Math Grade 10

DEPED COPYActivity 15: Solve the problem. Packaging is one important feature in producing quality products. A box designer needs to produce a package for a product in the shape of a pyramid with a square base having a total volume of 200 cubic inches. The height of the package must be 4 inches less than the length of the base. Find the dimensions of the product. Solution: Let __________= area of the base __________= height of the pyramid If the volume of the pyramid is V = 1 (base)(height), 3 then, the equation that will lead to the solution is 36 = ____________. The possible roots of the equation are:______________. Using synthetic division, the roots are_______________. Therefore, the length of the base of the package is ___________ and its height is _______________________. 95 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

SUMMARY/SYNTHESIS/GENERALIZATIONThis lesson was about solving polynomial equations and the Rational RootTheorem. You learned how to:  use the Fundamental Theorem of Algebra to determine the maximum number of real roots of a polynomial;  solve polynomial equations in factored form;  solve polynomial equations using the Rational Root Theorem; and  solve problems that can be modelled by polynomial equations.DEPED COPYGLOSSARY OF TERMSDegree of a Polynomial - the highest degree of a term in a polynomialFactor Theorem - the polynomial P(x) has x – r as a factor if and only if P(r) = 0Mathematical Model - a mathematical representation of some phenomena in real worldPolynomial - an algebraic expression of the form anxn + an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0, where an  0, and a0, a1, a2, …, an are real numbersLIST OF THEOREMS USED IN THIS MODULE:Rational Root Theorem - Let an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0 = 0 be apolynomial equation of degree n. If p , in lowest terms, is a rational qroot of the equation, then p is a factor of a0 and q is a factor of an.Remainder Theorem - If the polynomial P(x) is divided by (x – r), the remainder R is a constant and is equal to P(r).Synthetic Division - a short method of dividing polynomial expressions using only the coefficient of the terms 96 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYREFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Acelajado M. J., Beronque Y. B. & Co, F. F. (2005) Algebra Concepts and Processes. (3rd Ed.). Mandaluyong City: National Book Store. Coronel I. C., Coronel, A. C. & Munsayac, J. M. (2013) Growing up with Math. Quezon City: FNB Educational, Inc. Coronel I. C., Villano, L. M., Manalastas P. R., Marasigan J. A. & Coronel A. C. (2004) Advanced Algebra, Trigonometry and Statistic. Quezon City: Bookmark Inc. Coronel I. C., Villano L. M., Manalastas P. R., Marasigan J. A. & Coronel A. C. (1992) Mathematics and Integrated Approach (SEDP) Bookmark Inc. De Leon C. M. & Bernabe J. G. (2002) Elementary Algebra: Textbook for First Year. (Pilot Edition) Quezon City: JTW. Publishing Co. Dilao S. J., Orines, F. B. & Bernabe J. C. (2003). Advanced Algebra, Trigonometry and Statistics. JTW Publishing Co. Green J. W., Ulep S. A., Gallos F. L. & Umipig D. F. (n. a) Teaching Mathematics IV Volume I. Philippines – Australia Science and Mathematics Educational Project. Hadlay W. S., Pfluger J. & Coratto M. (2006) Algebra 1 Student Text. USA: Carnegie Learning Pittsburgh. Larson R. & Hostetler R. P. (2013). Algebra and Trigonometry. (8th Ed.) Cergage Learning Asia Ple ltd. Orines, F. B., Esparrago M. S. & Reyes N. V. (2008) Advanced Algebra, Trigonometry and Statistics. Quezon City: Phoenix Publishing House. Oronce, O. A. & Mendoza M. O. (2003) Exploring Mathematics Advanced Algebra and Trigonometry.(1st Ed). Manila: Rex Printing, Company, Inc. 97 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYQuimpo N. F. (2005) A Course in Freshman Algebra. Mathematics Department Loyola Schools. Ateneo de Manila University.Villaluna T. T. & Van Zandt G. E.(2009) Hands-on, Minds-on Activities in Mathematics IV: Advanced Algebra, Trigonometry and Statistics. Philippines: St. Jude Thaddeus Publication.Website Links as References and as Sources of Learning Activities:http://www.mathsisfun.com/algebra/polynomials-division-long.htmlhttp://www.youtube.com/watch?v=qd-T-dTtnX4http://www.purplemath.com/modules/polydiv2.htmhttps://www.brightstorm.com/math/algebra-2/factoring/rational-roots-theorem/http://www.youtube.com/watch?v=RXKfaQemtii 98 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY 10 Mathematics Learner’s Module Unit 2 This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected]. We value your feedback and recommendations. Department of Education Republic of the Philippines All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Mathematics – Grade 10Learner’s ModuleFirst Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice wherein the work is created shall be necessary for exploitation of such work for profit.Such agency or office may, among other things, impose as a condition the payment ofroyalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,trademarks, etc.) included in this book are owned by their respective copyright holders.DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seekingpermission to use these materials from their respective copyright owners. All means havebeen exhausted in seeking permission to use these materials. The publisher and authors donot represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS andonly within the agreed framework may copy this Learner’s Module. Those who have notentered in an agreement with FILCOLS must, if they wish to copy, contact the publisher andauthors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02)439-2204, respectively.Published by the Department of EducationSecretary: Br. Armin A. Luistro FSCUndersecretary: Dina S. Ocampo, PhDDEPED COPY Development Team of the Learner’s ModuleConsultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, andRosemarievic Villena-Diaz, PhDAuthors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz,Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines,Rowena S. Perez, and Concepcion S. TernidaEditor: Maxima J. Acelajado, PhDReviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P.Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd,Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A.Tudlong, PhDIllustrator: Cyrell T. NavarroLayout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. PagulayanManagement and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr.,Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr.Printed in the Philippines by REX Book StoreDepartment of Education-Instructional Materials Council Secretariat (DepEd-IMCS)Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600Telefax: (02) 634-1054, 634-1072E-mail Address: [email protected] All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Table of ContentsUnit 2Module 3: Polynomial Functions ............................................................ 99 Lessons and Coverage........................................................................ 100 Module Map......................................................................................... 100 Pre-Assessment .................................................................................. 101 Learning Goals and Targets ................................................................ 105 Activity 1 .................................................................................... 106 Activity 2 .................................................................................... 107 Activity 3 .................................................................................... 108 Activity 4 .................................................................................... 108 Activity 5 .................................................................................... 110 Activity 6 .................................................................................... 111 Activity 7 .................................................................................... 112 Activity 8 .................................................................................... 115 Activity 9 .................................................................................... 115 Activity 10 .................................................................................. 118 Activity 11 .................................................................................. 119 Activity 12 .................................................................................. 121 Activity 13 .................................................................................. 122 Activity 14 .................................................................................. 123 Summary/Synthesis/Generalization........................................................... 125 Glossary of Terms ...................................................................................... 125 References Used in this Module ................................................................. 126Module 4: Circles .................................................................................... 127 Lessons and Coverage........................................................................ 127 Module Map......................................................................................... 128 Pre-Assessment .................................................................................. 129 Learning Goals and Targets ................................................................ 134 Lesson 1A: Chords, Arcs, and Central Angles .......................................... 135 Activity 1 .................................................................................... 135 Activity 2 .................................................................................... 137 Activity 3 .................................................................................... 138 Activity 4 .................................................................................... 139 Activity 5 .................................................................................... 150 Activity 6 .................................................................................... 151 Activity 7 .................................................................................... 151 Activity 8 .................................................................................... 152 Activity 9 .................................................................................... 152 Activity 10 .................................................................................. 155 Activity 11 .................................................................................. 155 Activity 12 .................................................................................. 157 Activity 13 .................................................................................. 159 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Summary/Synthesis/Generalization ........................................................... 160 Lesson 1B: Arcs and Inscribed Angles....................................................... 161 Activity 1 .................................................................................... 161 Activity 2 .................................................................................... 162 Activity 3 .................................................................................... 163 Activity 4 .................................................................................... 164 Activity 5 .................................................................................... 167 Activity 6 .................................................................................... 168 Activity 7 .................................................................................... 169 Activity 8 .................................................................................... 170 Activity 9 .................................................................................... 172 Activity 10 .................................................................................. 174 Activity 11 .................................................................................. 175 Activity 12 .................................................................................. 176 Summary/Synthesis/Generalization ........................................................... 177 Lesson 2A: Tangents and Secants of a Circle ............................................ 178 Activity 1 .................................................................................... 178 Activity 2 .................................................................................... 179 Activity 3 .................................................................................... 180 Activity 4 .................................................................................... 188 Activity 5 .................................................................................... 189 Activity 6 .................................................................................... 192 Activity 7 .................................................................................... 194 Activity 8 .................................................................................... 197 Summary/Synthesis/Generalization ........................................................... 198 Lesson 2B: Tangent and Secant Segments ................................................. 199 Activity 1 .................................................................................... 199 Activity 2 .................................................................................... 200 Activity 3 .................................................................................... 200 Activity 4 .................................................................................... 201 Activity 5 .................................................................................... 204 Activity 6 .................................................................................... 205 Activity 7 .................................................................................... 206 Activity 8 .................................................................................... 207 Activity 9 .................................................................................... 208 Activity 10 .................................................................................. 210 Summary/Synthesis/Generalization ........................................................... 211 Glossary of Terms ....................................................................................... 212 List of Theorems and Postulates on Circles ............................................... 213 References and Website Links Used in this Module.................................. 215 Module 5: Plane Coordinate Geometry ............................................... 221 Lessons and Coverage ........................................................................ 222 Module Map ......................................................................................... 222 Pre-Assessment................................................................................... 223 Learning Goals and Targets ................................................................. 228 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYLesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof .......................................................... 229 Activity 1 .................................................................................... 229 Activity 2 .................................................................................... 230 Activity 3 .................................................................................... 231 Activity 4 .................................................................................... 232 Activity 5 .................................................................................... 241 Activity 6 .................................................................................... 242 Activity 7 .................................................................................... 242 Activity 8 .................................................................................... 243 Activity 9 .................................................................................... 245 Activity 10 .................................................................................. 248 Activity 11 .................................................................................. 250 Summary/Synthesis/Generalization........................................................... 251 Lesson 2: The Equation of a Circle............................................................ 252 Activity 1 .................................................................................... 252 Activity 2 .................................................................................... 253 Activity 3 .................................................................................... 254 Activity 4 .................................................................................... 263 Activity 5 .................................................................................... 265 Activity 6 .................................................................................... 265 Activity 7 .................................................................................... 266 Activity 8 .................................................................................... 267 Activity 9 .................................................................................... 267 Activity 10 .................................................................................. 269 Summary/Synthesis/Generalization........................................................... 270 Glossary of Terms ...................................................................................... 270 References and Website Links Used in this Module ................................. 271 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYI. INTRODUCTION You are now in Grade 10, your last year in junior high school. In this level and in the higher levels of your education, you might ask the question: What are math problems and solutions for? An incoming college student may ask, “How can designers and manufacturers make boxes having the largest volume with the least cost?” And anybody may ask: In what other fields are the mathematical concepts like functions used? How are these concepts applied? Look at the mosaic picture below. Can you see some mathematical representations here? Give some. As you go through this module, you are expected to define and illustrate polynomial functions, draw the graphs of polynomial functions and solve problems involving polynomial functions. The ultimate goal of this module is for you to answer these questions: How are polynomial functions related to other fields of study? How are these used in solving real-life problems and in decision making? 99 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYII. LESSON AND COVERAGE This is a one-lesson module. In this module, you will learn to:  illustrate polynomial functions  graph polynomial functions  solve problems involving polynomial functions The Polynomial Functions Illustrations of Polynomial Functions Graphs of Polynomial Functions Solutions of Problems Involving Polynomial Functions 100 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

III. PRE-ASSESSMENTPart 1Let us find out first what you already know related to the content of thismodule. Answer all items. Choose the letter that best answers eachquestion. Please take note of the items/questions that you will not be ableto answer correctly and revisit them as you go through this module forself-assessment.1. What should n be if f(x) = xn defines a polynomial function? A. an integer C. any numberDEPED COPY B. a nonnegative integer D. any number except 02. Which of the following is an example of a polynomial function? A. f(x)  4  3x 1 C. f (x)  7x  2x6 x3 B. f (x)  3  3 x2 D. f (x)  x3  3x  5 2x 2 23. What is the leading coefficient of the polynomial function f (x)  2x  x3  4 ? A. 1 C. 3 B. 2 D. 44. How should the polynomial function f (x)  2x  x3  3x5  4 be writtenin standard form? C. f (x)  4  2x  x3  3x5 A. f (x)  x3  2x  3x5  4 B. f (x)  4  3x5  2x  x3 D. f (x)  3x5  x3  2x  45. Which of the following could be the graph of the polynomial function y  x3  4x2  3x 12? y yy y x B. C. xxA. D. 101 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

6. From the choices, which polynomial function in factored form represents the given graph? A. y  (x  2)(x 1)(x 1) B. y  (x 1)(x 1)(x  2) C. y  x(x  2)(x 1)(x 1) D. y  x(x 1)(x 1)(x  2)7. If you will draw the graph of y  x(x  2)2 , how will you sketch it with respect to the x-axis? A. Sketch it crossing both (-2,0) and (0,0). B. Sketch it crossing (-2,0) and tangent at (0,0). C. Sketch it tangent at (-2,0) and crossing (0,0). D. Sketch it tangent at both (-2,0) and (0,0).8. What are the end behaviors of the graph of f (x)  2x  x3  3x5  4 ? A. rises to the left and falls to the right B. falls to the left and rises to the right C. rises to both directions D. falls to both directions9. You are asked to illustrate the sketch of f (x)  x3  3x5  4 using its properties. Which will be your sketch? y y yyDEPED COPY x xA. B. x x C. D. 102 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

10. Your classmate Linus encounters difficulties in showing a sketch of the graph of y  2x3  3x2  4x  6. You know that the quickest technique is the Leading Coefficient Test. You want to help Linus in his problem. What hint/clue should you give? A. The graph falls to the left and rises to the right. B. The graph rises to both left and right. C. The graph rises to the left and falls to the right. D. The graph falls to both left and right.11. If you will be asked to choose from -2, 2, 3, and 4, what values for a and n will you consider so that y = axn could define the graph below?DEPED COPY A. a = 2 , n = 3 B. a = 3 , n = 2 C. a = - 2 , n = 4 D. a = - 2 , n = 312. A car manufacturer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.00125x4 + x – 3, where x represents the number of cars sold. What is the profit when x = 300? A. Php 101.25 C. Php 3,000,000.00 B. Php 1,039,500.00 D. Php 10,125,297.0013. A demographer predicts that the population, P, of a town t years from now can be modeled by the function P(t) = 6t4 – 5t3 + 200t + 12 000.What will the population of the town be two (2) years from now? A. 12 456 C. 1 245 600 B. 124 560 D. 12 456 00014. Consider this Revenue-Advertising Expense situation:The total revenue R (in millions of pesos) for a company is related toits advertising expense by the functionR  1 x3  600x2 , 0  x  400 100 000where x is the amount spent on advertising (in ten thousands ofpesos). 103 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Currently, the company spends Php 2,000,000.00 for advertisement. If you are the company manager, what best decision can you make with this business circumstance based on the given function with its restricted domain? A. I will increase my advertising expenses to Php 2,500,000.00 because this will give a higher revenue than what the company currently earns. B. I will decrease my advertising expenses to Php 1,500,000.00 because this will give a higher revenue than what the company currently earns. C. I will decrease my advertising expenses to Php 1,500,000.00 because lower cost means higher revenue. D. It does not matter how much I spend for advertisement, my revenue will stay the same.Part 2Read and analyze the situation below. Then, answer the question andperform the tasks that follow. Karl Benedic, the president of Mathematics Club, proposed a project:to put up a rectangular Math Garden whose lot perimeter is 36 meters. Hewas soliciting suggestions from the members for feasible dimensions of thelot. Suppose you are a member of the club, what will you suggest to KarlBenedic if you want a maximum lot area? You must convince him through amathematical solution.Consider the following guidelines: 1. Make an illustration of the lot with the needed labels. 2. Solve the problem. Hint: Consider the formulas P = 2l + 2w for perimeter and A = lw for the area of the rectangle. Use the formula for P and the given information in the problem to express A in terms of either l or w. 3. Make a second illustration that satisfies the findings in the solution made in number 2. 4. Submit your solution on a sheet of paper with recommendations. 104 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Rubric for Rating the Output:Score Descriptors 4 The problem is correctly modeled with a quadratic function, appropriate mathematical concepts are fully used in the 3 solution, and the correct final answer is obtained. The problem is correctly modeled with a quadratic function, 2 appropriate mathematical concepts are partially used in the 1 solution, and the correct final answer is obtained. The problem is not properly modeled with a quadratic function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained. The problem is not totally modeled with a quadratic function, a solution is presented but has incorrect final answer.DEPED COPY The additional two (2) points will be determined from the illustrationsmade. One (1) point for each if properly drawn with necessary labels.IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts on polynomial functions. Furthermore, you should be able to conduct a mathematical investigation involving polynomial functions. 105 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Start this module by recalling your knowledge on the concept ofpolynomial expressions. This knowledge will help you understand theformal definition of a polynomial function.Activity 1:Determine whether each of the following is a polynomial expression or not.Give your reasons. DEPED COPY1. 14x 6.  7. 3x 3  3x 2  9x  22. 5x 3  4 2x  x 8. x 3  2x  13. 2014x 9. 4x 100  4x 100 10. 1 – 16x2 314. x 4  3x 4  75. 123 2x 3 3x 4 4x 5 Did you answer each item correctly? Do you remember when anexpression is a polynomial? We defined a related concept below. A polynomial function is a function of the form P x an x n  an1x n1  an2x n2  ...  a1x  a0 , an  0,where n is a nonnegative integer, a0, a1, ..., an are real numbers calledcoefficients, an xn is the leading term, an is the leading coefficient,and a0 is the constant term. The terms of a polynomial may be written in any order. However, ifthey are written in decreasing powers of x, we say the polynomial function isin standard form. Other than P(x), a polynomial function may also be denoted by f(x).Sometimes, a polynomial function is represented by a set P of ordered pairs(x,y). Thus, a polynomial function can be written in different ways, like thefollowing. 106 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

f ( x) an x n  an 1x n1  an 2x n2  ...  a1x  a0 or y an x n  an 1x n 1  an 2x n 2  ...  a1x  a0 Polynomials may also be written in factored form and as a product ofirreducible factors, that is, a factor that can no longer be factored usingcoefficients that are real numbers. Here are some examples. a. y = x4 + 2x3 – x2 + 14x – 56 in factored form is y = (x2 + 7)(x – 2)(x + 4) b. y = x4 + 2x3 – 13x2 – 10x in factored form is y = x(x – 5)(x + 1)(x + 2) c. y = 6x3 + 45x2 + 66x – 45 in factored form is y = 3(2x – 1)(x + 3)(x + 5) d. f(x) = x3 + x2 + 18 in factored form is f(x) = (x2 – 2x + 6)(x + 3) e. f(x) = 2x3 + 5x2 + 7x – 5 in factored form is f(x) = (x2 + 3x + 5)(2x – 1) Activity 2:DEPED COPYConsider the given polynomial functions and fill in the table below. Polynomial Function Polynomial Degree Leading Constant Function in Coefficient Term1. f ( x ) = 2 – 11x + 2x2 Standard Form2. f ( x) 2x 3  5  15x 333. y = x (x2 – 5)4. y x x  3x  35. y  (x  4)(x  1)(x 1)2 After doing this activity, it is expected that the definition of a polynomialfunction and the concepts associated with it become clear to you. Do thenext activity so that your skills will be honed as you give more examples ofpolynomial functions. 107 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 3:Use all the numbers in the box once as coefficients or exponents to form asmany polynomial functions of x as you can. Write your polynomial functions instandard form.1 –2 7 2 1 3 4 6 How many polynomial functions were you able to give? Classify eachaccording to its degree. Also, identify the leading coefficient and the constantterm.DEPED COPY In this section, you need to revisit the lessons and your knowledge on evaluating polynomials, factoring polynomials, solving polynomial equations, and graphing by point-plotting. Your knowledge of these topics will help you sketch the graph of polynomial functions manually. You may also use graphing utilities/tools in order to have a clearer view and a more convenient way of describing the features of the graph. Also, you will focus on polynomial functions of degree 3 and higher, since graphing linear and quadratic functions were already taught in previous grade levels. Learning to graph polynomial functions requires your appreciation of its behavior and other properties. Activity 4:Factor each polynomial completely using any method. Enjoy working withyour seatmate using the Think-Pair-Share strategy. 1. (x – 1) (x2 – 5x + 6) 2. (x2 + x – 6) (x2 – 6x + 9) 3. (2x2 – 5x + 3) (x – 3) 4. x3 + 3x2 – 4x – 12 5. 2x4 + 7x3 – 4x2 – 27x – 18 Did you get the answers correctly? What method(s) did you use? Now,do the same with polynomial functions. Write each of the following polynomialfunctions in factored form: 108 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

6. y  x3  x2 12x 7. y  x4 16 8. y  2x 4 8x3  4x 2  8x  6 9. y  x5 10x3  9x10. y  2x4  7x3  4x2  27x 18 The preceding task is very important for you since it has something todo with the x-intercepts of a graph. These are the x-values when y = 0,thus, the point(s) where the graph intersects the x-axis can be determined. To recall the relationship between factors and x-intercepts, considerthese examples:DEPED COPYa. Find the intercepts of y  x3  4x2  x  6 .Solution:To find the x-intercept/s, set y = 0. Use the factored form. That is,y = x3 – 4x2 + x + 6 Factor completely.y = (x + 1)(x – 2)(x – 3) Equate y to 0.0 = (x + 1)(x – 2)(x – 3)x + 1 = 0 or x – 2 = 0 or x – 3 = 0 Equate each factor to 0 x = –1 x=2 x = 3 to determine x. The x-intercepts are –1, 2, and 3. This means the graph will passthrough (-1, 0), (2, 0), and (3, 0). Finding the y-intercept is more straightforward. Simply set x = 0in the given polynomial. That is, y = x3 – 4x2 + x + 6 y = 03 – 4(0)2 + 0 + 6 y=6The y-intercept is 6. This means the graph will also pass through (0,6). 109 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

b. Find the intercepts of y  x4  6x3  x2  6xSolution:For the x-intercept(s), find x when y = 0. Use the factored form.That is, Factor completely. y = x4 + 6x3 – x2 – 6x Equate y to 0. y = x(x + 6)(x + 1)(x – 1) 0 = x(x + 6)(x + 1)(x – 1)x = 0 or x + 6 = 0 or x + 1 = 0 or x – 1 = 0 Equate each factor to x = –6 x = –1 x = 1 0 to determine x.DEPED COPYThe x-intercepts are -6, -1, 0, and 1. This means the graph will passthrough (-6,0), (-1,0), (0,0), and (1,0). Again, finding the y-intercept simply requires us to set x = 0 inthe given polynomial. That is, y  x 4  6x 3  x 2  6x y  (04)  6(03)  (02)  6(0) y 0 The y-intercept is 0. This means the graph will pass also through (0,0). You have been provided illustrative examples of solving for the x- andy- intercepts, an important step in graphing a polynomial function. Remember,these intercepts are used to determine the points where the graph intersectsor touches the x-axis and the y-axis. But these points are not sufficient todraw the graph of polynomial functions. Enjoy as you learn by performing thenext activities.Activity 5:Determine the intercepts of the graphs of the following polynomial functions: 1. y = x3 + x2 – 12x 2. y = (x – 2)(x – 1)(x + 3) 3. y = 2x4 + 8x3 + 4x2 – 8x – 6 4. y = –x4 + 16 5. y = x5 + 10x3 – 9x You have learned how to find the intercepts of a polynomial function.You will discover more properties as you go through the next activities. 110 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 6:Work with your friends. Determine the x-intercept/s and the y-intercept ofeach given polynomial function. To obtain other points on the graph, find thevalue of y that corresponds to each value of x in the table.1. y = (x + 4)(x + 2)(x – 1)(x – 3) x-intercepts: __ __ __ __ y-intercept: __ x -5 -3 0 2 4 yList all your answers above as ordered pairs.DEPED COPY2. y = –(x + 5)(2x + 3)(x – 2)(x – 4) x-intercepts: __ __ __ __ y-intercept: __ x -6 -4 -0.5 3 5 yList all your answers above as ordered pairs.y = –x(x + 6)(3x – 4) x-intercepts: __ __ __ y-intercept: __ x -7 -3 1 2 yList all your answers above as ordered pairs.3. y = x2(x + 3)(x + 1)(x – 1)(x – 3) x-intercepts: __ __ __ __ __ y-intercept: __x -4 -2 -0.5 0.5 2 4yList all your answers above as ordered pairs In this activity, you evaluated a function at given values of x. Notice thatsome of the given x-values are less than the least x-intercept, some are betweentwo x-intercepts, and some are greater than the greatest x  intercept. Forexample, in number 1, the x-intercepts are -4, -2, 1, and 3. The value -5 isused as x-value less than -4; -3, 0, and 2 are between two x-intercepts; and 4is used as x-value greater than 3. Why do you think we should considerthem? 111 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

In the next activity, you will describe the behavior of the graph of apolynomial function relative to the x-axis.Activity 7:Given the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3), complete thetable below. Answer the questions that follow.Value of Value of Relation of y value to 0: Location of the point (x, y): x y y > 0, y = 0, or y < 0? above the x-axis, on the y 0 -5 144 x-axis, or below the x-axis? y=0 above the x-axis -4 0 -3 on the x - axis -2 DEPED COPY 0 1 2 3 4Questions: 1. At what point(s) does the graph pass through the x-axis? 2. If x  4 , what can you say about the graph? 3. If  4  x  2 , what can you say about the graph? 4. If  2  x  1, what can you say about the graph? 5. If 1 x  3 , what can you say about the graph? 6. If x  3 , what can you say about the graph? Now, this table may be transformed into a simpler one that will instantlyhelp you in locating the curve. We call this the table of signs. The roots of the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3) arex = –4, –2, 1, and 3. These are the only values of x where the graph will crossthe x-axis. These roots partition the number line into intervals. Test values arethen chosen from within each interval. 112 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

The table of signs and the rough sketch of the graph of this functioncan now be constructed, as shown below.The Table of Signs x  4 Intervals 1 x  3 x 3 -5 4  x  2 2  x  1 2 4 Test value – + + -3 0 x 4 ++ x 2 – – + + + x 1 – – – + + x 3 – – – – + + – + – +y  (x  4)(x  2)(x 1)(x  3)Position of the curve above below above below above relative to the x-axisDEPED COPYThe Graph of y  (x  4)(x  2)(x 1)(x  3) We can now use the information from the table of signs to construct apossible graph of the function. At this level, though, we cannot determine theturning points of the graph, we can only be certain that the graph is correctwith respect to intervals where the graph is above, below, or on the x-axis. The arrow heads at both ends of the graph signify that the graph indefinitely goes upward. 113 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Here is another example: Sketch the graph of f (x)  x(x  2)(3x  4)Roots of f(x): 4 -2, 0, 3Table of Signs: Intervals x  2 2  x  0 0x 4 x4 3 3 -3 Test value + -1 1 2 –x – – – +– + x+2 + + 3x – 4 above –f(x) = –x(x + 2)(3x – 4) Position of the curve below relative to the x-axisDEPED COPY ++ –– –+ below aboveGraph: In this activity, you learned how to sketch the graph of polynomialfunctions using the intercepts, some points, and the position of the curvesdetermined from the table of signs. The procedures described are applicablewhen the polynomial function is in factored form. Otherwise, you need toexpress first a polynomial in factored form. Try this in the next activity. 114 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Activity 8: For each of the following functions, give (a) the x-intercept(s) (b) the intervals obtained when the x-intercepts are used to partition the number line (c) the table of signs (d) a sketch of the graph 1. y = (2x + 3)(x – 1)(x – 4) 2. y = –x3 + 2x2 + 11x – 2 3. y = x4 – 26x2 + 25 4. y = –x4 – 5x3 + 3x2 + 13x – 10 5. y = x2(x + 3)(x + 1)4(x – 1)3 Post your answer/output for a walk-through. For each of these polynomial functions, answer the following: a. What happens to the graph as x decreases without bound? b. For which interval(s) is the graph (i) above and (ii) below the x-axis? c. What happens to the graph as x increases without bound? d. What is the leading term of the polynomial function? e. What are the leading coefficient and the degree of the function? Now, the big question for you is: Do the leading coefficient and degree affect the behavior of its graph? You will answer this after an investigation in the next activity. Activity 9: After sketching manually the graphs of the five functions given in Activity 8, you will now be shown polynomial functions and their corresponding graphs. Study each figure and answer the questions that follow. Summarize your answers using a table similar to the one provided. 115 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Case 1 y yThe graph on the right is defined byy = 2x3 – 7x2 – 7x + 12 yor, in factored form,y = (2x + 3) (x – 1) (x – 4).Questions: xa. Is the leading coefficient a positive or anegative number?b. Is the polynomial of even degree or odddegree?c. Observe the end behaviors of the graph onDEPED COPYboth sides. Is it rising or falling to the left or tothe right?Case 2The graph on the right is defined by x xy  x5  3x4  x3  7x2  4or, in factored form,y  (x 1)2(x 1)(x  2)2 .Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right?Case 3The graph on the right is defined byy  x4  7x2  6x or, in factored form,y  x(x  3)(x 1)(x  2).Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? 116 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Case 4 y xThe graph on the right is defined byy  x4  2x3  13x2 14x  24or, in factored form,y  (x  3)(x 1)(x  2)(x  4) .Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right?DEPED COPY Now, complete this table. In the last column, draw a possible graph forthe function, showing how the function behaves. (You do not need to placeyour graph on the xy-plane). The first one is done for you.Sample Polynomial Function Leading Degree: Behavior of the Possible Coefficient: Even or Graph: Sketch n  0 or Odd Rising or Falling n0 Left- Right- hand hand1. y  2x 3  7x 2  7x  12 n0 odd falling rising2. y x 5  3x 4  x 3  7x 2  43. y x 4  7x 2  6x4. y x 4  2x 3  13x 2  14x  24Summarize your findings from the four cases above. What do you observeif:1. the degree of the polynomial is odd and the leading coefficient is positive?2. the degree of the polynomial is odd and the leading coefficient is negative?3. the degree of the polynomial is even and the leading coefficient is positive?4. the degree of the polynomial is even and the leading coefficient is negative? 117 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Congratulations! You have now illustrated The Leading CoefficientTest. You should have realized that this test can help you determine the endbehaviors of the graph of a polynomial function as x increases or decreaseswithout bound. Recall that you have already learned two properties of the graph ofpolynomial functions; namely, the intercepts which can be obtained from theRational Root Theorem, and the end behaviors which can be identified usingthe Leading Coefficient Test. Another helpful strategy is to determine whetherthe graph crosses or is tangent to the x-axis at each x-intercept. This strategyinvolves the concept of multiplicity of a zero of a polynomial function.Multiplicity tells how many times a particular number is a zero or root for thegiven polynomial. The next activity will help you understand the relationship betweenmultiplicity of a root and whether a graph crosses or is tangent to the x-axis. Activity 10:Given the function y  (x  2)2(x 1)3(x 1)4(x  2) and its graph, complete thetable below, then answer the questions that follow. y xDEPED COPY Characteristic Behavior of GraphRoot or Zero Multiplicity of Multiplicity: Relative to x-axis at this Odd or Even Root: Crosses or Is Tangent to -2 -1 1 2 118 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Questions: a. What do you notice about the graph when it passes through a root of even multiplicity? b. What do you notice about the graph when it passes through a root of odd multiplicity? This activity extends what you learned when using a table of signs tograph a polynomial function. When the graph crosses the x-axis, it means thegraph changes from positive to negative or vice versa. But if the graph istangent to the x-axis, it means that the graph is either positive on both sidesof the root, or negative on both sides of the root. In the next activity, you will consider the number of turning points of thegraph of a polynomial function. The turning points of a graph occur when thefunction changes from decreasing to increasing or from increasing todecreasing values. Activity 11:DEPED COPYComplete the table below. Then answer the questions that follow.Polynomial Function Sketch Number of Degree Turning Points1. y  x4 y2. y  x4  2x2 15 x y x3. y  x5 y x 119 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Polynomial Function Sketch Number of 4. y  x5  x3  2x 1 Degree Turning y 5. y  x5  5x3  4x PointsDEPED COPY x y xQuestions: a. What do you notice about the number of turning points of the quartic functions (numbers 1 and 2)? How about of quintic functions (numbers 3 to 5)? b. From the given examples, do you think it is possible for the degree of a function to be less than the number of turning points? c. State the relation of the number of turning points of a function with its degree n. In this section, you have encountered important concepts that can helpyou graph polynomial functions. Notice that the graph of a polynomialfunction is continuous, smooth, and has rounded turns. Further, thenumber of turning points in the graph of a polynomial is strictly less than thedegree of the polynomial. Use what you have learned as you perform the activities in thesucceeding sections. 120 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY The goal of this section is to help you think critically and creatively as you apply the techniques in graphing polynomial functions. Also, this section aims to provide opportunities to solve real-life problems involving polynomial functions. Activity 12: For each given polynomial function, describe or determine the following, then sketch the graph. You may need a calculator in some computations. a. leading term b. end behaviors c. x-intercepts points on the x-axis d. multiplicity of roots e. y-intercept point on the y-axis f. number of turning points g. sketch 1. y  (x  3)(x 1)2(2x  5) 2. y  (x2  5)(x  1)2(x  2)3 3. y  x 3  2x 2  2x  4 4. y  x2(x2  7)(2x  3) 5. y  2x4  3x3 18x2  6x  28 In this activity, you were given the opportunity to sketch the graph of polynomial functions. Were you able to apply all the necessary concepts and properties in graphing each function? The next activity will let you see the connections of these mathematics concepts to real life. 121 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 13:Work in groups. Apply the concepts of polynomial functions to answer thequestions in each problem. Use a calculator when needed.1. Look at the pictures below. What do these tell us? Filipinos need to take the problem of deforestation seriously.DEPED COPY The table below shows the forest cover of the Philippines in relation toits total land area of approximately 30 million hectares.Year 1900 1920 1960 1970 1987 1998Forest Cover (%) 70 60 40 34 23.7 22.2Source: Environmental Science for Social Change, Decline of the Philippine ForestA cubic polynomial that best models the data is given byy 26x3  3500x2  391 300x  69 717 000 ; 0  x  98 1 000 000where y is the percent forest cover x years from 1900. y 80 70 60 50 40 30 20 10 O 10 20 30 40 50 60 70 80 90 x -10 122 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Questions/Tasks: a. Using the graph, what is the approximate forest cover during the year 1940? b. Compare the forest cover in 1987 (as given in the table) to the forest cover given by the polynomial function. Why are these values not exactly the same? c. Do you think you can use the polynomial to predict the forest cover in the year 2100? Why or why not? 2. The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made. Question/Task: Suppose you are chosen as the leader and you are tasked to lead in solving the problem. What will you do to meet the specifications needed for the box? Show a mathematical solution. Were you surprised that polynomial functions have real and practical uses? What do you need to solve these kinds of problems? Enjoy learning as you proceed to the next section. The goal of this section is to check if you can apply polynomial functions to real-life problems and produce a concrete object that satisfies the conditions given in the problem. Activity 14: Read the problem carefully and answer the questions that follow. You are designing candle-making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle with a square base. You want the height of the candle to be 2 inches less than the edge of the base. 123 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Questions/Tasks: 1. What should the dimensions of your candle mold be? Show a mathematical procedure in determining the dimensions. 2. Use a sheet of cardboard as sample material in preparing a candle mold with such dimensions. The bottom of the mold should be closed. The height of one face of the pyramid should be indicated. 3. Write your solution in one of the faces of your output (mold).Rubric for the Mathematical SolutionDEPED COPYPoint Descriptor 4 3 The problem is correctly modeled with a polynomial function, 2 appropriate mathematical concepts are used in the solution, 1 and the correct final answer is obtained. The problem is correctly modeled with a polynomial function, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained. The problem is not properly modeled with a polynomial function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained. The problem is not properly modeled with a polynomial function, a solution is presented but the final answer is incorrect.Criteria for Rating the Output:  The mold has the needed dimensions and parts.  The mold is properly labeled with the required length of parts.  The mold is durable.  The mold is neat and presentable.Point/s to Be Given: 4 points if all items in the criteria are evident 3 points if any three of the items are evident 2 points if any two of the items are evident 1 point if any of the items is evident 124 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYSUMMARY/SYNTHESIS/GENERALIZATION This lesson was about polynomial functions. You learned how to:  illustrate and describe polynomial functions;  show the graph of polynomial functions using the following properties: - the intercepts (x-intercept and y-intercept); - the behavior of the graph using the Leading Coefficient Test, table of signs, turning points, and multiplicity of zeros; and  solve real-life problems that can be modeled with polynomial functions. GLOSSARY OF TERMS Constant Function - a polynomial function whose degree is 0 Evaluating a Polynomial - a process of finding the value of the polynomial at a given value in its domain Intercepts of a Graph - points on the graph that have zero as either the x- coordinate or the y-coordinate Irreducible Factor - a factor that can no longer be factored using coefficients that are real numbers Leading Coefficient Test - a test that uses the leading term of the polynomial function to determine the right-hand and the left-hand behaviors of the graph Linear Function - a polynomial function whose degree is 1 Multiplicity of a Root - tells how many times a particular number is a root for the given polynomial Nonnegative Integer - zero or any positive integer Polynomial Function - a function denoted by P(x)  an xn  an1xn1  an2xn2  ...  a1x  a0 , where n is a nonnegative integer, a0, a1, ..., an are real numbers called coefficients but an  0, anxn is the leading term, an is the leading coefficient, and a0 is the constant term Polynomial in Standard Form - any polynomial whose terms are arranged in decreasing powers of x Quadratic Function – a polynomial function whose degree is 2 125 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYQuartic Function – a polynomial function whose degree is 4Quintic Function – a polynomial function whose degree is 5Turning Point – a point where the function changes from decreasing toincreasing or from increasing to decreasing valuesREFERENCES USED IN THIS MODULE:Alferez, M.S., Duro, MC.A. & Tupaz, KK.L. (2008). MSA Advanced Algebra. Quezon City, Philippines: MSA Publishing House.Berry, J., Graham, T., Sharp, J. & Berry, E. (2003). Schaum’s A-Z Mathematics. London, United Kingdom: Hodder & Stoughton Educational.Cabral, E. A., De Lara-Tuprio, E.P., De Las Penas, ML. N., Francisco, F. F., Garces, IJ. L., Marcelo, R.M. & Sarmiento, J. F. (2010). Precalculus. Quezon City, Philippines: Ateneo de Manila University Press.Jose-Dilao, S., Orines, F. B. & Bernabe, J.G. (2003). Advanced Algebra, Trigonometry and Statistics. Quezon City, Philippines: JTW Corporation.Lamayo, F. C. & Deauna, M. C. (1990). Fourth Year Integrated Mathematics. Quezon City, Philippines: Phoenix Publishing House, Inc.Larson, R. & Hostetler, R. P. (2012). Algebra and Trigonometry. Pasig City, Philippines: Cengage Learning Asia Pte. Ltd.Marasigan, J. A., Coronel, A.C. & Coronel, I.C. (2004). Advanced Algebra with Trigonometry and Statistics. Makati City, Philippines: The Bookmark, Inc.Quimpo, N. F. (2005). A Course in Freshman Algebra. Quezon City, Philippines.Uy, F. B. & Ocampo, J.L. (2000). Board Primer in Mathematics. Mandaluyong City, Philippines: Capitol Publishing House.Villaluna, T. T. & Van Zandt, GE. L. (2009). Hands-on, Minds-on Activities in Mathematics IV. Quezon City, Philippines: St. Jude Thaddeus Publications. 126 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYI. INTRODUCTION Have you imagined yourself pushing a cart or riding in a bus having wheels that are not round? Do you think you can move heavy objects from one place to another easily or travel distant places as fast as you can? What difficulty do you think would you experience without circles? Have you ever thought of the importance of circles in the field of transportation, industries, sports, navigation, carpentry, and in your daily life? Find out the answers to these questions and determine the vast applications of circles through this module. II. LESSONS AND COVERAGE: In this module, you will examine the above questions when you take the following lessons: Lesson 1A – Chords, Arcs, and Central Angles Lesson 1B – Arcs and Inscribed Angles Lesson 2A – Tangents and Secants of a Circle Lesson 2B – Tangent and Secant Segments 127 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

In these lessons, you will learn to:Lesson 1A  derive inductively the relations among chords, arcs, central angles, and inscribed angles;Lesson 1B  illustrate segments and sectors of circles;  prove theorems related to chords, arcs, central angles, and inscribed angles; andLesson 2A  solve problems involving chords, arcs, central angles, and inscribed angles of circles.  illustrate tangents and secants of circles;Lesson 2BDEPED COPY prove theorems on tangents and secants; and  solve problems involving tangents and secants of circles.Here is a simple map of the lessons that will be covered in this module: CirclesRelationships among Chords, Arcs, Applications ofChords, Arcs, Central and Central CirclesAngles, and Inscribed Angles Angles Arcs and Tangents and Inscribed Secants of Circles Angles Tangents and Secants Tangent and Secant Segments 128 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

III. PRE-ASSESSMENTPart IFind out how much you already know about the topics in this module.Choose the letter that you think best answers each of the followingquestions. Take note of the items that you were not able to answercorrectly and find the right answer as you go through this module.1. What is an angle whose vertex is on a circle and whose sides containchords of the circle?A. central angle C. circumscribed angleDEPED COPYB. inscribed angle D. intercepted angle2. An arc of a circle measures 30°. If the radius of the circle is 5 cm, whatis the length of the arc?A. 2.62 cm B. 2.3 cm C. 1.86 cm D. 1.5 cm3. Using the figure below, which of the following is an external secantsegment of M? TC EM IN OA. CO C. NOB. TI D. NI4. The opposite angles of a quadrilateral inscribed in a circle are _____.A. right C. complementaryB. obtuse D. supplementary5. In S at the right, what is mVSI if mVI = 140?A. 35 C. 140B. 75 D. 2306. What is the sum of the measures of the central angles of a circle withno common interior points?A. 120 B. 240 C. 360 D. 480 129 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

7. Catherine designed a pendant. It is a regular hexagon set in a circle.Suppose the opposite vertices are connected by line segments andmeet at the center of the circle. What is the measure of each angleformed at the center?A. 22.5 B. 45 C. 60 D. 728. If an inscribed angle of a circle intercepts a semicircle, then the angleis _________.A. acute B. right C. obtuse D. straight9. At a given point on the circle, how many line/s can be drawn that istangent to the circle?DEPED COPYA. one B. two C. three D. four10. What is the length of ZK in the figure on the right?A. 2.86 units C. 8 unitsB. 6 units D. 8.75 units11. In the figure on the right, mXY = 150 and mMN = 30. PWhat is mXPY ? NA. 60 MB. 90 X YC. 120 60°D. 180 120 cm12. The top view of a circular table shown on the right has a radius of 120 cm. Find the area of the smaller segment of the table (shaded region) determined by a 60 arc.  A. 2400  3600 3 cm2 B. 3600 3 cm2 C. 2400 cm2  D. 14 400  3600 3 cm2 130 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

13. In O given below, what is PR if NO = 15 units and ES = 6 units?A. 28 units ERB. 24 units PSC. 12 unitsD. 9 units O N14. A dart board has a diameter of 40 cm and is divided into 20 congruentsectors. What is the area of one of the sectors?DEPED COPYA. 20 cm2 C. 80 cm2B. 40 cm2 D. 800 cm215. Mr. Soriano wanted to plant three different colors of roses on the outer rim of a circular garden. He stretched two strings from a point external to the circle to see how the circular rim can be divided into three portions as shown in the figure below. 192° C A M 20° BWhat is the measure of minor arc AB?A. 64° B. 104° C. 168° D. 192°16. In the figure below, SY and EY are secants. If SY = 15 cm,TY = 6 cm, and LY = 8 cm. What is the length of EY ?A. 20 cm B. 12 cm C. 11.25 cm D. 6.75 cm ST Y EL 131 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

17. In C below, mAB = 60 and its radius is 6 cm. What is the area ofthe shaded region in terms of pi (  )? A 60°A. 6  cm 2 C. 10  cm 2B. 8  cm 2 D. 12  cm 2 CB 6 cm18. In the circle below, what is the measure of SAY if DSY is aDEPED COPYsemicircle and mSAD  70? SA. 20 C. 110B. 70 D. 150 DA Y19. Quadrilateral SMIL is inscribed in E. If mSMI  78 andmMSL  95, find mMIL .A. 78 C. 95 MB. 85 D. 102 78° I E S 95° L20. In M on the right, what is mBRO if mBMO  60?A. 120 C. 30B. 60 D. 15 B R M O 132 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Part IISolve each of the following problems. Show your complete solutions.1. Mr. Javier designed an arch made of bent iron for the top of a school’s main entrance. The 12 segments between the two concentric semicircles are each 0.8 meter long. Suppose the diameter of the inner semicircle is 4 meters. What is the total length of the bent iron used to make this arch? 0.8 m 4mDEPED COPY2. A bicycle chain fits tightly around two gears. What is the distance between the centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches, respectively, and the portion of the chain tangent to the two gears is 26.5 inches long?Rubric for Problem SolvingScore Descriptors 4 3 Used an appropriate strategy to come up with the correct solution 2 and arrived at a correct answer. 1 Used an appropriate strategy to come up with a solution, but a part of the solution led to an incorrect answer. Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer. Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution.Part IIIRead and understand the situation below, then answer the questions andperform what is required. The committee in-charge of the Search for the Cleanest and GreenestSchool informed your principal that your school has been selected as a regionalfinalist. Being a regional finalist, your principal would like to make your schoolmore beautiful and clean by making more gardens of different shapes. Hedecided that every year level will be assigned to prepare a garden of particularshape. In your grade level, he said that you will be preparing circular,semicircular, or arch-shaped gardens in front of your building. He furtherencouraged your grade level to add garden accessories to make the gardensmore presentable and amusing. 133 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

1. How will you prepare the design of the gardens?2. What garden accessories will you use?3. Make the designs of the gardens which will be placed in front of your grade level building. Use the different shapes that were required by your principal.4. Illustrate every part or portion of the garden including their measurements.5. Using the designs of the gardens made, determine all the concepts or principles related to circles.6. Formulate problems involving these mathematics concepts or principles, then solve.Rubric for DesignScore Descriptors DEPED COPY 4 The design is accurately made, presentable, and appropriate. 3 The design is accurately made and appropriate but not presentable. 2 The design is not accurately made but appropriate. 1 The design is made but not accurate and appropriate.Rubric on Problems Formulated and SolvedScore Descriptors 6 Poses a more complex problem with two or more correct possible 5 solutions, communicates ideas unmistakably, shows in-depth 4 comprehension of the pertinent concepts and/or processes, and 3 provides explanations wherever appropriate. 2 1 Poses a more complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes most significant parts of the solution, communicates ideas unmistakably, and shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. Poses a problem but demonstrates minor comprehension, not being able to develop an approach.Source: D.O. #73, s. 2012IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrateunderstanding of key concepts of circles and formulate real-life problemsinvolving these concepts, and solve these using a variety of strategies.Furthermore, you should be able to investigate mathematical relationships invarious situations involving circles. 134 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Start Lesson 1A of this module by assessing your knowledge of thedifferent mathematical concepts previously studied and your skills inperforming mathematical operations. These knowledge and skills willhelp you understand circles. As you go through this lesson, think of thisimportant question: “How do the relationships among chords, arcs, andcentral angles of a circle facilitate finding solutions to real-life problemsand making decisions?” To find the answer, perform each activity. If youfind any difficulty in answering the exercises, seek the assistance of yourteacher or peers or refer to the modules you have studied earlier. Youmay check your work with your teacher.DEPED COPYActivity 1:Use the figure below to identify and name the following terms related to A.Then, answer the questions that follow.1. a radius 5. a minor arc L J2. a diameter 6. a major arc AN3. a chord 7. 2 central angles E4. a semicircle 8. 2 inscribed angles s 135 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Questions: a. How did you identify and name the radius, diameter, and chord? How about the semicircle, minor arc, and major arc? inscribed angle and central angle? b. How do you describe a radius, diameter, and chord of a circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle?Write your answers in the table below.Terms Related to Circles Description1. radius2. diameter DEPED COPY3. chord4. semicircle5. minor arc6. major arc7. central angle8. inscribed anglec. How do you differentiate among the radius, diameter, and chord of a circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle? Were you able to identify and describe the terms related to circles?Were you able to recall and differentiate them? Now that you know theimportant terms related to circles, let us deepen your understanding offinding the lengths of sides of right triangles. You need this mathematicalskill in finding the relationships among chords, arcs, and central anglesas you go through this lesson. 136 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 2:In each triangle below, the length of one side is unknown. Determine thelength of this side.1. c=? 4. b = 16 a=6 a=? c = 20 b=8DEPED COPY2. c=? 5. c=5 b=? a=9 b = 15 a=3 6. b = ?3. a=9 c=? c = 14 a=7 b=9Questions: a. How did you find the missing side of each right triangle? b. What mathematics concepts or principles did you apply to find each missing side? In the activity you have just done, were you able to find the missing side of a right triangle? The concept used will help you as you go on with this module. 137 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 3: BUse the figures below to answer the questions that follow. PT OO Q F ACSR E DDEPED COPYFigure 1 Figure 21. What is the measure of each of the following angles in Figure 1? Use aprotractor. d. ROSa. TOPb. POQ e. SOTc. QOR2. In Figure 2, AF , AB , AC , AD , and AE are radii of A. What is themeasure of each of the following angles? Use a protractor.a. FAB d. EADb. BAC e. EAFc. CAD3. How do you describe the angles in each figure?4. What is the sum of the measures of TOP, POQ , QOR, ROS, and SOT in Figure 1? How about the sum of the measures of FAB, BAC, CAD, EAD, and EAF in Figure 2? 138 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.