# Algebra 2 Companion Book, Volume 2

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 6 - Polynomial Functions Ch 7 - Exponential and Logarithmic Functions Ch 8 - Rational and Radical Functions Ch 9 - Properties and Attributes of Functions

A companion book for Thinkwell's Algebra 2 online video course. Volume #2 covers Chapters 6 ‒ 9.

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 6: Polynomial Functions��������������������������������������������������������������������������������������������������5 6.1 Operations with Polynomials��������������������������������������������������������������������������������������7 6.1.1 Polynomials �����������������������������������������������������������������������������������������������9 6.1.2 Adding and Subtracting Polynomials �������������������������������������������������������� 15 6.1.3 Multiplying Polynomials ����������������������������������������������������������������������������20 6.1.4 Factoring Polynomials ������������������������������������������������������������������������������27 6.1.5 Dividing Polynomials ��������������������������������������������������������������������������������33 6.1.6 Synthetic Division and the Remainder and Factor Theorems ������������������ 39 6.1 Review Worksheet, Part 1 ��������������������������������������������������������������������������47 6.1 Review Worksheet, Part 2 ��������������������������������������������������������������������������49 6.2 Roots and Graphs of Polynomial Functions ������������������������������������������������������������51 6.2.1 Finding Real Roots of Polynomial Equations ��������������������������������������������53 6.2.2 Fundamental Theorem of Algebra ������������������������������������������������������������ 59 6.2.3 Investigating Graphs of Polynomial Functions �����������������������������������������65 6.2.4 Transforming Polynomial Functions ��������������������������������������������������������� 73 6.2.5 Curve Fitting with Polynomial Models ������������������������������������������������������80 6.2 Review Worksheet, Part 1 ��������������������������������������������������������������������������85 6.2 Review Worksheet, Part 2 ��������������������������������������������������������������������������88 Chapter 7: Exponential and Logarithmic Functions��������������������������������������������������������������������91 7.1 Exponential Functions and Logarithms��������������������������������������������������������������������93 7.1.1 Exponential Functions, Growth, and Decay ���������������������������������������������95 7.1.2 Inverses of Relations and Functions ������������������������������������������������������100 7.1.3 Logarithmic Functions ����������������������������������������������������������������������������106 7.1.4 Properties of Logarithms �������������������������������������������������������������������������112 7.1 Review Worksheet �������������������������������������������������������������������������������������118 7.2 Applying Exponential and Logarithmic Functions �������������������������������������������������� 123 7.2.1 Exponential and Logarithmic Equations and Inequalities �����������������������125 7.2.2 The Natural Base, E ������������������������������������������������������������������������������� 132 7.2.3 Transforming Exponential and Logarithmic Functions ��������������������������� 137 7.2.4 Curve Fitting With Exponential and Logarithmic Models ������������������������143 7.2 Review Worksheet ������������������������������������������������������������������������������������148

Chapter 8: Rational and Radical Functions������������������������������������������������������������������������������������153 8.1 Rational Functions��������������������������������������������������������������������������������������������������������155 8.1.1 Variation Functions ��������������������������������������������������������������������������������������157 8.1.2 Multiplying and Dividing Rational Expressions ���������������������������������������������165 8.1.3 Adding and Subtracting Rational Expressions ���������������������������������������������172 8.1.4 Rational Functions ����������������������������������������������������������������������������������������180 8.1.5 Solving Rational Equations and Inequalities ������������������������������������������������189 8.1 Review Worksheet, Part 1 ������������������������������������������������������������������������������197 8.1 Review Worksheet, Part 2 ������������������������������������������������������������������������������201 8.2 Radical Functions ��������������������������������������������������������������������������������������������������������203 8.2.1 Radical Expressions and Rational Exponents ���������������������������������������������205 8.2.2 Radical Functions ����������������������������������������������������������������������������������������212 8.2.3 Solving Radical Equations and Inequalities �������������������������������������������������220 8.2 Review Worksheet ������������������������������������������������������������������������������������������229 Chapter 9: Properties and Attributes of Functions�������������������������������������������������������������������������233 9.1 Functions and Their Graphs ����������������������������������������������������������������������������������������235 9.1.1 Multiple Representations of Functions ���������������������������������������������������������237 9.1.2 Piecewise Functions ������������������������������������������������������������������������������������243 9.1.3 Transforming Functions ��������������������������������������������������������������������������������248 9.1 Review Worksheet, Part 1 ������������������������������������������������������������������������������254 9.2 Functional Relationships ��������������������������������������������������������������������������������������������259 9.2.1 Operations with Functions ��������������������������������������������������������������������������� 261 9.2.2 Functions and Their Inverses ����������������������������������������������������������������������� 268 9.2.3 Modeling Real-World Data ����������������������������������������������������������������������������274 9.2 Review Worksheet ������������������������������������������������������������������������������������������281 Formulas & Symbols����������������������������������������������������������������������������������������������������������������������285

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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.

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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.

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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!

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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.)

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4

Chapter 6 Polynomial Functions

5

6.1

Operations with Polynomials

7

6.1.1 Polynomials Key Objectives • Find the degree of polynomials. • Write polynomials in standard form. • Identify the leading coefficient of polynomials.

• Classify polynomials. • Evaluate polynomials. Key Terms • A monomial is a number or a product of numbers and variables with exponents that are whole numbers. • A binomial is a polynomial with two terms. • A trinomial is a polynomial with three terms. • A polynomial is one monomial or the sum or difference of monomials. • The degree of a monomial is the sum of the exponents of the variables in the monomial. • The degree of a polynomial is equal to the degree of the monomial in the polynomial with the greatest degree. • A polynomial in one variable is in standard form when the terms are written in descending order by degree. • A leading coefficient is the coefficient of the first term of a polynomial in standard form. A polynomial can be classified by its degree and the number of terms, as shown in the tables below. For example, a polynomial with degree 1 and two terms is a linear binomial. Identifying the degree of a polynomial is discussed in Example 1.

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6.1.1 Polynomials (continued) Example 1 Classifying Polynomials In Example 1, Prof. Burger finds the degree of polynomials.

The degree of a monomial with exactly one variable is equal to the variable’s exponent. The degree of a monomial with more than one variable is equal to the sum of the variable’s exponents. The degree of a monomial with no variable is 0. The degree of a polynomial is equal to the degree of the monomial term in the polynomial with the greatest degree.

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6.1.1 Polynomials (continued)

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6.1.1 Polynomials (continued) Example 2 Evaluating Polynomials In Example 2, Prof. Burger demonstrates evaluating a polynomial in function notation for a given value of the variable.

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6.1.1 Polynomials - Worksheet

Example 1: Identify the degree of each monomial. 1. − 7 x 2. 4 x 2 y 3

3. 13

4. m 3 n 2 p

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. 5. 4 x + 2 x 2 − 7 + x 3 6. 3 x 2 + 5 x − 4

7. 5 x 2 − 4 x 3

8. 4 x 4 + 8 x 2 + 1 − 3 x

Example 2: 9.  The sum of the squares of the first n natural numbers is given by the polynomial function = + + n n n n ( ) . F 3 2

1 3

1 2

1 6

a . Evaluate F ( n ) for n = 5 and n = 10.

b . Describe what the values F (5) and F (10) represent.

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6.1.1 Polynomials - Practice

2.  Find the degree of the polynomial 6 x 3 y 6 + 2 xy + x 4 .

1.  Find the degree of the monomial − 6 x 2 y 5 .

3.  Classify the polynomial according to its degree and number of terms. 8 x

4.  Classify the polynomial according to its degree and number of terms. 7 b 3 + 3 b 2 − 7 b

5.  Evaluate g ( x ) = 1.8 x 3 − 0.0034 x + 0.5 for x = 1 and x = 2.

6.  A toy rocket is launched from a platform 42 feet above the ground at a speed of 91 feet per second. The height of the rocket in feet is given by the polynomial − 16 t 2 + 91 t + 42, where t is the time in seconds. How high will the rocket be after 3 seconds?

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6.1.2 Adding and Subtracting Polynomials Key Objectives • Add polynomials horizontally. • Write polynomial expressions for perimeter.

• Find the opposite of polynomials. • Subtract polynomials horizontally. • Write polynomial expressions for profit. • Evaluate polynomial expressions. Formulas • Perimeter of a Rectangle P = 2 L + 2 w , where L and w are the length and width of the rectangle, respectively. • Profit Formula P = R − C , where R is revenue and C is cost. Operations can be performed on polynomials, just as they can be performed on numbers. Example 1 Adding Polynomials Horizontally To add polynomials, combine like terms. Recall that like terms are two or more terms with the same variable raised to the same power. In Example 1, Prof. Burger adds polynomials by applying the Associative Property (to remove the parentheses) and then combining like terms.

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6.1.2 Adding and Subtracting Polynomials (continued) Example 2 Art Application

In Example 2, Prof. Burger writes a polynomial expression to represent the perimeter of a rectangle. The Distributive Property is used to simplify the expression. Recall that by the Distributive Property, a ( b + c ) = ab + ac.

Example 3 Finding the Opposite of a Polynomial To find the opposite of a polynomial, multiply the polynomial by − 1, or write the opposite of each term in the polynomial.

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6.1.2 Adding and Subtracting Polynomials (continued) Example 4 Subtracting Polynomials Horizontally

Recall that subtracting a number is the same as adding its opposite (e.g., 3 − 5 = 3 + − 5). To subtract a polynomial, add the opposite of that polynomial and then combine like terms, as demonstrated by Prof. Burger in Example 4.

In Example 5, Prof. Burger writes a polynomial expression for the profit generated by selling x model kits, using the fact that profit is equal to revenue minus cost. He then evaluates that expression for x = 300,000 to find the profit earned for selling 300,000 model kits.

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6.1.2 Adding and Subtracting Polynomials - Worksheet

Example 1: Add. 1. (5 n 3 + 3 n + 6) + (18 n 3 + 9)

2. (3.7 q 2 − 8 q + 3.7) + (4.3 q 2 − 2.9 q + 1.6)

3. ( − 3 x + 12) + (9 x 2 + 2 x − 18)

4. (9 x 4 + x 3 ) + (2 x 4 + 6 x 3 − 8 x 4 + x 3 )

Example 2: 5. Write a polynomial that represents the measure of angle ABD .

C

( 8 a 2 – 2 a + 5 ) °

A

(7a + 4)°

B

D

Example 3: Find the opposite of each polynomial. 6. 0.32 r 3 + 0.19 r 4

7. 9 y 2 − 6 x 2

8. − 4 + w − 2 z

Example 4: Subtract. 9. (6 c 4 + 8 c + 6) − (2 c 4 )

10. (16 y 2 − 8 y + 9) − (6 y 2 − 2 y + 7 y )

11. (2 r + 5) − (5 r − 6)

12. ( − 7 k 2 + 3) − (2 k 2 + 5 k − 1)

Example 5: 13.  The length of line segment AC is (15 x − y ) cm and the length of line segment AB is (8 x + 3 y − 1) cm. Write a polynomial that represents the length of line segment BC.

8 x + 3 y – 1

A

B

C

15 x – y

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6.1.2 Adding and Subtracting Polynomials - Practice

(5 c 5 − c 3 ) + ( c 5 + 4 c 3 − 3)

4 m 2 − 10 m 3 − 3 m 2 + 20 m 3

4.  Subtract.

3.  Subtract.

( m 2 + n 2 ) − (4 m 2 + 7 n 2 )

(10 d 4 − d 3 ) − ( d 4 + 5 d 3 − 1)

6.  A company distributes its product by train and by truck. The cost of

5.  Subtract.

( z 4 + 5 z 2 − 2 z ) − ( z 2 + 3 z 4 − z − 6)

distributing by train can be modeled as − 0.07 x 2 + 40 x − 105, and the cost of distributing by truck can be modeled as − 0.03 x 2 + 22 x − 180, where x is the number of tons of product distributed. Write a polynomial that represents the difference between the cost of distributing by train and the cost of distributing by truck.

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6.1.3 Multiplying Polynomials Key Objectives • Multiply polynomials. • Multiply special cases of polynomials. Example 1 Multiplying Polynomials

To multiply a monomial by a monomial, use the Product of Powers Property. To simplify a power of a monomial, use the Power of a Power Property. To multiply a monomial by a polynomial, use the Distributive Property and properties of exponents, as applicable.

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6.1.3 Multiplying Polynomials (continued)

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6.1.3 Multiplying Polynomials (continued) Example 2 Multiplying Polynomials: More Polynomial Tricks Recall that to multiply a binomial by a binomial, use the Distributive Property twice or use the FOIL method.

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6.1.3 Multiplying Polynomials (continued)

Example 3 Multiplying Polynomials: Special Cases

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6.1.3 Multiplying Polynomials (continued)

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6.1.3 Multiplying Polynomials - Worksheet

Example 1 and 2: Find each product. 1. − 4 c 2 d 3 (5 cd 2 + 3 c 2 d )

2. 3 x 2 (2 y + 5 x )

3. xy (5 x 2 + 8 x − 7)

4. 2 xy (3 x 2 − xy + 7)

5. ( x − y )( x 2 + 2 xy − y 2 )

6. (3 x − 2)(2 x 2 + 3 x − 1)

7. ( x 3 + 3 x 2 + 1)(3 x 2 + 6 x − 2)

8. ( x 2 + 9 x + 7)(3 x 2 + 9 x + 5)

9.  A businessman models the number of items (in thousands) that his company sold from 1998 through 2004 as N ( x ) = − 0.1 x 3 + x 2 − 3 x + 4 and the average price per item (in dollars) as P ( x ) = 0.2 x + 5, where x represents the number of years since 1998. Write a polynomial R ( x ) that can be used to model the total revenue for this company.

Example 3: Find each product. 10. ( x − 5 y ) 2

11. ( x + 2) 3

12. ( x − 3 y ) 3

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6.1.3 Multiplying Polynomials - Practice

2. Multiply.

1. Multiply.

( y 2 − 5 y + 10)( y + 3)

4 x 3 y (3 x 2 − 5 y )

4.  The number of items is modeled by h ( x ) = 0.42 x 2 + 0.3 x + 4, and the cost per item is modeled by

3.  Multiply.

( p 2 − 5 p + 2)( p 2 − p + 6)

r ( x ) = − 0.005 x 2 − 0.2 x + 7. The total cost is the product of the number of items and the cost per item. Write a polynomial q ( x ) that can be used to model the total cost.

6.  Multiply. ( z − 3) 3

5.  Expand the expression. (2 m − 3 n ) 3

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6.1.4 Factoring Polynomials Key Objectives • Use various methods to factor polynomials. Key Terms • Grouping is a method used in factoring that calls for terms to be grouped together to find common factors. Example 1 Factoring Polynomials by Grouping Some four-term polynomials can be factored into two binomials using the grouping method.

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6.1.4 Factoring Polynomials (continued)

Example 2 Factoring Polynomials Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring

the sum or difference of two cubes. Factoring the Sum of Two Cubes A 3 + B 3 = ( A + B )( A 2 − AB + B 2 ) Factoring the Difference of Two Cubes A 3 − B 3 = ( A − B )( A 2 + AB + B 2 )

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6.1.4 Factoring Polynomials (continued)

Example 3 Methods of Factoring

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6.1.4 Factoring Polynomials (continued)

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6.1.4 Factoring Polynomials - Worksheet

Example 1: Factor each expression. 1. x 3 + x 2 − x − 1

2. x 3 + 5 x 2 − 4 x − 20

3. 8 x 3 + 4 x 2 − 2 x − 1

4. 2 x 3 − 2 x 2 − 8 x + 8

5. 2 x 3 − 3 x 2 − 2 x + 3

6. 12 x 2 + 3 x − 24 x − 6

Example 2 and 3: Factor each expression. 7. 8 − m 6

8. 2 t 7 + 54 t 4

9. x 3 + 64

10. 27 + x 3

11. 4 t 5 − 32 t 2

12. y 3 − 125

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6.1.4 Factoring Polynomials - Practice

2. Factor.

1. Factor.

8 m 3 + 27

14 ab − 12 a − 21 b + 18

4.  Factor.

3.  Factor.

250 b 6 − 16 b 3

64 a 3 − 1

6.  Factor.

5.  Factor.

x 4 − 120 x 2 − 121

128 b 5 − 54 b 2

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6.1.5 Dividing Polynomials Key Objectives • Divide polynomials by monomials and by binomials. • Perform long division of polynomials. Example 1 Dividing a Polynomial by a Monomial To divide a polynomial by a monomial, first write division as a rational expression. Then divide each term in the polynomial (numerator) by the monomial (denominator).

Example 2 Dividing a Polynomial by a Binomial Use the following steps to divide a polynomial by a binomial. 1. Factor the numerator and denominator, if possible. 2. Divide out any common factors. 3. Simplify.

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6.1.5 Dividing Polynomials (continued) Example 3 Polynomial Long Division Use the following steps to divide polynomials using long division, as demonstrated by Prof. Burger in Example 3. 1. Write the binomial and the polynomial in long division form, with the expressions in standard form. 2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. 3. Multiply the first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms. 4. Subtract the product from the dividend. 5. Bring down the next term in the dividend. 6. Repeat Steps 2-5 as necessary until the difference is a 0 or until the degree of the remainder is less than the degree of the binomial.

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6.1.5 Dividing Polynomials (continued) Example 4 Long Division with a Remainder

When dividing polynomials, if the divisor (denominator) is a factor of the dividend (numerator), then the remainder of long division will be 0. If the divisor is not a factor of the dividend, then the remainder is not 0. The remainder can be written as a rational expression, as demonstrated by Prof. Burger in Example 4.

Example 5 Dividing Polynomials That Have a Zero Coefficient In Example 5, Prof. Burger demonstrates using long division when a place holder is needed for a term with a zero coefficient.

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6.1.5 Dividing Polynomials - Worksheet

Example 1: Divide. 1. (4 x 2 − x ) ÷ 2 x

2. (16 a 4 − 4 a 3 ) ÷ 4 a

3. (21 b 2 − 14 b + 24) ÷ 3 b

4. (18 r 2 − 12 r + 6) ÷ − 6 r

5. (6 x 3 + 12 x 2 + 9 x ) ÷ 3 x 2

6. (5 m 4 + 15 m 2 − 10) ÷ 5 m 3

Example 2: Divide. 7. − − + x x x 2 3 1 2

+ − − y y 6 11 10 3 2 2 y

a a a 2

12

8.

9.

− − − 4

x 2 + + +

p p p 2

10. − + − t t t 2

11. x

x 16 15 15

20

6 8 4

12.

− − + 4

Example 3: Divide using long division. 13. ( c 2 + 7 c + 12) ÷ ( c + 4)

14. (3 s 2 − 12 s − 15) ÷ ( s − 5)

x x x 2

x x x 2

5 14 7

4 12 2

15.

16.

+ − +

+ − −

36

6.1.5 Dividing Polynomials - Worksheet (continued)

Example 4: Divide using long division. 17. ( a 2 + 4 a + 3) ÷ ( a + 2)

18. (2 r 2 + 11 r + 5) ÷ ( r − 3)

19. ( n 2 + 8 n + 15) ÷ ( n + 4)

20. (2 t 2 − t + 4) ÷ ( t − 1)

21. (8 n 2 − 6 n − 7) ÷ (2 n + 1)

22. ( b 2 − b + 1) ÷ ( b + 2)

Example 5: Divide using long division. 23. (3 x − 2 x 3 − 10) ÷ (3 + x )

24. (3 p 3 − 2 p 2 − 4) ÷ ( p − 2)

25. ( m 2 + 2) ÷ ( m − 1)

26. (3 x 2 + 4 x 3 − 5) ÷ (5 + x )

27. (4 k 3 − 2 k − 8) ÷ ( k + 1)

28. ( j 3 + 6 j + 2) ÷ ( j + 4)

37

6.1.5 Dividing Polynomials - Practice

2. Divide.

1. Divide.

2

c

− + − c c

17 70 10

( − 20 y 4 + 10 y 2 − 15 y ) ÷ 5 y

4.  Divide using long division. ( y 2 − 11 y + 15) ÷ ( y − 4)

3.  Divide using long division. ( y 2 − 8 y + 18) ÷ ( y − 5)

6.  Divide using long division. (7 n 2 + 3 n 3 + 8) ÷ ( n + 2)

5.  Divide.

(3 x 3 + 2 x 2 + 10) ÷ ( x − 3)

38

6.1.6 Synthetic Division and the Remainder and Factor Theorems Key Objectives

• Use synthetic division to divide polynomials. • Use the Remainder and Factor Theorems. Example 1 Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear binomial by using only the coefficients. When using synthetic division, the polynomial must be written in standard form, using 0 as a coefficient for any missing terms, and the divisor must be in the form ( x − a ).

39

6.1.6 Synthetic Division and the Remainder and Factor Theorems (continued) Example 2 Calculating the Remainder By the Remainder Theorem, if the polynomial function P ( x ) is divided by x − a , then the remainder r is P ( a ).

40

6.1.6 Synthetic Division and the Remainder and Factor Theorems (continued) Example 3 Calculating the Remainder: Review

41

6.1.6 Synthetic Division and the Remainder and Factor Theorems (continued) Example 4 The Factor Theorem

42

6.1.6 Synthetic Division and the Remainder and Factor Theorems (continued)

Example 5 The Zeros of the Polynomial

43

6.1.6 Synthetic Division and the Remainder and Factor Theorems (continued)

44

6.1.6 Synthetic Division and the Remainder and Factor Theorems - Worksheet

Example 1: Divide by using synthetic division. 1. (7 x 2 − 23 x + 6) ÷ ( x − 3)

2. ( x 4 − 5 x + 10) ÷ ( x + 3)

3. ( x 2 + x − 42) ÷ ( x + 7)

Example 2 and 3: Use synthetic substitution to evaluate the polynomial for the given value. 4. P ( x ) = 2 x 3 − 9 x 2 + 27 for x = 2 5. P ( x ) = x 2 − x − 30 for x = − 8

1 3

6.

7. P ( x ) = 3 x 5 + 4 x 2 + x + 6 for x = − 1

3

2

= + + + P x x x x x ( ) 3 5 4 2 for

=

Example 4 and 5: Determine whether the given binomial is a factor of the polynomial P ( x ). 8. ( x + 1); P ( x ) = 2 x 4 + 2 x 3 − x 2 − 5 x − 4

9. ( x − 2); P ( x ) = 5 x 3 + x 2 − 7

10. (2 x − 4); P ( x ) = 2 x 5 − 4 x 4 + 2 x 2 − 2 x − 4

45

6.1.6 Synthetic Division and the Remainder and Factor Theorems - Practice

2. Evaluate the polynomial for the given values by using synthetic division. P ( x ) = 2 x 3 + 7 x 2 − 12 for x = − 3 and x = 3

1. Divide using the synthetic division method. (2 x 3 + 7 x 2 + 2 x − 12) ÷ ( x + 3)

4.  Identify the binomial that is a factor of the polynomial P ( x ) = 3 x 3 − 11 x 2 − 2 x + 24.

3.  Identify the binomial that is not a factor of the polynomial P ( x ) = x 3 + 7 x 2 + 4 x − 12.

○ x − 4 ○ x − 2 ○ x + 2 ○ x + 4

○ x + 6 ○ x − 1 ○ x − 4 ○ x + 2

6.  Write an expression for the height of a rectangle if the area is represented by 3 x 3 − x 2 − 6 x − 8 and the base is represented by x − 2.

5.  Find the factors of f ( x ), given that x = − 2 is a zero. f ( x ) = x 3 + x 2 − 14 x − 24

46

6.1 Review Worksheet, Part 1

6.1.1 Polynomials Identify the degree of each monomial. 1. x 8 2. 6 x 3 y

3. 8

4. a 4 b 6 c 3

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. 5. 3 x 3 + 2 x 4 − 7 x + x 2 6. 6 x − 4 x 4 + 5 7

7. 2 x 3 + 10 x − 9

8. 3 x 2 + 2 x 6 − 4 x 4 − 1

9.  The distance d , in centimeters, that a diving board bends below its resting position when you stand at its end is dependent on your distance x , in meters, from the stabilized point. This relationship can be modeled by the function d ( x ) = − 4 x 3 + x 2 . a. Evaluate d ( x ) for x = 1 and x = 2.

b. Describe what the values d (1) and d (2) represent.

6.1.2 Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. 10. (15 x 2 − 3 x + 11) + (2 x 3 − x 2 + 6 x + 1)

11. (12 x − 1 + 2 x 2 ) + ( x 2 + 4)

12. (3 x 2 − 5 x ) − ( − 4 + x 2 + x )

13. ( x 2 − 3 x + 7) − (6 x 2 + 4 x + 12)

47

6.1 Review Worksheet, Part 1 (continued)

Add or subtract. Write your answer in standard form. 14. ( x 2 − 3 x + 4) + ( x 3 + 3 x − 4)

15. ( x 2 − 3 x + 4) − (3 x + x 3 − 4)

16. (5 y 3 − 2 y 2 − 1) − ( y 2 − 2 y − 3)

17. (2 y 2 − 5 y + 3) + ( y 2 − 2 y − 5)

6.1.3 Multiplying Polynomials Find each product. 18. 7 x 3 (2 x + 3)

19. 3 x 2 (2 x 2 + 9 x − 6)

20. xy 2 ( x 2 + 3 xy + 9)

21. 2 r 2 (6 r 3 + 14 r 2 − 30 r + 14)

22. ( x − y )( x 2 − xy + y 2 )

23. (2 x + 5 y )(3 x 2 − 4 xy + 2 y 2 )

24. ( x 3 + x 2 + 1)( x 2 − x − 5)

25. (4 x 2 + 3 x + 2)(3 x 2 + 2 x − 1)

26.  A bottom for a box can be made by cutting congruent squares from each of the four corners of a piece of cardboard. The volume of a box made from an 8.5-by-11-inch piece of cardboard would be represented by V ( x ) = x (11 − 2 x )(8.5 − 2 x ), where x is the side length of one square. a. Express the volume as a sum of monomials.

X

Jill ’s FLOWERS

8.5 in.

X

11 in.

b. Find the volume when x = 1 inch.

48

6.1 Review Worksheet, Part 2

6.1.4 Factoring Polynomials Factor each expression. 1. 8 y 3 − 4 y 2 − 50 y + 25

2. 4 b 3 + 3 b 2 − 16 b − 12

3. 3 p 3 − 21 p 2 − p + 7

4. 3 x 3 + x 2 − 27 x − 9

5. 8 z 2 − 4 z + 10 z − 5

6. 5 x 3 − x 2 − 20 x + 4

7. 125 + z 3

8. s 6 − 1

9. 24 n 2 + 3 n 5

10. 6 x 4 − 162 x

11. 40 − 5 t 3

12. y 5 + 27 y 2

6.1.5 Dividing Polynomials Divide by using long division. 13. (20 x 2 − 13 x + 2) ÷ (4 x − 1) 14. ( x 2 + x − 1) ÷ ( x − 1)

15. ( x 2 − 2 x + 3) ÷ ( x + 5)

16. (2 x 2 + 10 x + 8) ÷ (2 x + 2)

17. (9 x 2 − 18 x ) ÷ (3 x )

18. ( x 3 + 2 x 2 − x − 2) ÷ ( x + 2)

19. ( x 4 − 3 x 3 − 7 x − 14) ÷ ( x − 4) 20. ( x 6 − 4 x 5 − 7 x 3 ) ÷ (2 x 3 )

21. (6 x 2 − 7 x − 5) ÷ (3 x − 5)

6.1.6 Synthetic Division and the Remainder and Factor Theorems Divide by using synthetic division. 22. ( x 2 + 5 x + 6) ÷ ( x + 1) 23. ( x 4 + 6 x 3 + 6 x 2 ) ÷ ( x + 5) 24. ( x 2 + 9 x + 6) ÷ ( x + 8)

49

6.1 Review Worksheet, Part 2 (continued)

+ − ÷ −    x x

  

1 2

25. (2 x 2 + 3 x − 20) ÷ ( x − 2)

26.

27. (4 x 2 + 5 x + 1) ÷ ( x + 1)

2

x (2 13 8)

Use synthetic substitution to evaluate the polynomial for the given value. 28. P ( x ) = 2 x 2 − 5 x − 3 for x = 4

29. P ( x ) = 4 x 3 − 5 x 2 + 3 for x = − 1

1 3

4 5

30.

31.

2

3

2

P x x x x ( ) 3 5

x

2 for

= − x x ( ) 25 16 for P x

= − − +

=−

=

Determine whether the given binomial is a factor of the polynomial P ( x ). 32. ( x − 3); P ( x ) = 4 x 6 − 12 x 5 + 2 x 3 − 6 x 2 − 5 x + 10

33. ( x − 8); P ( x ) = x 5 − 8 x 4 + 8 x − 64

34. (3 x + 12); P ( x ) = 3 x 4 + 12 x 3 + 6 x + 24

35.  An experimental electrical system has a voltage that can be modeled by V ( t ) = 0.5 t 3 + 4.5 t 2 + 4 t , where t represents time in seconds. The resistance in the system also varies and can be modeled by R ( t ) = t + 1. The current I is related to voltage and resistance by the equation = I V R . Write an expression that represents the current in the system.

50

6.2

Roots and Graphs of Polynomial Functions

51

6.2.1 Finding Real Roots of Polynomial Equations Key Objectives • Use factoring to solve polynomial equations. • Identify the multiplicity of roots. • Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations. Key Terms • The multiplicity of root r is the number of times ( x − r ) appears as a factor of a polynomial P ( x ). Example 1 Using Factoring to Solve Polynomials Recall that by the Zero Product Property, the roots or zeros of the polynomial equation P ( x ) = 0 can be found by setting each factor of P ( x ) equal to 0 and solving for x .

53

6.2.1 Finding Real Roots of Polynomial Equations (continued)

Example 2 Identifying Multiplicity Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. The multiplicity of root r of P ( x ) is the number of times that x − r is a factor of P ( x ).

54

6.2.1 Finding Real Roots of Polynomial Equations (continued) Example 3 Marketing Application Not all polynomials are factorable, but the Rational Root Theorem can help find all possible rational roots of a polynomial equation. By the Rational Root Theorem, if the polynomial P ( x ) has integer coefficients, then every rational root of the polynomial equation P ( x ) = 0 can be written in the form p / q , where p is a factor of the constant term of P ( x ) and q is a factor of the leading coefficient of P ( x ).

55

6.2.1 Finding Real Roots of Polynomial Equations (continued) Example 4 Identifying All of the Real Roots of a Polynomial Equation

56

6.2.1 Finding Real Roots of Polynomial Equations - Worksheet

Example 1: Solve each polynomial equation by factoring. 1. 2 x 4 + 16 x 3 + 32 x 2 = 0

2. x 4 − 37 x 2 + 36 = 0

3. 4 x 7 − 28 x 6 = − 48 x 5

4. 3 x 4 + 11 x 3 = 4 x 2

5. 2 x 3 − 12 x 2 = 32 x − 192

6. x 4 + 100 = 29 x 2

Example 2: Identify the roots of each equation. State the multiplicity of each root. 7. 2 x ( x + 1 )( x − 1)( x + 3)( x + 3) = 0

8. ( x + 2)( x − 2)( x + 2)( x − 2)( x + 2)( x − 2) = 0

Example 3: 9.  A cedar chest has a length that is 3 feet longer than its width and a height that is 1 foot longer than its width. The volume of the chest is 30 cubic feet. What is the width?

Example 4: Identify all of the real roots of each equation. 10. x 3 + 6 x 2 − 5 x − 30 = 0

11. 3 x 3 − 18 x 2 − 9 x + 132 = 0

12. 2 x 3 − 42 x + 40 = 0

13. x 4 − 9 x 2 + 20 = 0

57

6.2.1 Finding Real Roots of Polynomial Equations - Practice

2.  Solve the equation by factoring. x 3 − 16 x = x 2 − 16

1.  Solve the equation by factoring. 2 x 4 + 8 x 3 + 8 x 2 = 0

3.  Identify the roots of the equation and the multiplicities of the roots. ( x − 5) 2 ( x + 2) = 0

4.  Identify the roots of the equation and the multiplicities of the roots. 8( x − 2) 3 = 0

5.  Identify all of the real roots of x 3 − 3 x 2 − 33 x + 35 = 0.

6.  Identify all of the real roots of x 4 − 3 x 3 − 5 x 2 + 13 x + 6 = 0.

58

6.2.2 Fundamental Theorem of Algebra Key Objectives • Use the Fundamental Theorem of Algebra and its corollary to write a

polynomial equation of least degree with given roots. • Identify all of the roots of a polynomial equation.

The following statements about real roots of polynomial equations are equivalent. • A real number r is a root of the polynomial equation P ( x ) = 0. • P ( r ) = 0 • r is an x -intercept of the graph of P ( x ). • x − r is a factor of P ( x ). • When you divide the rule for P ( x ) by x − r , the remainder is 0. • r is a zero of P ( x ). Example 1 Writing Polynomial Functions Given Zeros

To write the simplest polynomial function given its zeros, use the fact that if r is a zero of P ( x ), then x − r is a factor of P ( x ) to write the zeros as binomial factors of P ( x ). Then multiply the factors to write the function in expanded form.

59

6.2.2 Fundamental Theorem of Algebra (continued) Example 2 Finding All Roots of a Polynomial Equation Polynomial functions, like quadratic functions, may have complex zeros that are not real numbers. The Fundamental Theorem of Algebra Every polynomial function of degree n ≥ 1 has at least one zero, where a zero may be a complex number. Every polynomial function of degree n ≥ 1 has exactly n zeros, including multiplicities. Using this theorem, any polynomial function can be written in factored form. To find all roots of a polynomial equation, use a combination of the Rational Root Theorem, the Irrational Root Theorem, and methods for finding complex roots, such as the Quadratic Formula.

60

6.2.2 Fundamental Theorem of Algebra (continued) Example 3 Writing a Polynomial Function with Complex Zeros

The real numbers are a subset of the complex numbers, so every real number is also a complex number. A real number a can be written as the complex number a + 0 i . But here the term complex root will only refer to a root of the form a + bi , where b ≠ 0. Complex roots, like irrational roots, come in conjugate pairs. Recall that the complex conjugate of a + bi is a − bi . Complex Conjugate Root Theorem If a + bi is a root of a polynomial equation with real number coefficients, then a − bi is also a root.

61

6.2.2 Fundamental Theorem of Algebra (continued) Example 4 Problem Solving Application

62

6.2.2 Fundamental Theorem of Algebra - Worksheet

Example 1: Write the simplest polynomial function with the given zeros. 1. 1 3 ,1, 2 2. − 2, 2, 3

1 2

3. − 2,

, 2

Example 2: Solve each equation by finding all roots. 4. x 4 − 81 = 0

5. 3 x 3 − 10 x 2 + 10 x − 4 = 0

6. x 3 − 3 x 2 + 4 x − 12 = 0

Example 3: Write the simplest polynomial function with the given zeros. 7. 1 − i and 2 8. + 1 5 and 3

9. i 2 , 2, and 2

Example 4: 10.  A grain silo is shaped like a cylinder with a cone-shaped top. The cylinder is 30 feet tall. The volume of the silo is 1152 π cubic feet. Find the radius of the silo.

x

x

30 ft

63

6.2.2 Fundamental Theorem of Algebra - Practice

2.  Solve by finding all roots. x 4 − 2 x 3 + x 2 − 8 x − 12 = 0

1.  Write the simplest polynomial function that has the given zeros. − 2, 4, 5

4.  Write the simplest polynomial function having integer coefficients with the given zeros. 4 i , 3, − 1

3.  Write the simplest polynomial function having integer coefficients with the given zeros. − 4,1, 5

5.  A tent is in the form of a right circular cylinder surmounted by a cone. The volume of the tent can be modeled by the function V ( r ) = 8 πr 2 + πr 3 , where r is the radius in feet. For what value of r does the tent have a volume of 311 ft 3 ?

6.  A toy is in the form of a cone mounted on a hemisphere. The volume of the toy is modeled by the function π π = + V r r r ( ) , 3 2 , where r is the radius in centimeters. For what value of r does the toy have a volume of 377 cm 3 ? 1 3 4 3

64

6.2.3 Investigating Graphs of Polynomial Functions Key Objectives • Use properties of end behavior to analyze, describe, and graph polynomial functions. • Identify and use maxima and minima of polynomial functions to solve problems. Key Terms • End behavior is the trend in the y -values of a function as the x -values approach positive and negative infinity. • A turning point is a point on the graph of a function that corresponds to a local maximum (or minimum) where the graph changes from increasing to decreasing (or vice versa). • For a function f , f ( a ) is a local maximum if there is an interval around a such that f ( x ) < f ( a ) for every x -value in the interval except a . • For a function f , f ( a ) is a local minimum if there is an interval around a such that f ( x ) > f ( a ) for every x -value in the interval except a . Polynomial functions are classified by their degree. The graphs of polynomial functions are classified by the degree of the polynomial. Each graph, based on the degree, has a distinctive shape and characteristics. End behavior is a description of the values of the function as x approaches positive infinity ( x → +∞ ) or negative infinity ( x → −∞ ). The degree and leading coefficient of a polynomial function determine its end behavior. Determining the end behavior is helpful when graphing polynomial functions.

65

6.2.3 Investigating Graphs of Polynomial Functions (continued) Example 1 Determining End Behavior of Polynomial Functions When determining end behavior, all terms in the polynomial expression can be ignored except the term with the greatest degree, because both the leading coefficient and the degree of the polynomial can be identified from this term.

Example 2 Using Graphs to Analyze Polynomial Functions

66

6.2.3 Investigating Graphs of Polynomial Functions (continued) Example 3 Graphing Polynomial Functions Use factoring, solving polynomial equations, and the Rational Root Theorem to graph a polynomial function.

Example 4 Determine Maxima and Minima with a Calculator

A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a local maximum or minimum. A graphing calculator can be used to graph and estimate maximum and minimum values. A polynomial of degree n has at most n − 1 turning points and at most n x -intercepts. If the function has n distinct roots, then it has exactly n − 1 turning points and exactly n x -intercepts.

67

6.2.3 Investigating Graphs of Polynomial Functions (continued)

Example 5 Industrial Application

68

6.2.3 Investigating Graphs of Polynomial Functions - Worksheet

Example 1: Identify the leading coefficient, degree, and end behavior. 1. P ( x ) = − 4 x 4 − 3 x 3 + x 2 + 4

2. Q ( x ) = − 2 x 7 + 6 x 5 + 2 x 3

3. R ( x ) = x 5 − 4 x 2 + 3 x − 1

4. S ( x ) = 3 x 2 + 6 x − 10

Example 2: Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient. 5. 6. 7. 8.

Example 3: Graph each function. 9. f ( x ) = x 2 − 5 x − 50

10. f ( x ) =− + + + f x x x x ( ) 3 2 3 2

25 12

69

6.2.3 Investigating Graphs of Polynomial Functions - Worksheet (continued)

Example 4: Graph each function on a calculator, and estimate the local maxima and minima. 11. f ( x ) = x 4 − 4 x 3 + 3 x + 5 12. f ( x ) = 2 x 3 − 3 x 2 − 6 x − 5

Example 5: 13.  Vera has 60 ft of fencing and wants to enclose a patio, using an existing wall for one side as shown. The area of the patio can be modeled by A ( x ) = 60 x − 3 x 2 , where x is in feet. Find the maximum area of the patio.

70

6.2.3 Investigating Graphs of Polynomial Functions - Practice

2. Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

1. Identify the leading coefficient, degree, and end behavior of P ( x ) = − 4 x 3 + 10 x 2 + 1.

4.  Graph f ( x ) = x 3 − 15 x 2 + 72 x − 112.

3.  Graph f ( x ) = x 3 + 5 x 2 + 2 x − 8.

71

6.2.3 Investigating Graphs of Polynomial Functions - Practice (continued)

6.  Graph on a calculator. f ( x ) = + − + f x x x x ( ) 3 2

5.  Graph f ( x ) = − x 3 + 5 x 2 − 3 x + 2 on a calculator and estimate the local maximum and minimum.

2 3

6

72

6.2.4 Transforming Polynomial Functions Key Objectives • Transform polynomial functions. • Write and describe transformations of polynomial functions.

The transformations performed on linear and quadratic functions can also be performed on polynomial functions.

Example 1 Translating a Polynomial Function

73

6.2.4 Transforming Polynomial Functions (continued) Example 2 Reflecting Polynomial Functions

Example 3 Compressing and Stretching Polynomial Functions

74

6.2.4 Transforming Polynomial Functions (continued) Example 4 Combining Transformations

75

6.2.4 Transforming Polynomial Functions (continued) Example 5 Bicycle Sales

76

6.2.4 Transforming Polynomial Functions - Worksheet

Example 1: For f ( x ) = x 4 − 8, write the rule for each function, and sketch its graph. 1. g ( x ) = f ( x ) + 4 2. h ( x ) = f ( x − 2) 3. j ( x ) = f (3 x )

1 2

4. k ( x ) = f ( x ) = − k x f x ( ) ( )

Example 2: Let f ( x ) = − x 3 + 3 x 2 − 2 x + 1. Write a function g that performs each transformation. 5. Reflect f ( x ) across the y -axis. 6. Reflect f ( x ) across the x -axis.

Example 3: Let f ( x ) = x 3 − 4 x 2 + 2. Graph f and g on the same coordinate plane. Describe g as a transformation of f . 7. = g x f x ( ) 8. g ( x ) = 3 f ( x ) 9. g ( x ) = f (2 x ) + 4

  

  

1 2

Example 4: Write a function that transforms f ( x ) = 4 x 3 + 2 in each of the following ways. Support your

solution by using a graphing calculator. 10. Compress vertically by a factor of 1 2

,, and move the y -intercept 2 units down.

11. Reflect across the y -axis, and compress horizontally by a factor of 1 2 ..

12. Move 2 units right, move 3 units down, and reflect across the x -axis.

77

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