The graph of y = f ( x ) is given below. Use a table to perform the reflection of y = f ( x ) across the x -axis. Graph the reflection.

6.  The graph of y = f ( x ) is given below. Use a table to stretch the function y = f ( x ) vertically by a factor of 2. Graph the transformation.

59

1.2.4 Introduction to Parent Functions Key Objectives • Identify parent functions from graphs and equations. • Identify transformations of parent functions. • Use parent functions to model real-world data sets and make estimates for unknown values. Key Terms • A parent function is the most basic function of a family of functions; the original function before a transformation is applied. Similar to the way that numbers are classified into sets based on common characteristics, functions can be classified into families of functions. The parent function is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function.

Example 1 Identifying Transformations of Parent Functions The graphs of functions in the same family have the same basic shape.

60

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