lead author chris kirkpatrick -...

Series Author and Senior ConsultantMarian Small

Lead AuthorChris Kirkpatrick

AuthorsMary Bourassa • Crystal Chilvers • Santo D’Agostino

Ian Macpherson • John Rodger • Susanne Trew

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page i

Catherine

Typewritten Text

Extra Practice Worksheets

Series Author and SeniorConsultantMarian Small

Lead AuthorChris Kirkpatrick

AuthorsMary Bourassa, Crystal Chilvers,Santo D’Agostino, Ian Macpherson,John Rodger, Susanne Trew

Contributing AuthorsDan Charbonneau, Ralph Montesanto,Christine Suurtamm

Technology ConsultantIan McTavish

Vice President, PublishingJanice Schoening

General Manager, Mathematics,Science, and TechnologyLenore Brooks

Publisher, MathematicsColin Garnham

Associate Publisher, MathematicsSandra McTavish

Managing Editor, MathematicsErynn Marcus

Product ManagerLinda Krepinsky

Program ManagerLynda Cowan

Developmental EditorsAmanda Allan; Nancy Andraos;Shirley Barrett; Tom Gamblin;Wendi Morrison, First FolioResource Group, Inc.; BobTempleton, First Folio ResourceGroup, Inc.

Contributing EditorsAnthony Arrizza, David Cowan,Beverly Farahani, David Gargaro,Elizabeth Pattison

Editorial AssistantsRachelle BoisjoliKathryn Chris

Executive Director, Content andMedia ProductionRenate McCloy

Director, Content and MediaProductionLinh Vu

Senior Content Production EditorDebbie Davies-Wright

CopyeditorPaula Pettitt-Townsend

ProofreaderJennifer Ralston

Production ManagerHelen Jager-Locsin

Senior Production CoordinatorKathrine Pummell

Design DirectorKen Phipps

Asset CoordinatorSuzanne Peden

Interior DesignMedia Services

Cover DesignCourtney Hellam

Cover Image© CanStock Images / Alamy

Production ServicesNesbitt Graphics Inc.

Director, Asset ManagementServicesVicki Gould

Photo/Permissions ResearcherDavid Strand

PrinterTranscontinental Printing Ltd.

COPYRIGHT © 2010 by NelsonEducation Limited.

ISBN-13: 978-0-17-633202-0ISBN-10: 0-17-633202-2

Printed and bound in Canada2 3 4 5 12 11 10 09

For more information contactNelson Education Ltd., 1120 Birchmount Road, Toronto,ON, M1K 5G4. Or you can visit our Internet site athttp://www.nelson.com

ALL RIGHTS RESERVED. No part ofthis work covered by the copyrightherein, except for any reproduciblepages included in this work, maybe reproduced, transcribed, orused in any form or by any means—graphic, electronic, or mechanical,including photocopying, recording,taping, Web distribution, orinformation storage and retrievalsystems—without the writtenpermission of the publisher.

For permission to use materialfrom this text or product, submit a request online atwww.cengage.com/permissions.Further questions aboutpermissions can be emailed [email protected]

Every effort has been made totrace ownership of all copyrightedmaterial and to secure permissionfrom copyright holders. In theevent of any question arising as tothe use of any material, we will bepleased to make the necessarycorrections in future printings.

Principles of Mathematics 10

00_Nelson_FM_ppi-v.qxd 10/16/09 8:44 AM Page ii

NEL iii

Paul AlvesDepartment Head of MathematicsStephen Lewis Secondary SchoolPeel District School BoardMississauga, ON

Anthony ArrizzaDepartment Head of MathematicsWoodbridge CollegeYork Region District School BoardWoodbridge, ON

Terri BlackwellSecondary Mathematics TeacherBurlington Central High SchoolHalton District School BoardBurlington, ON

Mark CassarPrincipalHoly Cross Catholic Elementary SchoolDufferin-Peel Catholic District School BoardMississauga, ON

Angela ConettaMathematics TeacherChaminade College SchoolToronto Catholic District School BoardToronto, ON

Tamara CoyleTeacherMother Teresa Catholic High SchoolOttawa Catholic School BoardNepean, ON

Justin de WeerdtMathematics Department HeadHuntsville High SchoolTrillium Lakelands District School BoardHuntsville, ON

Sandra Emms JonesMath TeacherForest Heights C.I.Waterloo Region District School BoardKitchener, ON

Beverly FarahaniHead of MathematicsKingston Collegiate and Vocational InstituteLimestone District School BoardKingston, ON

Richard GallantSecondary Curriculum ConsultantSimcoe Muskoka Catholic District School

BoardBarrie, ON

Jacqueline HillK–12 Mathematics FacilitatorDurham District School BoardWhitby, ON

Patricia KehoeItinerant Teacher, Student Success

DepartmentOttawa Catholic School BoardOttawa, ON

Michelle LangConsultant, Learning Services 7–12Waterloo Region District School BoardKitchener, ON

Angelo LilloHead of MathematicsSir Winston Churchill Secondary SchoolDistrict School Board of NiagaraSt. Catharines, ON

Susan MacRurySenior Mathematics TeacherLasalle Secondary SchoolRainbow District School BoardSudbury, ON

Frank MaggioDepartment Head of MathematicsHoly Trinity Catholic Secondary SchoolHalton Catholic District School BoardOakville, ON

Peter MatijosaitisRetiredToronto Catholic District School Board

Bob McRobertsHead of MathematicsDr. G W Williams Secondary SchoolYork Region District School BoardAurora, ON

Cheryl McQueenMathematics Learning CoordinatorThames Valley District School BoardLondon, ON

Kay MinterTeacherCedarbrae C.I.Toronto District School BoardToronto, ON

Reshida NezirevicHead of MathematicsBlessed Mother Teresa C.S.SToronto Catholic District School BoardScarborough, ON

Elizabeth PattisonMathematics Department HeadGrimsby Secondary SchoolDistrict School Board of NiagaraGrimsby, ON

Kathy PilonProgram LeaderSt. John Catholic High SchoolCatholic District School Board of Eastern

OntarioPerth, ON

Jennifer PortelliTeacherHoly Cross Catholic Elementary SchoolDufferin-Peel Catholic District School BoardMississauga, ON

Tamara PorterDepartment Head of MathematicsPrince Edward Collegiate InstituteHastings and Prince Edward District School

BoardPicton, ON

Margaret RussoMathematics TeacherMadonna Catholic Secondary SchoolToronto Catholic District School BoardToronto, ON

Scott TaylorDepartment Head of Mathematics, Computer

Science and BusinessBell High SchoolOttawa-Carleton District School BoardNepean, ON

Joyce TonnerEducatorThames Valley District School BoardLondon, ON

Salvatore TrabonaMathematics Department HeadMadonna Catholic Secondary SchoolToronto Catholic District School BoardToronto, ON

James WilliamsonTeacherSt. Joseph-Scollard Hall C.S.SNipissing-Parry Sound Catholic District School

BoardNorth Bay, ON

Charles WyszkowskiInstructorSchool of Education, Trent UniversityPeterborough, ON

Krista ZupanMath Consultant (Numeracy K–12)Durham Catholic District School BoardOshawa, ON

Reviewers and Advisory Panel

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page iii

NELiv Table of Contents

Table of Contents

Chapter 1: Systems of Linear Equations 2

Getting Started 4

1.1 Representing Linear Relations 8

1.2 Solving Linear Equations 15

1.3 Graphically Solving Linear Systems 21

Curious Math 29

Mid-Chapter Review 30

1.4 Solving Linear Systems: Substitution 33

1.5 Equivalent Linear Systems 41

1.6 Solving Linear Systems: Elimination 49

1.7 Exploring Linear Systems 57

Chapter Review 60

Chapter Self-Test 64

Chapter Task 65

Chapter 2: Analytic Geometry: Line Segments and Circles 66

Getting Started 68

2.1 Midpoint of a Line Segment 72

2.2 Length of a Line Segment 81

2.3 Equation of a Circle 88


2.4 Classifying Figures on a Coordinate Grid 96

2.5 Verifying Properties of Geometric Figures 104

2.6 Exploring Properties of Geometric Figures 111

Curious Math 114

2.7 Using Coordinates to Solve Problems 115

Chapter Review 122


Chapter Task 127

Chapter 3: Graphs of Quadratic Relations 128

Getting Started 130

3.1 Exploring Quadratic Relations 134

3.2 Properties of Graphs of Quadratic Relations 138

Curious Math 149

3.3 Factored Form of a Quadratic Relation 150


3.4 Expanding Quadratic Expressions 161

3.5 Quadratic Models Using Factored Form 169

3.6 Exploring Quadratic and Exponential Graphs 179

Chapter Review 183


Chapter Task 188

Chapters 1–3 Cumulative Review 189

Chapter 4: Factoring AlgebraicExpressions 192

Getting Started 194

4.1 Common Factors in Polynomials 198

4.2 Exploring the Factorization of Trinomials 205

4.3 Factoring Quadratics: 207


4.4 Factoring Quadratics: 217

4.5 Factoring Quadratics: Special Cases 225

Curious Math 232

4.6 Reasoning about Factoring Polynomials 233

Chapter Review 238


Chapter Task 243

ax2+ bx + c

x 2+ bx + c

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page iv

Catherine

Rectangle

Catherine

Rectangle

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Use this page as a guide to the content in the worksheets. Worksheets are in chapter order by section.

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

NEL vTable of Contents

Chapter 5: Applying Quadratic Models 244

Getting Started 246

5.1 Stretching/Reflecting Quadratic Relations 250

5.2 Exploring Translations of Quadratic Relations 259

5.3 Graphing Quadratics in Vertex Form 263


5.4 Quadratic Models Using Vertex Form 275

5.5 Solving Problems Using Quadratic Relations 285

Curious Math 296

5.6 Connecting Standard and Vertex Forms 297

Chapter Review 303


Chapter Task 307

Chapter 6: Quadratic Equations 308

Getting Started 310

6.1 Solving Quadratic Equations 314

6.2 Exploring the Creation of Perfect Squares 322

Curious Math 3246.3 Completing the Square 325


6.4 The Quadratic Formula 336

6.5 Interpreting Quadratic Equation Roots 345

6.6 Solving Problems Using Quadratic Models 352

Chapter Review 360


Chapter Task 364


Chapter 7: Similar Triangles and Trigonometry 368

Getting Started 370

7.1 Congruence and Similarity in Triangles 374

7.2 Solving Similar Triangle Problems 382


7.3 Exploring Similar Right Triangles 391

7.4 The Primary Trigonometric Ratios 394

7.5 Solving Right Triangles 400

Curious Math 407

7.6 Solving Right Triangle Problems 408

Chapter Review 415


Chapter Task 419

Chapter 8: Acute Triangle Trigonometry 420

Getting Started 422

8.1 Exploring the Sine Law 426

8.2 Applying the Sine Law 428


8.3 Exploring the Cosine Law 437

Curious Math 439

8.4 Applying the Cosine Law 440

8.5 Solving Acute Triangle Problems 446

Chapter Review 452


Chapter Task 455


Appendix A:REVIEW OF ESSENTIAL SKILLS AND KNOWLEDGE 459

Appendix B:REVIEW OF TECHNICAL SKILLS 486

Glossary 528

Answers 536

Index 600

Credits 604

6706_Nelson-Math10_FM_ppi-v.qxd 5/13/09 11:38 AM Page v

Catherine

Rectangle

Catherine

Rectangle

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Catherine

Rectangle

Chapter 1 Diagnostic Test STUDENT BOOK PAGES 4–7

1. Complete the table of values for the linear relation 3x + 2y = 18. x –2 4 0 10

y 12 6 –3 0 2. Graph the relation 5y – 4x = 60 by determining its

x- and y-intercepts.

3. a) Using a graphing calculator, enter the relation y = –3x + 7 into the equation editor as Y1. Press GRAPH to view the graph of the relation.

b) Determine the slope and y-intercept of the relation 12x + 4y = 16. Predict how the graph of this relation would compare with the graph of the relation in part a).

c) Write the relation in part b) in the form y = mx + b, enter it as Y2, and press GRAPH. Explain what you see.

4. a) To graph the equation 3x – 6y = 9, would you determine the x- and y-intercepts or would you determine the slope and y-intercept? Explain your choice.

b) You suspect that the equations 4y – 3x + 12 = 0 and 134

=−yx have identical graphs.

Describe a strategy you could use to decide whether your suspicion is correct.

5. Expand and simplify as necessary. a) 12x – 10 – 7x – 5 b) 4(–2x + 7) c) (2x – 3) – (5x – 4) d) 3(x + 7) – 5(6 – 2x)

6. Solve. a) x – 5 = –3 b) 26 = –2x – 8 c) 3x – 10 = –2x d) 3x + 5 = 5x + 17

7. Consider the equations y = m1x + 4 and y = m2x – 2. For what values of m1 and m2 will the graphs of these two equations be identical? For what values will the graphs be parallel? Explain your answers.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 1 Diagnostic Test | 1

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

1. Systems of Linear Equations

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

Chapter 1 Diagnostic Test Answers 1. x –2 4 2 0 8 6 10

y 12 3 6 9 –3 0 –6

2.

x-intercept: –15; y-intercept: 12

3. a)

b) slope: –3; y-intercept: 4; same slope,

different y-intercept (i.e., parallel) c) y = –3x + 4

The two lines are parallel because they have the same slope but different y-intercepts.

4. a) Answers may vary, e.g., I prefer x- and y-intercepts because the equation will not need to be rearranged.

b) Answers may vary, e.g., I could determine the x- and y-intercepts of both lines, and then compare them to see if they are equal.

5. a) 5x – 15 c) –3x + 1 b) –8x + 28 d) 13x – 9

6. a) x = 2 c) x = 2 b) x = –17 d) x = –6

7. The graphs will never be identical because they have different y-intercepts. They will be parallel only when m1 = m2

because the slopes are the same for parallel lines.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

If students have difficulty with the questions on the Diagnostic Test, it may be necessary to review the following topics: • expanding and simplifying algebraic expressions • graphing a linear relation, given the x- and y-intercepts • graphing a linear equation, given the y-intercept and the slope • solving linear equations

2 | Principles of Mathematics 10: Chapter 1 Diagnostic Test Answers

Catherine

Typewritten Text

Lesson 1.1 Extra Practice STUDENT BOOK PAGES 8–14

1. The ordered pair (c, c) satisfies the relation 3x – y = 8. What is the value of c?

2. Define variables for each situation, and write an equation to represent the situation. a) Alexia earns $7.00/h gardening for her aunt

and $8.50/h babysitting. Last month, she earned $98.50.

b) Dieter has a long-distance plan that costs $4.95/month, plus 4¢/min for long-distance calls within Canada and the U.S., and 5¢/min for calls to Europe. Last month, Dieter’s total spending on long-distance calls was $11.95.

3. Graph each equation for question 2.

4. Justin buys almonds at 2.25¢ per gram and dried apricots at 1.5¢ per gram to make a snack mix. He spends $6.00 in total. a) Write an equation to represent this situation,

using x for the amount of almonds, in grams, and y for the amount of dried apricots, in grams.

b) Use a graphing calculator to graph your equation. Remember to use appropriate window settings.

c) Set up and display a table of values for your equation to help you determine the amount of dried apricots Justin bought if he bought 250 g of almonds.

5. One year ago, Renata invested some of her income in two investment accounts: a guaranteed-income fund that pays a return of 4% per year and a high-growth account that, for the year just past, paid out 6%. Renata’s return on her investment is $360. Use two different strategies to represent Renata’s possible investments in each account.

6. Marta works for a company that makes floral arrangements for special events. She is negotiating a new plan for her travelling expenses and suggests 15¢/km as a fair rate. The company suggests an alternative plan of 12¢/km, with an additional flat payout of $12.50/month. In a typical month, Marta drives 700 km on company business. Which plan is better for Marta?

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 1.1 Extra Practice | 3

Lesson 1.1 Extra Practice Answers 1. c = 4

2. a) Let x represent the number of hours spent gardening, and let y represent the number of hours spent babysitting. 7.00x + 8.50y = 98.50

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) Let x represent the number of minutes spent on calls within Canada or the U.S., and let y represent the number of minutes spent on calls to Europe. 0.04x + 0.05y + 4.95 = 11.95

3. a)

b)

4. a) 2.25x + 1.50y = 600 b)

c)

5. Answers may vary, e.g., Table of values:

x 0 1500 3000 4500 6000 7500 9000

y 6000 5000 4000 3000 2000 1000 0

Graph:

Equation: 0.04x + 0.06y = 360 Graphing calculator:

6. The plan that Marta suggested is better.

4 | Principles of Mathematics 10: Lesson 1.1 Extra Practice Answers


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

1. DriveEasy, a car share-rental company, rents vehicles online on a sliding scale, as shown in the graph. a) How much does it cost to

rent a car for 1 h 45 min? b) For how long can you rent

a car if you have $18.00?

2. a) Write an equation for the linear relation in question 1.

b) Use your equation to calculate the answers for question 1.

c) Compare your original answers for question 1 with your answers for part b). Identify one advantage of each strategy.

3. This graph shows the relationship between the cost of a safari tour and the number of tourists.

a) Write an equation for the relation. b) Use your equation to determine the cost

of a safari for six people. c) How many tourists can go on a safari for

$1250? How many can go for 1850? d) Is it possible for a safari to cost $1600?

Explain.

4. Benoit drove at 85 km/h from Barrie to Sudbury. At this speed, his truck uses gas at a rate of 7.5 L/100 km. Benoit left Barrie with 40 L of gas in the tank, and his low fuel warning light came on when 4 L was left. Estimate how long after leaving Barrie the warning light came on.

5. Nita is comparing two job offers. Roxie’s Boutique is offering $3000/month plus 25% commission on sales, and Cherie Womenswear is offering $2400/month plus 40% commission on sales. Compare the two offers. Which job should Nita take? Explain your decision.

6. A train leaves Montréal for Toronto at 9:30 a.m., travelling at 120 km/h. At the same time, a train bound for Montréal leaves Toronto, travelling at 105 km/h. The distance from Montréal to Toronto by train is 520 km. a) Describe two strategies you could use to

estimate when the trains will pass each other. b) Use one of your strategies to estimate when

the trains will pass each other.


Lesson 1.2 Extra Practice Answers 1. a) about $8.60 b) about 4 h 20 min

2. a) y = 3.5x + 2.5 b) $8.63, 4 h 26 min

c) Answers may vary, e.g., using the graph to estimate is quicker, but using the equation is more accurate.

3. a) y = 800 + 150x

c) 3 tourists, 7 tourists b) $1700

d) no, because the corresponding x-value is not a whole number

4. Answers may vary, e.g., about 5 h 40 min

5. Roxie’s pays more than Cherie Womenswear for up to $4000/month in sales. If Nita thinks that she can sell more than $4000, she should choose Cherie Womenswear’s offer; otherwise she should choose Roxie’s offer.

6. a) Answers may vary, e.g., drawing a graph of the distances of the two trains from either Montréal or Toronto; creating a table of values of these distances and determining when these values are equal; writing expressions to represent the distances travelled by the trains, setting the sum of the two expressions for distance equal to 520, and solving for time.

b) Answers may vary, e.g., about 11:50 a.m.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Which ordered pair is a solution to the system of equations x + 2y = 6 and y = 8 – 3x? a) (–2, 2) b) (0, 8) c) (0.5, 6.5) d) (2, 2)

2. Which system of equations matches the graph shown?

a) 3x + y = 2 and 2x – 3y = 6 b) 3x – y = 2 and 2x + 3y = 12 c) 2x – 3y = 12 and y = 3x – 2 d) y = 3x – 2 and 2x + 3y = 6

3. a) Graph the system 3x – y = 5 and x + 2y = 4 by hand.

b) Solve the system using your graph. c) Verify your solution by substituting into the

given equations.

4. Use graphing technology to graph and solve the system 2x – 5y = 3 and x + y = 12.

5. A passenger airplane takes 4 h 15 min for a journey of 3600 km. The airplane travels at a cruising speed of 900 km/h. The mean speed is 600 km/h during takeoff and landing. Use a graphing strategy to estimate the amount of time that the airplane travels at cruising speed.

6. Anya works as a senior sales rep for a computer store. She earns $4500/month plus 5% commission on her monthly sales, but she is considering an offer of $3750/month plus 10% commission from another store. a) Which option is better if Anya’s monthly

sales average $12 000? Which option is better for sales of $20 000?

b) Write equations for Anya’s two options, and graph your equations.

c) How much in monthly sales would Anya have to make for both options to have the same value?

7. Five years ago, a high-school cafeteria charged $5.85 for three pieces of fruit and a chicken salad. Today, each piece of fruit costs 12% more, while a chicken salad costs 15% more. The new cost of three pieces of fruit and a chicken salad is $6.66. Determine the new prices of a piece of fruit and a chicken salad.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 1.3 Extra Practice Answers 1. d)

2. b)

3. a)

b) (2, 1) c) 3(2) – 1 = 5, 2 + 2(1) = 4

4.

5. 3 h 30 min

6. a) Anya’s current job; the other offer b) y = 4500 + 0.05x, y = 3750 + 0.10x

c) $15 000

7. 84¢, $4.14

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Chapter 1 Mid-Chapter Review Extra Practice STUDENT BOOK PAGES 30–32

1. Define variables x and y for each situation, and write an equation to represent the situation. a) Indra earns $20/h at her day job and $12/h at

her evening job. Last month, she earned $3600.

b) Laurent keeps a change jar for snack machines. The jar contains $15.75 in loonies and quarters.

c) Rebecca goes on a road trip, travelling at 100 km/h on six-lane highways and 80 km on other highways. She travels a total distance of 480 km.

2. Graph each equation for question 1.

3. Waterworld Rentals rents windsurfing boards for $32/day and regular surfboards for $20/day. Last Tuesday, Waterworld charged $960 for rentals. Choose two strategies to represent the possible combinations of windsurfing boards and regular surfboards.

4. This graph shows the scale of fares charged by Speedy Taxi Company.

a) What is the minimum fare for a Speedy Taxi trip?

b) What is the fare for a 12 km trip? c) How much extra does each kilometre cost? d) Write an equation to represent Speedy Taxi

fares. Use your equation to determine the fare for a 29 km trip.

5. A 1300 L water tank empties at the rate of 4 L/min. At 3:15 p.m., 170 L of the water is left in the tank. Estimate when the tank was last filled.

6. a) Graph the linear system 3x + 4y = 12 and y = 2x – 3 by hand.

b) Estimate the solution to this system. c) Check your answer for part b) using a

graphing calculator.

7. Elena runs an ice cream store. She sells a 1 L tub of vanilla ice cream for $2.50 and a 1 L tub of mocha ice cream for $3.30. She wants to create a mix called Creamy Coffee Swirl to sell for $3.00 per 1 L tub. How much of each flavour must Elena use for one tub?

8. The equations y = 4, 2x + 3y = 8, and 2x – y = 8 form the sides of a triangle.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

a) Graph the triangle, and determine the coordinates of the vertices.

b) Calculate the area of the triangle.

Chapter 1 Mid-Chapter Review Extra Practice | 9

Chapter 1 Mid-Chapter Review Extra Practice Answers 1. a) Let x represent hours worked at the day

job, let y represent hours worked at the evening job; 20x + 12y = 3600

b) Let x represent number of loonies, let y represent number of quarters; x + 0.25y = 15.75

c) Let x represent time driven on six-lane highways, let y represent time driven on other highways; 100x + 80y = 480

2. a)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b)

c)

3. Answers may vary, e.g., Let x represent Graph: the number of windsurfing boards. Let y represent the number of regular surfboards.

Equation: 32x + 20y = 960

Table of values: x 0 5 10 15 20 25 30

y 48 40 32 24 16 8 0

Graphing calculator:

4. a) $4.50 b) $7.50 c) 25¢ d) y = 0.25x + 4.50, $11.75

5. about 10:30 a.m.

6. a)

b) about (2.2, 1.4) c)

7. 0.375 L of vanilla and 0.625 L of mocha

8. a) b) 16 square units

(–2, 4), (4, 0), (6, 4)

10 | Principles of Mathematics 10: Chapter 1 Mid-Chapter Review Extra Practice Answers


1. Isolate the indicated variable in each equation. a) 3x + y = 7, y c) 4x + 3y = 12, x b) y – 3x + 2 = 0, x d) 20x – 4y = 10, y

2. Solve the linear system x + y = 5 and 3x – 2y = 25.

3. A hat maker at a fair sells two kinds of novelty hats. The banana-split hat sells for $3.50, and the chocolate-sundae hat sells for $4.25. At the end of the day, the hat maker has sold 76 hats and taken in $290 in revenue. How many chocolate-sundae hats were sold?

4. The hat maker in question 3 has to pay $130/day to rent the stall at the fair, and the materials to make each hat cost $2.25. Determine how many hats per day the hat maker must sell to break even if a) only banana-split hats are sold b) only chocolate-sundae hats are sold

5. The difference between two adjacent angles in a parallelogram is 42°. Determine the measures of all four angles in the parallelogram.

6. Without graphing, determine the intersection point of the line 2x + y = 8 and the line passing through (0, 6) and (9, 0).

7. A change jar contains $7.55 in nickels, dimes, and quarters. There are eight more nickels and dimes than quarters, and 50 coins altogether. How many of each coin are in the jar?

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 1.4 Extra Practice Answers 1. a) y = 7 – 3x

b) x = 31 y +

32

c) x = 3 – 43 y

d) y = 5x – 2.5

2. (7, –2)

3. 32

4. a) 104 b) 65

5. 111°, 69°, 111°, 69°

6. (1.5, 5)

7. 12 nickels, 17 dimes, 21 quarters

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. a) Add and subtract the equations in the linear system 5x – 3y = 6 and x + 2y = –4.

b) By graphing, verify that the two new equations have the same solution as the original linear system.

2. a) Multiply 3x – y = 5 by –2, and multiply

2x + 4y = 7 by 21 .

b) Make a prediction about the graphs of the two new equations, compared with the graphs of the original two equations.

c) Suggest and apply a strategy to check your prediction without graphing.

3. a) Multiply x + 3y = 5 by 2, and multiply 3x + 2y = 15 by 3.

b) Create another linear system by adding and subtracting your equations for part a).

c) By graphing, verify that your linear system for part b) is equivalent to the original linear system for part a).

4. The linear system 2x – 3y = 2 and 3y – x = 8 is equivalent to the linear system ax + 13y = –22 and 3x + by = –6. Determine the values of a and b.

5. A student committee sells 104 tickets for a concert. Student tickets cost $9, and non-student tickets cost $12.50. The total revenue from ticket sales is $1135.50. a) Write two equations for this situation: one

equation describing the number of tickets sold, and the other equation describing the revenue.

b) Multiply your equation for the number of tickets by 9. Then subtract this new equation from your revenue equation. How are your two new equations related to your equations for part a)?

c) Determine the number of student tickets that were sold. Explain your strategy.

6. a) Use a substitution strategy to solve the system y + 5x + 32 = 0 and 3y – 4x = 37.

b) Show that multiplying the first equation by 3 and then adding and subtracting the equations forms an equivalent system.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 1.5 Extra Practice Answers 1. a) adding: 6x – y = 2;

subtracting: 4x – 5y = 10 b)

2. a) –6x + 2y = –10, x + 2y = 3.5 b) The graph of –6x + 2y = –10 will be the

same as the graph of 3x – y = 5; the graph of x + 2y = 3.5 will be the same as the graph of 2x + 4y = 7.

c) Answers may vary, e.g., check intercepts: –6x + 2y = –10 and 3x – y = 5 both have

intercepts ⎟⎠⎞

⎜⎝⎛ 0,

35 and (0, –5); x + 2y = 3.5

and 2x + 4y = 7 both have intercepts (3.5, 0) and (0, 1.75).

3. a) 2x + 6y = 10, 9x + 6y = 45 b) adding: 11x + 12y = 55;

subtracting: –7x = –35 c)

4. a = –10, b = –6

5. a) x + y = 104, 9.00x + 12.50y = 1135.50 b) 9x + 9y = 936, 3.50y = 199.50; equivalent

to equations for part a) c) 47; answers may vary, e.g., I solved the

equation 3.50y = 199.50 to determine that y = 57, substituted this value into the equation x + y = 104, and solved for x.

6. a) (–7, 3) b) 6y + 11x = –59, 19x = –133; the solution is

again (–7, 3), so the systems are equivalent.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. For an Open House at a high school, a student-run stall is selling two types of juice drinks: Power Juice and Juice Cooler. The table shows the amounts of pure juice and water in each type.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Type of Drink Amount of Juice (mL)

Amount of Water (mL)

Power Juice 350 150

Juice Cooler 250 750

If the students have 5.5 L of juice and 7.5 L of water, how much of each type of drink can they make?

2. During her morning commute, Rebecca averaged 30 km/h in heavy city traffic and 90 km/h once she got onto the highway. She travelled 35 km in 30 min. How far did Rebecca drive while on the highway?

3. To eliminate x from each linear system, by what numbers would you multiply equations ① and ②? a) 2x – 3y = –1 ① c) 5x + y = 12 ①

4x + y = 9 ② –2x – 3y = 0 ② b) 7x – 5y = 8 ① d) 2x + 2y = 5 ①

y + 3x = 9 ② 3x – 2y = –3 ②

4. A linear system consists of the equations 3x – 2y + 10 = 0 and 5x + 4y = –13. a) Solve the system by eliminating x. b) Solve the system by eliminating y.

5. You need to solve the linear system 3x – 5y = 41 and 2y – 3x = –29. a) Explain which variable you would choose to

eliminate. b) Solve the system by eliminating the variable

you chose.

6. A local charity decides to invest $20 000 in two funds. After one year, the returns on the funds are 4% and 6%. If the total return on the charity’s investment is $1040, how much did the charity invest in the fund returning 4%?

7. Kara thinks of two numbers. She performs the following steps: Step 1: She doubles the first number and subtracts three times the second number. The result is 19. Step 2: She switches the numbers and repeats step 1. The result is –41. Use an elimination strategy to discover Kara’s two numbers.


Lesson 1.6 Extra Practice Answers 1. 5 L of Power Juice, 8 L of Juice Cooler

2. 30 km

3. a) 2 and 1 c) 2 and 5 b) 3 and 7 d) 3 and 2

4. a), b) (–3, 0.5)

5. a) I would eliminate x because I don’t need to multiply either equation by a number. I can just add the equations.

b) (7, –4)

6. $8000

7. 17, 5

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Chapter 1 Review Extra Practice STUDENT BOOK PAGES 60–63

1. Steve has a budget of $25 each month to spend on cell-phone calls and text messages. He pays 15¢/min for airtime and 25¢ per text message. a) Use a table to show Steve’s possible

combinations of calls and text messages in one month.

b) Draw a graph to represent Steve’s possible combinations.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

2. Carly buys a bulk lot of stuffed animals and resells them at the school fair. The graph shows Carly’s net profit or loss.

a) Estimate Carly’s profit or loss if she sells

37 stuffed animals. b) Use the graph to write an equation that

represents Carly’s profit or loss. c) Determine Carly’s exact profit or loss if she

sells 37 stuffed animals.

3. Which system of equations matches the graph shown?

a) 3x + y = 1 and 2x – 5y = 6 b) x + 3y = 1 and x – 2y = 3 c) 3x + y = 1 and 5y – 2x = 6 d) x + 3y = 1 and 2x – 5y = 6

4. a) Isolate y in both equations in the system 4y + 7x = 19 and 5x – 2y = 13.

b) Use graphing technology to solve the system, expressing your answers to the nearest hundredth.

5. A transversal crosses two parallel lines, creating angles that measure a and b as shown. Determine a and b if tripling the angle measure a creates the same measure as doubling the angle measure b.

6. Which linear system is not equivalent to the system y – 2x = 13 and x + 5y = 10? a) x – y = –8 and 3x + 2y = –9 b) 4x – 7y = 41 and y = 3 c) x = –5 and 3x + 5y = 0 d) x – y = –8 and x + y = –2

7. Jamal wants to make a 300 g serving of kiwi and blueberries that contains 75 mg of vitamin C. He knows that 1 g of kiwi contains 0.7 mg of vitamin C, and each gram of blueberries contains 0.1 mg of vitamin C. Use any appropriate strategy to determine how much of each type of fruit Jamal should include.

8. a) Solve the system 3x + 2y = –4 and 5x – 2y = 12 using a substitution strategy.

b) Use an elimination strategy to solve the same linear system.

c) Which strategy is more efficient for solving this system? Explain in terms of both strategies.

9. a) Given the equation 5y – 3x = 15, write another equation to create a linear system with each number of solutions.

i) none ii) one iii) infinitely many b) Verify your answers for part a) graphically.

Chapter 1 Review Extra Practice | 17

Chapter 1 Review Extra Practice Answers 1. a) b)

Calls Text Messages

Number of Minutes

Cost ($)

Number of Messages

Cost ($)

Total Cost ($)

0 0 100 25 25

20 3 88 22 25

40 6 76 19 25

60 9 64 16 25

80 12 52 13 25

100 15 40 10 25

120 18 28 7 25

140 21 16 4 25

160 24 4 1 25

2. a) loss of about $64 or $65

b) y = 4.75x – 240 c) loss of $64.25

3. a)

4. a) y = x47

419

− , y = 2

1325

−x

b)

(2.65, 0.12)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. a = 72°, b = 108°

6. b)

7. 75 g kiwi, 225 g blueberries

8. a), b) (1, –3.5) c) Elimination is more efficient because the two

equations can simply be added together. For substitution, isolating either variable involves dividing both sides of one of the equations by a number.

9. Answers may vary, e.g., a) i) 5y – 3x = 10 iii) 10y – 6x = 30 ii) y – 3x = 9 b)

18 | Principles of Mathematics 10: Chapter 1 Review Extra Practice Answers


1. Calculate each unknown side length. Round to two decimal places, if necessary. a) b)

2. Solve each equation. Round to one decimal place, if necessary.

a) x2 + 25 = 169 c) 55

13

2=

++

− xx

b) 43 (2x + 1) – 3(5 – x) = 16 d) 85 = 16 + x2

3. Determine the equation of the line that a) passes through (5, –3) and (8, 6)

b) has a slope of 21

− and passes through (4, 7)

c) is perpendicular to y = 2 – 3x and passes through (3, 1) d) is parallel to 4x + 2y = 7 and passes through (–1, –4)

4. Determine the point of intersection for each pair of lines. a) y = 3x + 5 b) 3y – x = 12

y = 21 x – 15 2x + 4y = 7

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. Calculate the radius of the semicircle.

6. Which of these statements is not true? a) A rectangle is a special type of parallelogram. b) A square is a special type of rhombus. c) A rhombus is a special type of square. d) A rhombus is a special type of kite.


Catherine

Typewritten Text

2. Analytic Geometry: Line Segments and Circles

Catherine

Typewritten Text

Catherine

Typewritten Text

Chapter 2 Diagnostic Test Answers 1. a) 1.5 cm

b) 6.23 m

2. a) 12 or –12 b) 6.7 c) –31 d) 8.3 or –8.3

3. a) y = 3x – 18

b) y = –21 x + 9

c) y = 31 x

d) y = –2x – 6

4. a) (–8, –19) b) (–2.7, 3.1)

5. about 17 cm

6. c)

If students have difficulty with the questions in the Diagnostic Test, you may need to review the following topics: • applying the Pythagorean theorem to determine side lengths of a right triangle • determining the equation of a line given two points on the line • determining the equation of a line given the y-intercept and the slope • determining the point of intersection for a linear system

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Determine the coordinates of the midpoint of each line segment.

2. A midpoint of a line segment is (3.5, –2), and one endpoint is (1, 7). a) Describe a strategy you could use to

determine the coordinates of the other endpoint.

b) Apply your strategy to determine these coordinates.

3. a) The points (11, 4) and (–3, 2) are the endpoints of a diameter of a circle. Determine the coordinates of the centre of the circle.

b) Another diameter of the same circle has endpoint (–1, 8). Determine the coordinates of the other endpoint.

4. Three of the vertices of rhombus ABCD are A(5, 6), B(–2, 5), and C(3, 0). a) What property of rhombuses can you use

to determine the coordinates of the fourth vertex, D?

b) Determine the coordinates of D.

5. Determine the equation of the perpendicular bisector of the line segment with each pair of endpoints. a) (3, 1) and (5, 5) b) (–3, 2) and (5, –2) c) (3, 3) and (7, 3)

6. The municipal councils of two cities agree that a new airport will not be closer to one city than to the other. The coordinates of the cities on a map are A(23, 17) and B(47, 25). Describe all the possible locations for the airport.

7. Quadrilateral PQRS has vertices at P(1, 7), Q(6, 8), R(7, 1), and S(3, –1). a) Determine the point of intersection for the

diagonals of this quadrilateral. b) Determine the midpoint of each diagonal. c) Is PQRS a parallelogram? Explain how you

know.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 2.1 Extra Practice Answers 1. midpoint of AB: (3, 4)

midpoint of CD: (3.5, 1.5) midpoint of EF: (–1, –0.5)

2. a) Use the midpoint formula to write an equation for each coordinate:

5.32

21 =+ xx

and 22

21 −=+ yy

.

Substitute x1 = 1 and y2 = 7 into these equations, and solve for x2 and y2.

b) (6, –11)

3. a) (4, 3) b) (9, –2)

4. a) Opposites sides of a rhombus are parallel. b) (10, 1)

5. a) y = x21

− + 5

b) y = 2x – 2 c) x = 5

6. The airport can be located at any point on the line with equation y = –3x + 126.

7. a) (4.5, 3.5) b) midpoint of PR: (4, 4)

midpoint of QS: (4.5, 3.5) c) No. The diagonals do not meet at a

common midpoint.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Determine the length of each line segment.

2. Calculate the distance between every possible pair of points in the diagram.

3. The coordinates of four possible sites for a landfill are given below. Which site is farthest from a town located at (4, 8)? a) (0, 5) b) (7, 12) c) (6, 3) d) (8, 4)

4. Suppose that you are given the coordinates of the endpoints of a line segment. Describe the most efficient strategy you could use to calculate both the slope and the length of the line segment.

5. The walls of a room in an art gallery need to be repainted. The coordinates of the corners of the room, in metres, are (3, 8), (7, 7), (7, 1), (2, 2), and (–1, 5). The room is 5 m high, and 1 L of paint covers 35 m2. How much paint will be needed, to the nearest tenth of a litre?

6. An optical fibre trunk runs in a straight line through points (5, 20) and (45, 10) on a grid. What is the minimum length of optical fibre that is needed to connect the trunk to buildings at (30, 35) and (10, 10), if 1 unit on the grid represents 1 m? Round your answer to the nearest tenth.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 2.2 Extra Practice Answers 1. PQ: 5.8 units; RS: about 4.1 units

TU: about 4.2 units

2. AB: about 4.5 units AC: about 8.5 units AD: about 3.6 units BC: about 6.1 units BD: about 6.1 units CD: about 7.1 units

3. d)

4. Answers may vary, e.g., determine x1 – x2 and y1 – y2. For the slope, divide the second value by the first value. For the length, square the values, add them, and take the square root.

5. 3.5 L

6. 29.1 m

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. This table of values gives points that lie on the same circle.

x –7 1 5

y ±1 ±5 ±7 ±1

a) Copy and complete the table. b) Sketch the circle. c) Write the equation of the circle.

2. For each equation of a circle: i) Determine the diameter. ii) Determine the x- and y-intercepts. a) x2 + y2 = 625 b) x2 + y2 = 0.16

3. A circle is centred at (0, 0) and passes through point (–3, 7). Write the equation of the circle.

4. Which of these points is not on the circle with equation x2 + y2 = 225? a) (14, 1) b) (12, –9) c) (0, 15) d) (9, 12)

5. a) Determine the radius of a circle that passes through

i) (0, –6) ii) (5, 12) b) Write the equation of each circle in part a). c) State the coordinates of two other points on

each circle.

6. A satellite transfers from a near-Earth orbit, with equation x2 + y2 = 51 122 500, in kilometres, to another orbit that is 35 km higher. How many kilometres longer, to two decimal places, is the second orbit than the first?

7. A circle is centred at (0, 0) and passes through point (16, –7). Determine the other endpoint of the diameter through (16, –7). Explain your strategy.

8. Points (a, –7) and (4, b) are on the circle with equation x2 + y2 = 50. Determine the possible values of a and b. Round to one decimal place, if necessary.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 2.3 Extra Practice Answers 1. a) x –7 –5 –1 1 5 7

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

y ±1 ±5 ±7 ±7 ±5 ±1

b)

c) x2 + y2 = 50

2. a) i) 50 units ii) x-intercepts: (25, 0), (–25, 0)

y-intercepts: (0, 25), (0, –25) b) i) 0.8 units ii) x-intercepts: (0.4, 0), (–0.4, 0)

y-intercepts: (0, 0.4), (0, –0.4)

3. x2 + y2 = 58

4. a)

5. a) i) 6 units ii) 13 units b) i) x2 + y2 = 36 ii) x2 + y2 = 169

c) Answers may vary, e.g., i) (0, 6), (–6, 0) ii) (13, 0), (–13, 0)

6. 219.91 km

7. (–16, 7); the diameter has midpoint (0, 0), so the coordinates of the other endpoint are the opposites of the coordinates of the given endpoint.

8. a = ±1, b = ±5.8



1. The midpoint of line segment AB is at M(–5, 2.5). If endpoint A is at (–1, –3), determine the coordinates of endpoint B.

2. Rectangle EFGH has vertices at E(1, 5), F(9, 7), and G(8, 11). Determine the coordinates of the fourth vertex, H.

3. A guide rope is laid out in three stages along the ascent of a vertical rock face. The rope is attached to the rock face at points (0, 0), (5, 3), (–1, 12), and (3, 17), in that order. Determine the total length of the rope, to the nearest tenth of a metre.

4. The Big Pumpkin, a restaurant and pie shop, is a short distance from a major highway. The highway passes, in a straight line, through (–4, –2) and (8, 4) on a map. The Big Pumpkin is located at (–1, 3). What is the shortest distance from The Big Pumpkin to the highway, to the nearest tenth of a kilometre, if 1 unit on the map represents 1 km?

5. Two points, A(0.5, 2.5) and B(3.5, 3.5), lie on a line. Point C(–0.5, 5.5) lies off to one side of the line. a) Use only the distance formula to show that

BC is not perpendicular to the line through A and B.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) Determine the distance from C to the line through A and B. Do not calculate a point of intersection.

6. Write the equation of a circle, centred at (0, 0), that models each situation. a) a G-training device for astronauts, with a

radius of 13 m b) a wheel rim with a diameter of 16 in

7. A design for a kitchen tile uses circles and squares, as shown. If the large square has side lengths of 125 mm, what is the equation of the small circle, assuming that its centre is at (0, 0)? Round to the nearest millimetre.


Chapter 2 Mid-Chapter Review Extra Practice Answers 1. (–9, 8)

2. (0, 9)

3. 23.1 m

4. 3.1 km

5. a) Answers may vary, e.g., 16.310 ==AC or about 3.2 units and 47.452 ==BC or about 4.5 units so BC is not the minimum distance from C to AB. Therefore, BC is not perpendicular to AB.

b) Answers may vary, e.g., the slope of 31

=AB

and the slope of AC = –3, which means that AC is perpendicular to AB. The distance from C to AB is the length of AC, which is about 3.2 units.

6. a) x2 + y2 = 169 b) x2 + y2 = 64

7. x2 + y2 = 1953, or x2 + y2 = 1954

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Quadrilateral ABCD has two sides that measure 5 units and two sides that measure 2.5 units. What types of quadrilateral could ABCD be? For each type, make a possible sketch of ABCD on a grid.

2. P(–3, 2), Q(2, 3), and R(–2, 7) are the vertices of a triangle. a) Show that +PQR is an isosceles triangle. b) Is +PQR also a right triangle? Justify your

answer.

3. a) Describe a strategy you could use to decide whether a quadrilateral is a rhombus, if you are given the coordinates of its vertices.

b) Use your strategy to show that polygon JKLM, with vertices at J(–5, 3), K(3, 1), L(5, –7), and M(–3, –5), is a rhombus.

4. In quadrilateral EFGH, EF = 7.5 cm and FG = 5.5 cm. The slopes of EF, FG, and GH are

4, 41 , and 4, respectively.

a) What must be true about the slope and length of EH if EFGH is a parallelogram?

b) Could the length of GH equal 7.5 cm if EFGH is an isosceles trapezoid? Explain.

5. Two vertices of +ABC, an isosceles right triangle, are located at A(2, 1) and B(5, –2). Determine all the possible locations of C if a) AB is a side of+ABC that is not the

hypotenuse b) AB is the hypotenuse

6. A rhombus is a special type of kite and also a special type of parallelogram. How would you apply this description of a rhombus if you were using the coordinates of the vertices of a quadrilateral to determine the type of quadrilateral? Include examples in your explanation.

7. The corners of a new park have the coordinates S(–5, 1), T(5, –4), U(1, –5), and V(–5, –2) on a map. What shape is the park? Include your calculations in your answer.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 2.4 Extra Practice Answers 1. Answers may vary, e.g.,

parallelogram

rectangle

kite

2. a) PQ = 10.526 ==PQ units,

PR = 10.526 ==PR units

b) No. Answers may vary, e.g., 51

=PQm ,

mPR = 5, and mQR = –1; none of the slopes are negative reciprocals.

3. a) Determine all four side lengths, and check whether they are equal.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) 25.8172 ===== JMLMKLJK units

4. a) 41

=HEm , EH = 5.5 cm

b) No. Since EF and GH are the parallel sides in the isosceles trapezoid, they cannot be the same length.

5. a) (5, 4), (–1, –2), (8, 1), or (2, –5) b) (2, –2) or (5, 1)

6. Answers may vary, e.g., I would use the distance formula to determine the lengths of all the sides. If two pairs of adjacent sides are congruent, then the quadrilateral is a kite; e.g., A(–3, 3), B(1, 5), C(7, 3), D(1, 1). I would use the slope formula to determine the slopes of all the sides. If the slopes of opposite sides are equal, then the quadrilateral is a parallelogram; e.g., E(1, 2), F(4, 5), G(8, 5), H(5, 2). If the quadrilateral is both a kite and a parallelogram, then it is a rhombus; e.g., J(1, 1), K(4, 6), L(7, 1), M(4, –4).

7. 18.1155 ==ST units, 12.417 ==TU units, 71.653 ==UV units, SV = 3 units;

since the side lengths are all different, the park is not a parallelogram, kite, or isosceles trapezoid.

21

−=STm , 41

=TUm , 21

−=UVm , mSV is

undefined; since the park has one parallel pair of sides, it is a (non-isosceles) trapezoid.



1. a) Rectangle ABCD has vertices at A(–3, 1), B(0, –2), C(5, 3), and D(2, 6). Show that the diagonals are the same length.

b) Give an example of a quadrilateral that is not a rectangle but has diagonals that are the same length.

2. a) Show that the diagonals of quadrilateral EFGH bisect each other at right angles.

b) Make a conjecture about the type of

quadrilateral in part a). Use analytic geometry to explain why your conjecture is either true or false.

3. Kite PQRS has vertices at P(–3, 3), Q(2, 3), R(1, –5), and S(–6, –1). Show that the midsegments of this kite form a rectangle.

4. Show that the diagonals of the kite in question 3 are perpendicular, and that diagonal PR bisects diagonal QS.

5. a) Show that points A(5, –5), B(–5, 5), C(1, 7), and D(–7, –1) lie on the circle with equation x2 + y2 = 50.

b) Show that AB is a diameter of the circle. c) Show that AB is the perpendicular bisector of

chord CD.

6. A trapezoid has vertices at J(–4, 4), K(0, 7), L(8, 3), and M(8, –2). a) Show that the line segment joining the

midpoints of JK and LM bisects the line segment joining the midpoints of JM and KL.

b) Show that the two line segments described in part a) are perpendicular.

c) Based on your answer for part b), make a conjecture about trapezoid JKLM. Use analytic geometry to determine whether your conjecture is true.

7. +XYZ has vertices at X(3, 5), Y(3, –2), and Z(–3, –4). Verify that the area of the parallelogram formed by joining the midpoints of the sides of +XYZ and the vertex X is half the area of +XYZ.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 2.5 Extra Practice Answers 1. a) 25.8172 ==AC units,

25.8172 ==BD units b) Answers may vary, e.g., a quadrilateral

with vertices at (0, 2), (4, 0), (0, –6), and (–4, 0): both diagonals have a length of 8 units, but the quadrilateral is not a rectangle.

2. a) 31

−=EGm and mFH = 3, so EG and FH are

perpendicular; MEG = (1.5, 2.5) = MFH, so EG and FH bisect each other.

b) Answers may vary, e.g., conjecture: EFGH is a rhombus. EF = FG = GH = EH = 5 units, so my conjecture is correct.

3. Answers may vary, e.g., MPQ = (–0.5, 3), MQR = (1.5, –1), MRS = (–2.5, –3), MPS = (–4.5, 1); the slopes of the four

midsegments are –2, 21 , –2, and

21 , so

adjacent midsegments are perpendicular. Therefore, the midsegments form a rectangle.

4. Answers may vary, e.g., mPR = –2 and

21

=QSm , so the diagonals are perpendicular.

The equation of the line through PR is y = –2x – 3, and MQS = (–2, 1) lies on this line (check by substitution). Therefore, PR intersects QS at its midpoint.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. a) For A(5, –5), 52 + (–5)2 = 50; for B(–5, 5), (–5)2 + 52 = 50; for C(1, 7), 12 + 72 = 50; for D(–7, –1), (–7)2 + (–1)2 = 50; the points lie on the circle.

b) MAB = (0, 0), which is the centre of the circle, so AB is a diameter.

c) Answers may vary, e.g., mAB = –1 and mCD = 1, so AB and CD are perpendicular. The equation of the line through AB is y = –x. MCD = (–3, 3), so MCD lies on this diagonal (check by substitution) and therefore AB intersects CD at its midpoint, i.e., AB is the perpendicular bisector of CD.

6. a) Answers may vary, e.g., MJK = (–2, 5.5) and MLM = (8, 0.5), so the equation of the line through MJK and MLM is

.29

21

+−= xy MKL = (4, 5) and

MJM = (2, 1), so the midpoint of MKL and

MJM is (3, 3), which lies on 29

21

+−= xy

(check by substitution). Therefore, MJKMLM intersects MKLMJM at its midpoint, i.e., MJKMLM bisects MKLMJM.

b) The slope of 21

−=LMJK MM and the slope

of MKLMJM = 2, so MJKMLM and MKLMJM are perpendicular.

c) Answers may vary, e.g., conjecture: trapezoid JKLM is isosceles. Since

MJKL mm =−=21 and JK = 5 = LM, my

conjecture is correct.

7. Answers may vary, e.g., the parallelogram has vertices at X(3, 5), MXY = (3, 1.5), MYZ = (0, –3), and MXZ = (0, 0.5). Let XMXY be the base of the parallelogram, so the base length is 3.5 units, the height is 3 units, and the area of the parallelogram is 10.5 square units. Let XY be the base of +XYZ, so the base length is 7 units, the height is 6 units, and the area is 21 square units. Therefore, the area of the parallelogram is half the area of +XYZ.



1. In +ABC, the altitude from vertex B meets AC at point D.

a) Determine the equation of the line that

contains AC. b) Determine the equation of the line that

contains BD. c) Determine the coordinates of point D. d) Determine the area of +ABC, to the nearest

whole unit.

2. A fountain is going to be constructed in a park, an equal distance from each of the three entrances to the park. The entrances are located at P(2, 9), Q(11, 6), and R(3, 2). Where should the fountain be constructed?

3. +XYZ has vertices at X(6, 5), Y(4, 1), and Z(–2, 1). a) Determine the equation of the perpendicular

bisector of XY. b) Points P, Q, and R are on the perpendicular

bisector from part a). Determine the coordinates of these points that make the distances PX, QY, and RZ as short as possible.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

c) Point S is also on the perpendicular bisector from part a). Determine the coordinates of S that make the distances SX, SY, and SZ equal.

4. A circle with radius r is centred at (0, 0). The circle has a horizontal chord, AB, with endpoints A(–h, k) and B(h, k) and a vertical diameter, CD, with endpoints C(0, –r) and D(0, r). a) Draw a diagram of the circle, showing AB,

CD, and their point of intersection, E. Include expressions for the lengths of the line segments in your diagram.

b) Use properties of the intersecting chords of a circle to verify that the coordinates of A and B satisfy the equation of the circle.

5. Classify the triangle that is formed by the lines y = 2x + 3, x + 2y = 16, and 3y – x – 4 = 0 a) by its angles b) by its sides

6. The arch of a bridge, which forms an arc of a circle, is modelled on a grid. The supports are located at (–15, 0) and (15, 0), and the highest part of the arch is located at (0, 9). What is the radius of the arch, if each unit on the grid represents 1 m?


Lesson 2.7 Extra Practice Answers 1. a) 23 +−= xy

b) 31

31

+= xy

c) (0.5, 0.5) d) 15 square units

2. (6, 6)

3. a) y = –0.5x + 5.5 b) P(5, 3), Q(5, 3), R(0.2, 5. 4) c) S(1, 5)

4. a)

b)

222

222

))(())(())(())((

rkh

krh

krkrhhDECEBEAE

=+

−=

+−==

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. a) right triangle b) isosceles

6. 17 m



1. Show that the line defined by x + 2y = 0 is the perpendicular bisector of line segment AB, with endpoints A(6, 2) and B(2, –6).

2. Rhombus CDEF has vertices at C(–2, 2), D(1, 6), and E(5, 9). a) Determine the coordinates of vertex F. b) Determine the perimeter of the rhombus.

3. A sonar ping travels outward from a submarine in a circular wave at 1550 m/s. Assuming that the submarine is located at (0, 0), write an equation for the sonar wave after 5 s.

4. +ABC has vertices as shown. Determine whether this triangle is isosceles, equilateral, or scalene.

5. a) A quadrilateral is formed by the lines y = x + 4, y – 2x + 3 = 0, 2y – x = 5, and a fourth line. Explain why this quadrilateral cannot be a parallelogram.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) If the quadrilateral in part a) is a trapezoid, what must be true about the fourth line?

6. a) Show that +PQR, with vertices at P(1, 4), Q(1, 9), and R(5, 1), is isosceles.

b) Suppose that M and N are the midpoints of PQ and PR. Explain, without calculations, why +MNP is also isosceles.

7. +XYZ has vertices as shown. Determine the orthocentre of this triangle.

8. +DEF has vertices at D(10, 0), E(–5, k), and F(–5, –k), where k is a positive number. The circumcentre of +DEF is at (0, 0). a) Determine the equation of the circle that

passes through D, E, and F. b) Determine the value of k. c) Determine whether +DEF is isosceles,

equilateral, or scalene. Explain your answer.


Chapter 2 Review Extra Practice Answers 1. Answers may vary, e.g., the slope of the line

defined by x + 2y = 0 is 21

− , and mAB = 2, so

AB is perpendicular to the line. Since MAB = (4, –2) lies on the line (check by substitution), the line is the perpendicular bisector of AB.

2. a) F(2, 5) b) 20 units

3. x2 + y2 = 60 062 500

4. scalene

5. Answers may vary, e.g., a) No two of the three given lines are parallel,

so it is impossible for both pairs of opposite sides of the quadrilateral to be parallel.

b) The slope of the fourth line must be 1, 2,

or 21 .

6. a) PQ = PR = 5 units, so +PQR is isosceles. b) Answers may vary, e.g.,

21

=MP , 21

=PQ , PR = NP, so +MNP is

isosceles.

7. (5, 3)

8. a) x2 + y2 = 100 b) 35 or about 8.66

c) equilateral; 32.17310 ==DE or about 17.3 units, 32.17310 ==EF or about 17.3 units, 32.17310 ==DF = or about 17.3 units

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Consider the following data. x –4 –3 –2 –1 0 1 2 3 4y 14 7 2 –1 –2 –1 2 7 14

a) Create a scatter plot, and draw a curve. b) Use your graph to determine the value of x when y = 2.5.

2. Expand and simplify each expression. a) 4(5x – 3) b) –2(4 – 3x) + 4(2x – 5) c) a2(2a + 6b – 7) + a(2a2 – 6ab + a)

3. Determine the value of y for each given value of x. a) y = 18 – 2x, when x = 4.5 b) y = x2 – x – 6, when x = –2

4. Camille works as a computer administrator. She earns a base salary of $500/week plus $25 for every computer she fixes. Make a table of values, construct a graph, and write an equation to represent Camille’s weekly earnings.

5. Determine the x- and y-intercepts for each relation. Then sketch the graph. a) 2x – 5y = 10 b) x – y – 9 = 0 c) x = 5

Catherine

Typewritten Text

Catherine

Typewritten Text

3. Graphs of Quadratic Relations


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 3 Diagnostic Test Answers 1. a) b) x = 2.1

2. a) 20x – 12 b) 14x – 28 c) 4a3 – 6a2

3. a) 9 b) 0

y = 25x +500

5. a) x = 5, y = –2 b) x = 9, y = –9 c) x = 5, no y-intercept

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • graphing a linear relation • solving a linear equation • multiplying a polynomial by a monomial

4. x y

0 500 1 525 2 550 3 575


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. For each parabola, determine i) the equation of the axis of symmetry ii) the coordinates of the vertex iii) the y-intercept iv) the zeros v) the maximum or minimum value a) d)

b) e)

c) f)

2. Sketch the graph of each quadratic relation below. Then determine i) the equation of the axis of symmetry ii) the coordinates of the vertex iii) the y-intercept iv) the zeros v) the maximum or minimum value a) y = x2 – 9 c) y = –x2 – 2 b) y = x2 – 4x + 4 d) y = –x2 – 4x

3. Two points, on opposite sides of the same parabola, are given. Determine the equation of the axis of symmetry of the parabola. a) (–1, –5), (–5, –5) b) (6.5, –4), (9.0, –4) c) (–6, 3), (5, 3)

d) (–216 , –7), (–

211 , –7)

4. Kim knows that (–2, 31) and (8, 31) lie on a parabola defined by the equation y = 3x2 – 18x – 17. What are the coordinates of the vertex?

5. A golf ball is hit from the ground. Its height above the ground, in metres, can be approximated by the relation h = 24t – 4t2, where t is the time in seconds. a) What are the zeros of this relation? b) When does the ball hit the ground? c) What are the coordinates of the vertex? d) Use the information from parts a) and c)

to draw the graph. e) What is the maximum height of the golf ball?

After how many seconds does the maximum height occur?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 3.2 Extra Practice Answers 1. a) i) x = 3 iv) 1, 5 ii) (3, 4) v) 4 (maximum) iii) –5 b) i) x = 0 iv) –2, 2 ii) (0, –4) v) –4 (minimum) iii) –4 c) i) x = –3 iv) 0, –6 ii) (–3, 5) v) 5 (maximum) iii) 0 d) i) x = –4 iv) –2, –6 ii) (–4, –4) v) –4 (minimum) iii) 12 e) i) x = 2 iv) none ii) (2, 6) v) 6 (minimum) iii) 10 f) i) x = –2 iv) –2 ii) (–2, 0) v) 0 (maximum) iii) –5

2. a)

i) x = 0 iv) –3, 3 ii) (0, –9) v) –9 (minimum) iii) –9

b)

i) x = 2 iv) 2 ii) (2, 0) v) 0 (minimum) iii) 4

c)

i) x = 0 iv) none ii) (0, –2) v) –2 (maximum) iii) –2

d)

i) x = –2 iv) –4, 0 ii) (–2, 4) v) 4 (maximum) iii) 0

3. a) x = –3 c) x = –0.5 b) x = 7.75 d) x = –4

4. (3, –44)

5. a) 0, 6 b) 6 s c) (3, 36) d)

e) 36 m; 3 s


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. For each quadratic relation below, determine i) the zeros ii) the y-intercept iii) the equation of the axis of symmetry iv) the coordinates of the vertex

a) y = –3x(x + 4) c) y = (x – 7)(x + 5) b) y = –(6 + x)(x – 2) d) y = –2(x – 4)(x + 6)

2. Sketch each quadratic relation in question 1.

3. A parabola has an equation of the form y = a(x – r)(x – s). Determine the value of a when a) the parabola has zeros at (–3, 0) and (7, 0) and

a y-intercept at (0, 7) b) the parabola has zeros at (4, 0) and (2, 0) and

a maximum value of 6 c) the parabola has an x-intercept at (–9, 0) and

its vertex at (–4, 75)

4. Determine the y-intercept, zeros, equation of the axis of symmetry, and vertex of each parabola. Determine an equation for the parabola. a)

b)

c)

d)

5. The x-intercepts of a parabola are 1 and 7, and the parabola passes through the point (3, 4). a) Determine an equation for the parabola. b) Determine the coordinates of the vertex.

6. Andrew threw a football. This graph shows the height of the football at different times during its flight.

a) After how many seconds did the football reach

its maximum height? b) After how many seconds did the football hit

the ground? c) Determine an equation for the graph.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 3.3 Extra Practice Answers 1. a) i) 0, –4 ii) 0 iii) x = –2 iv) (–2, 12) b) i) –6, 2 ii) 12 iii) x = –2 iv) (–2, 16) c) i) –5, 7 ii) –35 iii) x = 1 iv) (1, –36) d) i) –6, 4 ii) 48 iii) x = –1 iv) (–1, 50)

2. a)

b)

c)

d)

3. a) 31

−

b) –6 c) –3

4. a) y-intercept: 50; zeros: –10, 10; equation of the axis of symmetry: x = 0; vertex: (0, 50);

equation: y = 21

− (x – 10)(x + 10)

b) y-intercept: 60; zeros: 20, 60; equation of the axis of symmetry: x = 40; vertex: (40, –20);

equation: y = 201 (x – 20)(x – 60)

c) y-intercept: 3; zeros: –1, 3; equation of the axis of symmetry: x = 1; vertex: (1, 4); equation: y = –(x – 3)(x + 1)

d) y-intercept: 32; zeros: 4, 8; equation of the axis of symmetry: x = 6; vertex: (6, –4); equation: y = (x – 4)(x – 8)

5. a) –21 (x – 1)(x – 7)

b) (4, 4.5)

6. a) after 3 s b) after 6 s c) y = –2x(x – 6)


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Determine whether each relation is quadratic. Justify your answer. a) x –2 –1 0 1 2 y 11 3 3 11 27

b) y = 5x2 – x3

c) y = 4x – 3x2 + 1 d)

2. Examine each parabola. i) Determine the coordinates of the vertex. ii) Determine the zeros. iii) Determine the equation of the axis

of symmetry. iv) If you calculated the second differences,

would they be positive or negative? Explain. a) b)

3. For each quadratic relation below, determine i) the equation of the axis of symmetry ii) the coordinates of the vertex iii) the y-intercept iv) the zeros v) the maximum or minimum value Then sketch a graph of the relation.

a) y = –x(x + 6) b) y = x2 – 2x – 3

4. Two points, on opposite sides of the same parabola, are given. Determine the equation of the axis of symmetry of the parabola. a) (–1, 4), (15, 4) b) (2.5, –3.5), (8.5, –3.5) c) (–12, 12), (–3, 12)

5. A ball is thrown from the top of a building. Its height above the ground can be approximated by the relation h = 8 + 6t – 2t2, where h is the height in metres and t is the time in seconds. a) From what height is the ball thrown? b) How long is the ball in the air? c) What is the maximum height of the ball? d) When does the ball reach a height of 10 m?

6. Determine the y-intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. Then sketch a graph of the relation. a) y = (x + 3)(x – 5) c) y = –(x + 4)(x – 2) b) y = –2(x + 3)2 d) y = 3(x – 1)(x + 3)

7. The x-intercepts of a parabola are 2 and –12. The parabola crosses the y-axis at –108. a) Determine an equation for the parabola. b) Determine the coordinates of the vertex.

8. Determine an equation for this quadratic relation.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 3 Mid-Chapter Review Extra Practice Answers 1. a) yes; all the second differences are 8, so they are

constant and non-zero b) no; degree 3 c) yes; degree 2 d) yes; parabola

2. a) i) (3, 4) ii) 1, 5 iii) x = 3 iv) negative, because the parabola opens

downward b) i) (2, –4) ii) 0, 4 iii) x = 2 iv) positive, because the parabola opens

upward

3. a) i) x = –3 ii) (–3, 9) iii) 0 iv) –6, 0 v) 9 (maximum)

b) i) x = 1 ii) (1, –4) iii) –3 iv) –1, 3 v) –4 (minimum)

4. a) x = 7

b) x = 5.5 c) x = –7.5

5. a) about 8 m b) about 4 s c) about 12.5 m

d) about 0.4 s, about 2.6 s

6. a) y-intercept: –15; zeros: –3, 5; equation of the axis of symmetry: x = 1; vertex: (1, –16)

b) y-intercept: –18; zero: –3; equation of the axis

of symmetry: x = –3; vertex: (–3, 0)

c) y-intercept: 8; zeros: –4, 2: equation of the axis

of symmetry: x = –1; vertex: (–1, 9)

d) y-intercept: –9; zeros: –3, 1; equation of the

axis of symmetry: x = –1; vertex: (–1, –12)

7. a) y = 4.5(x – 2)(x + 12) b) (–5, –220.5)

8. y = –(x – 1)(x – 7)


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Expand and simplify each expression. a) (x + 3)(x + 4) d) (x + 4)(x – 2)

b) (x – 5)(x – 3) e) (x – 2)(x – 2) c) (x + 5)(x + 5) f) (x – 6)(x + 3)

2. Expand and simplify. a) (2x + 3)(x + 1) d) (x – 2)(3x + 1) b) (3x – 1)(2x – 1) e) (2x – 5)(x + 1) c) (8x – 3)(6x – 7) f) (4x – 3)(7x + 6)

3. Expand and simplify. a) (x + 5)(x – 5) d) (x – 4)(x + 4) b) (5x – 2)(5x + 2) e) (x + 3)2 c) (2x – 1)2 f) (–4x + 3)2

4. Write a simplified expression for the area of each figure. a)

b)

5. Expand and simplify each expression. a) 3(x – 4)(x + 7) b) –(x + 5)(2x – 1)

c) –3(1 – 2x)(1 + 2x) d) 2(3x + 7)2 e) (x – 2)(4x + 3) – (x + 3)2 f) 3(2x + 1)2 – 2(x – 5)2

6. Determine an equation for each parabola. Express the equation in standard form.

a)

b)

c)

7. The dimensions of a rectangle are (2x + 3) and (3x – 4). Kendra determined the expression 6x2 – 17x – 12 to represent the area. Is Kendra’s expression correct? Justify your answer.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 3.4 Extra Practice Answers 1. a) x2 + 7x + 12

b) x2 – 8x + 15 c) x2 + 10x + 25 d) x2 + 2x – 8 e) x2 – 4x + 4 f) x2 – 3x – 18

2. a) 2x2 + 5x + 3 b) 6x2 – 5x + 1 c) 48x2 – 74x + 21 d) 3x2 – 5x – 2 e) 2x2 – 3x – 5 f) 28x2 + 3x – 18

3. a) x2 – 25 b) 25x2 – 4 c) 4x2 – 4x + 1 d) x2 – 16 e) x2 + 6x + 9 f) 16x2 – 24x + 9

4. a) 6m2 + m – 1 b) 10x2 + 13x – 3

5. a) 3x2 + 9x – 84 b) –2x2 – 9x + 5 c) –3 + 12x2 d) 18x2 + 84x + 98 e) 3x2 – 11x – 15 f) 10x2 + 32x – 47

6. a) y = –x2 + 4x b) y = x2 + 6x + 5 c) y = –2x2 + 16x – 24

7. No, Kendra’s expression is not correct. The mistake is in the middle term. It should be x. When the expression for the area is expanded, the coefficients of the x terms are –8 and 9, and their sum is 1, not –17 as in Kendra’s expression. The correct expression is 6x2 + x – 12.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. A parabola passes through the points (–2, 14), (–1, 0), (0, –10), (1, –16), (2, –18), (3, –16), (4, –10), and (5, 0). Determine an equation for the parabola in factored form.

2. Kathy kicked a soccer ball straight up into the air. The height of the ball was recorded every 0.25 s, as given in the table below.

Time (s) Height (m) 0.00 0.0 0.25 3.0 0.50 4.8 0.75 5.8 1.00 6.2 1.25 5.7 1.50 4.6 1.75 2.7 2.00 0.0

a) Use the data in the table to create a scatter plot. Then draw a curve of good fit.

b) Determine an equation for the curve of good fit you drew.

3. Jamie and Grace are conducting an experiment on motion. They set up a motion detector to collect data for Jamie on his skateboard as he moved toward the motion detector. The time and distance recorded by the motion detector are given in the table below.

Time (s) Distance (m) 0.0 15.1 0.5 15.0 1.0 14.6 1.5 14.0 2.0 13.1 2.5 12.0 3.0 10.6 3.5 9.0 4.0 7.1 4.5 5.0 5.0 2.0

a) Use the data in the table to create a scatter plot. Then draw a curve of good fit.

b) Determine an equation for the curve of good fit you drew.

c) Verify your equation using quadratic regression.

4. A parachutist jumps from an altitude of 2000 m. This table shows the distance of the parachutist from the ground after each 3 s period.

Time (s) 0 3 6 9 12Height (m) 2000 1911 1650 1215 608

a) Create a scatter plot for the data, and draw a curve of good fit.

b) Determine an equation for your curve of good fit.

c) Use your equation to estimate the parachutist’s distance from the ground after 10 s.

5. A school custodian found a tennis ball on the roof of the school and threw it to the ground below. The table shows the height of the ball above the ground as it moved through the air.

Time (s) Height (m) 0.0 5.00 0.5 11.25 1.0 15.00 1.5 16.25 2.0 15.00 2.5 11.25 3.0 5.00

a) Create a scatter plot, and draw a curve of good fit.

b) Determine an equation for your curve of good fit.

c) Use your equation to predict the height of the ball at 1.25 s and at 2.75 s.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 3.5 Extra Practice Answers 1. y = 2(x + 1)(x – 5)

2. a)

b) y = –6.2x(x – 2)

3. a)

b) Answers may vary, e.g., y = –0.54x2 + 15.1 c)

4. a)

b) Answers may vary, e.g., y = –9.6(x – 14.4)(x + 14.4)

c) Answers may vary, e.g., about 1031 m

5. a)

b) y = –5(x + 0.3)(x – 3.3) c) At 1.25 s, the height is about 15.9 m.

At 2.75 s, the height is about 8.4 m.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. State whether each relation is quadratic. Justify your decision.

a) y = x2(x + 1) b) x –7 –6 –5 –4 –3

y –46 –33 –22 –13 –6

c) y = 3x(x – 5)

2. Discuss how you know the direction that a parabola opens if you are given an equation for the parabola.

3. Graph each quadratic relation, and determine i) the equation of the axis of symmetry ii) the coordinates of the vertex iii) the y-intercept iv) the zeros a) y = –x2 – 2x + 8 b) y = x2 + 5x

4. Can two parabolas have the same zeros but different minimum values? Explain your reasoning.

5. Alicia knows that (–6, 36) and (1, 36) lie on the parabola defined by y = –2x2 – 10x + 48. What are the coordinates of the vertex?

6. The zeros of a parabola are –9 and 3, and the y-intercept is –54. a) Determine an equation for the parabola. b) Determine the coordinates of the vertex.

7. Determine an equation for each parabola. a) The x-intercepts are –7 and 5, and the

y-coordinate of the vertex is 36. b) The x-intercepts are –2 and 6, and the

y-coordinate of the vertex is –16. c) The zeros are at –5 and 3, and the y-intercept

is –7.5. d) The vertex is at (–3, 0), and the y-intercept

is –6.

8. Expand and simplify each expression. a) (x – 2)(x + 2) d) (x – 5)(x – 2)

b) (5x – 1)(x + 3) e) (5x + 3)(4x – 2) c) (3x – 4y)(2x + 7y) f) (7x + 3y)(3x – 4y)

9. Expand and simplify. a) –3(x – 2)2 c) 2(x – 1)(3x + 2)

b) – (3x – y)(x – 4y) d) 3(5x – y)2

10. Determine an equation for the parabola. Express the equation in standard form.

11. A ball is tossed straight up into the air. Its height is recorded every 0.25 s, as given in the table below.

Time (s) Height (m) 0.00 1.5 0.25 3.5 0.50 4.9 0.75 5.7 1.00 5.7 1.25 5.2 1.50 4.1 1.75 2.4 2.00 0.0

a) Create a scatter plot, and then draw a curve of good fit.

b) Is your curve of good fit a parabola? Explain. c) Determine an equation for your curve of good

fit. Express your equation in standard form. d) Estimate the time when the ball reaches its

maximum height. Then use your equation to calculate its maximum height.

12. Evaluate each expression. Express your answer in rational form. a) 2–4 c) –5–2 e) –(–4)0

b) 2

31 −

⎟⎠⎞

⎜⎝⎛ d)

3

54 −

⎟⎠⎞

⎜⎝⎛− f)

3

21 −

⎟⎠⎞

⎜⎝⎛−−

13. Which do you think is greater: 2–3 or 3–2? Justify your decision.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 3 Review Extra Practice Answers 1. a) no; degree 3

b) yes; all the second differences are –2; first differences: –13, –11, –9, –7

c) yes; degree 2

2. If the value of a (the coefficient of x2) is positive, then the parabola opens upward. If the value of a is negative, then the parabola opens downward.

3. a) i) x = –1 ii) (–1, 9) iii) 8 iv) –4, 2

b) i) x = –2.5 ii) (–2.5, –6.25) iii) 0 iv) 0, –5

4. Two parabolas can have the same zeros but

different minimum values if the value of a, the coefficient of x2, is different.

5. (–2.5, 60.5)

6. a) y = 2(x – 3)(x + 9) b) (–3, –72)

7. a) y = –(x – 5)(x + 7) b) y = (x + 2)(x – 6)

c) y = 21 (x – 3)(x + 5)

d) y = 32

− (x + 3)2

8. a) x2 – 4 d) x2 – 7x + 10 b) 5x2 + 14x – 3 e) 20x2 + 2x – 6 c) 6x2 + 13xy – 28y2 f) 21x2 – 19xy – 12y2

9. a) −3x2 + 12x − 12 b) −3x2 + 13xy − 4y2 c) 6x2 − 2x − 4 d) 75x2 − 30xy + 3y2

10. y = 2x2 + 12x + 10

11. a)

b) Answers may vary, e.g., yes, the curve is a U shape.

c) Answers may vary, e.g., y = −5x2 + 9.25x + 1.5 d) Answers may vary, e.g., about 0.9 s; about

5.8 m

12. a) 161 c) –

251 e) –1

b) 9 d) –64

125 f) 8

13. 2–3 is greater because 2–3 = 81 and 3–2 =

91 . Both

have 1 as a numerator, but 2–3 has a lesser denominator so it is greater.

Catherine

Typewritten Text

Chapters 1–3 Cumulative Review Extra Practice | 377

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapters 1–3 Cumulative Review Extra Practice STUDENT BOOK PAGES 189–191

1. Which ordered pair is the solution to the linear system 2x + y = 15 and 5x – 6y = –22? A. (3, 11) B. (4, 7) C. (2, 11) D. (5, 5)

2. Jennifer has $3.25 in her piggy bank. She has eight more nickels than dimes. If n represents the number of nickels and d represents the number of dimes, which linear system models this situation? A. 10n + 5d = 325 C. 5n + 10d = 325

n – d = 8 n – d = 8 B. 10n + 5d = 8 D. 5n + 10d = 325

n + d = 325 n + 8 = d

3. Amy bought 9 tickets to a movie and spent $83. She bought a combination of child tickets and adult tickets. The child tickets cost $7 each, and the adult tickets cost $11 each. How many adult tickets did Amy buy? A. 5 B. 4 C. 6 D. 3

4. The endpoints of the diameter of a circle are M(15, –21) and N(–8, 9). Which point is the centre of the circle? A. (11.5, –15) B. (3.5, –6) C. (–7.5, 5) D. (–3.5, 7.5)

5. Line segment AB has endpoints A(7, –20) and B(–2, –9). Which pair of points forms a line segment that is the same length? A. (–20, 7) and (2, 9) B. (12, 6) and (–3, 5) C. (–7, 20) and (9, 2) D. (–12, –6) and (–3, 5)

6. What is the equation of a circle that has a diameter with endpoints (12, –7) and (–12, 7)? A. x2 + y2 = 95 B. x2 + y2 = 84 C. x2 + y2 = 193 D. x2 + y2 = 25

7. What are the coordinates of the vertex of the parabola for the relation y = 3(x – 5)(x + 7)? A. (1, –96) B. (–1, –108) C. (6, 39) D. (–1, –105)

8. What is the equation of the axis of symmetry of the parabola for the relation y = –2(x – 4)(x – 6)? A. x = –5 B. x = 10 C. x = –10 D. x = 5

9. A quadratic relation has zeros at x = –10 and x = 25, and passes through (10, 150). Which equation describes this relation? A. y = –0.5(x + 25)(x – 10) B. y = (x – 25)(x + 10) C. y = –2(x – 25)(x + 10) D. y = –0.5(x – 25)(x + 10)

10. Which expression is the product of (5x + 2) and (7x – 4)? A. 35x2 – 6x – 8 B. 35x2 + 6x + 8 C. 12x2 + 34x – 8 D. 35x2 + 34x – 6

11. What is the value of (–5)–3?

A. 125

1

B. –125

C. 125

1−

D. 151

−

378 | Principles of Mathematics 10: Chapters 1–3 Cumulative Review Extra Practice Answers

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapters 1–3 Cumulative Review Extra Practice Answers 1. B

2. C

3. A

4. B

5. D

6. C

7. B

8. D

9. D

10. A

11. C


1. State the number of factors. Then write the product. a) 3m × 2n b) (4a)2 c) –5x(–10)(x) d) 3(x + 1)

2. Simplify each expression. a) –2y(y – 3) b) 12w2 ÷ 4 c) 4x2 – 2x – 3x2 + 2x d) ab + 3a2 – (5ab + b2)

3. Expand and simplify. a) 2m(–m + 5) b) –x(3x2 + 8x) c) 3p2(2p2 – 3p + 4) d) (x + 5)(x – 3)

4. A parabola is defined by the equation y = (x + 4)(x – 2). Write the standard form of the equation by determining the value of each symbol: y = x2 + x + .

5. State the greatest common factor for each pair. a) 16 and 40 b) 56 and 35 c) 9x and 12 d) 24 and 18y

6. Determine the multiplication expression and product shown in each area diagram. a) b)

7. Each of three students drew a different rectangle that has an area of 36n2. Determine the missing dimension of each rectangle. a) height = 4n, length = b) length = 18n2, height = c) height = 12; length =

8. The perimeter of a square photograph frame is 8x – 4. Determine the length of the sides and the area of the square.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Catherine

Typewritten Text

4. Factoring Algebraic Expressions

Chapter 4 Diagnostic Test Answers 1. a) two factors; 6mn

b) two factors; 16a2 c) three factors; 50x2 d) two factors; 3x + 3

2. a) –2y2 + 6y b) 3w2 c) x2 d) 3a2 – 4ab – b2

3. a) –2m2 + 10m b) –3x3 – 8x2 c) 6p4 – 9p3 + 12p2 d) x2 + 2x – 15

4. y = x2 + 2x – 8

5. a) 8 b) 7 c) 3 d) 6

6. a) 5x(5x – 3) = 25x2 – 15x b) –n(–4n + 1) = 4n2 – n

7. a) length = 9n b) height = 2 c) length = 3n2

8. length = 2x – 1; area = 4x2 – 4x + 1

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • simplifying algebraic expressions by combining like terms • expanding algebraic expressions by using the distributive property • using exponent laws for multiplication and division • determining common factors • using a rectangular area model



1. State the greatest common factor for each pair of terms. a) 40x and 100 b) x3 and x4 c) 6b2 and 4 d) 8x2 and –6x e) 2x2y and 3xy2 f) 5(x + 2) and –3x(x + 2)

2. Determine the GCF of the terms in each expression. a) 12t2 – 14t + 10 b) 8x2 + 24x + 16 c) 9m2 – 6m + 18 d) y4 – y3 + y5 e) 9m3 – 6m2 + 12m f) xy3 + xy5 + x2y2

3. Determine the missing factor. a) 9xy = ( )(3x) b) –10x2 = 2x2( ) c) 14a3b = 2a2( ) d) 6x + 6y = ( )(x + y) e) 4a – 8b = ( )(a – 2b) f) 36x4 + 24x = (12x)( )

4. Factor each expression. a) 16a2 – 4a

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) 9x5 – 27x3 c) 15x2 – 5x + 10 d) 12y4 – 6y3 + 18y e) –42x5y + 21x4y f) 72mn2 + 40m2n3 + 64m3n

5. Determine the factors and the product shown in this area diagram. What is the GCF of the product?

6. Trisha started to simplify 5(n + 2) + n(n + 2) and wrote 5n + 10 + n2 + 2n. Jon divided out the GCF (n + 2) and wrote (n + 2)(5 + n). Do both strategies result in the same polynomial? Explain.

7. Factor each expression. a) t(t + 5) + 4(t + 5) b) 3a(b – 3) – 2b(b – 3) c) –(2y + 7) + 3x(2y + 7) d) y(2x + 1) + (y – 1)(2x + 1) e) (5x – 3)2 – 3(5x – 3) f) 7a(xy – x + 1) + 9(xy – x + 1)

8. Factor each polynomial. The polynomial in part a) has been started for you. a) 4xy + 4xz + y2 + yz = 4x(y + z) + y(y + z) b) 3ab + 3ac + 2b + 2c c) 6mn + 2mx – 3n – x d) y3 – yz + zy2 – z2 e) 3n2 + 3p + 2n2 + 2p f) 7x2y – 7xy + x3 – x2

9. A parabola is defined by y = 4x2 – 6x. a) Express the equation in factored form. b) Determine the zeros and the equation of the

axis of symmetry. c) Determine the coordinates of the vertex. d) Sketch a graph of the parabola.

10. a) Write three polynomials whose terms have a greatest common factor of 3y2.

b) Factor each polynomial from part a).


Lesson 4.1 Extra Practice Answers 1. a) 20

b) x3 c) 2 d) 2x e) xy f) (x + 2)

2. a) 2 b) 8 c) 3 d) y3 e) 3m f) xy2

3. a) 3y b) –5 c) 7ab d) 6 e) 4 f) 3x3 + 2

4. a) 4a(4a – 1) b) 9x3(x2 – 3) c) 5(3x2 – x + 2) d) 6y(2y3 – y2 + 3) e) 21x4y(–2x + 1) f) 8mn(9n + 5mn2 + 8m2)

5. 2x(3x + 5) = 6x2 + 10x; GCF: 2x

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

6. Yes. Answers may vary, e.g., the resulting polynomial is the same. Trisha expanded and can simplify by collecting like terms to get n2 + 7n + 10. Jon divided out the GCF (n + 2) and can multiply the two binomials to get 5n + n2 + 10 + 2n, which simplifies to n2 + 7n + 10.

7. a) (t + 5)(t + 4) b) (b – 3)(3a – 2b) c) (2y + 7)(–1 + 3x) d) (2x + 1)(2y – 1) e) (5x – 3)(5x – 6) f) (xy – x + 1)(7a + 9)

8. a) (y + z)(4x + y) b) (b + c)(3a + 2) c) (3n + x)(2m – 1) d) (y2 – z)(y + z) e) 5(n2 + p) f) x(x – 1)(7y + x)

9. a) y = 2x(2x – 3) b) zeros: x = 0 and x = 1.5; equation of the axis

of symmetry: x = 0.75 c) (0.75, –2.25) d)

10. Answers may vary, e.g., a) 6xy2 + 3y2, 6xy3 – 3y2z, 3ay2 + 3by2 + 3cy2 b) 3y2(2x + 1), 3y2(2xy – z), 3y2(a + b + c)



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

1. Each tile model represents an algebraic expression. Identify the expression and its factors. a)

b)

2. One factor is given, and one factor is missing. What is the missing factor? a) x2 + 12x + 35 = (x + 7)( ) b) x2 – 11x + 30 = (x – 5)( ) c) m2 + 24m – 52 = ( )(m + 26) d) t2 – 18t + 45 = ( )(t – 3) e) y2 + 5y – 36 = (y + 9)( ) f) c2 – 7cd + 12d2 = (c – 3d)( )

3. Factor each expression. a) x2 – 5x + 6 b) x2 + 2x – 3 c) a2 + 6a + 5 d) x2 – 9x – 10 e) b2 – 7b + 10 f) x2 + x – 20

4. Factor. a) 2x2 + 10x + 12 b) 3x2 + 9x – 12 c) 5x2 – 35x + 50 d) 4y2 – 16y – 48 e) 2b2 – 16b – 66 f) x3 – 13x2 + 42x

5. The area of a rectangular carpet can be represented by the trinomial x2 – x – 42. Determine the binomials that represent the length and width of the carpet.

If a quadratic expression x2 + bx + c can be factored, then it can be factored as (x + r)(x + s), where r + s = b and r × s = c. It is easier to factor an algebraic expression if you first divide out the greatest common factor.

6. Write three different quadratic trinomials of the form x2 + bx + c that have (x + 6) as a factor.

7. A parabola is defined by y = x2 – 2x – 63. a) Express the equation in factored form. b) Determine the zeros. c) Determine the coordinates of the vertex. d) Does the parabola open upward or

downward? Explain how you know.

8. Factor each expression. a) x2 – 5xy – 66y2 b) m2 + 12mn + 32n2 c) a2b2 – 10ab – 24 d) x2y2 + 15xy + 54 e) a4 + 7a2 + 12 f) 3x2y + 27xy + 60y

9. Marika says that you can factor 6 – 7x + x2 if you rearrange the terms. Do you agree? Explain.

10. a) Justin says that if you write and expand two binomials of the form (x + r)(x + s), where r and s are integers, you can always factor the resulting trinomial x2 + xs + rx + rs. Do you agree? Give two examples to support your answer.

b) Another student says that you can factor any trinomial of the form x2 + bx + c, where b and c are integers. Use two examples to show that this statement is false.


Lesson 4.3 Extra Practice Answers 1. a) x2 + 6x + 5 = (x + 1)(x + 5)

b) x2 + 2x – 8 = (x – 2)(x + 4)

2. a) x + 5 b) x – 6 c) m – 2 d) t – 15 e) y – 4 f) c – 4d

3. a) (x – 3)(x – 2) b) (x + 3)(x – 1) c) (a + 5)(a + 1) d) (x – 10)(x + 1) e) (b – 5)(b – 2) f) (x + 5)(x – 4)

4. a) 2(x + 2)(x + 3) b) 3(x + 4)(x – 1) c) 5(x – 5)(x – 2) d) 4(y – 6)(y + 2) e) 2(b – 11)(b + 3) f) x(x – 6)(x – 7)

5. x + 6, x – 7

6. Answers may vary, e.g., x2 + 7x + 6 = (x + 6)(x + 1) x2 + 11x + 30 = (x + 6)(x + 5) x2 + 4x – 12 = (x + 6)(x – 2)

7. a) y = (x – 9)(x + 7) b) zeros: 9, –7 c) vertex: (1, –64) d) The parabola opens upward. Answers may

vary, e.g., the vertex is below the zeros, so it is the lowest point on the graph. The value of a is positive.

8. a) (x – 11y)(x + 6y) b) (m + 4n)(m + 8n) c) (ab – 12)(ab + 2) d) (xy + 9)(xy + 6) e) (a2 + 4)(a2 + 3) f) 3y(x + 5)(x + 4)

9. Yes. When you rearrange the terms, you get x2 – 7x + 6 = (x – 1)(x – 6).

10. a) Yes, because x2 + xs + rx + rs = (x + s)(x + r); answers may vary, e.g., when r = –2 and s = –3, x2 + x(–3) + (–2)x + 6 = (x – 3)(x – 2), and when r = 7 and s = 5, x2 + 5x + 7x + 35 = (x + 5)(x + 7).

b) Answers may vary, e.g., x2 + 3x + 7 or x2 – 6x + 12

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Determine the GCF for each set of terms. a) 15x, 25x b) 3a3, 12a2 c) 18xyz, 24x2y d) 12m, 16n, 8mn e) 15x2, 20x3, 5x f) 6pt2, –8s2t2, –6s2t2

2. Determine the GCF of the terms in each expression. a) 3a(a – 7) + 4(a – 7) b) –8(1 + x) + 7x(1 + x) c) 2m(n + m) – (m + n) d) 2x(2x – 3y) + 4(2x – 3y) e) rs(t2 – 1) + a(–1 + t2) f) 13x2y(x + y + z) + 11xy2(x + y + z)

3. Factor each expression. a) 4x + 10 b) 6h2 + 9h c) 25x6 – 5x2 d) 3p3qr – 4p2q2r2 e) 6y2 – 9xy + 12x2y2 f) 8x4y4 – 4x3y3 + 4x2y2

4. a) Vic wrote the polynomial xa + xb + 2a – 2b and factored it as x(a + b) + 2(a – b). Can this expression be factored further? Explain.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b) Lyn wrote the polynomial xa + xb – 2a – 2b. Factor Lyn’s polynomial.

5. The area and length of each parallelogram is given. Determine the height. a)

b)

6. Each model represents an algebraic expression. Identify the expression and its factors. a) b)

7. One factor is given, and one factor is missing. What is the missing factor? a) x2 + 12x + 27 = (x + 3) ( ) b) c2 – c – 56 = (c – 8) ( ) c) x2 – 23x + 112 = ( )(x – 16) d) x2 + 6xy – 40y2 = ( )(x – 4y) e) 7mn2 – 84mn – 315m = ( )(n – 15)(n + 3) f) 45a2 + 90a + 225 = ( )(a2 + 2a + 5)

8. Factor, if possible. a) x2 + 12x + 35 b) x2 + 2x – 8 c) d2 + d – 30 d) x2 – 7x – 44 e) x2 + 3x – 24 f) b2 – 23b + 130

9. A rectangular prism has a volume that is represented by the expression 4n2 – 20n – 24. The height of the prism is 4.

a) Determine the area of the rectangular base. b) Determine the length and width of the

rectangular base.

10. Dima says that you can factor –x2 – 5x + 24 by dividing out the GCF –1. Do you agree? Explain.


Chapter 4 Mid-Chapter Review Extra Practice Answers 1. a) 5x

b) 3a2 c) 6xy d) 4 e) 5x f) 2t2

2. a) a – 7 b) 1 + x c) m + n d) 2(2x – 3y) e) t2 – 1 f) xy(x + y + z)

3. a) 2(2x + 5) b) 3h(2h + 3) c) 5x2(5x4 – 1) d) p2qr(3p – 4qr) e) 3y(2y – 3x + 4x2y) f) 4x2y2(2x2y2 – xy + 1)

4. a) No. The GCF of the terms is 1 because the binomials are not identical.

b) x(a + b) – 2(a + b) = (a + b)(x – 2)

5. a) 3a b) 12

6. a) x2 – 7x + 10 = (x – 2)(x – 5) b) x2 + x – 12 = (x – 3)(x + 4)

7. a) x + 9 b) c + 7 c) x – 7 d) x + 10y e) 7m f) 45

8. a) (x + 7)(x + 5) b) (x + 4)(x – 2) c) (d + 6)(d – 5) d) (x – 11)(x + 4) e) not possible f) (b – 10)(b – 13)

9. a) n2 – 5n – 6 b) The length and width are (n + 1) and (n – 6).

Either could be the length.

10. yes, because –(x2 + 5x – 24) = –(x + 8)(x – 3)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

1. Each area diagram represents a trinomial. Identify the trinomial and its factors. a)

b)

2. Determine the missing factor. a) 3x2 – 17x + 10 = (x – 5)( ) b) 6x2 + 5x – 6 = (3x – 2)( ) c) 2y2 – 5y – 25 = ( )(2y + 5) d) 5x2 – 12xy – 9y2 = (5x + 3y)( )

3. Factor each trinomial. a) 2x2 + 7x + 3 b) 2x2 – 11x + 15 c) 3t2 + t – 4 d) 2x2 – 13x + 21 e) 6a2 + 17a + 12 f) 8b2 – 10b – 7

4. Factor, if possible. a) 4x2 + 3x – 1 b) 3x2 + 5x + 1 c) 4x2 – 8x + 3 d) 4y2 + 7y – 5 e) 20x2 – 7x – 6 f) 5x2 – 12x – 6

5. Write three different quadratic trinomials of the form ax2 + bx + c, where a ≠ 1, that have (5x – 6) as a factor.

If a quadratic expression of the form ax2 + bx + c, where a ≠ 1, can be factored, then it can be factored as (px + r)(qx + s), where pq = a, rs = c, and ps + rq = b. It is easier to factor an algebraic expression if you first divide out the greatest common factor.

6. A parabola is defined by the equation y = 5x2 + 6x – 8. a) Express the equation in factored form. b) Determine the zeros. c) Determine the coordinates of the vertex. d) Sketch a graph of the parabola.

7. Factor each trinomial. a) 3y2 + 2yz – z2 b) 2x2 + 5xy + 2y2 c) 4b2 – 7bc – 2c2 d) 15x2 – 13xy + 2y2 e) 6q2 + 7qr + r2 f) 6x2 – 2xy – 8y2

8. Factor. a) 8x2 + 6x – 20 b) 18b2 + 15b – 18 c) 30x2 – 58x – 28 d) 18a2 + 24a – 6 e) 18x2 – 21xy + 6y2 f) 12x2 + 28xy – 24y2

9. The area of a rectangular field is represented by the expression 12n2 + 8n – 15. a) Determine the dimensions of the field. b) If the value of n is 20 m, what are the

dimensions of the field in metres?

10. Lindsay says that you can factor –4x2 + x + 18 by dividing out the GCF –1. Do you agree? Explain.


Lesson 4.4 Extra Practice Answers 1. a) 3x2 + 7x + 2 = (3x + 1)(x + 2)

b) 8n2 + 2n – 15 = (2n + 3)(4n – 5)

2. a) 3x – 2 b) 2x + 3 c) y – 5 d) x – 3y

3. a) (2x + 1)(x + 3) b) (x – 3)(2x – 5) c) (3t + 4)(t – 1) d) (2x – 7)(x – 3) e) (2a + 3)(3a + 4) f) (2b + 1)(4b – 7)

4. a) (4x – 1)(x + 1) b) not possible c) (2x – 1)(2x – 3) d) not possible e) (4x – 3)(5x + 2) f) not possible

5. Answers may vary, e.g., 5x2 – x – 6 = (5x – 6)(x + 1) 5x2 – 11x + 6 = (5x – 6)(x – 1) 10x2 – 17x + 6 = (5x – 6)(2x – 1)

6. a) y = (5x – 4)(x + 2) b) zeros: x = –2 and x = 0.8 c) vertex: (–0.6, –9.8)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

d)

7. a) (3y – z)(y + z) b) (x + 2y)(2x + y) c) (4b + c)(b – 2c) d) (5x – y)(3x – 2y) e) (6q + r)(q + r) f) 2(x + y)(3x – 4y)

8. a) 2(x + 2)(4x – 5) b) 3(2b + 3)(3b – 2) c) 2(5x + 2)(3x – 7) d) 6(3a2 + 4a – 1) e) 3(2x – y)(3x – 2y) f) 4(3x – 2y)(x + 3y)

9. a) 2n + 3, 6n – 5 b) 115 m by 43 m

10. yes, –(4x2 – x – 18) = –(4x – 9)(x + 2)



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

4. Derek says that you can factor –64m2 + 49 by rearranging the terms. Do you agree? Explain.

5. a) Leah’s parabola is defined by y = x2 – 121. Describe what the graph of her parabola will look like.

b) Determine the coordinates of the vertex of the parabola defined by y = 4x2 + 44x + 121.

6. Factor each polynomial. a) 169x2 – y2

There are special forms of polynomials. If a quadratic expression has the form a2 + 2ab + b2 or a2 – 2ab + b2, then it is a perfect-square trinomial and can be factored as a2 + 2ab + b2 = (a + b)2, or a2 – 2ab + b2 = (a – b)2 If a quadratic expression has the form a2 – b2, then it is a difference of squares and can be factored as a2 – b2 = (a + b)(a – b)

1. Each area diagram represents a perfect-square trinomial. Identify the trinomial and its factors. a)

b)

2. Determine whether each expression is a perfect-square trinomial. If it is, then factor it. a) x2 + 8x + 16 b) x2 – 14x + 49 c) x2 – 9x + 9 d) 4x2 – 20x + 25 e) 4x2 – 12x – 9 f) 9a2 – 24a + 16

3. Factor, if possible. a) x2 – 25 b) x2 – 49 c) x2 + 64 d) 4y2 – 9 e) 4x2 + 25 f) x4 – 1

b) 81a2 – 121b2 c) 64p2 – 169c2 d) 25x6 – 144y4 e) –36m2 + 196n2 f) 49 – (x – y)2

7. Use the pattern for difference of squares to calculate. Show your work. a) 382 – 322 b) 552 – 452

8. Factor, if possible. a) x2 – 196 b) p2 – 4pq + 4q2 c) 16b2 + 40b + 25 d) c2 + 100 e) 36x2 + 6x + 1 f) 16x2 + 24xy + 9y2

9. Determine possible integers for k that will make each expression a perfect-square trinomial. a) x2 + kx + 49 b) 9x2 + kx + 25 c) 4x2 – 12x + k

10. Zach says that you can factor –4x2 + 36 by dividing out the GCF –1. Do you agree? Explain.


Lesson 4.5 Extra Practice Answers 1. a) 9x2 + 6x + 1 = (3x + 1)2

b) n2 – 22n + 121 = (n – 11)2

2. a) yes, (x + 4)2 b) yes, (x – 7)2 c) no d) yes, (2x – 5)2 e) no f) yes, (3a – 4)2

3. a) (x + 5)(x – 5) b) (x + 7)(x – 7) c) not possible d) (2y + 3)(2y – 3) e) not possible f) (x2 + 1)(x2 – 1) = (x2 + 1)(x – 1)(x + 1)

4. Yes; 49 – 64m2 is a difference of squares and can be factored as (7 + 8m)(7 – 8m).

5. a) The zeros are x = 11 and x = –11, so two points on the graph are (11, 0) and (–11, 0). The vertex is at (0, –121), and the parabola opens upward.

b) Since the factored equation is y = (2x + 11)2, there is only one x-intercept, x = –5.5, which corresponds to the vertex. The vertex is at (–5.5, 0).

6. a) (13x + y)(13x – y) b) (9a – 11b)(9a + 11b) c) (8p + 13c)(8p – 13c) d) (5x3 + 12y2)(5x3 – 12y2) e) (14n + 6m)(14n – 6m) f) (7 + x – y)(7 – x + y)

7. a) (38 + 32)(38 – 32) = (70)(6) = 420 b) (55 + 45)(55 – 45) = (100)(10) = 1000

8. a) (x + 14)(x – 14) b) (p – 2q)2 c) (4b + 5)2 d) not possible e) not possible f) (4x + 3y)2

9. a) k = 14 or k = –14 b) k = 30 or k = –30 c) k = 9

10. No. The GCF is –4, so –4(x2 – 9) can be factored as –4(x + 3)(x – 3).

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. Explain why each expression is not factored fully. a) 12m2n – 6mn2 = 6m(2mn – n2) b) p4 – 9p2 + 20 = (p2 – 4)(p2 – 5) c) 64x2 – 16 = 8(8x2 – 2) d) 18y2z4 – 9y2z2 = 9y2(2z4 – z2) e) t3 – 25t = t(t2 – 25) f) 6x2 – 21xy + 8xz – 28yz

= 3x(2x – 7y) + 4(2xz – 7yz)

6. Factor each polynomial. a) 12x – 8x3 + 36x2 b) 5m(n – p) – 45(n – p) c) 4wxz + 28wyz

Strategies for factoring: • Divide out the common monomial factors. • Check for a difference of squares:

a2 – b2 = (a + b)(a – b) • Check for a perfect-square trinomial:

a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

• Check for this form: x2 + bx + c = (x + r)(x + s), where c = r × s and b = r + s

• Check for this form when a ≠ 1: ax2 + bx + c = (px + r)(qx + s), where c = rs, a = pq, and b = ps + qr

• Try a grouping strategy for four or more terms.

1. Factor using a grouping strategy. Show your work. a) y2 + y – x – xy b) cd + 3c + 9 + 3d c) b2 – 4b – br + 4r d) 4s2 + 8s + 6st + 12t e) 3x3y – 6x3 – 2y + y2 f) 4mn2 – 12m2n – 3np + 9mp

2. Factor each binomial. a) 4x2 – 81 b) 49x2 – 121 c) 16x2 + 64y2 d) 36x2 – 81y2 e) 16m2n – 80n f) 3a2 – 243

3. Determine each missing factor. a) 3cde + 4c2de – 9cde2 = cde( ) b) 5x3 – 30x2 + 45x = ( )(x – 3)2 c) 6x2 – 13xy – 5y2 = (2x – 5y)( ) d) 9x2 + 3xy – 20y2 = ( )(3x – 4y) e) 2x2y2 – 6xy2 – 56y2 = (2y2)(x + 4)( )

f) a4 – 7a2b2 + 12b4 = (a + 2b)(a – 2b)( )

4. The trinomial 3x2 + 39x – 90 represents the volume of a rectangular prism. Determine the dimensions of the rectangular prism.

d) a4 – 81 e) 12k3 – 27k f) 3xz + 4yz – 2z – 12x – 16y + 8

7. Factor, if possible. a) 4x2 + 121 b) x4 – 13x2 + 36 c) 64a2 – 30ab + 49b2 d) 4x2 – xy – 18y2 e) 25x2 + 30xy + 9y2 f) 8s2 + 12st – 3t2 – 2st

8. The surface area of a closed cylinder can be determined by adding the areas of the circular ends (πr2) to the area of the curved surface (2πrh). a) Write the algebraic expression for the surface

area. Then factor your expression. b) Estimate the surface area of a cylinder if the

radius is 2 cm and the height is 10 cm.

9. A parabola is defined by y = 25x2 – 120x + 144. Determine the coordinates of the vertex.


Lesson 4.6 Extra Practice Answers 1. a) y(y + 1) – x(y + 1) = (y + 1)(y – x)

b) c(d + 3) + 3(3 + d) = (d + 3)(c + 3) c) b(b – 4) – r(b – 4) = (b – 4)(b – r) d) 4s(s + 2) + 6t(s + 2) = (s + 2)(4s + 6t)

= 2(s + 2)(2s + 3t) e) 3x3(y – 2) + y(–2 + y) = (y – 2)(3x3 + y) f) 4mn(n – 3m) – 3p(n – 3m)

= (n – 3m)(4mn – 3p)

2. a) (2x + 9)(2x – 9) b) (7x + 11)(7x – 11) c) 16(x2 + 4y2) d) 9(2x + 3y)(2x – 3y) e) 16n(m2 – 5) f) 3(a + 9)(a – 9)

3. a) 3 + 4c – 9e b) 5x c) 3x + y d) 3x + 5y e) x – 7 f) a2 – 3b2

4. The factored form is 3(x + 15)(x – 2), so the dimensions are 3, x + 15, and x – 2.

5. a) The GCF of the terms is 6mn, so the factored form should be 6mn(2m – n).

b) The first binomial, which is a difference of squares, also needs to be factored: (p + 2)(p – 2)(p2 – 5)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

c) The GCF of the terms is 16, so the factored form should be 16(4x2 – 1) = 16(2x – 1)(2x + 1).

d) The GCF of the terms is 9y2z2, so the factored form should be 9y2z2(2z2 – 1).

e) The binomial, which is a difference of squares, also needs to be factored: t(t – 5)(t + 5)

f) The second binomial can be factored to get 3x(2x – 7y) + 4z(2x – 7y). Then divide out the binomial GCF to get (2x – 7y)(3x + 4z).

6. a) 4x(3 – 2x2 + 9x) b) 5(m – 9)(n – p) c) 4wz(x + 7y) d) (a2 + 9)(a + 3)(a – 3) e) 3k(2k – 3)(2k + 3) f) (3x + 4y – 2)(z – 4)

7. a) not possible b) (x + 3)(x – 3)(x + 2)(x – 2) c) not possible d) (4x – 9y)(x + 2y) e) (5x + 3y)2 f) (2s + 3t)(4s – t)

8. a) 2πr2 + 2πrh = 2πr(r + h) b) 2π(2)(2 + 10) = 48π, which is about

50 × 3 = 150 cm2

9. The factored form is y = (5x – 12)2. There is one zero, when x = 2.4, so the vertex is at (2.4, 0).



1. Determine the GCF for each set of terms. a) 10c3 and 25x2 b) 49t and –28 c) 30mn and 18m2n d) 8x, 12xy, and 28xz e) 4p2q2 and –6p3q3 f) 12c2d, 30cd2, and 15c2d

2. Determine the missing factor. a) 6(2y + 1) + x(2y + 1) = ( )(6 + x) b) –3c(d + e) – a(e + d) = (d + e)( ) c) 16w3x2 + 24wx4 – 8wx2 = 8wx2( ) d) 36xy – 12x2y = ( )(3 – x) e) 2a(3b – 5) – 3(5 – 3b) = (3b – 5)( ) f) w(x + y + z) – 11w2(x + y + z)

= (x + y + z) (w)( )

3. Each model represents an algebraic expression. Identify the expression and its factors. a) b)

4. Factor each expression.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

a) x2 + x – 6 b) d2 + 9d + 14 c) a2 – 13a + 36 d) 10m2 + 7m – 12 e) 2w2 – w – 6 f) 6x2 + xy – y2

5. A parabola is defined by y = 2x2 + 10x – 48. a) Express the equation in factored form. b) Determine the zeros, the equation of the axis

of symmetry, and the coordinates of the vertex.

6. Factor, if possible. a) x2 + 16 b) x2 – 100y2 c) 5d2 – 45 d) 4y(z + 3) + x(z – 3) e) 5a2 + 10ab – 15b2 f) 25b2 – 15b + 2

7. The area of the shaded region is represented by the expression ( )(x2 + 4x + 3). What is the missing factor? Show your work.

8. a) Determine possible integers for k that will allow the trinomial x2 + kx – 72 to be factored.

b) Determine possible integers for k that will make the expression kx2 – 40x + 16 a perfect-square trinomial.

9. Decide whether each polynomial has (3y – 5) as one of its factors. Justify your decision. a) 6xy + 9y – 10x – 15 b) 9y2 – 30y + 25 c) 12y2 – 10 – 14y d) 9y2 – 25

10. The profit on T-shirts sold by a school is determined by the quadratic relation P = –x2 + 30x + 400, where x is the number of T-shirts sold at $10 each and P is the profit in dollars. a) Determine the number of T-shirts that must

be sold to break even. b) What is the maximum profit that the school

can earn by selling T-shirts? Explain.


Chapter 4 Review Extra Practice Answers 1. a) 5

b) 7 c) 6mn d) 4x e) 2p2q2 f) 3cd

2. a) 2y + 1 b) –3c – a c) 2w2 + 3x2 – 1 d) 12xy e) 2a + 3 f) 1 – 11w

3. a) 2x2 – 6x + 4 = (2x – 2)(x – 2) b) x2 – x – 12 = (x – 4)(x + 3)

4. a) (x + 3)(x – 2) b) (d + 7)(d + 2) c) (a – 4)(a – 9) d) (2m + 3)(5m – 4) e) (w – 2)(2w + 3) f) (2x + y)(3x – y)

5. a) y = 2(x – 3)(x + 8) b) zeros: x = 3 and x = –8; equation of the axis

of symmetry: x = –2.5; vertex: (–2.5, –60.5)

6. a) not possible b) (x + 10y)(x – 10y)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

c) 5(d + 3)(d – 3) d) not possible e) 5(a + 3b)(a – b) f) (5b – 2)(5b – 1)

7. Answers may vary, e.g., the shaded region is represented by (2x + 3)2 – x2, a difference of squares. This expression can be factored to get (2x + 3 + x)(2x + 3 – x) = (3x + 3)(x + 3) = 3(x + 1)(x + 3), which simplifies to 3(x2 + 4x + 3). The missing factor is 3.

8. a) k = –71, 71, –34, 34, –21, 21, –14, 14, –6, 6, –1, or 1

b) k = 25

9. a) Yes. The expression can be factored by grouping: 3y(2x + 3) – 5(2x + 3) = (2x + 3)(3y – 5)

b) Yes. The factored form is (3y – 5)2. c) Yes. The factored form is 2(3y – 5)(2y + 1). d) Yes. The factored form is (3y + 5)(3y – 5).

10. a) The factored form is P = (–x + 40)(x + 10). The zeros occur when x = 40 and x = –10 (not possible), so the number of T-shirts that must be sold to break even is 40.

b) The vertex is the maximum point for (x, P) and has the coordinates (15, 625). The maximum profit is $625.



1. a) Complete this table of values for the relation y = 2x2 – 5. x –3 –2 –1 0 1 3

y 13 3 b) Plot the points, and sketch the graph.

2. a) Sketch the graph of the relation y = 2x – 3. b) Sketch the reflection of y = 2x – 3 in the x-axis. c) Determine the equation of the line you sketched for part b).

3. a) Determine the zeros of the relation y = 3(x – 1)(x – 5). b) Is the relation quadratic? Explain how you know. c) Sketch the relation. d) Determine the coordinates of the vertex of the relation, if there is a vertex, or explain

why this is impossible to do.

4. For each translation, determine a) the vertices of +ABC after a translation 3 units left and 2 units up b) the equation of the relation y = –x + 6 after a translation 2 units up c) the coordinates of the points (–2, 4), (–1, 1), (0, 0), and (1, 1) after a translation

that moves (2, 4) to (4, 7)

5. Ricardo performed the following calculation: 3(4 – 2)2 +7 = 3(2)2 + 7 = 62 + 7 = 36 + 7 = 43 What error did Ricardo make?

6. Braking distances for a pickup truck that is travelling at different speeds are shown in this table. Speed (km/h) 30 45 60 75 90 105 120

Braking Distance (m) 22 45 74 105 145 211 252

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

a) Use the data to create a scatter plot. b) Sketch a line of good fit for the data. How well does the line appear to fit the data? c) Sketch a curve of good fit for the data. How well does the curve appear to fit the data?

What kind of relation might the curve be?

7. Factor, if possible. a) x2 + 2x – 3 b) 4x2 – 9 c) 2x2 + 7x + 15 d) x2 – 6x + 8


Catherine

Typewritten Text

5. Factoring Algebraic Expressions

Catherine

Typewritten Text

Catherine

Typewritten Text

Chapter 5 Diagnostic Test Answers

1. a) x –3 –2 –1 0 1 2 3

y 13 3 –3 –5 –3 3 13

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b)

2. a), b)

c) y = –2x + 3

3. a) x = 1, x = 5 b) yes, because expanding gives

y = 3x2 – 18x + 15

c) d) (3, –12)

4. a) A → (–2, 8), B → (–1, 4), C → (2, 6) b) y = –x + 8 c) (0, 7), (1, 4), (2, 3), (3, 4)

5. Ricardo should have evaluated the exponent (2)2 and then multiplied his result by 3, rather than multiplying 2 by 3.

6. a)–c)

b) The line fits the data reasonably well. c) The curve fits the data reasonably well. The

curve might be a quadratic relation.

7. a) (x – 1)(x + 3) c) cannot be factored b) (2x – 3)(2x + 3) d) (x – 2)(x – 4)

If students have difficulty with the questions in the Diagnostic Test, you may need to review the following topics: • creating tables of values and using them to draw graphs • performing translations on the Cartesian coordinate plane • applying the order of operations • creating a scatter plot and drawing a line of good fit • factoring a quadratic relation

2 | Chapter 5 Diagnostic Test Answers


1. Match each graph with the correct equation. a) y = 0.5x2 c) y = –2x2

b) y = –x2 d) y = 31 x2

i) iii)

ii) iv)

2. Write the equation of each graph. a) b)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

3. Determine the equation of a quadratic model to represent the lower arch of this bridge.

4. Which of the following relations do not represent parabolas that are vertical compressions of the graph of y = x2? a) y = 0.1x2 c) y = –2x2

b) y = x2 d) y = 51

− x2

5. Sketch the parabolas for question 4 that are vertical compressions of the graph of y = x2.

6. What transformation(s) must be applied to the graph of y = x2 to create each parabola shown? a) b)

7. The graph of y = x2 is vertically compressed by

a factor of 32 and reflected in the x-axis. What

is the equation of the resulting graph?


Lesson 5.1 Extra Practice Answers 1. a) ii) c) i) b) iv) d) iii)

2. a) y = 3x2 b) y = 41

− x2

3. y = –0.125x2

4. b) and c)

5. a)

d)

6. a) reflection in the x-axis, vertical stretch by a factor of 5

b) vertical compression by a factor of 0.75

7. y = 2

32 x−

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. List, in order, the transformations you would apply to the graph of y = x2 to create the graph of each relation.

a) y = 2x2 + 1 c) y = 2

21 x – 4

b) y = (x – 3)2 d) y = 3(x + 3)2 – 6

2. Sketch the graph of each relation in question 1.

3. Identify the vertex of each parabola. a) y = (x – 3)2 + 2 c) y = 2x2 – 5

b) y = (x + 7)2 d) y = 43

41

31 2

−⎟⎠⎞

⎜⎝⎛ −x

4. Write the equation of the parabola created by applying each set of transformations to the graph of y = x2. a) reflection in the x-axis, vertical compression

by a factor of 0.25, translation 3 units up b) vertical stretch by a factor of 2, translation

2 units right and 4 units down c) reflection in the x-axis, vertical stretch by

a factor of 3.5, translation 3 units left and 2 units up

5. Which graph is the graph of the quadratic

relation y = 21

− (x + 3)2 + 5? Explain your

reasoning.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

a)

b)

c)

d)

6. The height, h, in kilometres, of a space tourism rocket during its unpowered flight phase is given by h = –0.005(t – 150)2 + 200, where t is the time in seconds after its rocket motor is shut down. a) What is the maximum height of the rocket? b) How long after shutdown does the rocket

reach its maximum height?


Lesson 5.3 Extra Practice Answers 1. a) vertical stretch by a factor of 2, translation

1 unit up b) translation 3 units right

c) vertical compression by a factor of 21 ,

translation 4 units down d) vertical stretch by a factor of 3, translation

3 units left and 6 units down

2. a)

b)

c)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

d)

3. a) (3, 2) c) (0, –5)

b) (–7, 0) d) ⎟⎠⎞

⎜⎝⎛ −

43,

41

4. a) y = –0.25x2 + 3 b) y = 2(x – 2)2 – 4 c) y = –3.5(x + 3)2 + 2

5. The graph of y = 21

− (x + 3)2 + 5 is created from

the graph of y = x2 by a reflection in the x-axis, a

vertical compression by a factor of 21 , and a

translation 3 units left and 5 units up. These transformations produce graph d).

6. a) 200 km b) 150 s or 2 min 30 s



1. Write the equation of each graph. a)

b)

2. Which of the following relations have parabolas that are stretches of the graph of y = x2? a) y = 1.25x2 c) y = –x2 b) y = 0.25x2 d) y = –3x2

3. Write the equation of the parabola created by applying each set of transformations to the graph of y = x2.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

a) reflection in the x-axis, vertical stretch by a factor of 1.5

b) vertical stretch by a factor of 3, translation 3 units down and 4 units left

c) reflection in the x-axis, vertical compression

by a factor of 31 , and translation 5 units up

4. Sketch the graph of each quadratic relation in question 3.

5. For each pair of values of h and k, describe the transformation(s) of y = x2 that would create the graph of y = (x – h)2 + k. a) h = –3, k = 0 c) h = –1, k = 3 b) h = 2, k = 5 d) h = 0, k = –2

6. For each pair of values of h and k in question 5, sketch the graph of y = (x – h)2 + k.

7. Describe the transformations you would apply to the graph of y = x2, in the order you would apply them, to create the graph of each relation. a) y = 3x2 + 2 c) y = –(x + 3)2 + 4

b) y = 21 (x + 2)2 – 7 d) y = –2(x – 1)2 – 4

8. Which equation represents the graph shown? Explain your reasoning.

a) y = (x + 3)2 – 4 c) y = 2(x + 3)2 – 4 b) y = 2(x – 3)2 – 4 d) y = –2(x – 3)2 – 4

9. Write the equation of the parabola that matches each description. a) The graph of y = x2 is stretched vertically by

a factor of 3 and then translated 3 units down.

b) The graph of y = x2 is compressed vertically

by a factor of 51 and then translated 3 units

right and 1 unit down. c) The graph of y = x2 is reflected in the x-axis

and then translated 2 units left.


Chapter 5 Mid-Chapter Review Extra Practice Answers 1. a) y = –3x2 b) y = 4x2

2. a) and d)

3. a) y = –1.5x2 b) y = 3(x + 4)2 – 3

c) y = 31

− x2 + 5

4. a)

b)

c)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

5. a) translation 3 units left b) translation 2 units right and 5 units up c) translation 1 unit left and 3 units up d) translation 2 units down

6. a)

b)

c)

d)

7. a) vertical stretch by a factor of 3, translation 2 units up

b) vertical compression by a factor of 0.5, translation 2 units left and 7 units down

c) reflection in the x-axis, translation 3 units left and 4 units up

d) reflection in the x-axis, vertical stretch by a factor of 2, translation 1 unit right and 4 units down

8. The graph is created from the graph of y = x2 with a vertical stretch by a factor of 2 and a translation 3 units left and 4 units down. These transformations produce the graph in part c).

9. a) y = 3x2 – 3 c) y = –(x + 2)2

b) y = 51 (x – 3)2 – 1



1. a) Write an equation to describe all possible parabolas with vertex (–2, 9).

b) A parabola with vertex (–2, 9) passes through point (4, –3). Determine the value of a for this parabola.

c) Write the equation of the parabola described for part b).

d) What transformations must be applied to the graph of y = x2 to obtain the parabola described for part b)?

e) Graph the parabola described for part b).

2. Write the equation of each parabola in vertex form. a)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b)

3. Determine the equation of the parabola created by applying each set of transformations to the graph of y = x2. a) reflection in the x-axis, translation 4 units

right b) vertical stretch by a factor of 2, translation

3 units left and 2 units down

c) vertical compression by a factor of 51 ,

translation 1 unit left and 4 units up

4. Write an equation of a parabola with each set of properties. a) vertex at (–3, 2), opens upward, narrower

than y = x2 b) vertex at (2, 0), opens downward, wider than

y = x2 c) equation of the axis of symmetry x = –2,

opens upward, two zeros, same shape as y = x2

d) vertex at (3, 5), passes through (1, –3)

5. Each table of values represents a parabola. Determine the vertex of the parabola, and write the equation of the parabola in vertex form.

a) x y b) x y

–2 –11 –3 –8.5

–1 –4 –2 –9.0

0 1 –1 –8.5

1 4 0 –7.0

2 5 1 –4.5

3 4 2 –1.0

6. A golfer drives from an elevated tee. The following table gives partial data for the path of the golf ball through the air.

Time (s) 0.0 0.5 1.0 1.5 2.0 2.5

Height (m) 2.5 4.2 6.2 7.6 8.3 7.1

a) Use the data to create a scatter plot, and draw a quadratic curve of good fit.

b) Determine an equation in vertex form to model this relation.

c) Use your model for part a) to predict what happens after 4 s.

d) Check the accuracy of your model using quadratic regression.


Lesson 5.4 Extra Practice Answers 1. a) y = a(x + 2)2 + 9

b) a = 31

−

c) y = 31

− (x + 2)2 + 9

d) reflection in the x-axis, vertical compression

by a factor of 31 , translation 2 units left and

9 units up e)

2. a) y = 2x2 – 5

b) y = 21 (x + 3)2

3. a) y = –(x – 4)2 b) y = 2(x + 3)2 – 2

c) y = 51 (x + 1)2 + 4

4. a) Answers may vary, e.g., y = 3(x + 3)2 + 2 b) Answers may vary, e.g., y = –0.5(x – 2)2 c) Answers may vary, e.g., y = (x + 2)2 – 4 d) y = –2(x – 3)2 + 5

5. a) vertex at (2, 5); y = –(x – 2)2 + 5

b) vertex at (–2, –9); y = 21 (x + 2)2 – 9

6. Answers may vary, e.g., a)

b) h = –1.6(t – 2)2 + 8.3 c) The golf ball hits the ground after 4 s. d) Quadratic regression gives

h = –1.41t2 + 5.62t + 2.19. The equation in part b) is h = –1.6t2 + 6.4t + 1.9 in standard form, so the model is reasonably accurate.

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Determine the maximum or minimum value of a quadratic relation of the form y = a(x – h)2 + k, given the information below. a) a = –1, vertex at (3, –2)

b) a = 31 , vertex at (–2, 0)

c) a = 3, vertex at (0, 4) d) a = –0.25, vertex at (2, –5)

2. Determine the equation of a quadratic relation in vertex form, given the information below. a) vertex at (0, 2), passes through (2, 4) b) vertex at (–2, 5), passes through (0, –3) c) vertex at (7, 4), passes through (4, 13)

3. Determine the equation of each parabola in vertex form. a)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b)

c)

4. Write each equation for question 3 in standard form and factored form.

5. A quadratic relation has zeros at x = –1 and x = 3, and a minimum value of –12. Determine its equation in vertex form.

6. Express each equation in factored form and vertex form. a) y = 2x2 + 12x + 16 c) y = –4x2 – 40x – 64

b) y = 31

− x2 + 2x d) y = 2x2 – x – 1

7. The graph of y = –2(x – 3)2 + 5 was translated so that its new zeros are at 0 and 4. Determine the translation that was applied to the original graph.

8. A canal is 112 m wide. The underside of a bridge across the canal is a parabolic arch, with its bases set back 4 m from each bank of the canal. The maximum height of the arch must be at least 32 m above the surface of the canal. Write an equation to represent an arch that satisfies these conditions. Use graphing technology to graph your equation.

9. A parabola has a y-intercept of 2 and passes through points (–2, –4) and (8, –14). Determine the vertex of the parabola.


Lesson 5.5 Extra Practice Answers 1. a) maximum: –2 c) minimum: 4

b) minimum: 0 d) maximum: –5

2. a) y = 21 x2 + 2

b) y = –2(x + 2)2 + 5 c) y = (x – 7)2 + 4

3. a) y = –0.5(x – 3)2 b) y = 2(x + 3)2 – 8 c) y = 3x2 – 12

4. a) y = –0.5x2 + 3x – 4.5; y = –0.5(x – 3)2 b) y = 2x2 + 12x + 10; y = 2(x + 1)(x + 5) c) y = 3x2 – 12; y = 3(x – 2)(x + 2)

5. y = 3(x – 1)2 – 12

6. a) y = 2(x + 2)(x + 4); y = 2(x + 3)2 – 2

b) y = 31

− x(x – 6); y = 31

− (x – 3)2 + 3

c) y = –4(x + 2)(x + 8); y = –4(x + 5)2 + 36

d) y = (2x + 1)(x – 1); y = 89

412

2

−⎟⎠⎞

⎜⎝⎛ −x

7. 1 unit left and 3 units up

8. Answers may vary, e.g., y = –0.01x2, or 2

2252 xy ⎟

⎠⎞

⎜⎝⎛−=

9. (2, 4)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.



1. Determine the axis of symmetry for a parabola that passes through (3, –1) and (–5, –1).

2. Use partial factoring to determine the vertex form of the quadratic relation y = x2 – 4x + 6.

3. For each quadratic relation, i) use partial factoring to determine two points

that are the same distance from the axis of symmetry

ii) determine the coordinates of the vertex iii) express the relation in vertex form iv) sketch the graph of the relation a) y = x2 – 8x + 13 c) y = –2x2 + 12x – 9

b) y = 21 x2 + x + 3 d) y = 3x2 – 9x – 2

4. Determine the axis of symmetry of the parabola y = 3x2 – 6x – 9 using each strategy. Then describe what you did. a) partial factoring b) factoring completely, if possible; if not

possible, explain why not

5. Write each relation in vertex form. a) y = (x – 3)(x + 1) c) y = 2x(x + 4) – 5 b) y = x2 – 4x d) y = x2 – 5x + 13

6. Determine the values of a and b in the relation y = ax2 + bx – 6 if the vertex of its graph is at (–2, 6).

7. Acme Inc. produces a patent line of solar batteries. Acme’s monthly profit, P, in thousands of dollars, is modelled by P = –40x2 + 320x – 420, where x is the price, in dollars, of each solar battery. What is Acme’s maximum monthly profit? What price should Acme charge to obtain this maximum profit?

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


Lesson 5.6 Extra Practice Answers 1. x = –1

2. y = (x – 2)2 + 2

3. a) i) (0, 13), (8, 13) ii) (4, –3) iii) y = (x – 4)2 – 3 iv)

b) i) (0, 3), (–2, 3)

ii) ⎟⎠⎞

⎜⎝⎛−

25,1

iii) y = 21 (x + 1)2 +

25

iv)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

c) i) (0, –9), (6, –9) ii) (3, 9) iii) y = –2(x – 3)2 + 9 iv)

d) i) (0, –2), (3, –2) ii) (1.5, –8.75) iii) y = 3(x – 1.5)2 – 8.75 iv)

4. a) Use partial factoring to rewrite the relation: y = 3(x2 – 2x) – 9 = 3x(x – 2) – 9. Two points on the graph are (0, –9) and (2, –9). Since these points are the same distance from the axis of symmetry, the equation of the axis of

symmetry is x = 2

20+ = 1.

b) Factor completely to rewrite the equation: y = 3(x2 – 2x – 3) = 3(x – 3)(x + 1). The relation has zeros at 3 and –1. Since the zeros are the same distance from the axis of symmetry, the equation of the axis of

symmetry is x = 2

)1(3 −+ = 1.

5. a) y = (x – 1)2 – 4 c) y = 2(x + 2)2 – 13 b) y = (x – 2)2 – 4 d) y = (x – 2.5)2 + 6.75

6. a = –3, b = –12

7. maximum profit: $220 000; price to obtain maximum profit: $4



1. State whether each parabola is a vertical stretch or compression (or neither) of the graph of y = x2. Also state whether there is a reflection. a) y = 2x2 c) y = –x2 b) y = 0.25x2 d) y = –3.5x2

2. Sketch each parabola for question 1.

3. Describe the transformations that are applied to the graph of y = x2 to obtain each parabola. a) y = (x – 1)2 c) y = (x – 3)2 – 4 b) y = x2 + 7 d) y = (x + 1)2 – 5

4. Sketch the graph of each quadratic relation. Start with a sketch of y = x2, and apply the necessary transformations in the correct order. a) y = 3x2 + 3 c) y = –(x + 1)2 – 2 b) y = 0.25(x – 3)2 d) y = –2(x + 4)2 + 7

5. Write the equation of each parabola in vertex form. a)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

b)

c)

6. Determine the equation of a quadratic relation in vertex form, given the information below. a) vertex at (3, 2), passes through (4, 4) b) vertex at (–2, 7), passes through (1, 4)

7. Express each relation in factored form and vertex form. Use partial factoring if complete factoring is not possible. a) y = 2x2 + 16x c) y = –x2 + 8x + 2 b) y = 3x2 – 12 d) y = 5x2 – 30x + 25

8. A boat tour company charges $11 for a harbour tour and averages 450 passengers on Saturdays. Over the past few months, the company has been experimenting with the price of a tour and has noticed that every increase of $1 in the price decreases the number of customers by 25. Use an algebraic model to determine the price that maximizes revenue.

9. A parabola passes through points (–1, 2), (5, 2), and (1, 4). a) Determine the coordinates of the vertex, and

write the equation of the parabola in vertex form.

b) Write the equation of the parabola in standard form.

10. Determine the values of a and c in the relation y = ax2 + 30x + c if the vertex of its graph is at (–5, 3).


Chapter 5 Review Extra Practice Answers 1. a) stretch, no reflection

b) compression, no reflection c) no stretch or compression, reflection in the

x-axis d) stretch, reflection in the x-axis

2. a)

b)

c)

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

d)

3. a) translation 1 unit right b) translation 7 units up c) translation 3 units right and 4 units down d) translation 1 unit left and 5 units down

4. a)

b)

c)

d)

5. a) y = (x – 3)2 – 3 b) y = 0.5(x + 4)2

c) y = –2(x – 1)2 + 1

6. a) y = 2(x – 3)2 + 2 b) y = 31

− (x + 2)2 + 7

7. a) y = 2x(x + 8); y = 2(x + 4)2 – 32 b) y = 3(x – 2)(x + 2); y = 3x2 – 12 c) y = –x(x – 8) + 2; y = –(x – 4)2 + 18 d) y = 5(x – 1)(x – 5); y = 5(x – 3)2 – 20

8. $14.50

9. a) (2, 4.25); y = –0.25(x – 2)2 + 4.25 b) y = –0.25x2 + x + 3.25

10. a = 3, c = 7



Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Consider the quadratic relation y = x2 – 6x + 3. a) Use partial factoring to locate two points with the same y-coordinate on the graph. b) Determine the coordinates of the vertex. c) Sketch the graph.

2. Sketch the graph of each quadratic relation. a) y = –(x – 3)(x + 1) c) y = –4(x + 2)2 + 5

b) y = 2x2 + 10x – 12 d) y = 31 (x + 2)(x – 4)

3. Factor each expression, if possible. a) x2 – 5x – 24 c) 9x2 – 9x – 10 b) x2 – 4x + 8 d) –2x2 + 7x + 4

4. Expand and simplify. a) (2x – 5)(x + 7) c) (2m – 3)(2m + 3) b) (3 – 2q)2 d) (5c – 3)(11 – c)

5. For each quadratic relation, determine the zeros, the y-intercept, the equation of the axis of symmetry, the vertex, and the equation in standard form. a) b)

6. For each quadratic relation, determine the equation of the axis of symmetry, the vertex, and the zeros (if any). a) y = –2(x + 4)(x + 8) c) y = 3x(x – 4) + 12

b) y = –(x + 1)2 + 9 d) y = 31 (x – 5)2 + 2

7. Use graphing technology to graph each quadratic relation. Then explain why the relation cannot be written as y = a(x – r)(x – s), where r and s are integers. a) y = 2x2 + 6x + 7 b) y = x2 – 8x + 5

Catherine

Typewritten Text

Catherine

Typewritten Text

Catherine

Typewritten Text

6. Quadratic Equations

Catherine

Typewritten Text


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 6 Diagnostic Test Answers 1. a) y = x(x – 6) + 3; (0, 3), (6, 3) c)

b) (3, –6)

2. The points that are labelled may vary, e.g., a) c)

b)

d)

3. a) (x + 3)(x – 8) b) cannot be factored c) (3x + 2)(3x – 5) d) –(2x + 1)(x – 4)

4. a) 2x2 + 9x – 35 b) 9 – 12q + 4q2 c) 4m2 – 9 d) –5c2 + 58c – 33

5. a) –3, 5; –7.5; x = 1; (1, –8); y = 0.5x2 – x – 7.5 b) 3, 4; –12; x = 3.5; (3.5, 0.25); y = –x2 + 7x – 12

6. a) x = –6; (–6, 8); –4, –8 b) x = –1; (–1, 9); 2, –4 c) x = 2; (2, 0); 2 d) x = 5; (5, 2); no zeros

7. a) b)

There are no x-intercepts, so the relation The x-intercepts are not integers, so r and s cannot be factored. are not integers.

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • graphing a quadratic relation given in standard form, factored form, and vertex form • using technology to graph a quadratic relation • determining the equation of the axis of symmetry, the vertex, and the zeros of a quadratic relation • expanding and simplifying expressions • factoring quadratic expressions


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Use each graph to determine the roots of the quadratic equation, where y = 0. a)

b)

2. Determine whether –3 is a root of each equation. a) –(x – 3)(x + 6) = 0 b) x2 + 5x + 6 = 0 c) 5x2 + 13x + 5 = 11 d) –4x2 = 36

3. Solve by factoring. a) 2x2 + x – 10 = 0 b) x2 – 2x – 9 = 6 c) 3x2 + 11x – 15 = 5 d) 3x2 – 6x = 0

4. Solve by graphing. Round your answers to two decimal places. a) x2 – 6x – 4 = 0 b) x2 – 7x + 5 = 2 c) –4x2 – 12x + 15 = 0 d) 32x – 4x2 = 52

5. The height, h, in metres, of a football can be modelled by h = –5t2 + 23t, where t is the time, in seconds, after kickoff. The football lands on the ground without being caught. How long is the football in the air?

6. Tracy runs a small business doing tax returns. If she charges a fee of F dollars per tax return, the amount of money she will earn, E, in dollars, each

tax season can be modelled by E = )480(51 FF − .

a) How much will Tracy earn if she charges $375 per tax return?

b) If Tracy wants to earn at least $9000, between what two amounts should she charge?

7. a) Use algebra to determine the points of intersection of the line y = 4 – 3x and the parabola y = 3x2 + 4x – 2.

b) Verify your answers for part a) using a graphing calculator.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 6.1 Extra Practice Answers 1. a) 3, –3 b) 5, –2

2. a) no b) yes c) yes d) no

3. a) 2, –221

b) 5, –3

c) 34 , –5

d) 0, 2

4. a) –0.61, 6.61 b) 0.46, 6.54 c) –3.95, 0.95 d) 2.27, 5.73

5. 4.6 s

6. a) $7875 b) between $127.75 and $352.25

7. a) ⎟⎠⎞

⎜⎝⎛ 2,

32 , (–3, 13)

b)


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Write each relation in vertex form by completing the square. a) y = x2 – 12x b) y = 3x2 + 6x + 9

2. a) Complete the square to write the relation y = –5x2 – 20x – 13 in vertex form.

b) Graph the relation.

3. Copy and complete this area model. Then express the relation y = x2 + 11x + 8 in vertex form.

4. Benoît has a home business making fancy gift boxes. If he sells each box for x dollars, his monthly profit, P, in dollars, can be modelled by

P = –50x2 + 350x – 520. a) What is the maximum profit that Benoît can

make per month? b) How much should he charge to make this

profit?

5. Kara wants to make a square and a rectangle with a piece of flexible wire that is 102 cm long. The length of the rectangle will be twice its width. Kara decides to represent the length of wire, measured in centimetres, that she will use for the square as 12x. a) Write expressions for the dimensions of the

square and the rectangle in centimetres. b) Show that the combined area of the square and

the rectangle, in square centimetres, is 17(x2 – 8x + 34).

c) Determine the minimum combined area that Kara can make. What are the lengths of wire that she must use for each shape?

6. a) Determine the vertex and the equation of the axis of symmetry of the quadratic relation y = 3x2 – 12x + 5 by completing the square.

b) How does doubling the coefficient of x in the relation affect the position of the axis of symmetry?

7. a) Continue the following pattern of perfect-square trinomials for three more terms: 100x2 – 20x + 1, 100x2 – 40x + 4, …

b) Express 100x2 – 140x + 65 as the sum of a perfect-square expression and the square of an integer.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 6.3 Extra Practice Answers 1. a) y = (x – 6)2 – 36

b) y = 3(x + 1)2 + 6

2. a) y = –5(x + 2)2 + 7 b)

3.

y = (x + 5.5)2 – 22.25

4. a) $92.50 b) $3.50

5. a) 3x by 3x, (34 – 4x) by (17 – 2x) b) Answers may vary, e.g.,

A = (3x)2 + (34 – 4x)(17 – 2x) A = 9x2 + 17(34) – 34(2x) – 17(4x) + 8x2 A = 17x2 – 17(8x) + 17(34) A = 17(x2 – 8x + 34)

c) 306 cm2; square: 48 cm; rectangle: 54 cm

6. a) (2, –7), x = 2 b) Answers may vary, e.g., after doubling, the axis

of symmetry is twice as far from the y-axis.

7. a) 100x2 – 60x + 9, 100x2 – 80x + 16, 100x2 – 100x + 25

b) (10x – 7)2 + 42


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Solve each quadratic equation by factoring. a) x2 – 13x + 30 = 0 b) x2 = 6(12 – x) c) (x + 1)2 = 16 d) x(x + 4) = 12

2. Solve each quadratic equation by factoring or by using a graphing calculator. Round your answers to two decimal places, if necessary. a) 2x2 + 9x – 5 = 0 b) x2 + 11 = 3x c) 3x2 – 5x = 7 d) 3(x2 – 5) = 4x

3. A rectangular theatre stage, with an area of 84 m2, is 5 m longer than it is wide. How long is the stage?

4. Determine the value of c that makes each expression a multiple of a perfect-square trinomial. a) x2 + 8x + c b) 2x2 + 12x + c c) 5x2 – 60x + c d) 7x2 – 42x + c

5. Complete the square to express each relation in vertex form. Then describe the transformations that must be applied to the graph of y = x2 to graph the relation. a) y = x2 + 14x + 29 b) y = 3x2 – 18x + 14

6. Armin charges $50/h to repair computers. If he charges for x hours per month, his monthly profit, P, in dollars, can be modelled by the relation P = –x2 + 180x – 4150. a) What is Armin’s maximum monthly profit? b) How many hours should he work each month

to achieve this profit?

7. A suborbital space plane is designed to give “space tourists” a brief experience of zero-G flight. According to the flight plan, when the engine shuts down at an altitude of 125 000 m, the space plane gains altitude at a speed of 1250 m/s. Its height, t seconds after engine shutdown, can be modelled by –5t2 + 1250t + 125 000. a) What is the maximum altitude of the space

plane? b) The period of freefall lasts from engine

shutdown to re-entry. Assuming that the space plane starts to re-enter the atmosphere at the same altitude of 125 000 m, how long is the space plane in freefall?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 6 Mid-Chapter Review Extra Practice Answers 1. a) 3, 10 b) 6, –12 c) 3, –5 d) 2, –6

2. a) 0.5, –5 b) no solutions c) about –0.91, about 2.57 d) about –1.67, 3

3. 12 m

4. a) 16 b) 18 c) 180 d) 63

5. a) y = (x + 7)2 – 20; translation 7 units left and 20 units down

b) y = 3(x – 3)2 – 13; vertical stretch by a factor of 3, translation 3 units right and 13 units down

6. a) $3950 b) 90 h

7. a) 203 125 m b) 250 s, or 4 min 10 s


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. a) Solve the equation 2x2 – 27x – 80 = 0 by factoring.

b) Solve the same equation using the quadratic formula.

c) Identify one advantage of each strategy over the other.

2. Solve each equation using the quadratic formula. a) x2 – 16 = 49 b) 3x(x – 5) = 5(15 – 3x)

3. Determine the roots of each equation. Round the roots to two decimal places, if necessary. a) 2x2 – 10 = 0 b) –3(x – 3)2 + 18 = 0 c) 3(x + 3)2 – 75 = 0 d) –0.5(x – 0.5)2 + 1 = 0

4. Solve each equation. Round your solutions to two decimal places, if necessary. a) 2x2 – 10x – 6 = 0 b) (x – 1)(2x + 3) = 0 c) 2(x2 + 1) = 7x d) 2x(2 – x) = 1 – x

5. Calculate the value of x, to two decimal places.

6. A baseball is bunted so that its height, h, in metres, after t seconds is modelled by h = –4.9t2 + 23.4t + 1.3. a) How long is the baseball in the air, to the

nearest tenth of a second? b) How long is the baseball above 25 m, to the

nearest tenth of a second?

7. A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving stones, 25 cm wide. The area of the ring of paving stones will be half the area of the flower bed. Determine the diameter of the flower bed, to the nearest centimetre.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 6.4 Extra Practice Answers 1. a)–b) 16, –2.5

c) Answers may vary, e.g., factoring makes it easier to check that the solutions are correct; the quadratic formula can be used directly, without checking all the factor pairs of 80.

2. a) about 8.06, about –8.06 b) 5, –5

3. a) 2.24, –2.24 b) 0.55, 5.45 c) 2, –8 d) 1.91, –0.91

4. a) 5.54, –0.54 b) 1, –1.5 c) 0.31, 3.19 d) 0.22, 2.28

5. 3.70

6. a) 4.8 s b) 1.9 s

7. 222 cm


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. a) Determine the roots of 2x2 – 9x – 35 = 0 using the strategy of your choice.

b) What do your results for part a) tell you about the graph of y = 2x2 – 9x – 35?

c) Predict whether the discriminant of the equation in part a) is positive, negative, or zero. Justify your prediction.

d) Confirm your prediction by calculating the value of D.

2. Calculate the discriminant of each equation, and state the number of real roots. a) x2 + 5x – 3 = 0 b) 4x(x – 1) + 1 = 0 c) 2x2 + 5 = 3x d) 3x(x + 5) = x2 + 8

3. How many times does each relation cross or touch the x-axis? Justify your answer in terms of the graph of the relation. a) y = (x – 3)2 – 7 b) y = –2x2 – 13 c) y = 5(x + 2)2

d) y = –(x + 10)2 + 2

4. The support rails of a suspension bridge over a river are modelled by the relation h = 22 + 0.002x2, where x is the distance, in metres, from the middle of the bridge and h is the height, in metres, above the river. Explain why, in terms of the bridge and the river, the discriminant of the relation must be negative.

5. Determine whether the vertex of each parabola lies above, below, or on the x-axis. Explain your answers. a) y = –x2 + 7x + 5 b) y = (x + 2)2 + 2x + 7 c) y = 4x2 – 20x + 25 d) y = –5x(x – 3) – 15

6. For what value(s) of k does the relation y = 4x2 – kx + 9 have each number of zeros? a) two zeros b) one zero c) no zeros

7. Suppose that a and c have opposite signs in the quadratic relation y = ax2 + bx + c. a) What does this tell you about the y-intercept

and the direction of opening of the graph of the relation?

b) Use sketches, without a grid, to show why the relation must have two roots.

c) Give an algebraic reason why the discriminant must be positive when a and c have opposite signs.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 6.5 Extra Practice Answers 1. a) 7, –2.5

b) The parabola crosses the x-axis twice. c) From parts a) and b), the parabola has two

roots. The discriminant must be positive. d) D = 361; my prediction is correct.

2. a) 37; two real roots c) –31; no real roots b) 0; one real root d) 289; two real roots

3. Answers may vary, e.g., a) 2; the vertex is below the x-axis, and the

graph opens upward. b) 0; the vertex is below the x-axis, and the

graph opens downward. c) 1; the vertex is on the x-axis. d) 2; the vertex is above the x-axis, and the

graph opens downward.

4. Answers may vary, e.g., the bridge is above the river, so the height of the rail is always positive. Therefore, the relation has no zeros.

5. Answers may vary, e.g., a) Above the x-axis; the relation opens

downward and, because the discriminant is positive, has two zeros.

b) Above the x-axis; the relation opens upward and, because the discriminant is negative, has no zeros.

c) On the x-axis; the discriminant is zero, so the relation has one zero.

d) Below the x-axis; the relation opens downward and, because the discriminant is negative, has no zeros.

6. a) k < –12 or k > 12 b) k = ±12 c) –12 < k < 12

7. Answers may vary, e.g., a) Either the y-intercept is positive and the

graph opens downward, or the y-intercept is negative and the graph opens upward.

b) Answers may vary slightly, e.g.,

In each case, the vertex is farther from the

x-axis than the y-intercept, so the graph must cross the x-axis twice.

c) Since a and c have opposite signs, ac < 0, so –4ac > 0. Therefore, b2 – 4ac > 0.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. For each relation, explain what each coordinate of the vertex represents and what the zeros represent. a) a relation that models the height, h, of a diver

who jumps upward from a board, after time t b) a relation that models the monthly profit, P,

earned for a given number of clients per month, x

c) a relation that models the area, A, of a rectangle with a fixed perimeter and width w

For questions 2 to 7, round your answers to two decimal places, if necessary.

2. The height, h, in metres, of a football is modelled by h = 1 + 5t(4 – t), where t is the time, in seconds, after the football is punted. a) What is the maximum height of the football? b) How long is the football in the air?

3. Carly is diving from the 10 m board at her local pool. During her descent underwater, her depth, d, in metres, is modelled by d = –7.5t2 + 14.3t, where t is the time, in seconds, after she enters the water. a) What is Carly’s maximum depth? b) How long does Carly take to reach this depth?

4. The sum of the squares of three consecutive positive odd integers is 515. Determine the integers.

5. Karsten, a tennis pro, is attempting a drop shot so that the ball just clears the net. The ball is 1.3 m above the ground when Karsten hits it. The ball reaches its maximum height, 3.1 m, after it has travelled 2.5 m horizontally. a) Write an equation to model this situation. b) The height of the net is 0.92 m in the middle of

the court. Assuming that Karsten’s shot is aimed down the centre line, how far away from the net must he be to make the shot?

6. Darya attempts a high jump of 2.22 m. Her centre of mass is initially at a height of 1.0 m, passes 1.8 m after 0.2 s, and passes 1.8 m again after 0.8 s. a) The height of Darya’s centre of mass, in

metres, is modelled by y = a(t – h)2 + k. Show that h = 0.5, 0.25a + k = 1, and 0.09a + k = 1.8.

b) Write the relation that models the height of Darya’s centre of mass in vertex form.

c) Does Darya clear the attempted height? Explain.

7. Rashid is planning six cubicles for an office. He wants each cubicle to have the same dimensions, and he has chosen an L-shaped layout. He has written the relation P = 9ℓ + 10w for the length of cubicle wall needed. If he has 72 m of cubicle wall to use, what dimensions will maximize the area of each cubicle?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 6.6 Extra Practice Answers 1. a) vertex: maximum height, h, and time, t, when

the maximum height is reached; first zero: no meaning; second zero: time when the diver enters the water

b) vertex: maximum profit, P, and number of clients per month, x, for the maximum profit; zeros: break-even points

c) vertex: maximum area, A, and width, w, for the maximum area; zeros: dimensions of the “rectangles” with zero length or width

2. a) 21 m b) 4.05 s

3. a) 6.82 m b) 0.95 s

4. 11, 13, 15

5. a) h = –0.29(x – 2.5)2 + 3.1 b) 5.24 m

6. a) h = 2

8.02.0 + = 0.5; 1 = a(0 – 0.5)2 + k, so

0.25a + k = 1; 1.8 = a(0.2 – 0.5)2 + k, so 0.09a + k = 1.8

b) h = –5(t – 0.5)2 + 2.25 c) Yes. The maximum height of Darya’s centre of

mass is greater than 2.22 m.

7. 4 m by 3.6 m

Chapter 6 Review Extra Practice Answers | 425

Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Solve each equation by factoring. Check your solutions with graphing technology. a) x2 + 12 = 7x b) 2x2 + 9x – 35 = 0 c) 17(x + 5) = 5(x2 + 9) d) 3x(x – 6) = 5(2 – x)

2. The quadratic relation y = 5x2 + 40x + c has a minimum value of 0. What is the value of c? Explain.

3. Write each relation in vertex form by completing the square. Then describe the transformations that must be applied to the graph of y = x2 to graph the relation. a) y = x2 – 12x + 50 b) y = 4x2 + 48x + 144 c) y = –0.25x2 + 3.5x + 6.75 d) y = –2x2 + 10x – 15

For questions 4 to 8, round your answers to two decimal places, if necessary.

4. A bridge with a parabolic inner arch is being constructed over a canal. The arch will just touch the water on each side of the canal, which is 27.2 m wide. The arch must have a clearance of 6.4 m at a horizontal distance of 9.5 m from either bank of the canal. a) Write a relation, in factored form, to model the

height, h, in metres, of the bridge at a horizontal distance of x metres from one bank of the canal.

b) Rewrite your relation for part a) in vertex form by expanding and completing the square.

c) How high above the water will the arch be, at its highest point, to the nearest hundredth of a metre?

5. Solve each equation using the quadratic formula. a) x2 – 7x + 9 = 0 b) 2.5x2 = 16x + 10.5 c) 2x(x + 3) = 14(1 – x) d) –2x(2x – 5) = –15

6. Kaneisha owns a business that produces and sells handmade carriage clocks. She models her monthly revenue, in dollars, as R = –0.2x2 + 50x, where x is the price, in dollars, of each clock. She models her monthly costs, in dollars, as C = 4250 – 13x. She calculates her monthly profit, in dollars, as P = R – C. Between what prices should Kaneisha charge to break even?

7. The height of a football, h, in metres, can be modelled by h = –5t2 + 25t + 1.5, where t is the time, in seconds, after it is punted. a) Write an equation for the time(s) at which the

football is c metres high. Determine an expression for the discriminant, D, of this equation in terms of c.

b) Use your expression for the discriminant to determine the maximum height of the football. Explain your reasoning.

c) How many realistic solutions does your equation from part a) have for each case below? Explain your reasoning.

i) 1.50 ≤ c < 32.75 ii) 0 ≤ c < 1.50 iii) c < 0

8. The hourly cost, C, in dollars, of running a printing press is modelled as C = 0.000 015x2 – 0.18x + 965, where x is the number of pages that are printed per hour. a) How many pages should be printed each hour

to minimize the cost? b) What rates of printing will keep the cost below

$800?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 6 Review Extra Practice Answers 1. a) 3, 4

b) –7, 2.5

c) –1.6, 5

d) 32

− , 5

2. 80; answers may vary, e.g., only a perfect-square trinomial could have a minimum value of zero because it has only one root, so 5x2 + 40x + c = 5(x2 + 8x + 16) = 5x2 + 40x + 80.

3. a) y = (x – 6)2 + 14; translation 6 units right and 14 units up

b) y = 4(x + 6)2; vertical stretch by a factor of 4, translation 6 units left

c) y = –0.25(x – 7)2 + 19; reflection in the x-axis, vertical compression by a factor of 0.25, translation 7 units right and 19 units up

d) y = –2(x – 2.5)2 – 2.5; reflection in the x-axis, vertical stretch by a factor of 2, translation 2.5 units right and 2.5 units down

4. a) h = −0.04x(x – 27.2) b) h = –0.04(x – 13.6)2 + 7.40 c) 7.40 m

5. a) about 1.70, about 5.30 b) 7, –0.6 c) about 0.66, about –10.66 d) about 3.55, about –1.05

6. between $97.87 and $217.13

7. a) –5t2 + 25t + 1.5 – c = 0; D = 655 – 20c b) 32.75 m; answers may vary, e.g., there is

exactly one solution for the time to reach maximum height, and the equation has exactly one solution for c = 32.75.

c) i) D > 0, so there are two solutions; both solutions are realistic, since the height is between the starting value of 1.5 m and the maximum height.

ii) D > 0, so there are two solutions; only one solution is realistic, since the height is less than the starting value of 1.5 m.

iii) D > 0, so there are two solutions; neither solution is realistic, since the height cannot be negative.

8. a) 6000 b) between 1000 and 11 000 pages per hour


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. What is the factored form of 12a2 – 7a – 10? A. (6a – 2)(2a + 5) B. (4a – 5)(3a + 2) C. (4a + 2)(3a – 5) D. (4a + 2)(3a – 2)

2. What is the factored form of 16c2 – 56c + 49? A. (4c – 7)2 B. (4c + 7)(4c – 7) C. (4c + 7)2 D. (8c – 7)(2c – 7)

3. What is the factored form of 81x2y2 – 49? A. (9xy – 7)2 B. (9x – 7y)(9x + 7y) C. (9xy + 7)2 D. (9xy – 7)(9xy + 7)

4. The following transformations were applied to the graph of y = x2: a reflection in the x-axis, a vertical

compression by a factor of 21 , and a translation

3 units right and 7 units down. What is the equation of the transformed graph?

A. y = 21 (x – 3)2 + 7

B. y = 21

− (x – 7)2 – 3

C. y = –2(x – 3)2 – 7

D. y = 21

− (x – 3)2 – 7

5. The vertex of a parabola is (–4, 5), and the parabola passes through (–6, –3). What is the equation of the parabola? A. y = –11(x + 4)2 + 5 B. y = –2(x + 4)2 + 5 C. y = 2(x – 4)2 + 5 D. y = –11(x – 5)2 – 4

6. What is the vertex form of the quadratic relation y = 4x(x – 5) – 10? A. y = 4(x + 2.5)2 + 15 B. y = 4(x – 2.5)2 – 15 C. y = 4(x – 2.5)2 – 35 D. y = 4(x – 2.5)2 + 35

7. Which values of x are solutions to the equation 8x2 – 2x – 3 = 0?

A. x = 41 , x =

23

−

B. x = 21

− , x = 43

C. x = 21 , x =

43

−

D. x = 41

− , x = 23

8. Which expression is not a perfect-square trinomial? A. 9x2

– 6x + 1 B. x2

+ 10x + 25 C. x2

– x + 1 D. 4x2

– 12x + 9

9. What is the minimum value of the quadratic equation y = 2x2 + 28x + 17? A. –81 B. 81 C. 32 D. –32

10. Which equation is equivalent to x2 + 8x + 15? A. y = (x + 8)2 – 1 B. y = (x + 4)2 – 1 C. y = (x + 4)2 + 1 D. y = (x + 2)2 + 11

11. Which value of b will make the quadratic equation 6x2 + bx + 5 have no real solutions? A. 11 B. 17 C. 4 D. 12


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapters 4–6 Cumulative Review Extra Practice Answers 1. B

2. A

3. D

4. D

5. B

6. C

7. B

8. C

9. A

10. B

11. C


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Express each ratio in simplest form.

a) 12 : 42 b) 18

5.4 c) 20 : –8 d) 6345

2. Solve each proportion.

a) x4

1512

= b) 32

45=

x c) x

5.2212

5.7= d)

2.156.98.4

=x

3. Determine the length of side a in each diagram. a) b)

4. As a shortcut to school, Jason walks across a rectangular field along the diagonal, instead of walking along two adjacent sides of the field. The dimensions of the field are 90 m by 120 m. How much shorter is Jason’s shortcut?

5. Determine the value of x in each diagram. a) c)

b) d)

Catherine

Typewritten Text

Catherine

Typewritten Text

7. Similar Triangles and Trigonometry

Catherine

Typewritten Text


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. a) 2 : 7

b) 41

c) 5 : –2

d) 75

2. a) 5 b) 30 c) 36 d) 7.6

3. a) about 8.2 m b) about 8.8 cm

4. 60 m

5. a) 30º b) 150º c) 123º d) 36º

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • solving proportions • applying the Pythagorean theorem to determine side lengths • using properties of angles in a triangle and angles formed by parallel lines and transversals

to determine angle measures


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. i) For each pair of triangles, determine whether the triangles are congruent, similar, or neither.

ii) If the triangles are congruent, identify the corresponding angles and sides that are equal. If the triangles are similar, identify the corresponding angles that are equal, and calculate the scale factor that relates the smaller triangle as a reduction of the larger triangle.

a)

b)

2. +ABC is similar to+PRQ. Determine the length of QR.

3. In+PQR, PQ = 8.0 cm, PS = 5.0 cm, and PT = 4.0 cm.

a) Which triangles are similar? How do you

know? b) Determine the length of PR.

4. +ABC is similar to+DEF. Determine the scale factor that relates the larger triangle as an enlargement of the smaller triangle.

5. Determine the value of each lower-case letter. a)

b)


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 7.1 Extra Practice Answers 1. a) i) similar ii) ,ABCDEF ∠=∠ ,BACEDF ∠=∠

;ACBDFE ∠=∠

scale factor: 32

===BCEF

ACDF

ABDE

b) i) congruent ii) ,PRQMON ∠=∠ ,RPQOMN ∠=∠

;PQRMNO ∠=∠ MN = PQ, MO = PR, NO = QR

2. 15 cm

3. a) +PST ~+PQR since ,PQRPST ∠=∠ ,PRQPTS ∠=∠ and QPRSPT ∠=∠

b) 6.4 cm

4. 23

5. a) b = 12 cm b) x = 4.5 cm, y = 3.3 cm


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. A section of roof rafters is shown below.

Determine the value of x to one decimal place.

2. A bridge is going to be built across the river at ED. Determine the width of the river where the bridge will be built.

3. On a sunny day, the shadow of a tower is 28.5 m long. A pole near the tower is 2.3 m high and casts a shadow that is 1.8 m long. Determine the height of the tower.

4. In a lumberjack competition, two poles of different heights are used for a pole-climbing event. The wires from the tops of the poles to the ground are parallel. The shorter pole is 6.0 m tall, and its wire is secured 8.0 m from the base. If the wire to the top of the taller pole is 16.0 m long, determine the height of the taller pole.

5. John uses a mirror to determine the height of a building. He knows that the angle of elevation is equal to the angle of reflection when a light is reflected off a mirror. What is the height of the building?

6. To determine the length of a lake, TR, the measurements shown in the diagram were made. How long is the lake?

When calculating side lengths and angle measures, round your answers to the number of decimal places in the given information when the required degree of accuracy is not stated.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 7.2 Extra Practice Answers 1. about 5 m

2. about 2.8 m

3. about 36.4 m

4. 9.6 m

5. about 16.7 m

6. about 435.2 m


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. If two triangles are similar, under what conditions would they also be congruent?

2. +MNO and+PQR are similar. Write a proportion for the corresponding side lengths.

3. The triangles below are similar. Determine the scale factor that relates the smaller triangle as a reduction of the larger triangle.


b)

c)

5. +ABC ~+DEF, BC = 5.2 cm, AC = 3.4 cm, DE = 8.2 cm, and EF = 10.6 cm. Determine the lengths of AB and DF.

6. The shadow of an apartment building is 106.0 m long. A 1.0 m pole near the building casts a shadow that is 1.8 m long. Determine the height of the building.

7. Determine the length of the lake, DE.

8. A 5 m ladder rests against a vertical wall, with its base 3 m from the wall. Another ladder, 12 m long, is resting against the wall, parallel to the first ladder. How far up the wall does each ladder reach, to one decimal place?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 7 Mid-Chapter Review Extra Practice Answers 1. If two triangles are similar and two corresponding

sides are equal, then the triangles are congruent.

2. Answers may vary, e.g.,

QRNO

PRMO

PQMN

==

3. 31

4. a) a = 4.5 cm, b = 4.0 cm b) c = 3.0 cm c) d = 5.0 cm, e = 7.5 cm

5. AB = 4.0 cm, DF = 6.9 cm

6. about 58.9 m

7. 72 m

8. 4.0 m, 9.6 m


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Determine each ratio to four decimal places.

a) sin P d) cos Q

b) cos P e) tan P c) tan Q f) sin Q

2. Solve for x, and express your answer to one decimal place.

a) sin 40º = 12x d) cos 60º =

34x

b) cos 47º = x

50 e) tan 15º = 43x

c) tan 65º = x

35 f) sin 60º = x

24

3. Determine the value of x. Then write the primary trigonometric ratios for θ. a)

b)

4. Determine the measure of θ to one decimal place. a)

b)

c)

5. Determine the measure of θ to one decimal place.

a) cos θ = 65 d) cos θ =

41

b) sin θ = 117 e) tan θ = 1

c) tan θ = 7

15 f) sin θ = 21


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 7.4 Extra Practice Answers

1. a) sin P = 5248 = 0.9231

b) cos P = 5220 = 0.3846

c) tan Q = 4820 = 0.4167

d) cos Q = 5248 = 0.9231

e) tan P = 2048 = 2.4000

f) sin Q = 5220 = 0.3846

2. a) 7.7 units b) 73.3 units c) 16.3 units d) 17.0 units e) 11.5 units f) 27.7 units

3. a) x = 39 cm

sin θ = 135

3915

=

cos θ = 1312

3936

=

tan θ = 125

3615

=

b) x = 40 cm

sin θ = 4140

cos θ = 419

tan θ = 940

4. a) 36.9º b) 46.4º c) 73.7º

5. a) 33.6º b) 39.5º c) 65.0º d) 75.5º e) 45.0º f) 30.0º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. For each triangle, i) state two trigonometric ratios that you could

use to determine x ii) determine x to the nearest unit a)

b)

2. Calculate the measures of A∠ and B∠ in each triangle using trigonometric ratios. Round your answers to the nearest degree. a)

b)

3. Solve each triangle. Round the measure of each angle to the nearest degree. Round the length of each side to one decimal place.

a)

b)

4. a) In+JKL, =∠J 31º, =∠K 90º, and k = 45.3 m. Calculate j.

b) In+RST, =∠R 80º, =∠T 90º, and s = 20.5 m. Calculate r.

5. a) In+ABC, =∠B 90º, b = 35.9 cm, and c = 16.2 cm. Calculate .C∠

b) In+DEF, =∠F 90º, d = 9.3 cm, and e = 13.2 cm. Calculate .E∠

6. A 10.5 m ladder is leaning against a wall. The foot of the ladder is 2.6 m from the wall. Determine the angle between the ladder and the ground.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 7.5 Extra Practice Answers

1. a) i) sin 60º = 15x , cos 30º =

15x

ii) x = 13 cm

b) i) tan 42º = 28x , tan 48º =

x28

ii) x = 25 cm

2. a) =∠A 52º, =∠B 38º b) =∠A 37º, =∠B 53º

3. a) =∠B 31º, a = 38.3 cm, c = 44.7 cm b) =∠D 58º, =∠F 32º, e = 40.8 cm

4. a) about 23.3 m b) about 116.3 m

5. a) about 27º b) about 55º

6. about 76º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. A runner estimates that the slope of a steep hill makes an angle of 50º with the ground. If the hill is 60 m high, what distance will the runner have to travel to get to the top of the hill?

2. Using sonar, a trawler captain detects a school of fish at a depth of 65.5 m. The angle of depression of the sounding is 18º. How far will the trawler have to travel to be directly above the school of fish?

3. A 25 m cellphone tower is supported by wires on opposite sides. The wires are anchored to the ground at a distance of 16 m from the foot of the tower. What is the angle of inclination that each wire makes with the ground?

4. The support for a shelf makes an angle of 48º with the wall. If the shelf is 32 cm wide, what is the length of the support? Round your answer to the nearest tenth of a centimetre.

5. A tree casts a shadow that is 20.0 m long when the angle of elevation of the Sun is 34º. Determine the height of the tree to one decimal place.

6. From the top of a building that is 55 m tall, the angle of depression to a car on the road is 35º. How far is the car from the base of the building?

7. An observer, who is 1.8 m tall, estimates that the angle of elevation of a cliff is 60º. The observer is 42.6 m from the base of the cliff. Determine the height of the cliff.

8. An equilateral triangle has side lengths of 29 cm. Determine the area of the triangle.

9. A satellite dish is mounted on the top of a building that is 100.0 m tall. The angle of elevation from the satellite dish to the top of a second building is 43º. The angle of depression to the base of the second building is 54º. How tall is the second building?

10. Determine the angle between the line 154

+= xy

and the x-axis, to the nearest degree.

11. A truck drives 15.9 km up a road, until it has gone 2.1 km vertically. If the road has a steady incline, what is the angle of elevation of the road to the nearest tenth of a degree?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 7.6 Extra Practice Answers 1. about 78 m

2. about 201.6 m

3. about 57º

4. 43.1 cm

5. 13.5 m

6. about 79 m

7. about 75.6 m

8. about 363 cm2

9. about 167.8 m

10. 39º

11. 7.6º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Determine whether the triangles in the diagram are similar. If they are similar, determine the scale factor that relates the smaller triangle as a reduction of the larger triangle.


b)

3. A 4.2 m ladder leans against a vertical wall, with its foot 1.2 m from the wall. A 5.6 m ladder leans against the wall, parallel to the first ladder. How far is the base of the second ladder from the wall?

4. On a sunny day, a tree casts a shadow that is 28.0 m long. Andrew, who is 1.9 m tall, is standing near the tree and casts a shadow that is 3.5 m long. What is the height of the tree?

5. In each triangle, i) determine the unknown side length ii) state the three primary trigonometric ratios

for A∠ iii) determine the measure of A∠ a) b)

6. Determine the value of x to one decimal place.

a) sin 25º = 6.15

x b) tan 65º = x

7.41

7. a) In+ABC, =∠B 90º, b = 12 cm, and c = 8 cm. Determine the measure of A∠ .

b) In+DEF, =∠E 90º, =∠F 35º, and d = 17.6 cm. Determine the length of side f.

8. Solve this triangle.

9. A tree casts a shadow that is 25.0 m long when the angle of elevation of the Sun is 36º. Determine the height of the tree to one decimal place.

10. A truck driver estimates that a country road rises 40 cm every 6 m along the road. What is the angle of elevation of the road? Round your answer to the nearest tenth of a degree.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 7 Review Extra Practice Answers 1. +ABC ~+EDC; ,EA ∠=∠ ;DB ∠=∠

scale factor: 61

2. a) x = 24 cm, y = 12 cm b) x = 6.6 cm, y = 85˚

3. 1.6 m

4. 15.2 m

5. a) i) 15 cm

ii) sin A = 1715 , cos A =

178 , tan A =

815

iii) =∠A 62˚ b) i) 25 cm

ii) sin A = 2524 , cos A =

257 , tan A =

724

iii) =∠A 74˚

6. a) 6.6 b) 19.4

7. a) about 48º b) about 12.3 cm

8. θ = 18º, a = 3.9 cm, b = 12.6 cm

9. 18.2 m

10. 3.8º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Determine the measures of the indicated angles in each diagram. a) b)

2. Determine the value of each trigonometric ratio to four decimal places. a) sin 33º b) cos 27º c) cos 75º d) sin 68º

3. Determine the measure of ∠ A to the nearest degree. a) sin A = 0.8660 c) sin A = 0.8494 b) cos A = 0.8660 d) cos A = 0.8390

4. Solve.

a) 6

158=

x c) x9

714

=

b) 121612

=x

d) 1824

18 x=

5. Determine the value of x in each triangle. Round your answers to one decimal place. a) b) c)

Catherine

Typewritten Text

Catherine

Typewritten Text

8. Acute Triangle Trigonometry

Catherine

Typewritten Text


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. a) a = 50º, b = 65º, c = 115º b) d = 52º, e = 45º, f = 83º

2. a) 0.5446 b) 0.8910 c) 0.2588 d) 0.9272

3. a) 60º b) 30º c) 58º d) 33º

4. a) 20 b) 9 c) 4.5 d) 13.5

5. a) 38.7º b) 36.9º c) 23.0 cm

If students have difficulty with the questions in the Diagnostic Test, it may be necessary to review the following topics: • applying properties of angles in a triangle and angles formed by parallel lines to

determine angle measures • solving proportions • determining and using the primary trigonometric ratios • applying the primary trigonometric ratios to determine side lengths and angle measures


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Solve. Determine x to one decimal place and θ to the nearest degree.

a) °

=° 71sin

5.743sinx

b) 3165sin61sin °

=°

x

c) 43

sin3755sin θ

=°

d) 130

73sin100sin °

=θ

2. Determine the indicated side lengths and angle measures.

a)

b)

3. a) In+ABC, a = 22 cm, =∠B 53º, and =∠C 43º. Determine the length of side b.

b) In+PQR, q = 9 cm, r = 7 cm, and =∠R 47º. Determine the measure of .Q∠

4. Solve each triangle. a)

b)

5. Solve each triangle. Determine each side length to one decimal place and each angle measure to the nearest degree. a) In+ABC, =∠B 42º, =∠C 59º, and

a = 20.0 cm. b) In+RST, r = 15 cm, s = 12 cm, and =∠S 48º.

6. Three towns, represented by points A, B, and C, are located so that B is 25.0 km from A, and C is 34.4 km from A. If =∠B 80º, determine the distance from B to C.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 8.2 Extra Practice Answers 1. a) 5.4

b) 29.9 c) 72º d) 47º

2. a) about 18.2 cm b) about 62º

3. a) about 18 cm b) about 70º

4. a) =∠A 35º, a = 7 cm, c = 10 cm b) =∠P 56º, =∠Q 44º, p = 421 m

5. a) =∠A 79º, b = 13.6 cm, c = 17.5 cm b) =∠R 68º, =∠T 64º, t = 14.5 cm

6. about 28.3 km


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Determine the indicated side lengths to one decimal place and the indicated angle measures to the nearest degree. a)

b)

c)

d)

2. Write three equivalent ratios using the sides and angles in acute triangle KLM. Record the answer in two different ways.

3. a) In+ABC, =∠A 47º, =∠B 65º, and b = 6.8 cm. Determine the length of side a.

b) In+DEF, =∠E 59º, =∠F 41º, and d = 93 cm. Determine the length of side f.

c) In+PQR, =∠Q 70º, p = 5 cm, and q = 7 cm. Determine the measure of .P∠

4. Solve each triangle. a) In+PQR, =∠P 70º, =∠R 25º, and q = 55 cm. b) In+DEF, =∠E 50º, e = 33 cm, and f = 41 cm.

5. In+ABC, c = 45.0 cm, =∠A 70º, and =∠B 60º. Determine the perimeter of+ABC.

6. A slide in a park is 9 m long, and its ladder is 7 m long. If the slide makes an angle of 48º with the ground, determine the angle that the ladder makes with the ground.

7. Two points, A and B, are located on opposite sides of a river. Point C is located 125 m from point B, on the same side of the river. If =∠ABC 71º and

=∠ACB 47º, determine the distance from A to B.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 8 Mid-Chapter Review Extra Practice Answers 1. a) b = 22.7 cm

b) a = 18.6 cm, b = 22.2 cm c) θ = 39º d) θ = 60º, α = 52º

2. M

mL

lK

ksinsinsin

==

mM

lL

kK sinsinsin

==

3. a) about 5.5 cm b) about 62 cm c) about 42º

4. a) =∠Q 85º, p = 52 cm, r = 23 cm b) =∠D 58º, =∠F 72˚, d = 37 cm

5. about 151.1 cm

6. about 73º

7. about 104 m


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. a) Record the cosine law for+JKL for determining the length of side j.

b) Record the cosine law for+STR for determining the measure of .R∠

2. Determine the length of each indicated side. a)

b)

3. Determine the measure of each indicated angle. a)

b)

4. Correct the mistake(s) in each of these for acute triangle XYZ. a) y2 = x2 + z2 – 2 xy cos Y b) x2 = y2 + z2 – yz cos X c) z2 = x2 + z2 + 2 xy cos Z d) y2 = x2 + y2 – 2 xy cos Y e) z2 = x2 + y2 + xy cos Y

5. Solve each triangle. a) In+ABC, =∠B 43º, a = 10.5 m, c = 11.2 m. b) In+DEF, d = 60 m, e = 50 m, f = 40 m. c) In+PQR, p = 10.0 m, r = 15.0 m, =∠Q 50º.

6. Determine the perimeter of+ABC if a = 43.0 cm, b = 28.0 cm, and =∠C 75˚.

7. A parallelogram has sides that are 12.0 cm and 18.0 cm long. The angle between these sides is 78º. Determine the length of the shorter diagonal.

8. A jogger travels 5.4 km directly east, and then turns and travels 6.3 km in a N57ºW direction. How far does the jogger have to travel to go directly back to the starting point?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 8.4 Extra Practice Answers 1. a) j2 = k2 + l2 – 2 kl cos J

b) r2 = s2 + t2 – 2 st cos R

2. a) about 6.2 cm b) about 6.8 cm

3. a) about 55º b) about 63º

4. a) y2 = x2 + z2 – 2 xz cos Y b) x2 = y2 + z2 – 2 yz cos X c) z2 = x2 + y2 – 2 xy cos Z d) y2 = x2 + z2 – 2 xz cos Y e) z2 = x2 + y2 – 2 xy cos Z

5. a) b = 8.0 m, =∠A 64º, =∠C 73º b) =∠D 83º, =∠E 56º, =∠F 41º c) q = 11.5 m, =∠P 42º, =∠R 88º

6. about 115.8 cm

7. about 19.4 cm

8. about 3.4 km


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Would you use the sine law or the cosine law for each of the following? a) The length of each side is given, and you need

to determine the measure of an angle. b) The lengths of two sides and the angle opposite

one of the sides are given, and you need to determine the measure of an angle.

c) The lengths of two sides and the contained angle are given, and you need to determine the length of the other side.

2. In+ABC, AB = 30.0 cm, =∠A 80º, and =∠B 55º. Determine the perimeter of+ABC.

3. An isosceles triangle has sides that are 4 cm, 10 cm, and 10 cm long. Determine the measure of the equal angles.

4. Louise works for a landscaping company. Her job involves measuring properties that are going to be landscaped. A triangular property has a 4.9 m side and a 5.8 m side, which meet at a 35º angle. Determine the perimeter of the property.

5. A surveyor is locating three points, M, N, and P, around an artificial pond. The distance from M to N is 728 m, and the distance from M to P is 638 m. The measure of ∠N is 57º. a) Determine the measure of ∠M. b) Determine the distance from N to P.

6. The buoys that mark a triangular course for a yacht race are located at points Y, T, and B. If YT = 5.5 km, =∠Y 55º, and =∠T 75º, determine the length of the course.

7. Todd and Scott leave the dining hall at a camp. They walk on two straight paths that diverge by 48º. Scott walks 580 m, and Todd walks 740 m. How far apart are they?

8. Two fishing boats leave the same dock at the same time. One boat travels at a speed of 15.0 km/h, and the other boat travels at a speed of 18.0 km/h. After 45 min, the boats are 14.0 km apart. Assuming that both boats are travelling in straight paths, what is the angle between their paths?


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Lesson 8.5 Extra Practice Answers 1. a) cosine law

b) sine law c) cosine law

2. about 106.6 cm

3. about 78º

4. about 14.0 m

5. a) about 50º b) about 583 m

6. about 18.3 km

7. about 556 m

8. about 68º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. Which of the following are not correct for acute triangle RST?

a) T

tS

ssinsin

=

b) sT

rR sinsin=

c) T

ts

Ssin

sin=

d) s sin S = t sin T e) r sin T = t sin R

2. Calculate the indicated side length or angle measure in each triangle. a)

b)

3. Solve+ABC, if =∠A 32º, =∠B 72º, and b = 72.4 cm.

4. Akila is making a triangular support to hang plants. She is using three lengths of pipe, which measure 34 cm, 25 cm, and 28 cm. Determine the angles in the triangular support.

5. Calculate the indicated side length or angle measure in each triangle. a)

b)

6. Solve+DEF, if d = 9 cm, e = 7 cm, and f = 10 cm.

7. Two planes are flying at the same altitude. They are 3000 m apart when they spot a raft on the sea below them. The angles of depression to the raft are 57º and 48º. Determine the distance from the raft to the closer plane.

8. A plot of land is the shape of an isosceles triangle, with equal sides that are 175 m long. The angle between the equal sides is 80º. Determine the length of the third side.

9. A parallelogram has adjacent sides that are 12 cm and 18 cm long. The shorter diagonal is 15 cm long. Determine the measures of all four angles in the parallelogram.


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapter 8 Review Extra Practice Answers 1. b), c), d)

2. a) b = 29 cm, c = 32 cm b) θ = 46º, α = 59º

3. =∠C 76º, a = 40.3 cm, c = 73.9 cm

4. about 54º, about 80º, about 46º

5. a) about 7.3 cm b) about 76º

6. =∠D 61º, =∠E 43º, =∠F 76º

7. about 2308 m

8. about 225 m

9. about 56º, about 124º, about 56º, about 124º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.


1. What is the value of x?

A. 12.0 cm B. 5.1 cm C. 7.1 cm D. 9.4 cm

2. A board that is 3.2 m long is leaning against a vertical wall, with its foot 2.0 m away from the wall. Another board, which is 4.8 m long, is leaning against the wall, parallel to the first board. How far is the foot of the second board from the wall? A. 1.0 m B. 4.0 m C. 4.8 m D. 3.0 m

3. What is the value of θ to the nearest degree?

A. 50º B. 40º C. 56º D. 35º

4. John is standing at the top of a hill. He observes that the angle of depression to the bottom of the hill is 40º. If the distance to walk down the hill is 15 m, what is the vertical distance from the top of the hill to the ground? A. 11.5 m B. 12.6 m C. 7.5 m D. 9.6 m

5. What is the length of side a?

A. 13.2 m B. 17.3 m C. 14.5 m D. 7.2 m

6. In+ABC, ∠A = 47º, b = 20.0 cm, and c = 14.0 cm. What is the length of side a? A. 34.1 cm B. 14.6 cm C. 15.9 cm D. 26.9 cm

7. What is the measure of θ?

A. 72º B. 31º C. 58º D. 70º

8. In+ABC, ∠A = 80º, ∠B = 40º, and a = 7.5 cm. Solve+ABC. A. b = 4.9 cm, c = 6.6 cm, ∠C = 40º B. b = 6.6 cm, c = 4.9 cm, ∠C = 60º C. b = 4.9 cm, c = 6.6 cm, ∠C = 60º D. b = 3.6 cm, c = 11.3 cm, ∠C = 40º

9. In+DEF, d = 4.6 m, e = 4.2 m, and f = 2.1 m. Solve+DEF. A. ∠D = 86º, ∠E = 34º, ∠F = 60º B. ∠D = 87º, ∠E = 66º, ∠F = 27º C. ∠D = 34º, ∠E = 86º, ∠F = 60º D. ∠D = 65º, ∠E = 88º, ∠F = 27º


Cop

yrig

ht ©

201

1 by

Nel

son

Edu

catio

n Lt

d.

Chapters 7–8 Cumulative Review Extra Practice Answers 1. C

2. D

3. B

4. D

5. A

6. B

7. A

8. C

9. B

lead author chris kirkpatrick -...

Documents