Precalculus Learner’s Material - Department of Education

PY Precalculus O C Learner’s Material...

PY Precalculus O C Learner’s Material E D This learning resource was collaboratively developed and EP 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. D 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, 2016. Precalculus Learner’s Material First Edition 2016 Republic Act 8293. Section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. 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Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publishers and authors directly. O Authors and publishers may email or contact FILCOLS at [email protected] or (02) 435-5258, respectively. Published by the Department of Education C Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD D Development Team of the Precalculus Learner’s Material Joy P. Ascano Jesus Lemuel L. Martin Jr. E Arnel D. Olofernes Mark Anthony C. Tolentino, Ph.D Reviewers: Jerico B. Bacani, Ph.D Richard B. Eden, Ph.D EP Dr. Flordeliza F. Francisco Mark Anthony J. Vidallo Carly Mae Casteloy Angela Dianne Agustin Cover Art Illustrator: Quincy D. Gonzales Team Leader: Ian June L. Garces, Ph.D. D Management Team of the Precalculus Learner’s Material Bureau of Curriculum Development Bureau of Learning Resources Printed in the Philippines by Sunshine Interlinks Publishing House, Inc. 3F Maine City Tower, 236 Tomas Morato Avenue, Brgy. South Triangle, Quezon City Department of Education-Bureau of Learning Resources (DepEd-BLR) Office Address: Ground Floor Bonifacio Building, DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (02) 634-1054, 634-1072, 631-4985 E-mail Address: [email protected] / [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, 2016.  Table of Contents To the Precalculus Learners 1 DepEd Curriculum Guide for Precalculus 2 PY Unit 1: Analytic Geometry 6 Lesson 1.1: Introduction to Conic Sections and Circles........ 7 1.1.1: An Overview of Conic Sections........................... 7 O 1.1.2: Definition and Equation of a Circle....................... 8 1.1.3: More Properties of Circles................................ 10 C 1.1.4: Situational Problems Involving Circles.................... 12 Lesson 1.2: Parabolas......................................... 19 D 1.2.1: Definition and Equation of a Parabola.................... 19 1.2.2: More Properties of Parabolas............................. 23 E 1.2.3: Situational Problems Involving Parabolas................ 26 Lesson 1.3: Ellipses........................................... 33 EP 1.3.1: Definition and Equation of an Ellipse..................... 33 1.3.2: More Properties of Ellipses............................... 36 1.3.3: Situational Problems Involving Ellipses................... 40 D Lesson 1.4: Hyperbolas....................................... 46 1.4.1: Definition and Equation of a Hyperbola.................. 46 1.4.2: More Properties of Hyperbolas........................... 50 1.4.3: Situational Problems Involving Hyperbolas............... 54 Lesson 1.5: More Problems on Conic Sections................ 60 1.5.1: Identifying the Conic Section by Inspection............... 60 1.5.2: Problems Involving Different Conic Sections.............. 62 iii 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, 2016.  Lesson 1.6: Systems of Nonlinear Equations.................. 67 1.6.1: Review of Techniques in Solving Systems of Linear Equations................................................ 68 1.6.2: Solving Systems of Equations Using Substitution......... 69 1.6.3: Solving Systems of Equations Using Elimination.......... 70 1.6.4: Applications of Systems of Nonlinear Equations.......... 73 Unit 2: Mathematical Induction 80 Lesson 2.1: Review of Sequences and Series................... 81 PY Lesson 2.2: Sigma Notation................................... 86 2.2.1: Writing and Evaluating Sums in Sigma Notation......... 87 2.2.2: Properties of Sigma Notation............................. 89 O Lesson 2.3: Principle of Mathematical Induction.............. 96 2.3.1: Proving Summation Identities............................ 97 C 2.3.2: Proving Divisibility Statements........................... 101 ? 2.3.3: Proving Inequalities...................................... 102 D Lesson 2.4: The Binomial Theorem........................... 108 2.4.1: Pascal’s Triangle and the Concept of Combination........ 109 E 2.4.2: The Binomial Theorem................................... 111 EP 2.4.3: Terms of a Binomial Expansion.......................... 114 ? 2.4.4: Approximation and Combination Identities............... 116 Unit 3: Trigonometry 123 D Lesson 3.1: Angles in a Unit Circle........................... 124 3.1.1: Angle Measure........................................... 124 3.1.2: Coterminal Angles....................................... 128 3.1.3: Arc Length and Area of a Sector......................... 129 Lesson 3.2: Circular Functions................................ 135 3.2.1: Circular Functions on Real Numbers..................... 136 3.2.2: Reference Angle.......................................... 139 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, 2016.  Lesson 3.3: Graphs of Circular Functions and Situational Problems......................................... 144 3.3.1: Graphs of y = sin x and y = cos x........................ 145 3.3.2: Graphs of y = a sin bx and y = a cos bx................... 147 3.3.3: Graphs of y = a sin b(x − c) + d and y = a cos b(x − c) + d..................................... 151 3.3.4: Graphs of Cosecant and Secant Functions................ 154 3.3.5: Graphs of Tangent and Cotangent Functions............. 158 3.3.6: Simple Harmonic Motion................................. 160 PY Lesson 3.4: Fundamental Trigonometric Identities............. 171 3.4.1: Domain of an Expression or Equation.................... 171 3.4.2: Identity and Conditional Equation....................... 173 O 3.4.3: The Fundamental Trigonometric Identities............... 174 3.4.4: Proving Trigonometric Identities......................... 176 C Lesson 3.5: Sum and Difference Identities..................... 181 3.5.1: The Cosine Difference and Sum Identities................ 181 D 3.5.2: The Cofunction Identities and the Sine Sum and Difference Identities...................................... 183 3.5.3: The Tangent Sum and Difference Identities............... 186 E Lesson 3.6: Double-Angle and Half-Angle Identities........... 192 EP 3.6.1: Double-Angle Identities.................................. 192 3.6.2: Half-Angle Identities..................................... 195 Lesson 3.7: Inverse Trigonometric Functions.................. 201 D 3.7.1: Inverse Sine Function.................................... 202 3.7.2: Inverse Cosine Function.................................. 205 3.7.3: Inverse Tangent Function and the Remaining Inverse Trigonometric Functions................................. 208 Lesson 3.8: Trigonometric Equations.......................... 220 3.8.1: Solutions of a Trigonometric Equation.................... 221 3.8.2: Equations with One Term................................ 224 3.8.3: Equations with Two or More Terms...................... 227 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, 2016.  Lesson 3.9: Polar Coordinate System......................... 236 3.9.1: Polar Coordinates of a Point............................. 237 3.9.2: From Polar to Rectangular, and Vice Versa............... 241 3.9.3: Basic Polar Graphs and Applications..................... 244 Answers to Odd-Numbered Exercises in Supplementary Problems and All Exercises in Topic Tests 255 References 290 PY O C E D EP D 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, 2016.  To the Precalculus Learners The Precalculus course bridges basic mathematics and calculus. This course completes your foundational knowledge on algebra, geometry, and trigonometry. It provides you with conceptual understanding and computational skills that are prerequisites for Basic Calculus and future STEM courses. Based on the Curriculum Guide for Precalculus of the Department of Edu- cation (see pages 2-5), the primary aim of this Learning Manual is to give you an adequate stand-alone material that can be used for the Grade 11 Precalculus course. PY The Manual is divided into three units: analytic geometry, summation no- tation and mathematical induction, and trigonometry. Each unit is composed of lessons that bring together related learning competencies in the unit. Each lesson is further divided into sub-lessons that focus on one or two competencies for effective learning. O At the end of each lesson, more examples are given in Solved Examples to reinforce the ideas and skills being developed in the lesson. You have the oppor- C tunity to check your understanding of the lesson by solving the Supplementary Problems. Finally, two sets of Topic Test are included to prepare you for the exam. D Answers, solutions, or hints to odd-numbered items in the Supplementary Problems and all items in the Topic Tests are provided at the end of the Manual to guide you while solving them. We hope that you will use this feature of the E Manual responsibly. Some items are marked with a star. A starred sub-lesson means the discussion EP and accomplishment of the sub-lesson are optional. This will be decided by your teacher. On the other hand, a starred example or exercise means the use of calculator is required. We hope that you will find this Learning Manual helpful and convenient to D use. We encourage you to carefully study this Manual and solve the exercises yourselves with the guidance of your teacher. Although great effort has been put into this Manual for technical correctness and precision, any mistake found and reported to the Team is a gain for other students. Thank you for your cooperation. The Precalculus LM Team 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, 2016.  K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT Grade: 11 Semester: First Semester Core Subject Title: Pre-Calculus No. of Hours/ Semester: 80 hours/ semester Pre-requisite (if needed): D Subject Description: At the end of the course, the students must be able to apply concepts and solve problems involving conic sections, systems of nonlinear equations, series and mathematical induction, circular and trigonometric functions, trigonometric identities, and polar coordinate system. CONTENT PERFORMANCE CONTENT LEARNING COMPETENCIES CODE EP STANDARDS STANDARDS Analytic The learners The learners shall be able The learners... Geometry demonstrate an to... understanding of... E 1. illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases.*** STEM_PC11AG-Ia-1 model situations 2. define a circle. STEM_PC11AG-Ia-2 2 key concepts of appropriately and solve problems accurately using D 3. determine the standard form of equation of a circle STEM_PC11AG-Ia-3 conic sections and conic sections and systems 4. graph a circle in a rectangular coordinate system STEM_PC11AG-Ia-4 systems of of nonlinear equations 5. define a parabola STEM_PC11AG-Ia-5 nonlinear 6. determine the standard form of equation of a parabola STEM_PC11AG-Ib-1 equations 7. 8. C graph a parabola in a rectangular coordinate system define an ellipse STEM_PC11AG-Ib-2 STEM_PC11AG-Ic-1 9. determine the standard form of equation of an ellipse STEM_PC11AG-Ic-2 10. graph an ellipse in a rectangular coordinate system STEM_PC11AG-Ic-3 11. 12. O define a hyperbola determine the standard form of equation of a hyperbola STEM_PC11AG-Id-1 STEM_PC11AG-Id-2 PY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - K to 12 Senior High School STEM Specialized Subject – Pre-Calculus December 2013 Page 1 of 4 electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.  K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT CONTENT PERFORMANCE CONTENT LEARNING COMPETENCIES CODE STANDARDS STANDARDS D 13. graph a hyperbola in a rectangular coordinate system STEM_PC11AG-Id-3 14. recognize the equation and important characteristics of the STEM_PC11AG-Ie-1 different types of conic sections 15. solves situational problems involving conic sections STEM_PC11AG-Ie-2 16. illustrate systems of nonlinear equations STEM_PC11AG-If-1 17. determine the solutions of systems of nonlinear equations using EP STEM_PC11AG-If-g-1 techniques such as substitution, elimination, and graphing*** 18. solve situational problems involving systems STEM_PC11AG-Ig-2 of nonlinear equations Series and Mathematical key concepts of series and E keenly observe and investigate patterns, and 1. illustrate a series STEM_PC11SMI-Ih-1 Induction mathematical formulate appropriate 2. differentiate a series from a sequence STEM_PC11SMI-Ih-2 3 induction and the mathematical statements 3. use the sigma notation to represent a series STEM_PC11SMI-Ih-3 Binomial and prove them using D 4. illustrate the Principle of Mathematical Induction STEM_PC11SMI-Ih-4 Theorem. mathematical induction 5. apply mathematical induction in proving identities STEM_PC11SMI-Ih-i-1 and/or Binomial Theorem. 6. illustrate Pascal’s Triangle in the expansion of 𝑥 + 𝑦 𝑛 for small STEM_PC11SMI-Ii-2 positive integral values of 𝑛 7. prove the Binomial Theorem STEM_PC11SMI-Ii-3 8. C determine any term of 𝑥 + 𝑦 𝑛 , where 𝑛 is a positive integer, STEM_PC11SMI-Ij-1 without expanding 9. solve problems using mathematical induction and the Binomial STEM_PC11SMI-Ij-2 O Theorem PY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - K to 12 Senior High School STEM Specialized Subject – Pre-Calculus December 2013 Page 2 of 4 electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.  K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT CONTENT PERFORMANCE CONTENT LEARNING COMPETENCIES CODE STANDARDS STANDARDS Trigonometry key concepts of 1. formulate and solve 1. illustrate the unit circle and the relationship between the linear circular functions, accurately situational and angular measures of a central angle in a unit circle STEM_PC11T-IIa-1 trigonometric problems involving 2. convert degree measure to radian measure and vice versa STEM_PC11T-IIa-2 D identities, inverse trigonometric circular functions 3. illustrate angles in standard position and coterminal angles STEM_PC11T-IIa-3 functions, and 4. illustrate the different circular functions STEM_PC11T-IIb-1 the polar coordinate 5. uses reference angles to find exact values of circular functions STEM_PC11T-IIb-2 system 6. determine the domain and range of the different circular functions STEM_PC11T-IIc-1 7. graph the six circular functions (a) amplitude, (b) period, and (c) STEM_PC11T-IIc-d-1 phase shift EP 8. solve problems involving circular functions STEM_PC11T-IId-2 2. apply appropriate 9. determine whether an equation is an identity or a conditional trigonometric identities in STEM_PC11T-IIe-1 equation E solving situational problems 10. 11. derive the fundamental trigonometric identities derive trigonometric identities involving sum and difference of STEM_PC11T-IIe-2 STEM_PC11T-IIe-3 angles 4 12. derive the double and half-angle formulas STEM_PC11T-IIf-1 D 13. simplify trigonometric expressions STEM_PC11T-IIf-2 14. prove other trigonometric identities STEM_PC11T-IIf-g-1 15. solve situational problems involving trigonometric identities STEM_PC11T-IIg-2 3. formulate and solve 16. illustrate the domain and range of the inverse trigonometric STEM_PC11T-IIh-1 accurately situational functions. problems involving 17. C evaluate an inverse trigonometric expression. STEM_PC11T-IIh-2 appropriate trigonometric 18. solve trigonometric equations. STEM_PC11T-IIh-i-1 functions 19. solve situational problems involving inverse trigonometric STEM_PC11T-IIi-2 functions and trigonometric equations 4. formulate and solve 20. O locate points in polar coordinate system STEM_PC11T-IIj-1 accurately situational 21. convert the coordinates of a point from rectangular to polar STEM_PC11T-IIj-2 problems involving the systems and vice versa polar coordinate system 22. solve situational problems involving polar coordinate system STEM_PC11T-IIj-3 ***Suggestion for ICT-enhanced lesson when available and where appropriate PY All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - K to 12 Senior High School STEM Specialized Subject – Pre-Calculus December 2013 Page 3 of 4 electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.  K to 12 BASIC EDUCATION CURRICULUM SENIOR HIGH SCHOOL – SCIENCE, TECHNOLOGY, ENGINEERING AND MATHEMATICS (STEM) SPECIALIZED SUBJECT Code Book Legend Sample: STEM_PC11AG-Ia-1 D LEGEND SAMPLE DOMAIN/ COMPONENT CODE Learning Area and Science, Technology, Strand/ Subject or Engineering and Mathematics Analytic Geometry AG EP Specialization Pre-Calculus First Entry STEM_PC11AG Series and Mathematical Induction SMI Grade Level Grade 11 E 5 Uppercase Domain/Content/ Trigonometry T Letter/s Component/ Topic D Analytic Geometry - Roman Numeral *Zero if no specific Quarter First Quarter I quarter C Lowercase Letter/s *Put a hyphen (-) in between letters to Week Week one O a indicate more than a specific week - illustrate the different types of conic sections: parabola, Arabic Number Competency 1 PY ellipse, circle, hyperbola, and degenerate cases All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - K to 12 Senior High School STEM Specialized Subject – Pre-Calculus December 2013 Page 4 of 4 electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2016.  Unit 1 Analytic Geometry PY O C E D EP San Juanico Bridge, by Morten Nærbøe, 21 June 2009, https://commons.wikimedia.org/wiki/File%3ASan Juanico Bridge 2.JPG. Public Domain. D Stretching from Samar to Leyte with a total length of more than two kilome- ters, the San Juanico Bridge has been serving as one of the main thoroughfares of economic and social development in the country since its completion in 1973. Adding picturesque effect on the whole architecture, geometric structures are subtly built to serve other purposes. The arch-shaped support on the main span of the bridge helps maximize its strength to withstand mechanical resonance and aeroelastic flutter brought about by heavy vehicles and passing winds. 6 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, 2016.  Lesson 1.1. Introduction to Conic Sections and Circles Learning Outcomes of the Lesson At the end of the lesson, the student is able to: (1) illustrate the different types of conic sections: parabola, ellipse, circle, hyper- bola, and degenerate cases; (2) define a circle; (3) determine the standard form of equation of a circle; PY (4) graph a circle in a rectangular coordinate system; and (5) solve situational problems involving conic sections (circles). Lesson Outline (1) Introduction of the four conic sections, along with the degenerate conics O (2) Definition of a circle (3) Derivation of the standard equation of a circle (4) Graphing circles C (5) Solving situational problems involving circles D Introduction We present the conic sections, a particular class of curves which sometimes E appear in nature and which have applications in other fields. In this lesson, we first illustrate how each of these curves is obtained from the intersection of a EP plane and a cone, and then discuss the first of their kind, circles. The other conic sections will be covered in the next lessons. 1.1.1. An Overview of Conic Sections D We introduce the conic sections (or conics), a particular class of curves which oftentimes appear in nature and which have applications in other fields. One of the first shapes we learned, a circle, is a conic. When you throw a ball, the trajectory it takes is a parabola. The orbit taken by each planet around the sun is an ellipse. Properties of hyperbolas have been used in the design of certain telescopes and navigation systems. We will discuss circles in this lesson, leaving parabolas, ellipses, and hyperbolas for subsequent lessons. Circle (Figure 1.1) - when the plane is horizontal Ellipse (Figure 1.1) - when the (tilted) plane intersects only one cone to form a bounded curve 7 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, 2016.  Parabola (Figure 1.2) - when the plane intersects only one cone to form an unbounded curve Hyperbola (Figure 1.3) - when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyper- bola) PY O Figure 1.1 Figure 1.2 Figure 1.3 We can draw these conic sections (also called conics) on a rectangular co- ordinate plane and find their equations. To be able to do this, we will present C equivalent definitions of these conic sections in subsequent sections, and use these to find the equations. There are other ways for a plane and the cones to intersect, to form what are D referred to as degenerate conics: a point, one line, and two lines. See Figures 1.4, 1.5 and 1.6. E EP D Figure 1.4 Figure 1.5 Figure 1.6 1.1.2. Definition and Equation of a Circle A circle may also be considered a special kind of ellipse (for the special case when the tilted plane is horizontal). As we get to know more about a circle, we will also be able to distinguish more between these two conics. 8 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, 2016.  See Figure 1.7, with the point C(3, 1) shown. From the figure, the distance of A(−2, 1) from pC is AC = 5. By the distance formula, the distance of B(6, 5) from C is BC = (6 − 3)2 + (5 − 1)2 = 5. There are other points P such that P C = 5. The collection of all such points which are 5 units away from C, forms a circle. PY O Figure 1.7 Figure 1.8 C Let C be a given point. The set of all points P having the same distance from C is called a circle. The point C is called the center of D the circle, and the common distance its radius. E The term radius is both used to refer to a segment from the center C to a point P on the circle, and the length of this segment. EP See Figure 1.8, where a circle is drawn. It has center C(h, k) and radius r > 0. A point P (x, y) is on the circle if and only if P C = r. For any such point then, its coordinates should satisfy the following. PC = r D p (x − h)2 + (y − k)2 = r (x − h)2 + (y − k)2 = r2 This is the standard equation of the circle with center C(h, k) and radius r. If the center is the origin, then h = 0 and k = 0. The standard equation is then x2 + y 2 = r 2. 9 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, 2016.  Example 1.1.1. In each item, give the standard equation of the circle satisfy- ing the given conditions. (1) center at the origin, radius 4 √ (2) center (−4, 3), radius 7 (3) circle in Figure 1.7 (4) circle A in Figure 1.9 (5) circle B in Figure 1.9 (6) center (5, −6), tangent to the y- axis Figure 1.9 PY (7) center (5, −6), tangent to the x-axis (8) It has a diameter with endpoints A(−1, 4) and B(4, 2). Solution. (1) x2 + y 2 = 16 O (2) (x + 4)2 + (y − 3)2 = 7 (3) The center is (3, 1) and the radius is 5, so the equation is (x − 3)2 + (y − 1)2 = 25. C (4) By inspection, the center is (−2, −1) and the radius is 4. The equation is (x + 2)2 + (y + 1)2 = 16. D (5) Similarly by inspection, we have (x − 3)2 + (y − 2)2 = 9. (6) The center is 5 units away from the y-axis, so the radius is r = 5 (you can E make a sketch to see why). The equation is (x − 5)2 + (y + 6)2 = 25. (7) Similarly, since the center is 6 units away from the x-axis, the equation is EP (x − 5)2 + (y + 6)2 = 36. (8) The center C is the midpoint of A and B: C = −1+4 , 4+2 = 32 , 3. The

q 2 q 2
2 radius is then r = AC = −1 − 32 + (4 − 3)2 = 29 4. The circle has 2 equation x − 23 + (y − 3)2 = 29 2
. D 4 1.1.3. More Properties of Circles After expanding, the standard equation  2 3 29 x− + (y − 3)2 = 2 4 can be rewritten as x2 + y 2 − 3x − 6y + 4 = 0, an equation of the circle in general form. 10 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, 2016.  If the equation of a circle is given in the general form Ax2 + Ay 2 + Cx + Dy + E = 0, A 6= 0, or x2 + y 2 + Cx + Dy + E = 0, we can determine the standard form by completing the square in both variables. Completing the square in an expression like x2 + 14x means determining the term to be added that will produce a perfect polynomial square. Since the coefficient of x2 is already 1, we take half the coefficient of x and square it, and we get 49. Indeed, x2 + 14x + 49 = (x + 7)2 is a perfect square. To complete PY the square in, say, 3x2 + 18x, we factor the coefficient of x2 from the expression: 3(x2 + 6x), then add 9 inside. When completing a square in an equation, any extra term introduced on one side should also be added to the other side. Example 1.1.2. Identify the center and radius of the circle with the given equa- O tion in each item. Sketch its graph, and indicate the center. (1) x2 + y 2 − 6x = 7 C (2) x2 + y 2 − 14x + 2y = −14 (3) 16x2 + 16y 2 + 96x − 40y = 315 D Solution. The first step is to rewrite each equation in standard form by complet- ing the square in x and in y. From the standard equation, we can determine the center and radius. E (1) EP x2 − 6x + y 2 = 7 x2 − 6x + 9 + y 2 = 7 + 9 (x − 3)2 + y 2 = 16 Center (3, 0), r = 4, Figure 1.10 D (2) x2 − 14x + y 2 + 2y = −14 x2 − 14x + 49 + y 2 + 2y + 1 = −14 + 49 + 1 (x − 7)2 + (y + 1)2 = 36 Center (7, −1), r = 6, Figure 1.11 (3) 16x2 + 96x + 16y 2 − 40y = 315 11 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, 2016.    2 5 2 16(x + 6x) + 16 y − y = 315 2     2 2 5 25 25 16(x + 6x + 9) + 16 y − y + = 315 + 16(9) + 16 2 16 16  2 5 16(x + 3)2 + 16 y − = 484 4  2  2 2 5 484 121 11 (x + 3) + y − = = = 4 16 4 2 Center −3, 54 , r = 5.5, Figure 1.12. 2
PY O C Figure 1.10 Figure 1.11 Figure 1.12 In the standard equation (x − h)2 + (y − k)2 = r2 , both the two squared D terms on the left side have coefficient 1. This is the reason why in the preceding example, we divided by 16 at the last equation. E 1.1.4. Situational Problems Involving Circles EP Let us now take a look at some situational problems involving circles. ? Example 1.1.3. A street with two lanes, each 10 ft wide, goes through a semicircular tunnel with radius 12 ft. How high is the tunnel at the edge of each lane? Round off to 2 decimal places. D 12 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, 2016.  Solution. We draw a coordinate system with origin at the middle of the highway, as shown. Because of the given radius, the tunnel’s boundary is on the circle x2 + y 2 = 122. Point P is the point on the arc just above the edge of a lane, so its x-coordinate is 10. We√need its y-coordinate. We then solve 102 + y 2 = 122 for y > 0, giving us y = 2 11 ≈ 6.63 ft. 2 Example 1.1.4. A piece of a broken plate was dug up in an archaeological site. It was put on top of a grid, as shown in Figure 1.13, with the arc of the plate passing through A(−7, 0), B(1, 4) and C(7, 2). Find its center, and the standard equation of the circle describing the boundary of the plate. PY O C D Figure 1.13 E EP D Figure 1.14 Solution. We first determine the center. It is the intersection of the perpendicular bisectors of AB and BC (see Figure 1.14). Recall that, in a circle, the perpen- dicular bisector of any
chord passes through4−0 the center. Since the midpoint M of AB is −7+12 , 0+4 2 = (−3, 2), and m AB = 1+7 = 12 , the perpendicular bisector of AB has equation y − 2 = −2(x + 3), or equivalently, y = −2x − 4. 13 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, 2016.  Since the midpoint N of BC is 1+7 4+2 = (4, 3), and mBC = 2−4 = − 13 ,
2 , 2 7−1 the perpendicular bisector of BC has equation y − 3 = 3(x − 4), or equivalently, y = 3x − 9. The intersection of the two lines y = 2x − 4 and y = 3x − 9 is (1, −6) (by solving a system of linear equations). We can take the radius as the distance of this point from any of A, B or C (it’s most convenient to use B in this case). We then get r = 10. The standard equation is thus (x − 1)2 + (y + 6)2 = 100. 2 More Solved Examples PY 1. In each item, give the standard equation of the circle satisying the given con- ditions. (a) center at the origin, contains (0, 3) (b) center (1, 5), diameter 8 (c) circle A in Figure 1.15 O (d) circle B in Figure 1.15 C (e) circle C in Figure 1.15 D (f) center (−2, −3), tangent to the y- axis E (g) center (−2, −3), tangent to the x- axis EP Figure 1.15 (h) contains the points (−2, 0) and (8, 0), radius 5 Solution: D (a) The radius is 3, so the equation is x2 + y 2 = 9. (b) The radius is 8/2 = 4, so the equation is (x − 1)2 + (y − 5)2 = 16. (c) The center is (−2, 2) and the radius is 2, so the equation is (x + 2)2 + (y − 2)2 = 4. (d) The center is (2, 3) and the radius is 1, so the equation is (x − 2)2 + (y − 3)2 = 1. (e) The center is (1, −1) and by the Pythagorean Theorem, √ the radius √ (see 2 2 Figure 1.16) is 2 + 2 = 8, so the equation is (x − 1)2 + (x + 1)2 = 8. Figure 1.16 14 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, 2016.  (f) The radius is 3, so the equation is (x + 2)2 + (y + 3)2 = 9. (g) The radius is 2, so the equation is (x + 2)2 + (y + 3)2 = 4. (h) The distance between (−2, 0) and (8, 0) is 10; since the radius is 5, these two points are endpoints of a diameter. Then the circle has center at (3, 0) and radius 5, so its equation is (x − 3)2 + y 2 = 25. 2. Identify the center and radius of the circle with the given equation in each item. Sketch its graph, and indicate the center. (a) x2 + y 2 + 8y = 33 (b) 4x2 + 4y 2 − 16x + 40y + 67 = 0 PY (c) 4x2 + 12x + 4y 2 + 16y − 11 = 0 Solution: (a) O x2 + y 2 + 8y = 33 x2 + y 2 + 8y + 16 = 33 + 16 x2 + (y + 4)2 = 49 C Center (0, −4), radius 7, see Figure 1.17. (b) D 4x2 + 4y 2 − 16x + 40y + 67 = 0 67 x2 − 4x + y 2 + 10y = − E 4 67 x2 − 4x + 4 + y 2 + 10y + 25 = − + 4 + 25 EP 4  2 2 2 49 7 (x − 2) + (y + 5) = = 4 2 Center (2, −5), radius 3.5, see Figure 1.18. D (c) 4x2 + 12x + 4y 2 + 16y − 11 = 0 11 x2 + 3x + y 2 + 4y = 4 9 11 9 x2 + 3x + + y 2 + 4y + 4 = + +4 4 4 4  2 3 x+ + (y + 2)2 = 9 2   3 Center − , −2 , radius 3, see Figure 1.19. 2 15 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, 2016.  Figure 1.17 Figure 1.18 Figure 1.19 3. A circular play area with radius 3 m is to be partitioned into two sections using PY a straight fence as shown in Figure 1.20. How long should the fence be? Solution: To determine the length of the fence, we need to determine the coordi- nates of its endpoints. From Figure 1.20, O the endpoints have x coordinate −1 and are on the circle x2 + y√2 = 9. Then 1 + y 2 = 9, or y = ±2 2. √ Therefore, the length of the fence is 4 2 ≈ 5.66 m. C Figure 1.20 4. A Cartesian coordinate system was used to identify locations on a circu- lar track. As shown in Figure 1.21, the circular track contains the points D A(−2, −4), B(−2, 3), C(5, 2). Find the total length of the track. E EP D Figure 1.21 Figure 1.22 Solution: The segment AB is vertical and has midpoint (−2, −0.5), so its perpendicular bisector has equation y = −0.5. On the other hand, the segment BC has slope −1/7 and midpoint (1.5, 2.5), so its perpendicular bisector has equation y − 2.5 = 7(x − 1.5), or 7x − y − 8 = 0. The center of the circle is the
intersection of y = −0.5 and 7x − y − 8 = 0; 15 that is, the center is at 14 , − 12. The radius of the circle is the distance from ther center to any of the points A, 2125 5√ B, or C; by the distance formula, the radius is = 170. Therefore, 98 14 16 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, 2016.  the total length of the track (its circumference), is 5√ 2×π× 170 ≈ 29.26 units. 14 Supplementary Problems 1.1 Identify the center and radius of the circle with the given equation in each item. Sketch its graph, and indicate the center. 1 1. x2 + y 2 = PY 4 2. 5x2 + 5y 2 = 125  2 2 3 3. (x + 4) + y − =1 O 4 4. x2 − 4x + y 2 − 4y − 8 = 0 C 5. x2 + y 2 − 14x + 12y = 36 6. x2 + 10x + y 2 − 16y − 11 = 0 D 7. 9x2 + 36x + 9y 2 + 72y + 155 = 0 8. 9x2 + 9y 2 − 6x + 24y = 19 E 9. 16x2 + 80x + 16y 2 − 112y + 247 = 0 EP Find the standard equation of the circle which satisfies the given conditions. √ 10. center at the origin, radius 5 3 D 11. center at (17, 5), radius 12 12. center at (−8, 4), contains (−4, 2) 13. center at (15, −7), tangent to the x-axis 14. center at (15, −7), tangent to the y-axis 15. center at (15, −7), tangent to the line y = −10 16. center at (15, −7), tangent to the line x = 8 17. has a diameter with endpoints (3, 1) and (−7, 6) 17 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, 2016.      9 3 18. has a diameter with endpoints , 4 and − , −2 2 2 19. concentric with x2 + 20x + y 2 − 14y + 145 = 0, diameter 12 20. concentric with x2 − 2x + y 2 − 2y − 23 = 0 and has 1/5 the area 21. concentric with x2 + 4x + y 2 − 6y + 9 = 0 and has the same circumference as x2 + 14x + y 2 + 10y + 62 = 0 22. contains the points (3, 3), (7, 1), (0, 2) 23. contains the points (1, 4), (−1, 2), (4, −3) PY 24. center at (−3, 2) and tangent to the line 2x − 3y = 1 25. center at (−5, −1) and tangent to the line x + y + 10 = 0 26. has center with x-coordinate 4 and tangent to the line −x + 3y = 9 at (3, 4) O 27. A stadium is shaped as in Figure 1.23, where its left and right ends are circular C arcs both with center at C. What is the length of the stadium 50 m from one of the straight sides? E D EP Figure 1.23 28. A waterway in a theme park has a D semicircular cross section with di- ameter 11 ft. The boats that are going to be used in this waterway have rectangular cross sections and are found to submerge 1 ft into the water. If the waterway is to be filled with water 4.5 ft deep, what is the maximum possible width of the Figure 1.24 boats? 4 18 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, 2016.  Lesson 1.2. Parabolas Learning Outcomes of the Lesson At the end of the lesson, the student is able to: (1) define a parabola; (2) determine the standard form of equation of a parabola; (3) graph a parabola in a rectangular coordinate system; and (4) solve situational problems involving conic sections (parabolas). PY Lesson Outline (1) Definition of a parabola (2) Derivation of the standard equation of a parabola O (3) Graphing parabolas (4) Solving situational problems involving parabolas C Introduction A parabola is one of the conic sections. We have already seen parabolas which D open upward or downward, as graphs of quadratic functions. Here, we will see parabolas opening to the left or right. Applications of parabolas are presented at the end. E 1.2.1. Definition and Equation of a Parabola EP Consider the point F (0, 2) and the line ` having equation y = −2, as shown in Figure 1.25. What are the distances of A(4, 2) from F and from `? (The latter is taken as the distance of A from A` , the point on ` closest to A). How about D the distances of B(−8, 8) from F and from ` (from B` )? AF = 4 and AA` = 4 p BF = (−8 − 0)2 + (8 − 2)2 = 10 and BB` = 10 There are other points P such that P F = P P` (where P` is the closest point on line `). The collection of all such points forms a shape called a parabola. Let F be a given point, and ` a given line not containing F. The set of all points P such that its distances from F and from ` are the same, is called a parabola. The point F is its focus and the line ` its directrix. 19 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, 2016.  Figure 1.25 PY O C D Figure 1.26 Consider a parabola with focus F (0, c) and directrix ` having equation y = −c. E See Figure 1.26. The focus and directrix are c units above and below, respectively, the origin. Let P (x, y) be a point on the parabola so P F = P P` , where P` is the point on ` closest to P. The point P has to be on the same side of the directrix