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14580FM.pgs 3/26/09 12:11 PM Page i Teacher’s Manual with Answer Key ALGEBRA 2 and TRIGONOMETRY Marilyn Davis AMSCO AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 14580FM.pgs 3/26/09 12:11 PM Page ii Author of the Teacher’s Manual Marilyn Davis Math Consultant Portions of this book were adapted from the following Amsco Publication: Teacher’s Manual/Mathematics B by Edward P. Keenan, Ann Xavier Gantert, and Isidore Dressler Please visit our Web site at: www.amscopub.com When ordering this book, please specify: N 159 CD or TEACHER’S MANUAL/ALGEBRA 2 AND TRIGONOMETRY ISBN: 978-1-56765-704-3 Copyright © 2009 by Amsco School Publications, Inc. No part of this Teacher’s Manual may be reproduced in any form without written permission from the publisher except by those teachers using the AMSCO textbook ALGEBRA 2 AND TRIGONOMETRY, who may reproduce or adapt portions of the manual in limited quantities for classroom use only. Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 14 13 12 11 10 09 14580FM.pgs 3/26/09 12:11 PM Page iii Contents About the Teacher’s Manual v Assessment vi Suggested Time Outlines Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 The Integers viii 1 The Rational Numbers 17 Real Numbers and Radicals 30 Relations and Functions 45 Quadratic Functions and Complex Numbers 63 Sequences and Series 79 Exponential Functions 91 Logarithmic Functions 104 Trigonometric Functions 118 More Trigonometric Functions 133 Graphs of Trigonometric Functions 145 Trigonometric Identities 162 Trigonometric Equations 172 Trigonometric Applications 183 Statistics 199 Probability and the Binomial Theorem 217 Summary of Formulas and Trigonometric Relationships 232 Table of Measures 242 Holistic Scoring Rubric 243 Conversion Chart 245 iii 14580FM.pgs 3/26/09 12:11 PM Page iv Contents Answer Keys For Enrichment Activities 246 For Extended Tasks 255 For Suggested Test Items 261 For SAT Preparation Exercises 269 For Textbook Exercises iv Chapter 1 271 Chapter 2 274 Chapter 3 277 Chapter 4 282 Chapter 5 291 Chapter 6 299 Chapter 7 303 Chapter 8 308 Chapter 9 312 Chapter 10 319 Chapter 11 324 Chapter 12 334 Chapter 13 343 Chapter 14 345 Chapter 15 349 Chapter 16 359 14580FM.pgs 3/26/09 12:11 PM Page v About the Teacher’s Manual This manual parallels the organization of the text and provides: ✔ suggestions for assessment. ✔ aims for each chapter. ✔ commentary on each individual section of the text. ✔ leading questions to stimulate classroom discussion. ✔ a variety of approaches to promote flexibility in problem solving. ✔ suggestions to maximize the effectiveness of specific examples and exercises. ✔ techniques for dealing with difficulties that students may encounter. ✔ reproducible Enrichment Activities and suggestions for more challenging aspects of topics in the text. ✔ appropriate hands-on activities. ✔ suggestions for Extended Tasks and investigations to be undertaken by students. ✔ supplementary material that reflects current thinking in mathematics education. ✔ a set of SAT Preparation Exercises for each chapter. ✔ questions to form the basis for chapter tests. ✔ an answer key for the Enrichment Activities, the Extended Tasks, the Suggested Test Items, the SAT Preparation Exercises, and the text. v 14580FM.pgs 3/26/09 12:11 PM Page vi Assessment As a student studies the content of the text associated with this manual, it is important that the student, teacher, parents, and administrators have tangible evidence of the student’s progress toward established goals. That evidence can be obtained in a variety of ways. Some of these tangibles, and the features of the student text and teacher’s manual that can aid in assembling that evidence, are listed. Homework Assignments The need for consistent reflection on and practice with the content of each day’s lesson has always been accepted as a necessary factor in promoting the student’s understanding and skills. Each day’s homework assignment should enable the student and the teacher to recognize progress as well as identify needs. The text provides both routine and challenging exercises at the end of each section for this purpose. Cumulative Reviews Cumulative reviews in the text allow the student to evaluate his/her skills within the format of the Regents Examination in Algebra 2 and Trigonometry. Use the Holistic Scoring Rubric in this manual to provide the student with insight into the number of credits his/her responses merit. Guide the student in formulating responses that will receive full credit on the examination. Measure student progress by scaling cumulative review scores using the Conversion Chart. Portfolio A portfolio is a record of a student’s progress. The selection of materials to be included in the portfolio presents opportunities for the student to reflect upon what he/she has done, improve faulty work, and take pride in work that is well done and in insightful information that has been gained. Significant examples of homework assignments, tests, and independent research or readings, as well as completed Enrichment Activities, Hands-On Activities, Bonus Questions, or Extended Tasks from the student text or this manual, are excellent materials for inclusion. Tests and Quizzes Tests give the student opportunities to demonstrate that he/she understands concepts and has the ability to make use of that understanding. This manual provides Suggested Test Items for each chapter. These questions can be used by the teacher when constructing a test based on the instructional goals established for the class. vi 14580FM.pgs 3/26/09 12:11 PM Page vii Independent and Group Projects Mathematical understanding and enthusiasm are increased by opportunities to explore in-depth topics of special interest to the students. Some of these topics may be suggested by the students themselves. Others are given as Explorations in the Cumulative Reviews of the chapters of the text or as Enrichment Activities in this manual. These Explorations and Activities lend themselves to either individual or group study. Journal A journal can be a useful tool to promote a studied reflection upon day-by-day progress. A journal may be a record of feelings, understandings, fears, insights, and questions. Putting these into words can help students to clarify ideas and identify concepts that need further explanation and exploration. The content of the journal can be the choice of the student, a response to a teacher’s question or prompt, or an explanation based on questions from the Writing About Mathematics section of the exercises. You may ask students to: • describe characteristics they have noticed about problems that can be solved using a particular method. • explain or justify steps for a computation or proof. • explain key properties or concepts in their own words. • write their own problems for a particular concept or method. • explain the possible results for a given situation (such as solving a quadratic equation). • give examples of how certain concepts are applied in the real world. • explain how they organized their work for more complex Applying Skills problems. • create a flowchart or other diagram for solving a particular problem. • explain why a given answer must be incorrect based on properties, number sense, or estimation. • explain why they chose one method of proof rather than another. • make a list of common errors they must watch for in their own work. vii 14580FM.pgs 3/26/09 12:11 PM Page viii Suggested Time Outlines Since teaching from a new text can create time problems, the timetable below is offered to assist you in planning your work. If a state or national test is given before the end of the school year, it may be difficult to cover the content for the year before the test is administered. To use time most effectively, check if any of the lessons will not be tested. Consider skipping non-tested lessons and returning to them after the test. You may also wish to identify lessons that are predominantly review of concepts from an earlier grade and reduce the amount of class dedicated to these lessons by assigning them as homework only. CHAPTER 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 Year 1.5 Years 2 Years Days Days Days The Integers 4–5 10–12 15–16 The Rational Numbers 4–5 10 15–16 Real Numbers and Radicals 4–6 10–11 15–16 Relations and Functions 13–15 19–20 25–26 Quadratic Functions and Complex Numbers 13–15 19–20 25–26 Sequences and Series 13–15 15–20 24–25 Exponential Functions 12–14 17–20 22–24 Logarithmic Functions 12–14 17–20 22–24 Trigonometric Functions 13–15 19–20 24–25 More Trigonometric Functions 8–9 15 20–22 Graphs of Trigonometric Functions 8–9 15 20–22 Trigonometric Identities 12–14 18–20 23–25 Trigonometric Equations 12–14 16–21 23–25 Trigonometric Applications 9–10 15–16 20–22 Statistics 9–10 15 22–24 Probability and the Binomial Theorem 9–10 15 20–22 155–180 245–270 335–360 Total viii 14580TM_C01.pgs 3/26/09 12:11 PM Page 1 CHAPTER 1 THE INTEGERS Aims • To review operations and properties of integers. • To perform operations with polynomial expressions. • To solve absolute value equations and inequalities. • To factor polynomial expressions. • To solve quadratic equations and inequalities by factoring. This chapter reviews properties and skills that should be familiar to students from previous mathematics courses. Integer solutions to all exercises in this chapter provide a built-in check for students looking to refresh their skills. Factoring is essential for success in algebra, and since the process of factoring reverses the process of multiplication, multiplication of polynomials is reviewed before factoring is explored. Factoring is applied to solve quadratic equations and inequalities. The procedure for inequalities will be new for many students and the need to consider the signs of the factors should be discussed in depth. CHAPTER OPENER In the golf example, par is equivalent to zero on a number line, that is, the reference point from which distances above and below are determined. Rosie’s standing is found by evaluating the expression 22 1 (22) 1 1 5 23 or 3 below par for the course. Nancy’s standing is 21 1 (21) 1 (21) 5 23, so she is also 3 below par. If par is 100 strokes, then both players shot 97. If par is 95 strokes, then both players shot 92. 1-1 WHOLE NUMBERS, INTEGERS, AND THE NUMBER LINE The number line, introduced with the set of counting numbers, is extended to include 0 and then all of the integers. Basic concepts including opposites, subtraction, and absolute value are reviewed. Emphasize that 2a means “the opposite of a,” and that the opposite of a negative number is a positive number. For Exercise 2, students can write b 2 a 5 2(a 2 b). Using substitution, a 2 b 5 2(a 2 b), so a 2 b 5 b 2 a. 1-2 WRITING AND SOLVING NUMBER SENTENCES Properties of equality and inequality are reviewed and applied to solve problems. Emphasize that to maintain the order of inequality when multiplying or dividing an inequality by some number c, c must be posi- 1 14580TM_C01.pgs 2 3/26/09 12:11 PM Page 2 Chapter 1 tive. Illustrate this multiplication and division property of inequality as follows. Given: 7 . 4 Let c 5 0. Is 7(0) . 4(0) or 0 . 0? No; 0 5 0. Given: 7 . 4 Let c 5 25. Is 7(25) . 4(25) or 235 . 220? No; 235 , 20. The examples will help students to remember to reverse the inequality when multiplying or dividing by a negative number. Example 2 illustrates the approach needed to solve an equation involving absolute value. If 6x 2 3 5 u, then the equation becomes u 5 15 and, by definition, u can equal either 15 or 215. This is the source of the two derived equations. When solving absolute value equations, students should be aware that the absolute value term has to be alone on one side of the equation. For example, the steps for solving 2x 1 3 2 4 5 7 are: 2x 1 3 2 4 5 7 2x 1 3 2 4 1 4 5 7 1 4 2x 1 3 5 11 2x1 3 5 11 2x1 323 5112 3 2x1 3 5 211 or 2x1 323 5 2112 3 2x 5 8 2x 5 214 x54 2x 5 27 Ask students if 3x 2 8 5 25 has a solution. (No; 3x 2 8 represents some number and absolute value cannot be negative.) 1-3 ADDING POLYNOMIALS When reviewing the addition of polynomials in this section, address common errors that students make, such as writing 3x2 2 x2 5 3 or failing to write the opposite of each term in an expression being subtracted. Have students recall the distributive property as they work with expressions: 3x2 2 x2 5 3x2 2 1x2 5 (3 2 1)x2 5 2x2 2(4c3 2 2c2 1 5) 5 2(4c3) 2 (22c2) 2 (5) 5 24c3 1 2c2 2 5 For Exercise 25, the solution is found by using cents to represent the values of the coins and the toll. If x is the number of quarters, then x 1 3 is the number of dimes and 25x 5 10(x 1 3). 1-4 SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES To isolate the absolute value expression in an equation, the same steps are used as for isolating a variable. During this process, the absolute value expression is treated as a single entity. To help students see the form, let A represent 4a 1 2 so that 4a 1 2 1 7 5 21 becomes A 1 7 5 21. The quantity A must be isolated before the equation can be solved, so A 1 7 2 7 5 21 2 7 and A 5 14. Call attention to the fact that an absolute value inequality of the form x 2 a 5 k, k . 0, has two solutions that separate the number line into three regions: The graph of the solution set of x 2 a , k is the portion of the number line between the solutions of x 2 a 5 k; the graph of the solution set of x 2 a . k is the union of the other two regions, that is, the portion of the number line to the left of the smaller solution and the portion of the number line to the right of the larger solution. To summarize for students, explain that to solve an absolute value inequality, you must change from absolute value to plain inequality. The way to handle the change depends on which direction the inequality points with respect to the absolute-value term. To solve for x: If ax 1 b * c, solve 2c * ax 1 b * c. If ax 1 b + c, solve ax 1 b + c and ax 1 b * 2c. For Exercise 29, be sure students understand that “as much as” means “less than or equal to” or “no more than.” 200 2 c # 28 14580TM_C01.pgs 3/26/09 12:11 PM Page 3 The Integers 1-5 MULTIPLYING POLYNOMIALS 1-6 When a polynomial is multiplied by a polynomial, each term of the first factor is multiplied by each term of the second factor. Thus, when multiplying a binomial by a binomial, there are 2 3 2 5 4 multiplications. The product, in simplest form, has four terms when no two terms of the product are like terms: (3a 2 1)(2b 1 1) 5 6ab 1 3a 2 2b 2 1 The product, in simplest form, has three terms when two terms of the product are like terms: (3a 2 1)(4a 1 1) 5 12a2 1 3a 2 4a 2 1 5 12a2 2 a 2 1 The product, in simplest form, has two terms when two terms of the product are opposites: 2. Keep in mind that the difference of two squares is the only binomial form that can be factored into two binomials. If the factoring is to be done with respect to the set of integers, the squares must be the squares of integers or even powers of the variables. 3. To factor a trinomial into binomial factors, write the terms of the trinomial in order of descending or ascending powers of the variables. ax2 (b) 5 2bx(2ax) a2 1 a 2 3 3a3 1 3a2 2 9a 2a2(a2 1 a 2 3) 5 2a4 1 2a3 2 6a2 3 1. Always look for a common monomial factor first. 2 a2 2 a 1 3 Suggest that to multiply a polynomial by a polynomial, students write each polynomial in standard form, that is, in descending order according to the values of their exponents. Some students may benefit from using the vertical form of multiplication, writing similar terms one under the other. For example, 4 To factor efficiently, students must understand the use of the distributive property in the multiplication of polynomials. Emphasize the following: 3 2a2 1 3a 2 1 5 9a2 2 4 3a(a2 1 a 2 3) 5 FACTORING POLYNOMIALS Example 2 illustrates the technique of factoring by grouping, which may be useful when four or more terms are given. To group, we take the terms two at a time and look for common factors for each of the pairs on an individual basis. Factoring by grouping works only if a new common factor appears, the exact same one in each term. As noted in the text, a polynomial of four terms can be factored into two binomials if the product of the first and last terms equals the product of the two middle terms. For example, in the polynomial ax2 2 bx 2 ax 1 b: (3a 1 2)(3a 2 2) 5 9a2 1 6a 2 6a 2 4 21(a2 1 a 2 3) 5 3 2 2a 1 5a 2 4a 2 10a 1 3 In Enrichment Activity 1-5: On the Ins and Outs, students use products of polynomials to analyze a number pattern and then use algebra to generalize the pattern. ? abx2 5 abx2 ✔ However, the pairs of terms ax2 2 bx and 2ax 1 b do not yield a common binomial factor. We must change the order of the middle terms. ax2 2 ax 2 bx 1 b 5 ax(x 2 1) 2 b(x 2 1) 5 (x 2 1)(ax 2 b) Factoring is an opposite operation of multiplication. Therefore, we can factor a trinomial into the product of two binomials by reversing the steps of the multiplication process. 14580TM_C01.pgs 4 3/26/09 12:11 PM Page 4 Chapter 1 Multiplication: (2x 2 3)(3x 1 5) 5 2x(3x 1 5) 2 3(3x 1 5) degree polynomials and provides additional practice that leads to a general formula for determining the sum and difference of two cubes. 5 6x2 1 10x 2 9x 2 15 5 6x2 1 x 2 15 1-7 Factoring: 6x2 1 x 2 15 5 6x2 1 10x 2 9x 2 15 5 2x(3x 1 5) 2 3(3x 1 5) 5 (3x 1 5)(2x 2 3) Note that the only step that is not easily reversed is the last step in the multiplication. There is a unique sum of 10x and 29x, but there is not a unique way of writing 1x as the sum of two terms. To choose the correct pair of terms, we must observe that the product 6x2(215) is the same as the product 10x(29x). Therefore, the first step of this factoring method is to find two terms whose sum is the middle term and whose product is the same as the product of the first and last terms. Method 2 presents the approach that uses checking pairs of factors. Students may find these tips helpful: 1. If the last term of the trinomial is positive, the two binomials contain the same operation. • If the middle term is positive, the two binomials each contain addition as their operation. • If the middle term is negative, the two binomials each contain subtraction as their operation. 2. When the last term of the trinomial is negative, the binomials have opposite operations. In Exercises 37 and 38, suggest that students substitute P for the expression in parentheses to identify the factoring technique. Factor in terms of P, then substitute and simplify. Enrichment Activity 1-6: Factoring the Sum and Difference of Two Cubes applies the method of finding a binomial factor to higher- QUADRATIC EQUATIONS WITH INTEGRAL ROOTS The procedure for solving a quadratic equation by factoring rests on the multiplication property of 0, which allows us to apply the principle ab 5 0 if and only if a 5 0 or b 5 0. Students may sometimes try incorrectly to apply the principle in solving an equation such as x2 2 6x 5 7 by factoring to get x(x 2 6) 5 7, and then setting each factor equal to 7. Emphasize that in order to apply the principle, one side of the equation must be 0. Thus, the first step in solving a quadratic equation is to remove all terms from one side of the equation. Then, combine all like terms before factoring. Point out that when solving an equation whose highest-power term has a negative coefficient, we generally make it positive by multiplying both sides of the equation by 21. This makes the factoring process easier. For example: 2x2 1 5x 1 6 5 0 21(2x2 1 5x 1 6) 5 21 ? 0 x2 2 5x 2 6 5 0 (x 2 6)(x 1 1) 5 0 x–650 x56 x1150 x 5 21 The numbers 6 and 21 both satisfy the original equation 2x2 1 5x 1 6 5 0. In Examples 2 and 3, point out that since the 2 that was factored out is an expression not containing a variable, we do not have to set it equal to zero. In Exercise 2, students are asked to justify the extension of the zero-factor property to three factors. 14580TM_C01.pgs 3/26/09 12:11 PM Page 5 The Integers 1-8 QUADRATIC INEQUALITIES When a quadratic inequality is solved algebraically by factoring, two cases must always be considered. If the product is positive (. 0), then the factors are either both positive or both negative. If the product is negative (, 0), then either the first factor is negative and the second factor is positive or the first factor is positive and the second factor is negative. If (ax 1 b)(cx 1 d) . 0, then: (ax 1 b) , 0 and (cx 1 d) , 0 or (ax 1 b) . 0 and (cx 1 d) . 0. If (ax 1 b)(cx 1 d) , 0, then: 2(ax 1 b) , 0 and (cx 1 d) . 0 or (ax 1 b) . 0 and (cx 1 d) , 0. Another common method of finding the solution set of a quadratic inequality is to indicate the signs of the factors of the polynomial using a number line. The sign graph for the inequality x2 2 2x 2 15 , 0 of Example 1 is shown below. The sign of each factor in each interval is considered. (x 2 5) (x 1 3) 23 5 2 2 2 2 2 2 2 2 2 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 The product is negative if one factor is negative and one factor is positive. The graph shows this to be the case for 23 , x , 5. Graphing quadratic inequalities on the coordinate plane and determining the solutions from the graphs will be examined in Chapter 5. Have students practice solving quadratic inequalities using their calculators. Note that the inequality symbols can also be entered from the TEST menu. 5 EXTENDED TASK For the Teacher: This task is intended to reinforce students’ understanding of distance and absolute value. The definition of “walking distance” models distance in a city with a grid of perpendicular streets and dense construction that does not allow for diagonal shortcuts. Students may need to review the midpoint and distance formulas and the property of the perpendicular bisector of a segment. SAT PREPARATION EXERCISES Students should be given experience that will promote ease in dealing with unusual problems presented in a variety of ways, thus increasing their chances for success in standardized testing. This experience should be made available throughout the students’ mathematics education. In keeping with the philosophy that the classroom teacher is uniquely qualified to offer such training on an ongoing basis, this Teacher’s Manual provides a set of SAT Preparation Exercises for each of the 16 chapters of the book. These exercises, which are of the type and level of difficulty students will encounter in actual SATs, include Multiple-Choice and Student-Produced Response Questions. Each set of problems reflects the basic concepts of the chapter, but many of these problems are quite challenging. Students may be asked to attempt individually to solve two or three problems each night, or students may work in groups. No matter how these problems are assigned, students should not be discouraged if they find them to be difficult. In time, students will begin to learn how to approach these exercises and how to find correct solutions. Teachers may wish to adapt the following answer sheet for use with the SAT Preparation Exercises to provide students with practice gridding answers. 14580TM_C01.pgs 6 3/26/09 12:11 PM Page 6 Chapter 1 SUGGESTED TEST ITEMS A set of test questions is provided in this Teacher’s Manual for each chapter of the text. Because of the variables in testing situations, such as the ability level of a class and the length of a class period, these sets of questions need not be used as formal tests. However, they are formatted as such to be readily reproducible. The Bonus items vary in nature, some requiring special insight into text material, others needing a fresh and unorthodox approach to problem solving. Students should have the opportunity to work on these problems outside of class. 14580TM_C01.pgs 3/26/09 12:11 PM Page 7 Name Class Date Answer Sheet SAT Preparation Exercises (Chapter ____ ) Use a No. 2 pencil. Fill in the circle completely. If you erase, erase completely. Incomplete erasures may be read as answers. I. MULTIPLE-CHOICE QUESTIONS A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E II. STUDENT-PRODUCED RESPONSE QUESTIONS / / . . . / / . . . / / . . . / / . . . 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 / / . . . . / / . . . . 0 0 0 0 0 0 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 8 8 9 9 . . . . / . / . . . 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 / / / / / / . . . . . . . . 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 Copyright © 2009 by Amsco School Publications, Inc. . . . . 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 14580TM_C01.pgs 3/26/09 12:11 PM Page 8 Name Class Date ENRICHMENT ACTIVITY 1-5 On the Ins and Outs Look at this sequence of consecutive integers: 4, 5, 6, 7 The product O of the outside pair is 4 3 7 5 28. The product I of the inside pair is 5 3 6 5 30. The difference between the inside product and the outside product is I 2 O 5 2. 1. Find I 2 O for each of these sequences. a. 9, 10, 11, 12 b. 18, 19, 20, 21 c. 30, 30 1 1, 30 1 2, 30 1 3 d. x, x 1 1, x 1 2, x 1 3 2. What pattern did you observe in Exercise 1? 3. Find I 2 O for each of these sequences. a. 7, 10, 13, 16 b. 20, 23, 26, 29 c. 101, 104, 107, 110 d. x, x 1 3, x 1 6, x 1 9 4. What pattern did you observe in Exercise 3? 5. What pattern would you expect to see in the differences of inside and outside products for numbers that differ by 4? 6. Find I 2 O for each of these sequences. a. x, x 1 2, x 1 4, x 1 6 b. x, x 1 5, x 1 10, x 1 15 c. x, x 1 6, x 1 2(6), x 1 3(6) d. x, x 1 k, x 1 2k, x 1 3k 7. Using the answer to Exercise 6d, explain how the difference between the inside and outside products is related to the numbers in the sequence. 8. If the difference between the inside and outside product is 162, what is the difference between consecutive terms? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 9 Name Class Date ENRICHMENT ACTIVITY 1-6 Factoring the Sum and Difference of Two Cubes We know that when the sum and the difference of the same two terms are multiplied, the product is a binomial because two of the terms of the product have a sum of 0. For example: (2x 1 5)(2x 2 5) 5 4x2 2 10x 1 10x 2 25 5 4x2 2 25 In a similar way, the product of a binomial and a trinomial whose product has six terms can be written as a binomial when two pairs of similar terms have a sum of 0. For example: (x 1 2)(x2 2 2x 1 4) 5 x3 2 2x2 1 4x 1 2x2 2 4x 1 8 5 x3 1 8 We notice that the product is the sum of two cubes: x3 is the cube of x and 8 is the cube of 2. In two of the terms whose sum is 0, 2 and 22 are the coefficients of x2, and 4 and 24 (the square of 2 and its opposite) are the coefficients of x. The pattern that we observe in the product shown above can help us to factor both the sum of two cubes and the difference of two cubes. Example Factor x3 1 27. Method 1 1. Since 27 is the cube of 3, use 3 and 23 as the coefficients of x2, and 9 and 29 as the coefficients of x: x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27 2. Factor out the greatest common factor of each group of three terms: x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27 5 x(x2 2 3x 1 9) 1 3(x2 2 3x 1 9) 3. Factor out the common trinomial factor (x2 2 3x 1 9): x3 1 27 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27 5 x(x2 2 3x 1 9) 1 3(x2 2 3x 1 9) 5 (x2 2 3x 1 9)(x 1 3) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 10 Name Class Date Method 2 1. Again, use 3 and 23 as the coefficients of x2, and 9 and 29 as the coefficients of x, but arrange the terms in a different order: x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27 2. Factor out the greatest common factor of each pair of terms: x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27 5 x2(x 1 3) 2 3x(x 1 3) 1 9(x 1 3) 3. Factor out the common binomial factor (x 1 3): x3 1 27 5 x3 1 3x2 2 3x2 2 9x 1 9x 1 27 5 x2(x 1 3) 2 3x(x 1 3) 1 9(x 1 3) 5 (x 1 3)(x2 2 3x 1 9) The difference of two cubes can be factored in a similar way. Exercises In 1–5, factor each binomial. 1. x3 2 8 5 x3 2 23 5 x3 2 2x2 1 2x2 2 4x 1 4x 2 8 2. x3 1 64 5 x3 1 43 5 x3 1 4x2 2 4x2 2 16x 1 16x 1 64 3. x3 2 64 5 x3 2 43 5 x3 2 4x2 1 4x2 2 16x 1 16x 2 64 4. x3 1 125 5 x3 1 53 5 x3 1 5x2 2 5x2 2 25x 1 25x 1 125 5. x3 2 8y3 5 x3 2 (2y)3 5 x3 2 x2(2y) 1 x2(2y) 2 x(2y)2 1 x(2y)2 2 (2y)3 6. Derive a pattern for the factors of a3 1 b3. 7. Derive a pattern for the factors of a3 2 b3. In 8–11, use the patterns derived in Exercises 6 and 7 to factor each binomial. 8. 8x3 1 y3 9. x3 2 8y3 10. 125 2 27d3 11. 64x3 1 27y3 12. Factor a4 2 b4 5 a4 2 a3b 1 a3b 2 a2b2 1 a2b2 2 ab3 1 ab3 2 b4 by first finding the common factor of each pair of terms. Use the same method to factor the resulting polynomial factor of four terms. 13. Use the factors of the difference of two squares twice to factor a4 2 b4. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 11 Name Class Date EXTENDED TASK Going for a Walk Assume the walking distance between two points on the coordinate plane is the length of the shortest path between the points that consists only of horizontal and vertical segments. 1. Find at least three ways to travel from (1, 1) to (7, 5). What is the walking distance for each? 2. Find the walking distance between each pair of points. a. (2, 3) and (11, 9) b. (25, 0) and (8, 26) c. (1, 27) and (14, 215) 3. a. Find all the integral points that are a walking distance of 6 units from the point (6, 6). Graph the points. b. If all the points from part a were connected, what shape is formed? c. Describe the set of all points whose walking distance from (6, 6) is greater than 6. d. Describe the set of all points whose walking distance from (6, 6) is less than 6. 4. a. Use the distance formula to find the straight line distance from the origin to (6, 6). b. What is the walking distance from the origin to (6, 6)? c. Which distance is greater? 5. a. When are the straight-line distance and walking distance between two points equal? b. When is the straight-line distance greater than the walking distance? 6. Graph the set of all the points that are the same walking distance from (8, 9) and (6, 5). Explain how to locate these points. 7. How would you locate all of the points that are the same straight-line distance from (8, 9) and (6, 5)? 8. Use absolute value notation to write the distance between each pair of points. a. (7, 3) and (2, 3) b. (5, 22) and (24, 22) c. (x1, y) and (x2, y) 9. Use absolute value notation to write the distance between each pair of points. a. (3, 5) and (3, 20) b. (24, 28) and (24, 15) c. (x, y1) and (x, y2) 10. Use absolute value notation to write the walking distance between each pair of points. a. (2, 4) and (10, 10) b. (25, 1) and (211, 7) Copyright © 2009 by Amsco School Publications, Inc. c. (x1, y1) and (x2, y2) 14580TM_C01.pgs 3/26/09 12:11 PM Page 12 Name Class Date Algebra 2 and Trigonometry: Chapter One Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–4, find the value of each given expression. 1. 6 1 213 2. 214 1 (2(29)) 3. 10 2 23 2 3 4. 2(25 1 8) 5. Find all integers n such that: a. n 5 2n b. n 5 n c. n 5 24 In 6–12, solve each equation or inequality. Each variable is an element of the set of integers. 6. 5x 1 8 5 43 7. 19 2 3c 5 31 8. 27 2 4y 5 29 9. 2x 1 6 5 20 10. 6m 1 9 5 15 11. 8k 1 7 . 23 12. 5 , 4x 1 1 # 37 In 13–15, write the sum or difference of the given polynomials in simplest form. 13. (n3 2 5n2 1 2n) 1 (22n2 2 n 1 2) 14. (3c2d 2 2cd2 1 5) 1 (c2d2 2 3c2d 1 2cd2) 15. (10x3 2 3x2 2 2x) 2 (5x2 2 7x 2 3) 16. Subtract 2y2 2 y 1 1 from the sum of y 1 2 and y2 2 4y 1 3. 17. Luis spent 28 minutes more on his math homework than his science homework. If the total time he spent on both subjects was 152 minutes, how much time did he spend on each subject? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 13 Name Class Date 18. A 40-foot rope is cut into two pieces, one of which is 2 feet shorter than 6 times the length of the other. How long is each piece of rope? 19. Mrs. Rudin is five times as old as her daughter Kate. In 9 years, Mrs. Rudin will be three times as old as Kate. How old is each now? In 20–22, write the solution set of each inequality if the variable is an element of the set of integers. 20. 4 2 x 1 1 # 3 21. 2x 2 3 . 5 22. 2x 1 3 2 5 # 10 23. Describe the solution set for the inequality 2x 1 5 1 6 $ 24 if x is an element of the set of integers. 24. Describe the solution set for the inequality 3x 2 5 1 4 , 2 if x is an element of the set of integers. In 25–29, perform the indicated operations and write the result in simplest form. 25. 22x(x 1 3)2 26. 2y [y 2 (3y 1 2)] 27. 3x2(x 2 2)(3x 2 5) 28. 2x 2 [x 1 3(x 2 1) 2 5] 29. 5x 1 4(x2 2 x 1 4) 30. The length of a rectangle is 3 more than twice the width x. a. Express the area of the rectangle in terms of x. b. Find the area if the rectangle is 21 inches long. In 31–36, factor each polynomial completely. 31. 5a 1 ab 1 5b 1 b2 32. 10x2 2 3x 2 18 33. x2 2 9y2 34. 4n3 2 5n2 1 n 35. 8x5 2 2x3 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 14 Name Class Date 36. x3 2 3x2 1 4x 2 12 In 37–40, solve and check each equation. 37. x2 1 x 5 30 38. y2 2 3y 2 4 5 0 39. x2 1 5x 2 14 5 0 40. (x 1 4)2 5 16 41. The product of two consecutive positive integers is 132. Find the integers. 42. If the sides of a square are each increased by 4 inches, the area becomes 121 square inches. Find the perimeter of the original square. In 43–46, write the solution set of each inequality if x is an element of the set of integers. 43. x2 2 7x 1 12 . 0 44. x2 1 2x # 3 45. x2 2 6x . 0 46. x2 , 2x 1 35 Bonus I: Show that a2 2 b2 2 c2 1 2bc can be factored as (a 2 b 1 c)(a 1 b 2 c). Bonus II: Solve (x 2 2)(x 1 3)(x 1 5) , 0. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C01.pgs 3/26/09 12:11 PM Page 15 Name Class Date SAT Preparation Exercises (Chapter 1) I. MULTIPLE-CHOICE QUESTIONS In 1–14, select the letter of the correct answer. 1. If x 2 y 5 8, then which of the following must equal 64? (B) x2 1 y2 (A) x2 2 y2 2 2 (C) x 2 2xy 1 y (D) x2 1 2xy 1 y2 (E) none of these 2. From (x 2 1)2 subtract (x 2 3)(x 2 1). The result is (A) 2(x 2 1) (B) 22(x 2 1) (C) 4(x 2 1) (D) 24(x 2 1) (E) x(x 2 1) 3. If 2x 2 5 5 78, what is the value of 2x 1 7? (A) 71 (B) 80 (C) 85 (D) 90 (E) 95 4. For what integer n is it true that 4(n 2 15) 5 n? (A) 220 (B) 212 (C) 0 (D) 5 (E) 20 5. If ▫x▫ 5 x 1 2 and a 5 3b 1 1, then ▫a▫ is equivalent to (A) 3b 1 1 (B) 3b 1 1 1 2 (C) 3b 1 1 1 2 (D) 3b 1 3 (E) 3b 1 3 6. What is the sum of the solutions to the equation 25x 2 5 5 35? (A) 214 (B) 26 (C) 22 (D) 2 (E) 14 7. At the Kids Carnival, children won a prize for guessing within 10 the number of jellybeans in a jar. If the jar contained 86 jellybeans, how many winning guesses are a multiple of 3? (A) 4 (B) 5 (C) 7 (D) 9 (E) 10 Copyright © 2009 by Amsco School Publications, Inc. 8. If 4 2 x , 4x 1 4, which of the following must be true? (A) x . 0 (B) x 5 0 (C) x , 0 (D) x . 22 (E) x , 22 9. What are all the values of x for which x2 1 9x 1 18 , 0? (A) 26 , x , 3 (B) 26 , x , 23 (C) x , 23 (D) x , 3 (E) x , 26 10. Which of the following sets of x-values gives all the solutions of x2 2 12x 5 220? (A) {210, 22} (B) {210, 2} (C) {10, 22} (D) {10, 2} (E) {220, 1} 11. What are all the values of w for which w2 1 2w $ 24? (A) 26 # w # 4 (B) w # 24 or w $ 6 (C) w # 26 or w $ 4 (D) 24 # w # 6 (E) w $ 24 or w $ 6 12. Which equation has two distinct integral solutions? (A) x2 1 25 5 0 (B) (x 2 5)2 5 0 2 (D) x 2 5 5 25 (C) (x 1 5) 5 0 (E) x2 2 25 5 0 13. If x 5 2 is one solution of x2 1 5x 2 c 5 0, then the other solution is (A) 214 (B) 27 (C) 7 (D) 10 (E) 14 14. If az 1 bz 5 2z, then what is the value of a 1 b? (A) 21 (B) 0 (C) 1 (D) 2 (E) 3 14580TM_C01.pgs 3/26/09 12:11 PM Page 16 Name Class II. STUDENT-PRODUCED RESPONSE QUESTIONS In 15–20, you are to solve the problem. 2 15. If x 2 3x 1 1 5 0, then what is the value of 2x2 2 6x 1 19? 16. A number increased by 6 gives the same result as that number multiplied by 19. What is the number? 17. If x 2 y 5 16 and x2 2 y2 5 96, then what is the value of x? Copyright © 2009 by Amsco School Publications, Inc. Date 18. If x2 1 y2 5 34 and xy 5 15, what is the value of (x 2 y)2? 19. What is the sum of the solutions to the equation 113 2 m 5 9? 20. A certain number is increased by 17 and the result is doubled. The final answer is 6 more than 3 times the original number. What was the original number? 14580TM_C02.pgs 3/26/09 12:11 PM Page 17 CHAPTER 2 THE RATIONAL NUMBERS Aims • To perform arithmetic operations with rational expressions and write expressions in lowest terms. • To simplify complex fractional expressions. 450 hour 5 12 hour 1 18 hour 5 30 60 1 3,600 where 450 is the number of seconds in 18 hour. Since 450 seconds 5 7 minutes 30 seconds, add 7 to 30 the minutes expression to get 37 60 1 3,600. Ask students to express 35 4 8. Instead of 438, they 30 should find 4 1 22 60 1 3,600. 5 8 • To solve rational equations and inequalities. • To use ratio and proportion to solve applied problems. This chapter defines a rational expression and applies the rules for addition, subtraction, multiplication, and division of fractions to these operations on rational expressions. Throughout the chapter, attention is called to the need to delete from the domain of any expression the values of the variable that make the denominator equal to 0. Students will understand the importance of this detail only if they are asked to state all restrictions when working with a fraction that contains a variable in the denominator. This careful approach to operations with rational expressions will help students to understand the meaning of an extraneous root when it is introduced at the end of the chapter. CHAPTER OPENER The result for 21 4 8 can be made clear to students by showing that 2-1 RATIONAL NUMBERS We need rational numbers to represent parts of a whole. You may wish to show how to locate the rational number ba on a number line by first dividing the line segment from 0 to 1 into b equal parts and then marking off a of these parts. This construction will help you to emphasize the conditions that a and b must be integers and that b cannot be 0. Since it is impossible to divide a line segment into 0 equal parts, the symbol 20 is not a representation of a rational number. In fact, 20 is not a symbol for any number. Neither a calculator nor a computer will give an exact representation of a repeating decimal value. All these devices work with infinitely repeating decimals rounded to the number of digits allowed by the operating system. A repeating decimal is sometimes called a periodic decimal. Emphasize that the if and 17 14580TM_C02.pgs 18 3/26/09 12:11 PM Page 18 Chapter 2 only if statement before Example 2 is equivalent to the two statements: • Every rational number can be written as an infinitely repeating decimal. • Every infinitely repeating decimal can be written as a rational number. Students should become fluent with the method presented in this section for writing a repeating decimal as a rational number. An alternate method is to write the repeating decimal as a sum and then use the formula for the sum of an infinite geometric series (Sec2 tion 6-7). After students find 0.18 5 11 , have them find the fraction for 0.18 5 17 90 . Other similar comparisons can be assigned with regard to the exercises. For example, Exercise 19 asks students to convert 0.83 to the fraction 5 6 ; have students compare the result for 0.83 5 83 99. Note that a sufficient number of the repeating digits must be entered in a calculator before the decimal-to-fraction function will return the desired fraction. For the example 0.18 above, the required entry is .188888888888. 2-2 SIMPLIFYING RATIONAL EXPRESSIONS Because variables represent real numbers, operations with rational numbers and rational expressions are very similar. To write a fraction in simplest form, divide both the numerator and the denominator by their greatest common factor (GCF). A similar approach is applied for rational expressions, also called algebraic fractions. Students should understand that cancellation produces the same result as using the inverse of the indicated operation. When the problem involves multiplication, canceling one of the factors is a division. When the problem involves addition, canceling one of the terms is a subtraction. 15 10 3(5) 5 2(5) 5 32 x175317 x53 Canceling a term from the numerator and the denominator of a fraction is a common error. If this error occurs when algebraic fractions are being simplified, offer numerical examples that are comparable. Algebraic: y 1 1 y 1 1 y 1 2 1 is in simplest form, and y 1 2 2 2. Numerical: 5 5 2 2 1 1 1 1 1 2 1 2 1 1 5 67, but 55 1 1 2 2 2. 1 1 5 34, but 22 1 1 2 2 2. Each cancellation shown above and to the right must be incorrect because each yields the value 12, which is not equivalent to the known value. We can cancel only factors common to both the numerator and the denominator. In 5 1 1 5 1 2 5 67 , observe that 5 is not a factor of 3 1 1 either 6 or 7. In 22 1 2 5 4, although 2 is a factor of 4, note that 2 is not a factor of both 3 and y 1 1 4. In y 1 2, note that y is not a factor of either y 1 1 or y 1 2. Any non-zero number divided by itself is 1. But 00 is indeterminate. Throughout this chapter, call attention to the conditions under which the original rational expression and the reduced rational expression are equivalent. Emphasize the method for factoring out 21 in the numerator or denominator to help simplify rational expressions. Finally, point out that one of the reasons for reducing rational expressions is that numerical substitutions are much easier when the expression is simplified. For example, x2 1 13x x 1 13 x(x 1 13) 5 x 1 13 5 x and, to find the value of the original expression for x 5 297.4653, it is clearly easier to work with the simplified form. 14580TM_C02.pgs 3/26/09 12:11 PM Page 19 The Rational Numbers 2-3 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS In practice, students will generally find it easier to use Method 2, factoring and cancelling common factors before multiplying resulting numerators and denominators. When simplifying computation by using cancellation of a factor in the numerator and denominator of a fraction, students need to be reminded that: 1. When a cancellation represents the division of a non-zero number by itself, the resulting factor is 1. x 2 3 x 2 3 5 1 where x 3. 2. Only in the multiplication of two fractions can the same factor be canceled from the numerator and the denominator of two different fractions. y x y x ? z 5 z where x 0, z 0. y x In x 1 z , the x’s cannot be canceled. 3. When two fractions are correctly multiplied and the product is reduced to lowest terms, the simplified result is equivalent to the product of the given fractions only for values of the variable for which the given fractions are defined. 2 x 2 3 x ? x x2 3 5 x where x 0, 3. 4. When all the visible factors have been canceled from a numerator and denominator, there still exists a factor of 1. By writing 1 as a factor, we see that 1 1 3 (x 2 1) 3 (x 2 1) (x 2 2) 1 1 5x22 1 By not writing 1, a student may factor and cancel correctly but overlook the numerator: 3(x 2 1) 3(x 2 1)(x 2 2) 2x22 Emphasize that before any factors can be canceled, a division problem must be changed to multiplication by the reciprocal of the divisor. In the following example, x 2 2 cannot be canceled when the fractions are written with the division sign between them, but x can be 19 canceled when the division is changed to multiplication by the reciprocal. x x 2 2 4 x(x 2 2) 4 x 5 x 2 2 ? x(x 42 2) 4 5 2 (x 2 0, 2) (x 2 2) 2-4 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS When finding the least common denominator of two or more algebraic fractions, encourage students to write each polynomial denominator in the same order (ascending or descending). It is easier to recognize the relationship of binomials such as x 2 2 and 2 2 x if they are written as x 2 2 and 2x 1 2 or as x 2 2 and 21(x 2 2). In Example 1, emphasize the use of the distributive property in the numerator when x2 2 6x 1 9 is subtracted from x2 2 3x; students often use the opposite of only the first term and will incorrectly write x2 1 3x2(x2 26x1 9) 5 x2 1 3x2x2 26x1 9. In the text example on page 54, the numerator is expressed as a binomial (a 1 1), but the denominator is left as the product of factors (2(a 1 1)(a 2 1)). When the numerator is simplified, it may contain a factor common to the denominator and, thus, the rational expression can be further simplified. Ask students which approach they think is easier for Example 2. Most will choose the first solution since the common denominator for each expression is convenient to work with and fewer steps are required to complete the problem. 2-5 RATIO AND PROPORTION A ratio is a comparison by division. Explain that every rational number is a ratio, but since the terms of a ratio do not have to be integers, not every ratio is a rational number. For example, in a right triangle with legs of length 2 and 14580TM_C02.pgs 20 3/26/09 12:11 PM Page 20 Chapter 2 3 and hypotenuse of length !13, the ratio of 2 the shortest side to the longest is !13 , which is not rational. A key idea of this section is that equal ratios are formed when each term of a ratio is multiplied by the same positive number. In Example 2, the ratio of the length of the side to the length of the base is 5 : 2 and if x is any positive number, then 5x : 2x is an equivalent ratio. The discussion of proportion with students should include the point that if the proportion a c b 5 d is true, then the reciprocal proportion b a 5 dc is also true. This idea is useful for solving 5 1 an equation such as x 2 2 5 4. After you invert the proportion, the denominator of the left term is 1. At that point, all you need to do is add 2 to each side. 1 x 2 2 2 5 54 S x 2 5 45 1 x 2 2 5 45 x 5 245 Proportions are a specific type of fractional equation. In Exercises 16–19, cross multiplication will produce quadratic equations that must be solved. Emphasize that both solutions must be tested in the original proportion. There are no extraneous roots in these exercises. Use Enrichment Activity 2-5: Investigating Ratios and Growth Rate in Leaves to give students the opportunity to connect mathematics to science and the real world. Have students work in small groups or individually. Students may need to look up the meaning of leaf blade and petiole. 2-6 COMPLEX RATIONAL EXPRESSIONS Continue to call attention to the fact that the denominator of a fraction cannot be 0. In a complex fraction, three of the four possible parts cannot be zero. Thus: a In bc , b, c, and d cannot be 0. d For example, in the fraction x24 x x21 x22 x, x 2 1, and x 2 2 cannot be equal to 0. Therefore, x cannot be equal to 0, 1, or 2. In Exercise 1, students need to simplify the fraction to a21 a a2 2 1 a2 and consider carefully the circumstances for which the three key parts described above will be zero. In addition to excluding 0 and 1, they must also realize that 21 will also result in a2 2 1 equaling 0. In Exercises 23 and 24, remind students to follow the correct order of operations. 2-7 SOLVING RATIONAL EQUATIONS Students often confuse the procedures involved in simplifying the sum of two fractions and those used to solve an equation that has the sum of two fractions on one side. In simplifying the sum of two fractions, each fraction is changed to an equivalent fraction by multiplying by a fraction equal to 1. 1 x 1 1x2 1 5 5 1 x 1 ? xx 2 2 1 1 x 2 1?x x 2 1 x x(x 2 1) 1 x(x 2 1) 1 x 2x 2 1 5 x(x 2 1) In solving an equation that has the sum of two fractions on one side, the equation is changed to an equivalent one in which neither side contains fractions. 3 1 1x2 1 5 2 3 1 2x(x 2 1) A x1 1 x 2 1 B 5 2x(x 2 1) A 2 B 2(x 2 1) 1 2x 5 x(x 2 1)(3) 1 x 4x 2 2 5 3x2 2 3x When both sides of an equation are multiplied by the same non-zero number, the resulting equation is equivalent to the original; that 14580TM_C02.pgs 3/26/09 12:11 PM Page 21 The Rational Numbers is, it has the same roots as the original. Therefore, 4x 2 2 5 3x2 2 3x has the same roots as 3 1 1 x 1 x 2 1 5 2 if x 0 and if x 2 1 0 or x 1. Since the roots of 4x 2 2 5 3x2 2 3x are not 0 or 1, the roots of both equations are the same, 2 and 13 . Summarize for students by explaining that whenever a variable appears in any denominator, you must check the possible solutions in the original equation. When checking, if a possible solution makes any denominator equal to 0, that value is not a solution to the equation. Such values are extraneous solutions introduced by multiplying by a LCD that contains a variable. Example 5 illustrates the solution of a standard “motion” problem. time time total traveling 1 traveling 5 time for at first speed at second speed trip Motion problems may also state that two parts of a trip were different distances traveled at different speeds, but the time for each part was the same. All the information in the problem is usually needed to construct a solution, so students should read carefully, look for needed facts, and make sure they answer the question that was asked. 2-8 SOLVING RATIONAL INEQUALITIES Students may find the second method used in the examples preferable since it does not require thinking about when an expression is positive or negative. The method involves these steps: 21 1. Identify the excluded values. 2. Solve the corresponding equation. 3. Use the excluded values and solutions to partition the number line. 4. Test values in each interval. Method 1 also requires that the student take into account both the interval on which the expression is positive or negative and the interval that yields the proposed solution in each case. In Example 1, the solution a . 4 applies if a . 0, so a . 4 is the interval where both conditions are satisfied. The solution a , 4 applies if a , 0, so a , 0 is the interval where both conditions are satisfied. You may wish to suggest that students work at least one exercise from 8–14 using both methods. EXTENDED TASK For the Teacher: This extended task is intended to let students see that rational equations are applied in the workplace. This is a task where students could work independently or in small groups. Since electronics is a technical discipline, this task allows students to see the connections between mathematics and other disciplines. It would be possible to use this as a science/math integrated activity where students could construct circuits modeling each of the five circuits given. Assessment of students’ work should include the ability to follow directions, neatness and accuracy of diagrams, understanding of circuits, accuracy of arithmetic computations, solution of rational equations, and the ability to express themselves in written form. 14580TM_C02.pgs 3/26/09 12:11 PM Page 22 Name Class Date ENRICHMENT ACTIVITY 2-5 Investigating Ratios and Growth Rate in Leaves As you study mathematics, you should see that it has many connections with the real world. This investigation is designed to help you see how rational numbers are connected to science and to nature. As you know from your previous studies, a rational number is one that can be expressed in fractional form where both the numerator and the denominator are integers and the denominator cannot be zero. If the numerator is zero, then the fraction is equal to 0. If the denominator is 1, the fraction is equal to a whole number. If the numerator and denominator are both non-zero and the denominator is not equal to 1, the fraction may be expressed as a terminating or repeating decimal. This investigation requires you to find some measures, to write some ratios from these measures in both fractional and decimal form, to make a scatter plot of these ratios, and to draw a conclusion about the growth rate of leaves. Follow the procedure outlined below to complete this task. 1. Select 10 to 15 leaves from a single tree or bush. Be sure to choose leaves that are different sizes, from the smallest ones to the largest ones that you can find. Pick them from different branches, but from the same tree or bush. Do not damage the tree or bush in removing the leaves you select. 2. Lay one of your leaves flat and, with your ruler, measure the longest part of the blade of the leaf. Do not include the petiole. Express your measure to the nearest millimeter or to the nearest tenth of a centimeter. Fold the leaf in half, making the point of the blade match the base of the blade. Again, ignore the petiole. Crease carefully, unfold, and measure the width of the leaf at the crease. 3. Express the ratio of the length to the width of your leaf in fractional form. Using your calculator, express each ratio as a decimal. Round off appropriately. This length/width ratio is a rational number. 4. Repeat steps 2 and 3 for the rest of your leaves. 5. Construct a data table and record your measurements and ratios. 6. What can you say about your ratios? 7. Make a scatter plot of the length vs. width of the leaves you measured. What shape do your points make (for example, a tight ball, a pencil shape, and so on)? 8. Now find the average length-to-width ratio of all the leaves you selected and measured. 9. Using the data table, your scatter plot, and the average length-to-width ratio, what can you conclude about the growth rate of the leaves on the tree or bush you selected? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 23 Name Class Date EXTENDED TASK Electronic Technician: Applying Rational Equations in the Workplace In previous mathematics classes, you have solved and applied various types of equations. You have developed techniques that enable you to do this algebraically. Many of these same algebraic techniques are used in the solution of rational equations; that is, equations involving rational expressions. Rational equations occur frequently in the workplace. In this extended task, you will see how rational equations are used in the electronics industry. The Setting Ed works as a technician for a small electronics firm. He repairs, tests, and builds prototypes of electronic equipment and products. He needs to use his knowledge of resistors, capacitors, and inductors, as they are essential components used in electronic circuits. He often needs to build resistors to reduce current in a circuit to a desirable level. Sometimes he connects the resistors in series and other times he needs to connect them in parallel. On occasion it is necessary for him to combine them into series-parallel groupings. There are also times when Ed needs to combine these three basic units to form complete electronic circuits. Resistance in a circuit is measured in ohms. In electronics, the values of resistance normally encountered are quite high. Often, thousands, and occasionally even millions, of ohms are used. The Task a. Use any resource available to you to find out the difference between parallel and series circuits. Explain this difference in words. b. Draw a diagram of a circuit containing two or more resistors connected in series. c. Draw a diagram of a circuit containing two or more resistors connected in parallel. d. Locate the formulas for the total resistance in a series circuit and the total resistance in a parallel circuit. State these formulas in algebraic form, and explain, in your own words, the meanings of these formulas. e. In each of the diagrams on the following page: (1) State whether the circuit is a series circuit, a parallel circuit, or a combination series-parallel circuit. (2) Using the values given for RT, the total resistance of the circuit, and the values for all resistors in the circuit except one, compute the missing resistance. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 24 Name Class 1. Date RT 5 15,000 ohms R1 R1 5 5,000 ohms R2 5 3,000 ohms R2 R3 5 ? R3 RT 5 8 ohms 2. R1 R1 5 15 ohms R3 R2 R2 5 ? ohms R3 5 40 ohms RT 5 400 ohms 3. R2 R1 R3 R4 R1 5 1,000 ohms R2 5 1,000 ohms R3 5 3,000 ohms R4 5 ? 4. RT 5 6,500 ohms R1 R1 5 2,000 ohms R2 5. R3 R5 R2 R3 5 18,000 ohms RT 5 15.8 ohms R1 5 1.6 ohms R4 R1 R2 5 ? ohms R2 5 4.0 ohms R3 5 6.0 ohms R3 R6 R4 5 3.8 ohms R5 5 10.0 ohms R6 5 ? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 25 Name Class Date Algebra 2 and Trigonometry: Chapter Two Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. 5 1. Express 12 as a repeating decimal. In 2–4, write each decimal as a common fraction. 2. 0.625 3. 0.4 4. 0.12 In 5–8, write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. 2x3y2z 5. 4x2yz2 3x 1 xy 6. y 1 3 2 19x 2 14 7. 3x 12 2 3x y2 2 10y 1 24 8. y2 2 5y 1 4 In 9–17, perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. 2a 1 2b 9. a 5a 1 b ? 15a2 2 3 x 2 1 10. x2 3x 2 2x 1 1 ? 3x 12c2 11. c 8c 1 2 4 c2 1 3c 1 2 3 1 1 1 12a 2 3a 12. 4a 3 1 13. a2 2 9 4 4a 1 12 1 5 x 1 4 14. 3x 2x 1 2 2 3x 1 3 1 6 1 15. 4x x2 2 1 1 x 1 1 4y 5 16. y2 2 25 2 y 1 5 17. A 3 2 x1 B A 1 1 9x2 12 1 B Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 26 Name Class Date 2 2 18. The length of the base of a triangle can be expressed as 10x x2 2 9 feet. The altitude 3 to that base can be expressed as xx 1 2 1 feet. Express the area of the triangle in terms of x. In 19–22, solve each proportion for the variable. 19. 4x 61 5 5 72 3 20. a5 5 a 2 2 1 7 21. 6x 10 5 2x 61 9 2 x 1 1 22. xx 2 1 4 5 x 1 10 23. At a picnic, the ratio of adults to children was 7 : 4. There were 15 more adults than children. How many people were at the picnic? In 24–27, simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. 24. 1 x2x 1 2 x12 3 25. ab 1 b2 1 2 26. x 2 2 x2 2 4 27. x21 4 x 4 11x1 x In 28–31, solve each equation and check. 28. x1 1 13 5 x5 3 3x 29. x 2 1 2 x 1 1 5 1 30. x 1 12 x 5 27 8 2 4 31. x 2 3 2 x 1 3 5 x2 2 9 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 27 Name Class Date 32. An express train travels 150 miles in the same time that a freight train travels 100 miles. If the express goes 20 miles per hour faster than the freight, find the rate each train travels. 5 . Find the 33. The sum of two numbers is 10. The sum of their reciprocals is 12 numbers. In 34–36, solve and check each inequality. 34. 5y 2 3 7 . 15y 2 2 28 2 4 35. b 2 , bb 2 b 2 6 x 1 3 36. 2x .2 3 2 6 Bonus: Simplify each complex fraction. 1 a. 2x 1 1 1 2x 1 2x 1 b. m1 1 m 1 m 11 1 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C02.pgs 3/26/09 12:11 PM Page 28 Name Class Date SAT Preparation Exercises (Chapter 2) 8. A I. MULTIPLE-CHOICE QUESTIONS In 1–14, select the letter of the correct answer. y 3 , then BC equals If AC 5 23 and BD 5 2x 1 9 5 (A) 3 1 2x (B) 4x6x 2 x 1 3 (C) 2x3x (D) 3 2x (E) x1 1 y1 1 2. If x 5 (1 2 x)(5 2 x) , then which of the following is the largest number? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 3. If x 1 y 5 10 and x 2 y 5 5, then x2 2 y2 2x D C 1 16 1 1. If x1 1 y1 5 1, then x 1 y equals 1 (A) xy (B) xy (C) xy (D) x B (E) 1 13 2x 9. D y x C x y is equal to (A) 10 3 (D) 5 (B) 2 (E) 10 y (C) 3 x A B 1 4. If x2 22 1 5 18 2 10 and x . 0, then x is (A) 18 (B) 9 (C) 4.5 (D) 3 (E) 2 The ratio of the shaded area to the area of square ABCD is 1 1 5. The average of x 1 2 and x 2 2 is (A) (x 1 y) 2 (A) x1 (B) x12 (D) x2 x2 4 (E) x2 2x 2 4 7. (A) 0 (B) 1 (D) x1 y x y 2 x (E) not determinable (E) (x 2 y) 2 xy (x 1 y) 2 xy x x2 1 y2 (B) (x 1 y) 2 x2 2 y2 2xy (C) x2 1 y2 (D) (x 1 y) 2 (E) not determinable 10. (C) 2 x12 l equals (A) 21 (C) 2xy (C) x2 12 4 6. If xy 5 1, then 2x 12 y 1 y 21 2x is y 2 (B) xy (D) (x 1 y)(x 2 y) xy x11 If the volume of the rectangular box is x2 1 4x 1 3, then the dimension l is 1 (A) x 1 3 3 (C) xx 1 1 2 (E) x 1 2 Copyright © 2009 by Amsco School Publications, Inc. (B) x 1 3 2 (D) xx 1 1 3 14580TM_C02.pgs 3/26/09 12:11 PM Page 29 Name Class 3 11. Solve 13 1 m62 5 m . (A) 23 and 26 (C) 3 and 26 (E) 3 and 6 (B) 29 and 22 (D) 2 and 29 12. If x4 1 4 1 x4 1 4 1 x4 1 4 5 0, then the value of x is (A) 24 (D) 1 4 (B) 22 (C) 21 (E) 1 x x 13. If A 100 2 1 B A 100 1 1 B 5 kx2 2 1, then the value of k is (A) 0.1 (D) 0.0001 (B) 0.01 (E) 0.00001 (C) 0.001 n 14. If 36 5 36 n , then which of the following could equal 72? (A) n2 (B) 72n (D) 2n 1 36 (C) n2 4 36 (E) 3n 2 36 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 15–22, you are to solve the problem. 15. If x 1 x1 5 30, then x2 1 12 is equal to x what number? Copyright © 2009 by Amsco School Publications, Inc. Date 16. If a 24 3b 5 b2, what is the value of ba? 17. The base of a triangle is 5 inches less than the altitude. The area of the triangle is 42 square inches. Find the base in inches. 18. A fraction is equivalent to 34. If 3 is subtracted from the numerator and 3 is subtracted from the denominator, the new fraction formed equals 23. Find the numerator and denominator of the original fraction. 19. Lucia rode her bicycle at a constant rate along a flat road 15 miles long. When the road became a hill, she reduced her speed by 4 mph for the last 3 miles of her trip. The trip took Lucia 2 hours. What was her rate of speed, in mph, on the flat part of the road? 20. If kxl 5 1 2 1 X22Y ? x , what 1 12x is the value of 5 2 21. If 2x 32 1 2 3x 4 5 6, what is x ? 22. The ratio of boys to girls at the basketball game was 7 : 5. There were 12 more boys at the game than girls. How many boys and girls were at the game? 14580TM_C03.pgs 3/27/09 12:38 PM Page 30 CHAPTER 3 REAL NUMBERS AND RADICALS Aims • To define the set of irrational numbers and the set of real numbers. • To graph inequalities. • To simplify radical expressions. • To add, subtract, multiply, and divide radical expressions. • To rationalize a denominator containing a radical. • To solve radical equations. While most numbers students deal with on a daily basis are rational, and even though there is an infinite number of rationals, mathematicians have proved that most of the points on the number line correspond to irrational numbers. In this chapter, the primary focus is on irrational numbers that are radicals. However, students should be aware that many other irrationals belong to the set of real numbers. CHAPTER OPENER There are many variations of the proof that !2 is not a rational number, but the basic approach is similar to that used by Euclid: assume the opposite of what is to be proved 30 and show that it leads to a contradiction. That p is, assume !2 is a rational number, so !2 5 q . p p Reduce q so that q 5 ba is in simplest form; thus, a and b are not both even integers. If 2 2 !2 5 ba, then A !2B 2 5 A ba B and 2 5 ba2. Then 2b2 5 a2 and 2b2 is even since it has a factor of 2. Since a2 is equal to 2b2, then a2 is even. However, if the square of an integer is even, then this integer is even (this result must be previously proven). Thus, a is even. Since a is even, we may write a 5 2k, where k is some integer. Then, 2b2 5 (2k) 2 By substitution 2b2 5 4k2 2 2 b 5 2k 2 By squaring 2k By dividing each side by 2 Since b is equal to 2k2, then b2 is even. Again, using the fact that if the square of an integer is even, then the integer is even, it follows that b is even. This contradicts the assumption that a and b were not both even. Therefore, the assumption that there exists a rational number whose square is 2 has led to a contradiction. Thus, the real number whose square is 2 cannot be a rational number. A more advanced version of the proof uses the result that any perfect square must have an even number of prime factors and any number that is twice a perfect square must have an odd number of prime factors. 14580TM_C03.pgs 3/27/09 12:38 PM Page 31 Real Numbers and Radicals 3-1 THE REAL NUMBERS AND ABSOLUTE VALUE From the definition of an irrational number, we may conclude that a number is irrational if and only if it can be expressed as an infinite nonrepeating decimal. Students should realize that it is often difficult to demonstrate that a decimal expansion does not repeat. The fact that p, which has been calculated to millions of decimal places, has not yet been shown to repeat does not mean the decimal expansion will not begin to repeat at a later decimal place. Therefore, proving that a number is irrational usually requires reasoning that is independent of decimal expansions. Students have graphed inequalities in previous courses and have solved absolute value inequalities in Chapter 1. The geometric approach used to graph x , 3 and x . 3 provides a basis of understanding from which to proceed to the more formal algebraic procedure. Call attention to the fact that an absolutevalue equality of the form x 5 k, k . 0, has two solutions that separate the number line into three regions. The graph of the solution set of x , k is the region between the solutions of x 5 k, and the graph of x . k is the union of the other two regions, that is, the region to the left of the smaller solution and the region to the right of the larger solution. If k is positive, then the equation x 5 k and the inequalities x . k and x , k have solutions that are proper subsets of the set of real numbers. These cases are explained and summarized in the text. After students understand these concepts, ask them to describe the solution set for each of the three cases where k is negative. (If k , 0, then x . k is true for all real numbers since x is non-negative and every non-negative number is greater than every negative number. If k , 0, then x 5 k and x , k have no real solution, since no nonnegative number is equal to or less than a negative number.) 3-2 31 ROOTS AND RADICALS Students are familiar with square-root radicals and with the use of the calculator to find a rational approximation when such a radical is irrational. This section extends the study of radicals to include other roots and explains how to evaluate radicals using a graphing calculator. In Chapter 7, students will learn that nth roots can also be found using a fractional exponent. Radical expressions that have indexes of 2, 4, 6, or any even number are said to be even roots. Radical expressions that have indexes of 3, 5, 7, or any odd number are said to be odd roots. When the index is even, the radicand must be non-negative for the radical to be a real number. However, an odd root of a positive number is a positive number, and an odd root of a negative number is a negative number. Students should understand that the solution of x 2 5 49 is different from that of x 5 !49: x2 5 49 x 5 6 !49 x 5 67 x 5 !49 x57 There are two square roots of 49, and both are real numbers. x is the principal square root of 49. The solutions of the following equations are the same: x3 5 8 3 x5 ! 8 3 x5 ! 8 x52 x52 There are three cube roots of 8, but only one is a real number. When "x2 is evaluated for any non-zero real number, the sign of the answer is positive. For example, x 5 2: "x2 5 "22 5 !4 5 2 x 5 22: "x2 5 " (22) 2 5 !4 5 2 14580TM_C03.pgs 32 3/27/09 12:38 PM Page 32 Chapter 3 Recall that the absolute value of any real number x, or x, is also a positive number for any non-zero value of x. We can conclude that "x2 5 x for any real number x. Enrichment Activity 3-2: A Square-Root Algorithm presents a method that was used to approximate square-root radicals. The activity will help students to appreciate the power and convenience of a calculator. 3-3 SIMPLIFYING RADICALS When simplifying radicals, students often have difficulty finding the largest perfect-square factor. If this happens, encourage them to try factors that are not squares. Start dividing the radicand by its smallest natural-number factors and test to see whether the other factor is a perfect square. For example: 3-4 ADDING AND SUBTRACTING RADICALS Any two numbers that are elements of the same set of numbers can be added. In the set of real numbers, the sum of 5 and !2 is 5 1 !2. Note that the real number 5 1 !2 is in simplest form; it cannot be written as a monomial. If we wish to simplify a number or an algebraic expression that is written as the sum of two terms, we must be able to express each term as the product of a rational number and a common factor. Compare the addition of two fractions to the addition of like monomial terms and radicals. Point out how the distributive property is used in each case. 1. Fractions: 3 7 108 5 2 ? 54 1 27 5 17 (3) 1 17 (2) 5 17 (3 1 2) 5 17 (5) 5 57 108 5 3 ? 36 Since 54 is not a perfect square, 2 ? 54 is not a useful pair of factors. However, 3 ? 36 gives us the largest perfect-square factor, and !108 in simplest form is 6 !3. Point out that it is helpful to know the square of the integers from 1 to 25 and the cubes of the integers from 1 to 10. Fractional radicands will be reconsidered in Section 3-7 as an application of rationalizing the denominator. The approach shown in this section, utilizing the idea of equivalent fractions, enables students to simplify fractional radicands before the multiplication rule for radicals is introduced. Point out that to simplify #q , we multiply the radicand np qn21 qn21 , p q by which results in the rational denom- inator q. 2. Like monomials: 3a 1 2a 5 a(3) 1 a(2) 5 a(3 1 2) 5 a(5) 5 5a 3. Like radicals: 3 !2 1 2 !2 5 !2(3) 1 !2(2) 5 !2(3 1 2) 5 !2(5) 5 5 !2 4. Unlike radicals that can be simplified: !12 2 1 4 !3 5 !42 ? 3 1 4 !3 2 !3 2 1 5 1 1 4 !3 5 (1 1 4) !3 5 5 !3 In Example 1, the terms !12 and #13 can be simplified and combined; likewise, the terms !20 and 2!45 can be simplified and 14580TM_C03.pgs 3/27/09 12:38 PM Page 33 Real Numbers and Radicals combined. However, the resulting terms 73 !3 and 2!5 do not share a common radicand and cannot be combined. Similar procedures are used with subtraction. Students often make the mistake !a 1 b 5 !a 1 !b . A simple counter example is provided by !16 1 !9. !16 1 !9 5 4 1 3 5 7 !16 1 9 5 !25 5 5 and 7 2 5. Students may also use their calculators to check examples that do not contain perfect squares, such as !19 1 !31 2 !19 1 31. 3-5 MULTIPLYING RADICALS Emphasize that two radicals to be multiplied can be combined under a single radical sign only if they have the same index. Additionally, the multiplication is valid only if the radicals represent real numbers. Point out the use of the multiplication properties and compare the operations with similar operations involving variable quantities. Expression with variables: 3x ? 2y 5 3(x ? 2)y 5 3(2x)y 5 (3 ? 2)(xy) 5 6xy Associative prop. of mult. Commutative prop. of mult. Associative prop. of mult. Closure under mult. Expression with radicals: 3!2 ? 2 !5 5 3A !2 ? 2B !5 Associative prop. of mult. 5 3A2 !2B !5 Commutative prop. of mult. 5 (3 ? 2)A !2 ? !5B Associative prop. of mult. 5 6!10 Closure under mult. Call attention to Example 2, part c. In general, if a and b are rational numbers, then A !a 1 !bB A !a 2 !bB 5 a 2 b , a rational number. (Students may recognize the similarity to the method for factoring the difference 33 of two squares.) This method of obtaining a rational number is the basis for the work in Section 3-7. In Enrichment Activity 3-5: A Radical Sequence, students add and multiply radical expressions to investigate the properties of a sequence that is both Fibonacci-like and geometric. If 1, k, k 2 , . . . is a Fibonacci-like sequence, then 1 1 k 5 k2 or k2 2 k 2 1 5 0. Using the quadratic formula in Chapter 5, the solutions to the equation are 1 62 !5. This activity shows the relationship between the golden ratio and the Fibonacci sequence. 3-6 DIVIDING RADICALS Demonstrate that in a radical of the form #ba , the numerator and denominator of the fraction may be simplified separately or together. For example, !2 5 !25 5 52 !2 5 52 !4 !2 !2 25 5 or #50 8 5 #4 5 2 50 #8 5 !50 !8 Both approaches are shown in Examples 1 and 2. In Section 3-7, students will see that a third method can also be applied: 50 #8 5 5 5 5 !50 !8 ? !8 !8 !50 !8 8 !400 8 20 5 8 5 2 Students should practice the different methods to become fluent, but in future work they can choose the approach they find easiest. 3-7 RATIONALIZING A DENOMINATOR To rationalize a denominator is to remove all radicals from the denominator. Before calculators, denominators were rationalized 14580TM_C03.pgs 34 3/27/09 12:38 PM Page 34 Chapter 3 because it is easier to divide by an integral value than by a radical. Today we rationalize denominators to have a standard way of expressing values within the mathematics and science communities. If a denominator is composed of a single radical term, then to rationalize a denominator, multiply both the numerator and the denominator of the fraction by: • the denominator, or • a radical that will result in the radicand in the denominator having a power equal to the index. The second method is illustrated in Example 3. The first method could be used in this case, but the work is more tedious: 6 !8 6 !8 5 !8 ? !8 5 6 !8 8 5 6 ? 28!2 5 128!2 5 3 !2 2 When the denominator of the rational expression is a binomial that contains a radical, we rationalize the denominator by multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed. When a binomial is multiplied by its conjugate, the products of the outer and inner terms will sum to zero. In Example 4, a linear equation with irrational coefficients is solved. The equation is not a radical equation (Section 3-8) because there are no terms with a variable in the radicand. Exercises 36 and 38 present a higher level of difficulty. You may wish to assign these exercises as group work or use them as bonus questions on a test. 3-8 SOLVING RADICAL EQUATIONS When solving a radical equation, the equation derived by squaring both sides is not equiva- lent to the given equation. In other words, the derived equation does not always have the same solution as the original equation. If a number is a root of the original equation, that number must also be a root of the derived equation, but the converse of this statement is not true. It is helpful to demonstrate why an extraneous root occurs. In Example 3, the derived equation x2 2 2x 1 1 5 15 2 7x is the result of squaring both sides of the equation x 2 1 5 !15 2 7x . However, the same derived equation is also the result of squaring the equation 1 2 x 5 !15 2 7x. The roots of x2 2 2x 1 1 5 15 2 7x are 27 and 2; 2 is the root of x 2 1 5 !15 2 7x, and 27 is the root of 1 2 x 5 !15 2 7x. A common mistake made by students is to incorrectly square the left side of (x 2 1) 2 5 A !15 2 7xB 2 Some students will incorrectly write x2 2 1 5 15 2 7x or x 2 1 5 15 2 7x Remind them of the FOIL method and have them write out (x 2 1)2 5 (x 2 1)(x 2 1). You may wish to show the solution of an equation involving two radicals in which squaring both sides does not eliminate the radical at first. For example: !2x 1 3 5 !x 1 1 1 1 1st Squaring: 2x 1 3 5 x 1 1 1 2 !x 1 1 1 1 2x 1 3 5 x 1 2 1 2 !x 1 1 x 1 1 5 2 !x 1 1 2nd Squaring: x2 1 2x 1 1 5 4(x 1 1) x2 1 2x 1 1 5 4x 1 4 x2 2 2x 2 3 5 0 (x 2 3)(x 1 1) 5 0 x2350 x53 x1150 x 5 21 14580TM_C03.pgs 3/27/09 12:38 PM Page 35 Real Numbers and Radicals Since both roots make the given equation true, the solution set is {21, 3}. EXTENDED TASK For the Teacher: This extended task is designed to help students make connections among various branches of mathematics. Geometry, arithmetic operations, and algebra are all addressed in the task. Students are expected to know the following from previous work: 1. The relationship that the altitude drawn to the hypotenuse of a right triangle is the mean proportional between the segments formed on the hypotenuse. 35 2. The locus of the vertices of a right triangle with a given segment as hypotenuse is a circle with the given hypotenuse as a diameter. 3. The construction to find the midpoint of a line segment. 4. The construction of a line perpendicular to a given line at a given point on the line. 5. How to solve problems involving percent. 6. How to measure the length of a line segment to a stated accuracy. 7. How to convert fractions to decimals. 14580TM_C03.pgs 3/27/09 12:38 PM Page 36 Name Class Date ENRICHMENT ACTIVITY 3-2 A Square-Root Algorithm An algorithm is a process or a method used in calculation. Follow the steps of the algorithm shown in the example below to approximate the square root of 2,210. . "22 10 . STEP 1. Write the number, placing a decimal point after the number and above the number. Starting at the decimal point, separate the number into pairs of two digits. Treat any number with an odd number of digits as a number with a leading 0; for example, treat 729 as 07 29 . XX XX 4 STEP 2. Find the largest perfect square that is less than or equal to the first pair of digits. Write that square below the first group of digits and its square root above the digits. STEP 3. Subtract the square from the first pair of digits, and bring down the next pair of digits. STEP 4. Double the 4 above the radicand and write this value with a blank line next to it as shown. STEP 5. Find the largest single digit d so that 8d times d is less than or equal to 609. 87 3 7 5 609 88 3 8 5 704 ✘ (Too large) . "X 22 X 10 . 16 4 . "22 10 . 216 610 4 . "22 10 . 216 (8_) 610 4 7. "22 10 . 216 (87) 610 XX XX XX Write 7, the value of d, on top of the blank and above the pair X 10 . STEP 6. Calculate 87 3 7 and write the product below 610. Subtract and bring down the next pair of digits. (In this case the next pair of digits is 00 .) X 4 7. "22 10 .00 216 610 (87) 2609 100 XX X 4 7. STEP 7. Double the 47 above the radicand and write this value with a blank line next to it as shown. X.00 X "X 22 10 216 610 2609 (94_) 100 (87) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 37 Name Class Date 4 7. 0 STEP 8. Find the largest single digit d so that 94d times d is less than or equal to 609. X.00 X "X 22 10 216 610 2609 (940) 100 (87) 940 3 0 5 0 941 3 1 5 941 ✘ (Too large) Write 0, the value of d, on top of the blank and above the pair 00 . STEP 9. Calculate 940 3 0 and write the product below 100. Subtract and bring down the next pair of digits. X XX XX (87) (940) XX XX STEP 10. Double the 470 above the radicand to obtain 940. The digit d such that 9,40d 3 d is less than or equal to 10,000 is 1. (87) (940) (9401) To the nearest tenth, !2,210 < 47.0. Note: If at any time the difference is 0 and all pairs of digits of the original number have been used, the process is complete. In such a case, the exact square root has been found: 3 5. "1,225. 29 → 352 5 1,225 (65) 325 2325 0 XX Exercises In 1–4, find the exact square root of each given number using the square-root algorithm. 1. 3,249 2. 5,184 3. 8,281 4. 1,521 In 5–8, find the square root of each given number to two decimal places and round the result to the nearest tenth. 5. 7 6. 17 Copyright © 2009 by Amsco School Publications, Inc. 7. 35 8. 88 4 7. 0 "22 10 .00 00 216 610 2609 100 20 10000 4 7. 0 1 "22 10 .00 00 216 610 2609 100 20 10000 14580TM_C03.pgs 3/27/09 12:38 PM Page 38 Name Class Date ENRICHMENT ACTIVITY 3-5 A Radical Sequence A Fibonacci-like sequence is one in which each term is the sum of the previous two terms. A geometric sequence is one in which the ratio of consecutive terms (common ratio) is a non-zero constant. Can a sequence have both properties? Consider the sequence below: 1, 1 12 !5, 3 12 !5, . . . 1. Show that the sum of the first two terms is equal to the third term of the sequence. 2. Find the fourth term by adding the second and third terms. 3. Find the fifth and sixth terms of the sequence. Write your answers in simplest form. 4. Why is the sequence 1, r, r2, r3, . . . a geometric sequence? What is the common ratio? 5. In the sequence above, what is the ratio of the second term to the first? Call this ratio r. 6. Show that the third term of the sequence is r 2. 7. Show that the fourth term of the sequence is r 3. 8. Show that the fifth term of the sequence is r4. 9. Show that the sixth term of the sequence is r 5. 10. Use your calculator to find an approximate value of 1 12 !5. Use the library or Internet to research the meaning of this ratio. What is the name of this ratio? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 39 Name Class Date EXTENDED TASK Finding Square Roots Geometrically Our real number system is composed of two major subsets, the set of rational numbers and the set of irrational numbers. The word rational is derived from “ratio.” Therefore, a rational number is any number that can be expressed as the quotient, or ratio, of two integers. The rational numbers include all integers, zero, and all positive and negative fractions. Fractional rational numbers may be expressed as either terminating or non-terminating decimals. In contrast, an irrational number cannot be expressed as the ratio of two integers. Therefore, irrational numbers do not have exact values. They can be approximated by a rational number and that approximation can be stated as precisely as we choose. In a right triangle, the measure of the altitude drawn to the hypotenuse from the vertex of the right angle is the mean proportional between the measures of the two segments of the hypotenuse. Therefore, in right triangle ACD below, if DB is the altitude drawn to the hypotenuse, then (DB)2 5 (AB)(BC) or DB 5 !(AB)(BC). Therefore, if AB 5 1 and BC 5 3, then DB 5 !3. D A B C You can use this knowledge to construct the square root of any whole number. Exercise I Follow the steps given in the procedure below to construct the square root of 3. Use a straightedge and compass to make your constructions. 1. Draw line segment ABC, such that AB 5 1 inch and BC 5 3 inches. a. What is the sum of AB and BC? b. What is the product of AB and BC? 2. Find, by construction, the midpoint, M, of AC. 3. Construct the locus of the vertices of all right triangles that have AC as hypotenuse. 4. Construct a line perpendicular to ABC at B. Label the point D where the perpendicular intersects the locus in step 3. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 40 Name Class Date 5. Is DB 5 !3 inches? Explain your response. 1 6. Use a ruler to find the length of DB to the nearest 16 of an inch expressed as a decimal. 7. Use your calculator to find !3. 8. Find the percent of difference between your constructed !3 and the calculator value for !3. 9. What accounts for this difference in step 8? Exercise II 1. Use the same procedure to find the geometric representation of !5 and of !7. 1 2. Measure each line segment representation to the nearest 16 of an inch and express as a decimal. 3. Find the ratio !5 : !7. 4. Express !5 as a fraction with a rational denominator. !7 5. Use your calculator to find the value of the numerical expression you obtained in step 4. 6. How close is your result in step 3 to the value obtained in step 5? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 41 Name Class Date Algebra 2 and Trigonometry: Chapter Three Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–5, tell whether each number is rational or irrational. 1. 37 2. 0.14 3. p 1 6 3 4. !216 5. 0.01002000300004 . . . In 6–9, find and graph the solution set of each inequality. 6. x , 5 7. a 2 3 $ 7 8. 2x 2 6 1 4 # 12 9. 3x 1 1 1 2 $ 28 In 10–13, evaluate each expression. 10. !441 3 11. 2!264 81 12. #121 4 13. !0.0016 In 14–17, write each radical in simplest radical form. All variables represent positive numbers. 14. !180 15. "147b4 3 16. " 56a6 17. #25x5 72y 3 In 18–32, perform the indicated operation and simplify. 18. !80 1 !45 19. 4!24 1 !54 20. !75 2 !3 21. 26!50 1 4 !98 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 42 Name Class Date 22. A2 1 !3B A2 2 !3B 23. A5 1 !2B 2 24. 2!6A3!6B 25. 3!2A3 1 !2B 26. !2A !18 2 !8B 27. 4 1 2!20 28. 123 !18 !2 29. 3 12 !5 6 30. !5 2 !2 2 1 31. !2 !2 1 1 32. !x 42 y In 33–36, solve each equation and check. 33. !2a 2 3 5 5 34. x !7 5 6 2 x 35. !3b 1 1 5 b 2 1 36. 2x 5 2!3x 2 2 4 4 37. Determine whether # 59 is less than, greater than, or equal to !330. Justify your answer without the aid of a calculator. 38. The perimeter of a right triangle is 12 meters. The length of the longer leg is 3 times the length of the shorter leg. Find, to the nearest tenth, the measure of each side of the triangle. 39. The length of a rectangle is x and the width is !3x 1 1. If the width is 1 less than the length, what are the dimensions of the rectangle? Bonus: Each of two congruent circles drawn in the interior of a square is tangent to two adjacent sides of the square and to the other circle, as shown in the diagram. If the measure of a side of the square is 16, find the radius of each circle. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C03.pgs 3/27/09 12:38 PM Page 43 Name Class Date SAT Preparation Exercises (Chapter 3) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If (x 2 3) 2 "x2 2 6x 1 9 (A) 1 (D) 5 7. If 1 2 x , x, then (A) 212 , x , 0 (C) x . 12 (E) x , 22 or x . 12 5 2 and x . 0, then x 5 (B) 3 (E) 6 (C) 4 2. In a storage room for vegetables, the Fahrenheit temperature F is controlled so that it does not vary from 45° by more than 6°. Which of the following best expresses the possible range of temperatures? (A) F 2 45 # 6 (B) F 2 45 . 6 (C) F 2 45 5 6 (D) F 2 6 # 45 (E) F # 39 or F $ 51 3. If 0 , y , x, which statement must be true? (A) !x 2 !y 5 !x 2 y (B) !y 1 !y . !2y (C) x !y 5 y !x (D) x !x . y !y (E) !x 1 !y , !x 1 y 4. If m is a positive number and m2 5 7, then m3 is equal to (A) !7 (B) 3 !7 (C) 14 (D) 7 !7 (E) 21 5. How many solutions does the equation !x 1 10 5 4 !x have? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 6. If !x 1 6 5 x, then which of the following statements is true? I. x 5 3 is a root. II. x 5 9 is a root. III. The sum of the roots is 13. (A) I only (B) II only (C) I and II only (D) II and III only (E) I and III only Copyright © 2009 by Amsco School Publications, Inc. (B) 0 , x , 12 (D) x , 212 8. If 2 !x 1 3 5 8, then !4x 1 12 is (A) 1 (B) 5 (C) 8 (D) 13 (E) 15 9. 4 5 16 if 1 (A) x 5 !x 1 !x (B) x 5 !x 2 x2 1 (C) x 5 x2 1 !x (D) x 5 x 2 8 (E) 10. x 5 x 2 10 A E D C B Equilateral ADE has sides of !2 and AC 5 !3. What is the perimeter of DEBC? (A) 3 !3 2 2 !2 (B) 3 !3 2 2 !2 1 2 (C) 2 !3 2 3 !2 1 3 (D) 3 !3 2 !2 1 3 (E) !3 2 2 !2 1 3 11. If !x 1 !y 5 !10 and x 2 y 5 5 !2, then !x 2 !y is (A) !10 (B) 2 !10 (C) !5 (D) 2 !5 (E) 5 !2 14580TM_C03.pgs 3/27/09 12:38 PM Page 44 Name 12. Class B A Date 14. 15. If AB 5 !3 and BC 5 !5, then the area of circle O is (A) 2p (B) 2!2p (C) 4p (D) 8p (E) 4!2p 13. A R S is a “product-sum x 4 x 1 is a “product-sum tandem” if x equals (A) 1 (B) 2 (D) 4 (E) 5 (C) 3 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. If 4 !2x 1 1 5 20, then what is the value of x? B D 4 tandem” if R and S are (A) 2 and 6 (B) 4 and 4 (C) 2 and 2 (D) 2 and 4 (E) 1 and 7 C O 2 17. If x 2 12 # 4 and y 2 8 # 5, what is the greatest possible value of x 2 y? C The diameters AC and BD are perpendicular. If AC 5 2!3 and mA 5 mB 5 mC 5 mD 5 30, then the perimeter of the polygon is (A) 8 1 2 !3 (B) 8 1 4 !3 (C) 4 1 4 !3 (D) 16 (E) 12 Questions 14 and 15 refer to the following definition. A “product-sum tandem” is a figure like the one below in which A !PB A !QB 5 !R 1 !S. P Q For example, 3 6 R 8 2 S is a “product-sum tandem” because A !3BA !6B 5 !8 1 !2. Copyright © 2009 by Amsco School Publications, Inc. 18. What is the positive solution for x 5 3 1 !9 2 x? 3 19. If x 5 A !8 B 2, what is the value of x? 20. Evaluate A8 2 !2B A8 1 !2B . 21. If x 2 y 5 16 and !x 2 !y 5 2, determine the value of !x 1 !y. 22. The diagonal of a square is 10 !2 units. How many units long is the perimeter of the square? 23. Evaluate !12 1 !27 2 !75. 3 24. If x 5 " !729, what is x 1 5? 14580TM_C04.pgs 3/26/09 12:12 PM Page 45 CHAPTER 4 RELATIONS AND FUNCTIONS Aims CHAPTER OPENER • To identify relations and functions and determine the domain and range of a function. If the trucker must drive for 8 hours a day, then d 5 8r. When r 5 50, d 5 400. When r 5 60, d 5 480. The ratio dr is constant. • To use functional notation and evaluate functions for given values. • To find the composition of functions and the inverse of a function. • To perform transformations with functions. • To determine the center-radius form for the equation of a circle. • To understand direct and inverse variation. The concept of a function is basic to the study of advanced mathematics. This chapter presents formal definitions of relation and function and illustrates these ideas with mathematical and real-world examples. The Cartesian graph is the most useful tool for understanding the properties of different functions. Several familiar algebraic functions are reviewed in this chapter and more complex relationships are also presented. The key idea of function composition is explored and mastery of this topic is essential for future work in calculus. 400 50 5 480 60 5 8 As the rate of travel increases, the distance also increases. The variables are directly related. If a trucker must drive 600 miles a day, then 600 5 rt. When r 5 50, t 5 12. When r 5 60, t 5 10. The product rt is constant. 50 3 12 5 60 3 10 5 600 As the rate of travel increases, the time required to travel a fixed distance decreases. The variables are inversely related. A similar relationship occurs when a fixed amount of work must be completed. In the classic example, the more workers painting a house, the fewer number of hours required to complete the job. 4-1 RELATIONS AND FUNCTIONS When a relation is finite, all ordered pairs can be tested to describe the relation. When a relation is infinite, it is useful to use an equation, an inequality, or a graph to indicate how 45 14580TM_C04.pgs 46 3/26/09 12:12 PM Page 46 Chapter 4 ordered pairs of the relation may be found or how it can be determined whether or not a given ordered pair is a member of the relation. Every function is a relation. However, not every relation is a function. A many-to-one relation such as {(1, 5), (2, 5), (3, 5), (4, 5)} is a function, but a one-to-many relation such as {(1, 1), (1, 21), (4, 2), (4, 22)} is not a function. Ask students to suggest some real-world relations and decide if they are functions. For example, the relation that makes the correspondence between a child and its mother is a function, but the relation that makes the correspondence between a mother and her children is not (one mother can correspond to several children). Be sure students understand the definition of onto. A function is onto if each element of B is mapped by at least one element of the domain A. When the relation is finite, inspection of the ordered pairs will determine whether there are two pairs with the same first element. When the relation is infinite, it is not possible to list all of the ordered pairs for inspection. Therefore, the vertical line test is one useful way of determining whether a given relation is a function. Since the coordinates of all points on a vertical line have the same first element, a relation having more than one point on the same vertical line has more than one point with the same first element and therefore is not a function. When determining the domain of an algebraic function, it is helpful to recall that the sum, difference, or product of any two real numbers is a real number. Therefore, any function for which the function value is defined using only addition, subtraction, or multiplication can have as its domain the set of real numbers. The domain of any polynomial function, for example, is the set of real numbers. Often the domain can easily be determined by identifying the real numbers that must be excluded. If a function {(x, y) : y 5 f(x)} defines y in terms of x, it is necessary to eliminate from the domain the values of x that make: 1. the denominator of a fraction equal to 0. 2. the radicand of an even root less than 0. 4-2 FUNCTION NOTATION Six common function notations are given in this section. The notation y 5 f(x) is used to show that y is a function of the variable x or that the value of y depends on the value of x. For this reason, the x-value is referred to as the independent variable and the y-value is referred to as the dependent variable. Be sure students understand that the parentheses in the symbol f(x) do not indicate multiplication. For the function f(x) 5 2x 1 3, f(2) 5 7 or y 5 7. Note that f(2) 5 7 tells us both elements of the pair, (2, 7), but y 5 7 tells us just the second element. Explain that x is a “dummy” variable that can be replaced by any symbol or expression that can represent a number in the domain of the function. For example: f(x) 5 2x 1 3 f(a) 5 2a 1 3 f(a 1 2) 5 2(a 1 2) 1 3 5 2a 1 7 f(3x 2 1) 5 2(3x 2 1) 1 3 5 6x 1 1 You may wish to have students work Exercise 16 using the sales tax rate that applies to your locality. For Exercise 17, students should plan the size of the intervals they will use on the x-axis (number of muffins) and y-axis (profit) before they start to draw. 4-3 LINEAR FUNCTIONS AND DIRECT VARIATION Be sure students understand that all linear equations, other than equations of the form x 5 a, will be functions. The graph of x 5 2 is shown at the top of the next page. Note that it is not a function since it does not pass the vertical line test. Equations of the form y 5 a are functions since they pass the vertical line test. 14580TM_C04.pgs 3/26/09 12:12 PM Page 47 Relations and Functions y 1 4-4 y 1 O 1 x O 1 x One-to-one functions are important because they are the only functions with inverses (Section 4-8). A function is one-toone if there is exactly one input for every output. In other words, if two output values are the same, the two input-values must be the same. The function y 5 x3 is one-to-one; the function y 5 x3 2 x is not since the output 0, for example, could result from an input of 1 or an input of 21. The horizontal line test shows which functions are one-to-one. It is helpful to point out that any function with all even exponents cannot be one-to-one. Students should be familiar with transformations of linear functions from previous algebra courses. An alternative way of describing the appearance of the graph of af(x) from that stated in the text is that as |a| increases, the steepness of the graph increases. However, the terminology “stretching” and “compression” are consistent with the transformation approach. A direct variation y 5 kx is a relationship that indicates that one quantity is a multiple of the other. The graph of a direct variation is always a line through the origin. An equation of a direct variation is a special case of the slope-intercept form of a line, y 5 mx 1 b. When m 5 k and b 5 0, y 5 mx 1 b becomes y 5 kx. In Hands-On Activities 1 and 2, students explore transformations by working directly from given graphs. Have students work individually or in pairs for these activities, then lead a discussion of the work. Ask students to explain how they used the given graph to stretch the graph of f(2x), 2f(x), and so on. Point out that Exercises 12 and 14 in HandsOn Activity 2 have two possible answers. (Both 2f(x) and g(x) can be described as the graph of f(x) reflected in the x-axis or the y-axis.) 47 ABSOLUTE VALUE FUNCTIONS When x is positive or zero, the absolute value function looks like the graph of y 5 x. When x is negative, the absolute value function looks like the graph of y 5 2x. Point out that absolute value equations of the form x 5 ay 1 b are not functions. For example, the equation x 5 y contains two points whose coordinates are (3, 3) and (3, 23), indicating that an x-value corresponds to more than one y-value. Be sure students understand the graphing method used in Example 2. Ask them what graphs they would draw to solve 2x 1 4 $ 8. (y 5 2x 1 4, y 5 8) Exercises 13–16 have students explore the transformations of the graph of y 5 x. If students understand the relationship between the algebraic form and the graphic form, they will be able to draw many absolute value graphs directly from the given equation without having to plot ordered pairs. Similarly, they will be able to give the equation of an absolute value graph by inspection. These skills will provide considerable advantage for the SAT and other standardized tests. 4-5 POLYNOMIAL FUNCTIONS Emphasize that the coefficients an, an21, . . . represent real numbers; the coefficient an is non-zero, and the exponents of the variable terms are all whole numbers. The degree of a polynomial is determined by the greatest exponent of the variable terms. The section reviews the properties of second-degree polynomials; that is, quadratic functions. Students should understand that the graphs of y 5 x2, y 5 (x 2 h)2, y 5 x2 1 k all have the same shape. The difference is their position. When the equation of a parabola is expressed in the vertex form y 5 (x 2 h)2 1 k, the vertex is (h, k) and the equation of the axis of symmetry is x 5 h. As the values of h and k change, the graph of y 5 (x 2 h)2 1 k is the graph of y 5 x2 translated |h| units left or right 14580TM_C04.pgs 48 3/26/09 12:12 PM Page 48 Chapter 4 and k units up or down. These transformations are further explored in Exercises 21 and 22. The x-intercept of a polynomial function is the x-coordinate of the point where the graph crosses the x-axis. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. A polynomial function has exactly one y-intercept, but it can have several x-intercepts depending on the degree of the sign polynomial. The x-intercepts are also called roots, zeros, or solutions of the polynomial, depending on the context of the problem. The x-intercepts are often where the graph goes from positive (above the x-axis) to negative (below the x-axis) or vice versa. Evendegree functions of the form y 5 x n are tangent to the x-axis at the origin. Other evendegree functions may or may not intersect the x-axis, depending on its location in the coordinate plane. An odd-degree function always crosses the x-axis at least once, which means that an odd-degree function always has at least one real root. In the section, it is pointed out that if a is a root of f(x), then (x 2 a) is a factor of f(x); the converse is also true. A turning point is where the graph changes direction. The maximum number of turning points for a function of degree n is n 2 1. The maximum number of x intercepts for a function of degree n is n, but there may be fewer intercepts than n depending on the function. Have students graph y 5 x8 1 1 on their calculators. They will see the graph has no intercepts and only one turning point. If y 5 f(x) is a polynomial function such that f(a) , 0 and f(b) . 0, then the function has at least one real root between a and b. Intuitively, students should understand that the only way for a polynomial function to change from positive to negative is to go through 0. Polynomial functions are continuous; the domain of any polynomial is the set of real numbers, so nothing is omitted that would allow a jump. Although it is easy to locate the real roots of higher-degree polynomial functions using a calculator, students should be aware that for many years, mathematicians such as Descartes worked to establish methods for identifying the roots of polynomial functions. Some of their conclusions will be explored in the Enrichment Activities. The calculator can determine rational roots and very close approximations of irrational roots because of the work done by these mathematicians. Additional concepts about roots, factors, and remainders will be considered in Chapter 5 in the context of quadratic functions. Enrichment Activity 4-5: The Method of Finite Differences considers how a polynomial function can be constructed if we are given the values of the function for consecutive values of x. 4-6 THE ALGEBRA OF FUNCTIONS Many functions arise as combinations of other functions. For example, suppose an airplane is flying over the Grand Canyon. Then at any time, t, the height, h(t), of the airplane above the bottom of the canyon is the sum of the height, f(t), of the airplane above the rim of the canyon and the depth, g(t), of the canyon directly below the airplane and rim. h(t) 5 f(t) 1 g(t) Let f and g be functions. The domains of f 1 g, f 2 g, and fg consist of those values of x where both f(x) and g(x) are defined, that is, the intersection of both domains. Because division by 0 is excluded, the quotient gf is the function whose domain consists of all numbers x in the domains of both f and g where g(x) 0. Emphasize that the domain of a combined function must be determined from the separate domains of f and g, rather than from the rule for the combined function. Consider this example: Let f(x) 5 "4 2 x2 and g(x) 5 !x 2 1. The domain of f is 22 # x # 2 and the domain of g is x $ 1. Therefore, the domain of the 14580TM_C04.pgs 3/26/09 12:12 PM Page 49 Relations and Functions product fg is the set of numbers in both domains, or 1 # x # 2. The rule for fg is: fg(x) 5 f(x)g(x) 5 "4 2 x2"x 2 1 5 "(4 2 x2)(x 2 1) (1 # x # 2) However, note that the expression "(4 2 x2)(x 2 1) is also meaningful for 2` , x # 22 . This is true because (4 2 x2)(x 2 1) $ 0 for x # 22. The domain must be based on the original functions, not the combined function. The same is true for f 1 g, f 2 g, and gf . Note that any polynomial g(x) 5 anxn 1 an 2 1xn 2 1 1 c 1 a1x 1 a0 can be thought of as a combination of constant functions and the identity function I(x) 5 x (see Section 4-8). g(x) 5 a fI(x)g n 1 a fI(x)g n21 1 c n n21 1 a1I(x) 1 a0 or g 5 anIn 1 an21In21 1 c 1 a1I 1 a0 In the last part of the section, transformations in the plane are interpreted by applying function arithmetic. 4-7 COMPOSITION OF FUNCTIONS A new function can be formed by using the output of one function as the input to a second function. This is called function composition. Two ways of writing the composition of two functions are given in the text. The symbol f(g(x)) indicates more clearly than (f + g)(x) the order in which the functions are to be evaluated. (In this case, first g, then f.) Emphasize to students that we start with a value of x and proceed to the left, first evaluating the function written closer to x. With the notation f(g(x)), indicate that we work from the inside out. Students should be aware of domain restrictions inherent in function composition. Consider, for example, the composition f(g(x)), where f(x) 5 x2 and g(x) 5 !x. Since f(g(x)) 5 fA !xB 5 A !xB 5 x 2 49 students might erroneously conclude that the domain of the composition is the set of real numbers. It is, in fact, the set of non-negative real numbers. The domain of f + g must be a subset of the domain of g. In this example, the domain of the composition is the same as the domain of g. As illustrated in Examples 1 and 3, function composition is not a commutative operation; that is, it is not always true that f(g(x)) 5 g(f(x)). Use Enrichment Activity 4-7: The Difference Quotient to prepare students for later work finding derivatives. The difference quotient can be thought of as the composition of some function f(x) and another function g(x) 5 x 1 h. 4-8 INVERSE FUNCTIONS Be sure that students understand that f 21 denotes the inverse of the function f and does not mean 1f. To reinforce students’ understanding of the meaning of an inverse function, use different one-to-one functions to illustrate the concept. Explain that the function f(x) 5 2x 2 4 is formed by multiplying the variable x by 2 and then subtracting 4. The inverse function is found by performing the inverse operations, in the opposite order. The inverse f21(x) is found by adding 4 (the inverse of subtracting 4) to x and then dividing the result by 2 (the inverse of multiplying by 2): 4 5 12x 1 2 f21 (x) 5 x 1 2 The composition of a function and its inverse function is the identity function, I(x) 5 x. Show this by applying the composition using specific values: f21 (f(x)) 5 I(x) 5 x f f21 h 12 (2) 1 2 5 3 f f21 h 12 (26) 1 2 5 21 f f21 h 12 (24) 1 2 5 0 3 h 2 21 h 26 0 h 24 f21 x h 2x 2 4 h 12 (2x 2 4) 1 2 5 x f 14580TM_C04.pgs 50 3/26/09 12:12 PM Page 50 Chapter 4 f(f21 (x)) 5 I(x) 5 x 21 f f h 5 h 2(5) 2 4 5 6 f21 f h 0 h 2(0) 2 4 5 24 f21 f h 2 h 2(2) 2 4 5 0 21 f f h 12x 1 2 h 2 A 12x 1 2 B 2 4 5 x The sets of ordered pairs f 5 {(23, 9), (22, 4), (21, 1), (0, 0), (1, 1), (2, 4), (3, 9)} and g 5 {9, 23), (4, 22), (1, 21), (0, 0), (1, 1), (4, 2), (9, 3)} are inverse relations. Two relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a). Have students explain in their own words the difference between inverse functions and inverse relations. Students are already familiar with the formula for converting Fahrenheit temperatures to Celsius, C(x) 5 59 (x 2 32) , and Celsius to Fahrenheit, F(x) 5 95x 1 32. Have students verify that these are inverse functions. 6 24 0 x 4-9 CIRCLES The distance and midpoint formulas are reviewed in this section since they must be applied to problems involving circles. Call attention to the fact that the equation of a circle can be written if any one of the following are given: 1. the coordinates of the center and length of the radius. 2. the coordinates of the center and the coordinates of one point on the circle. 3. the coordinates of the endpoints of a diameter. Next, they must complete the square twice, once for each variable. As shown, work first with the variable x, then with the variable y. Make sure that the same constants are added to both sides of the equation each time. For Exercise 28b, students must determine if the height of the arch at a distance of 3 feet from the center is at least 6 feet. 4-10 INVERSE VARIATION When two quantities vary inversely, it means that as one quantity increases, the other quantity decreases, and vice versa. Refer to the situation described in the chapter opener. When rate of travel is constant, distance varies directly as time. When the distance is constant, rate varies inversely as time. You may wish to refer to some of the relationships in Section 4-3. By keeping a different quantity constant, many of the same examples that were used to illustrate direct variation can be used to illustrate inverse variation. x y The proportion x1 5 y2 is only one of sev2 1 eral true proportions that can be formed given x1 y1 5 x2 y2. Ask students to suggest others. Note that in many applications of inverse variation, only positive values are acceptable replacements. However, when graphing the function xy 5 c, which expresses inverse variation, both positive and negative values are used. The resulting hyperbola is called a rectangular hyperbola. Hands-On Activity Instructions: 1. Draw the graph of xy 5 6. Example 3 illustrates the method for writing the center-radius form of the equation of a circle given the standard form. Suggest that students should begin by placing all terms containing like variables together: 2. Fold the paper so that one branch of the hyperbola coincides with the other. Open the paper. x2 1 3x 1 y2 2 4y 2 14 5 0 4. On a separate piece of graph paper, draw the graph of xy 5 26 and repeat the directions. Then, move the constant to the right side of the equation: x2 1 3x 1 y2 2 4y 5 14 3. Fold the paper again so that each branch coincides with itself. 14580TM_C04.pgs 3/26/09 12:12 PM Page 51 Relations and Functions Discoveries: 1. The folds are lines of symmetry for the graphs. The equations of these lines are y 5 x and y 5 2x. 2. When the constant is positive and the axis of symmetry is y 5 2x, the image of a point on one branch of the hyperbola is a point on the other branch. 3. When the constant is positive and the axis of symmetry is y 5 x, the image of a point on either branch of the hyperbola is a point on the same branch. 4. When the constant is negative and the axis of symmetry is y 5 x, the image of a point on one branch of the hyperbola is a point on the other branch. 5. When the constant is negative and the axis of symmetry is y 5 2x, the image of a point on either branch of the hyperbola is a point on the same branch. 6. The graphs of both xy 5 6 and xy 5 26 have point symmetry. The origin is the point of symmetry. 51 EXTENDED TASK For the Teacher: This activity is intended to help students develop an understanding of inverse variation and see that the hyperbola is the graphic representation of an inverse variation. Have students work in groups of three on this activity. One student can pull the spring balance, a second can observe and read the measurements, and the third can record the information. Accuracy and neatness should be encouraged throughout the activity. You may wish to do this activity in connection with the science teacher at the same time the science class is studying the concept of the balance. Materials for this inquiry might be obtained from the science teacher or from an elementary school teacher. 14580TM_C04.pgs 3/26/09 12:12 PM Page 52 Name Class Date ENRICHMENT ACTIVITY 4-5 The Method of Finite Differences If you are given the function f(x) 5 x2 1 2x 1 4, you can find the values of the function for different values of x: f(1) 5 1 1 2 1 4 5 7 f(2) 5 4 1 4 1 4 5 12 f(3) 5 9 1 6 1 4 5 19 f(4) 5 16 1 8 1 4 5 28 f(5) 5 25 1 10 1 4 5 39 Suppose you are given the values of a polynomial function, but not the function itself. The method of finite differences can be used to construct the original function. Follow these steps: 1. Write the function values with some space between values. 2. Find the differences between successive terms. This is called the first difference. If these differences are not all equal to the same value, repeat to find the next set of differences. 12 7 19 7 5 2 28 9 2 39 11 2 first difference constant 3. Stop when the differences are constant. In this case, the second difference is constant. The function is of the same degree as the constant difference, so the function we are looking for is a second-degree function of the form: f(x) 5 ax2 1 bx 1 c If constant differences did not appear until the third line, then we would be finding a third-degree function. 4. Use the given information to write three equations in three unknowns: a, b, and c. Since f(1) 5 7, a 1 b 1 c 5 7 (I.) f(2) 5 12, 4a 1 2b 1 c 5 12 (II.) f(3) 5 19, 9a 1 3b 1 c 5 19 (III.) The number of equations is one more than the degree of the function. For example, if the function were a third-degree polynomial, we would need to solve a system of four equations in four unknowns. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 53 Name Class Date 5. Solve the system using any method. For example, using elimination: II–I. 4a 1 2b 1 c 5 12 2 a1 b1c57 3a 1 b 55 So III–II. 9a 1 3b 1 c 5 19 2 4a 1 2b 1 c 5 12 5a 1 b 57 5a 1 b 5 7 2 3a 1 b 5 5 2a 5 2 a51 Substituting the value for a in the equations above gives: b52 c54 The function is f(x) 5 x2 1 2x 1 4, which we know is correct. Note that to use the method, the given function values must be successive terms. In 1–4, the values given correspond to f(1), f(2), f(3), f(4), and f(5) of some polynomial function. Use the method of finite differences to find the function. 1. 6, 6, 8, 12, 18 2. 4, 15, 30, 49, 72 3. 216, 211, 0, 17, 40 4. 1, 7, 21, 49, 97 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 54 Name Class Date ENRICHMENT ACTIVITY 4-7 The Difference Quotient The expression below is called the difference quotient. f(x 1 h) 2 f(x) h The difference quotient is the basis of an important concept in calculus. Graphically, the difference quotient represents the slope of a secant line for the function f(x). y e line f(x 1 h) in tl en f(x 1 h) 2 f(x) h ng 5 ta change in f(x) change in x sec ant f(x) 5 x 2 f(x) x x1h It is reasonable to think that as h gets smaller and smaller, the slope of the secant line approaches the slope of the line tangent to the curve at f(x). 1. a. Form the difference quotient for f(x) 5 x2 and simplify it. b. Evaluate the difference quotient if h 5 0. 2. a. Write the expression in simplest form: (x 1 h)3 b. Form the difference quotient for f(x) 5 x3 and simplify it. c. Evaluate the difference quotient if h 5 0. 3. a. Write the expression in simplest form: (x 1 h)4 b. Form the difference quotient for f(x) 5 x4 and simplify it. c. Evaluate the difference quotient if h 5 0. 4. a. Write the expression in simplest form: (x 1 h)5 b. Form the difference quotient for f(x) 5 x5. c. Evaluate the difference quotient if h 5 0. Copyright © 2009 by Amsco School Publications, Inc. x 14580TM_C04.pgs 3/26/09 12:12 PM Page 55 Name Class 5. a. Summarize your results by completing the table below. Function Difference Quotient when h 5 0 f(x) 5 x2 f(x) 5 x3 f(x) 5 x4 f(x) 5 x5 b. Describe any patterns you observe in the table. 6. Predict the difference quotient when h 5 0 for each function: a. f(x) 5 x6 b. f(x) 5 x9 c. f(x) 5 xn Copyright © 2009 by Amsco School Publications, Inc. Date 14580TM_C04.pgs 3/26/09 12:12 PM Page 56 Name Class Date EXTENDED TASK The Inverse Variation Hyperbola This extended task is designed to help you understand the concept of inverse variation by having you carry out an experiment that illustrates this type of variation. You should work in groups of three for this task. You will need the following materials: 50 100 150 200 250 • a numbered balance (see diagram) containing pegs equally spaced from the fulcrum from which weights can be suspended • several weights (for example, 50 g, 200 g, 500 g) • a spring balance Activity 1 1. Measure the distance, in centimeters, between pegs on the numbered balance and between the fulcrum and the first peg. 2. Suspend a weight of any size on one of the pegs to the left of the fulcrum, and determine its distance from the fulcrum. 3. Place your spring balance on the first peg to the right of the fulcrum, and exert a force on the spring sufficient to balance the weight. 4. Repeat this procedure for each peg to the right of the fulcrum on your numbered balance. 5. Organize and record your data in a table, showing the force required to balance the given weight and the distance from the fulcrum for each peg to the right of the fulcrum. 6. Examine your data to determine the relationship between the distance from the fulcrum and the force you exerted on the spring in order to balance the given weight. Activity 2 1. Repeat this entire experiment using a different weight on the left-hand side of the numbered balance. You may place the new weight on the same peg or on a different one. 2. Does a similar relationship exist? Activity 3 State in words and then write an algebraic expression for the relationship that you found in these two experiments. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 57 Name Class Date Activity 4 1. On the same set of axes, construct a graph for each set of data you collected. 2. Describe the curves that you obtained in step 1. Activity 5 1. Using the construction in Activity 4, reflect your curves over the line y 5 2x. 2. Describe in writing the reflected curves drawn for part 1. 3. What is the equation for the object and image curves? This task illustrates inverse variation. In your own words, define inverse variation and summarize, in writing, what you have learned from performing this task. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 58 Name Class Date Algebra 2 and Trigonometry: Chapter Four Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–5, in each case: a. State the domain of the relation. b. State the range of the relation. c. State whether the indicated set of ordered pairs defines y as a function of x. If not, explain. 1. {(22, 3), (21, 2), (0, 1), (1, 2), (2, 3)} 2. x2 1 y2 5 9 3. y 5 !9 2 x 4. y 3 3 x O 5. y 1 O 1 x 6. If f(x) 5 3x2 2 2x, find: a. f(21) b. f(5) 8. If h(x) 5 7. If g(x) 5 2x 12 1, find: a. g(0) b. g(23) 1 4x 1 1, what real number is not an element of the domain of h? 9. A constant function is defined by the equation y 5 26. What are the domain and range of this function? 10. x varies directly as y. If x is 6 when y is 24, find x when y is 40. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 59 Name Class Date 11. The cost of roses, c, is directly proportional to the number of roses purchased, r. One rose costs three dollars. a. Write an equation for the relationship described. b. Is the relationship a direct variation? c. Find the cost of nine roses. 12. Let f(x) 5 8x. The graph of g(x) is the graph of f shifted 4 units down and 6 units to the right. Write an expression for g(x). 13. A polynomial function of degree three intersects the x-axis at (26, 0), (22, 0), and (3, 0) and intersects the y-axis at (0, 236). If y 5 p(x), find p(x). 14. If f(x) 5 x2 1 x 2 6 and g(x) 5 x 2 2, find the following: a. (f 1 g)(x) b. (g 2 f)(x) c. (fg)(21) 15. If f(x) 5 x 1 1 and g(x) 5 !x, find each of the following: a. f(g(9)) b. g(f(9)) c. What is the smallest integer in the domain of g(f(x))? 16. If f(x) 5 x2 and g(x) 5 3x 2 1, find the rule for each composition: a. f + g(x) b. g + f(x) 1 17. If r(x) 5 2x 2 1 and s(x) 5 x 1 2 , find each of the following: a. r(s(3)) b. s(r(3)) In 18–20, write the inverse function in the form y 5 mx 1 b. 18. y 5 2x 2 1 19. y 5 x 1 7 20. y 5 5 2 12x 21. Assume that f(x) has an inverse. a. If the graph of f(x) lies in the first quadrant, in which quadrant does the graph of f21 (x) lie? b. If the graph of f(x) lies in the second quadrant, in which quadrant does the graph of f21(x) lie? 22. For each of the following, determine if the given function has an inverse. a. f(x) 5 {(22, 24), (3, 27), (5, 3), (26, 0)} b. f(x) 5 {(24, 2), (5, 3), (0, 2), (3, 7)} Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 60 Name Class Date 23. a. Find the center-radius form of the circle whose equation is x2 1 6x 1 y2 2 2y 1 6 5 0. b. Determine the center and radius of the circle. 24. Write an equation for a circle that has its center at (24, 3) and passes through (21, 7). 25. If y varies inversely as x and y 5 4 when x 5 12, find y when x 5 20. In 26–29, draw each required graph on a separate piece of graph paper. 26. Use a graph to solve the following: a. 2x 2 1 5 5 b. 2x 2 1 , 5 c. 2x 2 1 . 5 27. a. Graph y 5 x2 2 5x 1 4. b. Use the graph to find the solution set of x2 2 5x 1 4 $ 0. 28. a. Draw the graph of y 5 x2 1 2. b. On the same set of axes, draw the reflection in the line y 5 x of y 5 x2 1 2. c. Is the reflection drawn in part b a function? 29. a. Write an equation for the relationship between x and y if y varies inversely as x and y 5 3 when x 5 2. b. Sketch the graph. Bonus: For what values of x does f(x) 5 f21(x) if f(x) 5 x 21 1.5? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C04.pgs 3/26/09 12:12 PM Page 61 Name Class Date SAT Preparation Exercises (Chapter 4) I. MULTIPLE CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If f is a one-to-one function, f(g(4)) 5 8 and f(4) 5 8, then g(4) equals (A) 1 (B) 2 (D) 8 (E) 16 (C) 4 x2 2 4 2. For which values of x is f(x) 5 1 2 x 2 5 not defined? (A) 1 only (B) 22 and 2 (C) 21 and 4 (D) 24 and 6 (E) 26 and 5 3. y 1 O 1 x If the solid curve is the graph of f(x) 5 |x|, then the broken curve could be the graph of the function (A) g(x) 5 x 1 3 (B) g(x) 5 x 2 3 (C) g(x) 5 x 2 3 (D) g(x) 5 x 1 3 (E) g(x) 5 3x 4. If f(x) 5 2x2 2 x2 , which has the smallest value? (A) f(22) (B) f(21) (C) f A 212 B (E) f(1) (D) f A 12 B 1 5. If f(x) 5 x 2 2 and g(x) 5 2x 1 1, then f(g(3)) 2 g(f(3)) equals (A) 0 (B) 3 (D) 5 (E) 7 (C) 4 Copyright © 2009 by Amsco School Publications, Inc. 6. If f(x) 5 x 1 1, then f(f(f(x))) equals (A) x 1 3 (B) 3x 1 3 (C) x3 1 3 (D) 3x 1 1 (E) x3 1 1 7. A circle is tangent to the y-axis and has its center at (28, 5). The equation of the circle is (A) x2 1 y2 5 25 (B) (x 1 8)2 1 (y 2 5)2 5 64 (C) (x 2 8)2 1 y2 5 25 (D) (x 2 8)2 1 (y 1 5)2 5 64 (E) x2 1 64 1 y2 1 25 5 0 8. If f 5 {(22, 6), (21, 1), (0, 8), (1, 21), (2, 4)}, which statement is true? (A) f(1) 5 f(21) (B) f(0) 5 f(22) 1 f(2) (C) 2f(2) 5 f(0) (D) f(f(21)) 5 f(f(1)) (E) f(22) 5 f(0) 2 f(2) 9. If f(x) 5 f(2x) for all x in the domain, which equation could define f? (A) f(x) 5 2x (B) f(x) 5 x2 2 5 (C) f(x) 5 3x 1 1 (D) f(x) 5 x3 1 x (E) f(x) 5 x2 2 x3 10. If f(x) 5 3x 2 5 and g(x) 5 x 1 10, which of the following is equivalent to 3x 1 25? (A) f(x) 1 g(x) (B) f(x) 2 g(x) (C) 6g(x) 2 f(x) (D) f(g(x)) (E) g(f(x)) 11. If f(x) 5 3x and g(x) 5 5x 1 1, what is g(f(4))? (A) 61 (B) 63 (D) 241 (E) 252 (C) 83 14580TM_C04.pgs 3/26/09 12:12 PM Page 62 Name Class 12. II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. y C B E 1 Date O 1 16. If f(x) 5 16 2 x2 and the domain is 24 # x # 5, what is the largest value in the range? x D 17. What is the sum of the coordinates of the center of the circle whose equation is A x2 1 y2 2 6x 2 10y 1 30 5 0? In the figure above, which lettered point, other than the origin, lies in the interior of the circle x2 1 y2 5 16? (A) A (B) B (D) D (E) E (C) C 13. If it takes 4 workers 6 hours each to pave a road, how many hours will it take 12 workers to complete the same job? (A) 2 (B) 3 (D) 8 (E) 18 14. (C) 6 y 18. If f 5 {(0, 2), (2, 4), (4, 2)}, what is the value of f(f(f(0))) 1 f(f(f(2)))? 19. If f(x) 5 2 !x, what is the value of f(2) ? f(18) ? 20. If 4 painters can paint a house in 9 days, how many days would it take 6 painters to paint the house? 21. If f(x) 5 3x 2 4, what is the value of f21(11)? 22. If f(x) 5 x2 1 x3 and g(x) 5 !x, then f(g(0.01)) is equal to what number? 23. If f(x) 5 2x 1 1 and 3 2 f(x) 5 x, what is the value of x? 1O 1 24. y f(x) x g(x) 1O 1 x Let f(x) be defined by the graph above. If f(24) 1 f(2) 5 f(k), then k is equal to (A) 23 (B) 21 (D) 1 (E) 3 (C) 0 15. If f(x) 5 4x 2 2, then the inverse function f21(x) is (A) x4 1 2 (B) x 1 12 2 (C) x 1 4 (D) x 2 12 2 (E) x 2 4 Copyright © 2009 by Amsco School Publications, Inc. Let f(x) and g(x) be defined by the graphs above. What is the value of 3f(g(3))? 14580TM_C05.pgs 3/26/09 12:12 PM Page 63 CHAPTER 5 QUADRATIC FUNCTIONS AND COMPLEX NUMBERS Aims • To use the discriminant to determine the nature of the roots of a quadratic equation. • To solve quadratic equations using the quadratic formula. • To solve linear-quadratic systems. • To define the imaginary unit i and simplify powers of i. • To add, subtract, multiply, and divide complex numbers. • To apply the formulas for the sum and product of the roots of a quadratic equation. • To solve higher-order polynomial equations using factoring and/or quadratic form. This chapter introduces a new number, !21 or i, and the set of complex numbers that can be formed using i. Students learn to solve quadratic equations using the completing-thesquare procedure or the quadratic formula. The discriminant provides information about the roots of a quadratic equation and, if the roots are known, an equation with those roots can be written. Quadratic methods can be applied to higher-order polynomial equations that have special forms. The chapter opener leads into a discussion of the procedure for completing the square and the derivation of the quadratic formula. 5-1 REAL ROOTS OF A QUADRATIC EQUATION Students have learned that if x2 5 a, where a is a real number, then x 5 6 !a. This property can be used to solve simple quadratic equations such as x2 5 2. The purpose of the completing-the-square procedure is to transform a given trinomial into an equivalent expression that is the square of a binomial so that it is possible to take the square root of both sides of the equation. Emphasize that the coefficient of the squared term must be 1 before the steps for completing the square are applied. This section relates the roots of a quadratic equation, ax2 1 bx 1 c 5 0, to the points where the graph of y 5 ax2 1 bx 1 c intersects the xaxis. In the coordinate plane, every point represents a pair of real numbers. Therefore, it is possible to locate graphically only real roots. The form y 5 a(x 2 h)2 1 k is the most useful way of writing a quadratic function so that it may be graphed. Students should recognize that x 5 h is the equation of the axis of symmetry for the parabola and (h, k) are the coordinates of the vertex. Note that if k is positive and the parabola opens upward, 63 14580TM_C05.pgs 64 3/26/09 12:12 PM Page 64 Chapter 5 the graph cannot intersect the x-axis, so a(x 2 h)2 1 k 5 0 has no real roots. In Example 4, be sure students observe that since 1 was subtracted inside the parentheses, the 1 must be multiplied by the coefficient outside the parentheses, 2, when it is separated from the trinomial that is the perfect square binomial. In Exercise 37, it may be necessary to remind students that consecutive odd integers can be represented as 2n 1 1 and 2n 1 3. Students must remember to divide to make the coefficient of the n2 term equal to 1 before completing the square. 5-2 THE QUADRATIC FORMULA Emphasize that only quadratic equations with rational roots can be solved by factoring in the set of integers. However, any quadratic equation can be solved by using the quadratic formula. Even if a quadratic is factorable, in cases where the numbers are large and there are many possible factor pairs, it is often preferable to simply use the formula. When all the coefficients of a quadratic equation have a common factor, divide both sides of the equation by the common factor before using the quadratic formula. Calculation is greatly simplified with smaller values of a, b, and c. Remind students to be careful when they use the quadratic formula. The entire numerator of the quadratic formula must be divided by 2a. Only when both terms in the numerator and the denominator have a common factor may that common factor be divided out. In Example 1, point out that 2 62 !6 are the exact roots. Students may use their calculators to find rational approximations of these roots; the roots are 2.225 and 20.225 to the nearest thousandth. In Exercise 25, students explore a graphical representation of the quadratic formula. 5-3 THE DISCRIMINANT If a, b, and c, are rational numbers, the roots of the equation ax2 1 bx 1 c 5 0 are determined by the value of the discriminant, b2 2 4ac, according to the table in the text. If a, b, and c are irrational numbers, however, the table does not apply. For example, the equation x2 2 2 !2x 1 1 5 0 has an irrational coefficient. Although the discriminant is 4, a positive perfect square, the roots of the equation are not rational, but are the irrational numbers !2 6 1. If a, b, and c are imaginary numbers, the rules again do not apply. For example, the discriminant for the equation x2 2 4ix 2 5 5 0 is 4, but the roots of the equation are the complex numbers 2i 6 1. The graphs in this section illustrate the relationship between the nature of the roots determined by the discriminant and the number of times the graph of the related function intersects the x-axis. The example below shows that the same parabola can be used to solve many different equations that have real roots or to decide that the equation has no real roots. Example: Sketch the graph of y 5 2x2 2 2x 1 3, and from the graph determine the roots of the following equations: a. 2x2 2 2x 1 3 5 0 b. 2x2 2 2x 2 1 5 0 c. 2x2 2 2x 1 8 5 0 d. 2x2 2 2x 2 2 5 0 y55 y y54 y50 O x y 5 25 a. The equation 2x2 2 2x 1 3 5 0 can be expressed as the intersection of the parabola y 5 2x2 2 2x 1 3 and the line y 5 0. Answer: The roots are 23 and 1. b. The equation 2x2 2 2x 2 1 5 0 can be expressed as 2x2 2 2x 1 3 5 4 and, thus, can be expressed as the intersection of the parabola y 5 2x2 2 2x 1 3 and the line y 5 4. Answer: The root is 21 (two equal roots). 14580TM_C05.pgs 3/26/09 12:12 PM Page 65 Quadratic Functions and Complex Numbers c. The equation 2x2 2 2x 1 8 5 0 can be expressed as 2x2 2 2x 1 3 5 25 and, thus, can be expressed as the intersection of the parabola y 5 2x2 2 2x 1 3 and the line y 5 25. Answer: The roots are 24 and 2. d. The equation 2x2 2 2x 2 2 5 0 can be expressed as 2x2 2 2x 1 3 5 5 and, thus, can be expressed as the intersection of the parabola y 5 2x2 2 2x 1 3 and the line y 5 5. There are no points of intersection because there are no real roots. Answer: There are no real roots. The roots are imaginary. 65 number whose square is negative, we must define a set of numbers to which the terms positive and negative do not apply, that is, a set of numbers that are not part of the real number system. The set of imaginary numbers is such a set. Call attention to the way in which multiplication is defined for imaginary numbers. For example, !216 ? !24 2 !64; that is: !a ? !b 2 !ab if a , 0 or b , 0 !216 5 !16 ? !21 5 4i But !24 5 !4 ? !21 5 2i and Thus, !216 ? !24 5 4i ? 2i 5 8i2 5 28 In general: 5-4 THE COMPLEX NUMBER SYSTEM Imaginary Numbers In the study of numbers, our first encounter was with the set of counting numbers or natural numbers. As our mathematical knowledge progressed, we asked the question, “Is it possible to have a number less than 0?” and we discovered the set of integers. Later, in answer to the question, “Is it possible to have numbers between the integers, for example, between 1 and 2?” we discovered the set of rational numbers. Finally, in response to the question, “Is it possible to have a number whose square is 2?” the irrational numbers were defined and were combined with the rational numbers to form the real number system. The study of imaginary numbers can begin with the question, “Is it possible to have a number whose square is negative?” Most students will answer “No.” The next question might be, “What properties of the real numbers make it impossible to have a real number whose square is negative?” By the trichotomy property, which is one of the order properties of the real numbers, all non-zero numbers are either greater than 0 (positive) or less than 0 (negative). In the set of real numbers, the product of two positive numbers is positive and the product of two negative numbers is also positive. Therefore, the square of any nonzero real number is positive. In order to have a !2a 5 !a ? !21 5 i !a if a $ 0 Pure imaginary numbers are multiplied by expressing them in terms of i and then replacing i2 by its equal, 21, after multiplication is complete. Complex Numbers The correspondence between the set of real numbers and the points on a horizontal line is familiar to students. If we begin with the real number line, it is possible to use transformations to motivate the development of the complex number plane. The product of any real number, a, and 21 is the opposite of a, 2a. On the real number line, the image of any real number, a, under a reflection in the origin is the opposite of a, 2a. Therefore, multiplication by 21 can be associated with a reflection in the origin. A reflection in the origin is also a rotation of 180° about the origin. Since i ? i 5 21, we might identify multiplication by i as the transformation that accomplishes in two steps what multiplication by 21 does in one step. Therefore, if multiplication by 21 can be associated with a rotation of 180° about the origin, multiplication by i can be associated with a rotation of 90° about the origin. If a positive real number, a, is rotated 90° about the origin, its image, a point above the real number line on the line perpendicular to the real number line at the origin, should have 14580TM_C05.pgs 66 3/26/09 12:12 PM Page 66 Chapter 5 as its coordinate the product ai. If a negative real number, 2a, is rotated 90° about the origin, the image, a point below the real number line on the line perpendicular to the real number line at the origin, should have as its coordinate the product 2ai. The real number line and the line that is perpendicular to it at the origin are then the reference lines or axes that are used to locate any complex number, a 1 bi. The point of intersection of the real axis and the imaginary axis is 0 1 0i. This number is defined to be the real number 0, not imaginary and not both real and imaginary. Since 0i is not considered to be an imaginary number and since 2ai 1 ai 5 0i 5 0, a real number, the set of pure imaginaries is not closed under addition. Students should be aware that two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. 5-5 OPERATIONS WITH COMPLEX NUMBERS To add or subtract complex numbers, combine like terms; that is, combine the real parts and combine the imaginary parts. In the previous section, students modeled addition of complex numbers on a coordinate plane. The diagonal of the parallelogram that has sides representing the two complex numbers represents the sum of the two complex numbers. Since the additive inverse of c 1 di is 2c 2 di, subtraction can be defined as it is in the set of real numbers: (a 1 bi)(c 1 di) 5 (a 1 bi)(c) 1 (a 1 bi)(di) Distributive property 5 ac 1 bi(c) 1 adi 1 bi(di) Distributive property 5 ac 1 bci 1 adi 1 bdi2 Commutative property 5 ac 1 bci 1 adi 1 bd(21) Definition of i 5 ac 1 bd(21) 1 adi 1 bci Commutative property 5 ac 1 bd(21) 1 (ad 1 bc)i Distributive property 5 (ac 2 bd) 1 (ad 1 bc)i Substitution After students have observed that the procedure for obtaining the product of two complex numbers is similar to that for the multiplication of two binomials, they will be able to compute the product without memorizing the definition. Transformations can be used to visualize the product of two complex numbers. For h example, to find (2 1 i)(3 1 2i), let OA represent 3 1 2i and the transformations represent 2 1 i: (2 1 i)(3 1 2i) 5 2(3 1 2i) 1 i(3 1 2i) STEP 1: 2(3 1 2i) 5 6 1 4i h h 2 ? OA 5 OAr yi A9 A x O a 1 bi 2 (c 1 di) 5 a 1 bi 1 (2c 2 di) STEP 2: i(3 1 2i) 5 3 1 2i2 5 3i 1 2(21) 5 fa 1 (2c)g 1 fb 1 (2d)gi When we define multiplication and division in the set of complex numbers, we want the familiar properties (commutative, associative, and distributive) to be valid. Therefore, multiplication is defined using these properties: 5 22 1 3i h h i ? OA 5 OB yi A9 B A O x 14580TM_C05.pgs 3/26/09 12:12 PM Page 67 Quadratic Functions and Complex Numbers STEP 3: (2 1 i)(3 1 2i) 5 2(3 1 2i) 1 i(3 1 2i) h h 5 2 ? OA 1 i ? OA h h 5 OAr 1 OB h 5 OC 5 4 1 7i yi C B A9 A O x Students should be familiar with complex conjugates. Conjugates are often used to change a fraction whose denominator is a complex number to an equivalent fraction whose denominator is a real number. Also, beginning in Section 5-6, students will see that, given a quadratic equation with real coefficients, complex roots occur in conjugate pairs. In previous discussions, students have learned that binomials of the form a2 1 b2 cannot be factored in the set of real numbers. Now we see that these binomials can be factored in the set of complex numbers: a2 1 b2 5 (a 1 bi)(a 2 bi) x2 1 4 5 (x 1 2i)(x 2 2i) 25 1 y2 5 (5 1 yi)(5 2 yi) The same principle is used to find the multiplicative inverse of a complex number and the quotient of two complex numbers. To write a 1 bi a complex number of the form ca 1 di or c 1 di in the form x 1 yi, it is necessary to change the fraction to an equivalent one whose denominator is a real number. This is accomplished by multiplying numerator and denominator by the conjugate of the denominator. Point out that the division process for complex numbers is converted to a multiplication process. This is analogous to the situation with 67 fractions: division of fractions is converted to the process of multiplication by the reciprocal (multiplicative inverse) of the fraction. Finally, it is interesting for students to consider cases where a 1 bi and its conjugate a 2 bi are multiplicative inverses. This occurs when a 2 1 b 2 5 1. For example, 5 12 A 135 1 12 13i B A 13 2 13i B 5 1. Challenge students to find other examples. See Enrichment Activity 5-5: Complex Number Operations, Vectors, and Transformations. Multiplication by a complex number involves three basic transformations: two dilations and a rotation of 90°, as shown in the text. By investigating graphs produced by powers of i and powers of a 1 bi (a 0, b 0), students will discover patterns and learn why multiplication by a complex number is a transformation called a spiral similarity. (This activity sheet may be used at any time after Sections 5-4 and 5-5 have been taught.) 5-6 COMPLEX ROOTS OF A QUADRATIC EQUATION Any equation of the form ax2 1 bx 1 c 5 0 has, in the set of complex numbers, roots that may be real or imaginary. Review the conditions under which the quadratic equation has real roots. If the roots are not real, then they are imaginary. If the coefficients a, b, and c are real numbers, the roots will be real numbers when the discriminant is greater than or equal to 0 and imaginary when the discriminant is less than 0. Only when the coefficients are real numbers can the discriminant be used in this way to determine the nature of the roots. Enrichment Activity 5-6: Quaternions can be assigned anytime after Section 5-6. An advanced and abstract topic, the extension of the complex numbers to a four-tuple system is presented to give students an opportunity to work with the special rules that apply to this system. Quaternions satisfy all field postulates except commutativity. 14580TM_C05.pgs 68 3/26/09 12:12 PM Page 68 Chapter 5 5-7 SUM AND PRODUCT OF THE ROOTS OF A QUADRATIC EQUATION Students should be familiar with the two methods, the reverse factoring technique and the use of the sum and products of the roots, for determining an equation from the roots. Notice that in both methods, as demonstrated in the text, an equation is written whose leading coefficient a is 1. Any multiple of the equation in the text will also be a solution. For example, to write a quadratic equation whose roots are 10 and 22, we may use the reverse factoring technique: (x 2 10)(x 1 2) 5 0 S x2 2 8x 2 20 5 0 Any constant, however, may also be used as a factor. In this way: 4(x 2 10)(x 1 2) 5 0 S 4x2 2 32x 2 80 5 0 1 2 (x 2 10)(x 1 2) 5 0 S 12x2 2 4x 2 10 5 0 Any equation of the form kx2 2 8kx 2 20k 5 0 (k 0) is an acceptable quadratic whose roots are 10 and 22. Similarly, when using the sum and the product of the roots, we can choose a to be any value. In most cases, we let a 5 1 or we choose the equation with coefficients that are relatively prime. However, there are times when a value of a larger than 1 will give the simpler equation. Example: Write a quadratic equation whose roots are 21 and 213. Solution: The sum of the roots is 2ba 5 12 2 13 5 16 The product of the roots is ac 5 A 12 B A 213 B 5 216 Let a 5 6. Then 2b6 5 16 Also, 6c 5 216 2b 5 1 or c 5 21 or b 5 21 Answer: A quadratic equation with roots 21 and 213 is 6x2 2 x 2 1 5 0. 5-8 SOLVING HIGHER DEGREE POLYNOMIAL EQUATIONS Cubic equations can be solved with the quadratic formula if a quadratic factor can be found. Review the special formulas for cubics: Sum of two cubes: a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2) Difference of two cubes: a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2) When the polynomial has three terms, it is sometimes effective to break up a term so that the expression can be factored by grouping. For example, in Exercise 10, f(x) 5 x4 1 5x2 1 4 5 x4 1 x2 1 4x2 1 4 Rewrite 5x2 as x2 1 4x2 5 x2(x2 1 1) 1 4(x2 1 1) 5 (x2 1 4)(x2 1 1) Another method to apply to Exercise 10 is to use the quadratic form. Equations that can be written in the form a[f(x)]2 1 b[f(x)] 1 c are said to be in quadratic form. Then, f(x) 5 x4 1 5x2 1 4 5 (x2) 2 1 5(x2) 1 4 5 (x2 1 4)(x2 1 1) Each factor is then set equal to zero and the imaginary roots are easily found. x2 1 4 5 0 x2 5 24 x 5 6 !24 x 5 64i x2 1 1 5 0 x2 5 21 x 5 6 !21 x 5 6i Sometimes one or both of the quadratic form factors will again be factorable. Some students find it easier to identify the quadratic form if they use a “dummy” variable. In the example above, let u 5 x2, so the equation becomes f(a) 5 u2 1 5u 1 4. 14580TM_C05.pgs 3/26/09 12:12 PM Page 69 Quadratic Functions and Complex Numbers When the trinomial is factored and solutions are found, it is necessary to complete the work by substituting x2 for u and then solving for x. Students should be aware that when synthetic substitution is used and the number being tested is a root, the non-zero numbers in the last step are the coefficient of the reduced polynomial factor. So, for f(x) 5 x4 2 3x3 1 x2 2 2x 1 3 and f(1), we have 1 23 1 1 22 1 22 21 3 1 21 23 22 23 0 Then, (x 2 1) is a factor and the other factor is (x3 2 2x2 2 x 2 3). Point out that when synthetic substitution is used, a 0 must be written as the coefficient of any missing term. 5-9 SOLUTION OF SYSTEMS OF EQUATIONS AND INEQUALITIES Students have solved systems of linear equations graphically and algebraically. Similar methods can be applied to solve systems involving quadratic equations. The number of real solutions of the system is equal to the number of intersections of the graphs. The graphical procedure for solving a system of equations may result in an approximate answer when the coordinates of the intersec- 69 tion points are irrational. Exact answers may be obtained algebraically using the substitution method for a quadratic-linear system. When using substitution, the objective is to obtain a single equation containing only one variable. Note that students will sometimes solve for one variable and think that they have found the answer. Remind them that the solution, if one exists, consists of one or more ordered pairs. When a parabola is graphed in the coordinate plane, the points that lie on the parabola satisfy an equation of the form y 5 ax2 1 bx 1 c. Points in the solution sets of the related inequalities, y . ax2 1 bx 1 c and y , ax2 1 bx 1 c, lie, respectively, above and below the parabola. The first inequality specifies that the value of y is greater than the value of the trinomial. Therefore, such a point is located above a point with the same x-coordinate that has a value of y equal to the trinomial that lies on the parabola. Students often equate “above the parabola” with “inside the parabola,” since both phrases are descriptions of the points of the solution set when a is positive. When a is negative, however, the two descriptions are not the same. The graph of y . 2x2 makes this clear. In Example 4, after the solution of 2x2 1 3x 1 4 # 0 is identified, have students use the graph to solve 2x 2 1 3x 1 4 . 4. They should look for the interval in which the parabola is above the line y 5 4, which is 0 , x , 3. 14580TM_C05.pgs 3/26/09 12:12 PM Page 70 Name Class Date ENRICHMENT ACTIVITY 5-5 Complex Number Operations, Vectors, and Transformations 1. a. On one set of axes representing the complex plane, draw each of the following complex numbers as a vector: yi A h (1) A 5 5 1 3i (shown as vector OA in the diagram) (2) B 5 i ? A (3) C 5 i ? B (4) D 5 i ? C (5) E 5 i ? D O b. Without performing the operation, predict the value of F if F 5 i ? E. c. Write at least one observation relating the graph drawn in part a to geometric transformations of the plane. d. List as many specific symmetries as possible found in the graph in part a. A special transformation of the plane called a spiral similarity occurs when any point on the plane is multiplied by a complex number a 1 bi, where a 0 and b 0. In effect, this transformation is the sum of two images; one is a dilation of a, and the other is a rotation of 90° followed by a dilation of b, or Da 1 (Db + R908) . Exercises 2 and 3 involve spiral similarities. (Note that the sum of two images is not the same as the composition of two transformations.) 2. a. On a sheet of graph paper, draw a set of axes centered on the page. Draw the h vector OA, where A 5 3 1 i, and label points O and A. b. To demonstrate spiral similarity using multiplication by (1 2 i), draw and label vectors from O to points B, C, D, E, F, G, and H, where: B 5 (1 2 i) ? A C 5 (1 2 i) ? B D 5 (1 2 i) ? C F 5 (1 2 i) ? E G 5 (1 2 i) ? F H 5 (1 2 i) ? G E 5 (1 2 i) ? D c. To see the spiral similarity in this graph, draw segments AB, BC, CD, DE, EF, FG, and GH. 3. Spiral similarity involves powers. Let us start with A 5 1 1 i and multiply repeatedly by the same complex number, 1 1 i. a. Draw a set of axes centered on a sheet of graph paper. Then, on this graph, draw and label the appropriate vectors for each indicated complex number: A 5 (1 1 i) B 5 (1 1 i)2 C 5 (1 1 i)3 D 5 (1 1 i)4 E 5 (1 1 i)5 F 5 (1 1 i)6 G 5 (1 1 i)7 H 5 (1 1 i)8 I 5 (1 1 i)9 Copyright © 2009 by Amsco School Publications, Inc. x 14580TM_C05.pgs 3/26/09 12:12 PM Page 71 Name Class Date b. On this graph, draw segments AB, BC, CD, DE, EF, FG, GH, and HI. c. A pattern should be seen on the graph. If the pattern continues, name the points, using A, B, C, . . . , through Z 5 (1 1 i)26, that will lie on: (1) the positive ray of the x-axis (2) the negative ray of the x-axis (3) the yi-axis 4. Spiral similarity occurs when a point on the plane is multiplied by a complex number a 1 bi. Consider the following cases: Case 1: a 5 0, b 5 0 Case 2: a 0, b 5 0 Case 3: a 5 0, b 0 Case 4: a 0, b 0 In each case, describe the transformation that occurs. What restrictions are needed for a true spiral similarity to exist? 5. Let A 5 2 1 3i, B 5 the conjugate of A, C 5 A 1 B, and D 5 A ? B. a. Write the complex numbers B, C, and D. b. On one set of axes representing the complex plane, draw and label vectors h h h h OA, OB, OC, and OD. c. Of points A, B, C, and D, tell which ones lie on the x-axis, and explain why. d. If A 5 a 1 bi, where a 0 and b 0, and if B 5 the conjugate of A, C 5 A 1 B, and D 5 A ? B, which of points A, B, C, and D, if any, would lie on the x-axis? Explain your answer. e. If P 5 a 1 bi, where a 0 and b 0, and if Q 5 the conjugate of P, find all integral values of a and b such that P 1 Q 5 P ? Q. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C05.pgs 3/26/09 12:12 PM Page 72 Name Class Date ENRICHMENT ACTIVITY 5-6 Quaternions Extensions of the number system beyond the complex numbers are called hypercomplex numbers. Mathematicians have shown that it is impossible for all the field properties to be true in such systems. One hypercomplex system, developed by the Irish mathematician William Rowan Hamilton (1805–1865), is called quaternions or four-fold numbers. This system satisfies all properties except commutativity of multiplication. A quaternion is a number of the form (a, b, c, d) where a, b, c, and d are real numbers. Using the ordered four-tuples 1 5 (1, 0, 0, 0) i 5 (0, 1, 0, 0) j 5 (0, 0, 1, 0) k 5 (0, 0, 0, 1) we can describe a quaternion as follows: (a, b, c, d) 5 a(1, 0, 0, 0) 1 b(0, 1, 0, 0) 1 c(0, 0, 1, 0) 1 d(0, 0, 0, 1) (expanded form) 5 a?1 1 b?i 1 c?j 1 d?k 5 a 1 bi 1 cj 1 dk (ijk form) The number a of a quaternion is called its real or scalar part and the sum bi 1 cj 1 dk is called its vector part. Quaternions have useful applications in fields such as physics, computer graphics, and electrical engineering. 1. Write each quaternion in ijk form and expanded form. a. (4, 6, 2, 3) b. (27, 0, 5, 8) c. (0, 0, 6, 9) d. (23, 0, 0, 0) 2. Write the ordered four-tuple that corresponds to each quaternion. a. 8 1 2j 1 3k b. 4i 1 7j 2 k c. 8 1 j d. 26i 1 2k To add quaternions, add their corresponding parts: (a 1 bi 1 cj 1 dk) 1 (w 1 xi 1 yj 1 zk) 5 (a 1 w) 1 (b 1 x)i 1 (c 1 y)j 1 (d 1 z)k 3. Write each sum as an ordered four-tuple. a. (5 1 7i 2 3j 1 2k) 1 (11 2 3i 1 9j 1 4k) b. (27 1 0i 1 5j 1 8k) 1 (0 1 4i 1 7j 2 k) c. (4 1 6i 1 2j 1 3k) 1 (23 2 6i 2 2j 2 3k) d. (6j 1 9k) 1 (8 1 j) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C05.pgs 3/26/09 12:12 PM Page 73 Name Class Date Multiplication uses these rules: i2 5 j2 5 k2 5 21 ij 5 2ji 5 k ik 5 2ki 5 2j jk 5 2kj 5 i It is easy to remember the multiplication rules using the figure on the right where the points correspond to the quaternions i, j, k. The product of two adjacent quaternions is equal to the third if the movement from the first to the second is clockwise and is equal to the third with a minus sign if the movement is counterclockwise. Thus, multiplication is not commutative. The quaternion (1, 0, 0, 0) or 1 is the multiplicative identity since it can be shown that for all Q, i k 1 ? Q 5 Q ? 1 5 Q. 4. Find each quaternion product. a. 3i ? 2k b. 22j ? 5k c. 4i ? 6j d. 3k ? 7k e. i ? j ? k f. j ? j ? k g. j ? k ? j h. i ? j ? i 5. Use the multiplication rules and the distributive property to find the product (2 1 3i 1 4j 1 5k)(3 1 7i 1 2j 1 8k). The quaternions a 1 bi 1 cj 1 dk and a 2 bi 2 cj 2 dk are called conjugates. The product of quaternion conjugates is a real number. (a 1 bi 1 cj 1 dk)(a 2 bi 2 cj 2 dk) 5 a2 1 b2 1 c2 1 d2 6. What is the sum of a 1 bi 1 cj 1 dk and its conjugate? What kind of number is this sum? 7. Write the conjugate of each quaternion. Then find the conjugate product. a. 6 1 7i 1 2j 2 k b. 3 1 5j 1 2k c. 8i 1 3j d. 5i 2 9j 1 4k 8. In the complex number system, the equation x2 1 1 5 0 has two solutions. How many solutions does x2 1 1 5 0 have in the quaternion system? What are the solutions? Copyright © 2009 by Amsco School Publications, Inc. j 14580TM_C05.pgs 3/26/09 12:12 PM Page 74 Name Class Date Algebra 2 and Trigonometry: Chapter Five Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. 1. Without graphing, describe the translation, reflection, and or scaling that must be applied to y 5 x2 to obtain the graph of y 5 2x2 2 12x 1 6. In 2 and 3, find the exact solution for each equation by completing the square. 2. x2 2 6x 2 40 5 0 3. x2 1 3x 2 18 5 0 In 4–9, write each number in terms of i, perform the indicated operations, and write the answer in a 1 bi form. 4. !225 1 2!236 5. !28 1 !232 6. A3 !23BA !227B 7. A5 !22B A22 !240B 8. A4 !25B 2 9. !21A1 1 !21B In 10–12, simplify each expression. 10. i16 11. i59 12. i45 In 13–18, perform the indicated operation(s) and express the result in a 1 bi form. 13. (3 2 2i)(4 2 i) 14. (3 2 2i) 2 (4 2 i) 15. (3 2 2i)2 16. i(3 1 i) 2 5i 17. (4 2 bi) 4 (1 2 i) 18. 7 1 9i 2 (i3)3 19. Write the multiplicative inverse of 3 1 3i in a 1 bi form. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C05.pgs 3/26/09 12:12 PM Page 75 Name Class Date In 20–22, express the roots of each equation in a 1 bi form. 20. x2 2 2x 1 10 5 0 21. x2 1 10x 1 29 5 0 22. 2x2 5 2x 2 1 In 23–28, in each case: a. Find the value of the discriminant. b. Using the discriminant, determine if the roots of the quadratic equation are (1) rational and unequal, (2) rational and equal, (3) irrational and unequal, or (4) imaginary. 23. 2x2 2 7x 1 1 5 0 2 24. x 2 x 2 2 5 0 2 25. x 2 5x 5 2 a. b. a. b. a. b. 2 a. b. 2 a. b. a. b. 26. 2x 1 2x 1 3 5 0 27. 4x 2 4x 1 1 5 0 2 28. x 1 2 5 2x 2 29. For what value of k will the roots of 4x 1 4x 1 k be equal? 30. Find the sum of the roots of 2x2 2 8x 1 3 5 0. 31. Find the product of the roots of 2x2 2 8x 1 3 5 0. 32. Write a quadratic equation with integer coefficients that has roots 6 and 29. 33. Write a quadratic equation with integer coefficients that has roots 7 2 3i and 7 1 3i. 34. The product of two consecutive positive odd integers is 323. Find the integers by writing and solving a quadratic equation. 35. Find the length of the side of a square whose diagonal is 12 feet longer than the length of a side. Express the answer exactly and to the nearest hundredth. In 36–38: a. Solve each quadratic equation. b. Check both roots. (Show your work on a separate piece of paper.) 36. 2x2 2 5x 1 2 5 0 a. 37. x2 2 2x 1 5 5 0 a. 38. x2 5 2x 1 2 a. In 39–41, for each quadratic-linear system of equations: a. Solve graphically. Draw each graph on a separate piece of graph paper. b. Solve algebraically. 39. y 5 x2 1 2x 1 1 40. x2 1 y2 5 5 41. y 5 x2 2 x y5x13 2x 1 y 5 5 y5x22 b. b. b. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C05.pgs 3/26/09 12:12 PM Page 76 Name Class Date In 42–44, for each inequality: a. Write the solution set of the inequality. b. Graph the solution set on a number line. 42. x2 2 6x . 0 a. b. 2 43. x 1 x 2 b # 0 a. b. 2 44. 4x , 1 a. b. Bonus I: a. Write a quadratic equation whose roots are !2 6 1. b. What is the value of the discriminant of the equation whose roots are !2 6 1? c. Explain why the equation, whose discriminant is a perfect square, has irrational roots. Bonus II: Suppose you picked an integer at random from 1 through 20 as a value for c in the equation y 5 x2 1 8x 1 c. What is the probability the resulting equation will have imaginary roots? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C05.pgs 3/26/09 12:12 PM Page 77 Name Class Date SAT Preparation Exercises (Chapter 5) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. (2i)3(4i)2 equals (A) (4i)5 (B) (24i)5 7 (D) (2i) (E) (22i)7 (C) (8i)5 2. (4 1 2i)2 2 (3 1 2i)2 equals (A) 7 1 4i (B) 5 1 3i (C) 2 2 i (D) 1 1 4i (E) 1 3. If (3 1 4i)(5 2 2i) 5 k, then (4 1 4i)(5 2 2i) equals (A) k 1 1 (B) k 1 4 1 4i (C) k 1 5 2 2i (D) k(4 1 4i) (E) k(5 2 2i) 4. If xy 5 4i 1 6, x2 is a real number, and y 5 22 1 b, then b could be (A) 43 (B) 3 (D) 43i (E) 3i 8. If a 1 bi is the distance from a 1 bi to 0 1 0i in the complex number plane, then how many complex numbers a 1 bi satisfy a 1 bi 5 5? (A) 0 (B) 1 (C) 2 (D) 4 (E) Infinitely many 9. If x is a number in the first quadrant of the complex number plane, then x2 is in which quadrant? (A) I (B) II (C) III (D) IV (E) Cannot be determined 10. If x25 5 4,096 1 4,096i and x23 5 2,048 2 2,048i, then x2 is (A) 2 (B) 22 (C) 2i (D) 22i (E) 2 1 2i 11. Which number is included in the shaded area? yi (C) 23i 5. If the average of 3 1 i, 2 2 2i, and x is 1 1 i, then x is (A) 22 1 4i (B) 21 1 2i (C) 1 1 i (D) 3 1 3i (E) 5 2 i 6. If x 1 5i is a factor of 2x2 1 kx 1 15, then k is (A) 10i (B) 7i (C) 3i (E) 23i (D) 27i i 1 (A) 23 1 0.5i (C) 21 1 2.5i (E) 3 1 0.5i x (B) 21.5 2 4i (D) 1 1 3.5i 12. Which number is included in the shaded area? 7. If (a 1 bi)n is an odd integer, which of the following must be an odd integer? yi I. (a 1 bi) n11 i II. (a 1 bi) 2n 1 x 2 III. (a 1 bi) n (A) I only (C) III only (E) II and III (B) II only (D) I and II Copyright © 2009 by Amsco School Publications, Inc. (A) 22 1 2.5i (C) 1 2 2i (E) 2 2 2.5i (B) 21 1 i (D) 1.5 1 2i 14580TM_C05.pgs 3/26/09 12:12 PM Page 78 Name 13. Class II. STUDENT-PRODUCED RESPONSE QUESTIONS In questions 16–24, you are to solve the problem. y 1 O 1 Date x 16. What is the value of 35i72 1 7i34? 17. What is the value of (((1 1 i)i 1 1)i 1 1)i 1 1? 18. What is the value of f (25i) 22g 22? Which equation is best represented by the graph above? (A) y 5 2(x 1 3)2 (B) y 5 2(x 2 3)2 2 (C) y 5 x 2 3 (D) y 5 2x2 2 3 2 (E) y 5 2x 1 3 14. Which parabola intersects the x-axis in two distinct points? (A) y 5 (x 1 7)2 (B) y 5 (x 2 7)2 2 (C) y 5 x 2 49 (D) y 5 x2 1 49 2 (E) y 5 2x 2 7 15. A parabola with an equation of the form y 5 ax2 1 bx 1 c has the point (22, 1) as its vertex. If (24, 5) also lies on the parabola, which of the following is another point on this parabola? (A) (24, 3) (B) (21, 3) (C) (0, 5) (D) (2, 1) (E) (4, 5) Copyright © 2009 by Amsco School Publications, Inc. 19. What is the distance between the points where the graph of y 5 22x2 2 x 1 15 crosses the x-axis? 20. To the nearest hundredth, what is the distance between the points where the graph of y 5 x2 2 6x 1 7 crosses the x-axis? 21. If 28(2 2 3i)2 5 x(3 1 2i)2, then what is the value of x? 22. Let a and b be real numbers, ax 1 by 5 10, x 5 2 1 2i, and y 5 4 1 3i. Evaluate b 2 a. 23. What is the value of (3 2 i)2(3 1 i)2? 24. If (3 1 7i)(5 2 yi) 5 43 1 23i, then what is the value of y? 14580TM_C06.pgs 3/26/09 12:12 PM Page 79 CHAPTER SEQUENCES AND SERIES 6 Aims • To know and apply sigma notation. • To identify an arithmetic or geometric sequence and find the formula for its nth term. • To recognize and find the sum of arithmetic sequences and series. • To recognize and find the sum of geometric sequences and series. In this chapter, students learn to represent and analyze patterns using algebraic methods. Sequences and series are examples of recursive processes and such processes have many applications. For example, fractals are created using a recursive geometric process. In addition to the fascinating visual patterns found in fractals, number patterns can be identified. In business, the computation of compound interest, amount of annuities, amortization of debts, and depreciation of assets involve concepts of geometric sequences and series. f(5) 5 777.51 f(10) 5 322.37 f(6) 5 689.17 f(11) 5 227.21 f(7) 5 599.51 f(12) 5 130.62 f(8) 5 508.50 f(13) 5 32.58 f(9) 5 416.13 f(14) 5 33.07 The purchase will be paid off at the end of 14 months. Point out that as the balance decreases, less interest is added on, and the balance decreases more rapidly after each payment. Ask students to determine the total amount paid for the computer ($1,333.07). 6-1 SEQUENCES In the text, a sequence of seven terms is shown to be a one-to-one correspondence between the terms of the sequence and the first seven positive integers. The correspondence between the terms 8, 4, 2, 1, 21, 41, 18 and the integers 1, 2, 3, 4, 5, 6, 7 can be written as the set of ordered pairs U (1, 8) , (2, 4), (3, 2), (4, 1), A 5, 12 B , A 6, 14 B , A 7, 18 B V . Therefore, we can define CHAPTER OPENER this finite sequence as a function whose Have students complete the payment schedule. The balances at the end of each month are: domain is {1, 2, 3, 4, 5, 6, 7} and whose rule is f(1) 5 1,118 f(3) 5 950.29 f(2) 5 1,034.77 f(4) 5 864.54 f(x) 5 A 12 B . The positive integer with which each element of the sequence is paired is indicated by the position of the term in a listing of the terms of the sequence or by the subscript of the term name. For example, a3 5 2 indix21 79 14580TM_C06.pgs 80 3/26/09 12:12 PM Page 80 Chapter 6 cates that (3, 2) is a pair of the function whose domain is the set of positive integers and whose rule is an 5 8 A 12 B . Be sure students understand that a finite sequence is a function whose domain includes only the first n positive integers; the domain of an infinite sequence is all the positive integers. When discussing Example 1, emphasize the difference between the general expression for an and the recursive rule for an. As will be shown in later sections, a general rule expresses each term of the sequence using the first term a1 and the common difference or common ratio. The recursive definition for a sequence describes how to find the nth term from the terms before it and must include an initial value. For example, the sequence 4, 7, 10, 13, 16, . . . can be described in two ways: n21 an 5 4 1 (n 2 1)3 General formula or an 5 an21 1 3 where a1 5 4 6-2 ¶ Recursive formula ARITHMETIC SEQUENCES Have students graph the ordered pairs for any of the arithmetic sequences in the section. Mention that the graph represents a discrete function whose domain consists of distinct values and there is no continuity between those values. The range is the set of numbers representing the terms of the sequence. For any arithmetic sequence, the points graphed will be along a straight line. A linear function can be written in the form y 5 mx 1 b. Ask students to compare this slope-intercept form with the expression for the nth term of an arithmetic sequence, an 5 a1 1 (n 2 1)d. Students should be able to identify that with a change in the order of the terms and factors, y 5 an, m 5 d, x 5 n 2 1, and b 5 a1. Explain that the general formula can be used to find the nth term of an arithmetic sequence or to find the position of a given term in an arithmetic sequence. The formula can also be used to solve problems such as the following: Example: Determine how many numbers between 8 and 1,621 are divisible by 6. Solution: The first number between 8 and 1,621 that is divisible by 6 is 12. Let a1 5 12. Then by repeatedly adding 6, we could list the successive numbers that are divisible by 6, so d 5 6. a1 1 (n 2 1)d # 1,621 12 1 (n 2 1)6 # 1,621 12 1 6n 2 6 # 1,621 6n 1 6 # 1,621 6n # 1,615 n # 269 (remainder 1) There are 269 numbers between 8 and 1,621 that are divisible by 6. In Enrichment Activity 6-2: Arithmetic Sequences students identify arithmetic sequences after partitioning the set S[9] 5 {1, 2, 3, 4, 5, 6, 7, 8, 9} and discover an interesting property of S[9], that it is the smallest set such that any separation into two sets yields at least one set with at least one three-term arithmetic sequence. 6-3 SIGMA NOTATION A series is the sum of a sequence. A series may be finite or infinite depending on whether the sequence it is based on is finite or infinite. At this time, some students may find the idea of an infinite sequence having a sum puzzling, but this concept will be made clear in Section 6-7. Students who feel that an infinite arithmetic series cannot have a sum are correct. The summation symbol, , which is the uppercase Greek letter sigma, gives a compact way to write a series from the general term of the corresponding sequence. The letter n used in the summation notation is called the index of summation or simply the index. In Example 1, the 1 below the summation symbol is called the lower limit and the 10 above the symbol is called the upper limit. If a summation symbol is written without any upper and lower limits, it means that all the given data are to be summed. For example, a formula used to find 14580TM_C06.pgs 3/26/09 12:12 PM Page 81 Sequences and Series the arithmetic mean, x– (read x bar), of a set of data is x 5 Sx n , where n is the number of data values. Students should be aware that different summation notation can represent the same 6 Example: Find the eighth term of a geometric sequence if a4 5 16 and r 5 0.5. Solution: First find the value of a1. a4 5 a1r421 16 5 a1(0.5) 3 4 16 5 0.125a1 series. For example, a (k 2 2) 2 and a k2 k53 16 0.125 k51 both represent the same sum 1 1 4 1 9 1 16. 6-4 ARITHMETIC SERIES Sn 5 1 an) Substitute a1 1 (n 2 1)d for an. Sn 5 n2 fa1 1 (a1 1 (n 2 1)d)g 5 n2 f2a1 1 (n 2 1)dg This version of the formula can be used when the value of the last term is not known. 6-5 GEOMETRIC SEQUENCES A geometric sequence, like an arithmetic sequence, is a function whose domain is the set of positive integers. However, the graph of the terms is not linear. In Chapter 7, students will learn to recognize that the points of the graph follow the same pattern as the exponential function y 5 abx. (For the general exponential function, the value of b must be positive, but for a geometric sequence an 5 a1r n–1, the value of r may be either positive or negative.) • If an , an21, then r , 1. • If an . an21, then r . 1. • If the signs of a geometric sequence alternate, then r , 0. The following example shows how to find a given term when you know the common ratio and one term of the geometric sequence, but not a1. 5 a1 128 5 a1 Now use the formula an 5 a1rn21 to find a8. a8 5 128(0.5) 821 Two versions of the formula for Sn are given before Example 1. Students should understand how one is derived from the other. n 2 (a1 81 5 128(0.5) 7 51 An alternative solution is to start with the given fourth term and multiply by the common ratio to find a5 to a8. 6-6 GEOMETRIC SERIES In the formula for the sum of a geometric series, it is necessary to restrict the value of r to values not equal to 1 so that division by 0 does not occur. An example of a geometric series where r 5 1 is 5, 5, 5, . . . . The sum of the first n terms is 5n. In general, the sum of a geometric series with r 5 1 is na1. In Exercise 1, students are asked to justify the equivalent expression a 2 ar Sn 5 11 2 rn . Since an 5 a1r n21, we can multiply each side by r to get anr 5 a1r n. By substituting anr for a1r n in the formula for the sum of a geometric series, we have Sn 5 a1 (1 2 rn) 1 2 r a 2 a rn 5 11 2 r1 a 2 ar 5 11 2 rn This version of the formula is useful for finding the sum if the first and last terms are given, but the value of n is not given. In Exercises 15–22, sigma notation is used to represent the sum of a geometric series. When n is small, suggest that students find the sum in two ways: (1) by using the formula and (2) by writing out all the terms of the sum and performing the actual addition. 14580TM_C06.pgs 82 6-7 3/26/09 12:12 PM Page 82 Chapter 6 INFINITE SERIES All the geometric series that students have worked with so far have been finite. The series n11 1 1 1 12 1 14 1 18 1 16 1 c 1 A 12 B 1c is an infinite geometric series, as indicated by the three dots at the end that signify the series continues indefinitely in the same manner. We can see that the series above approaches the value 2 as more terms are added. (The first six partial sums are 1, 1.5, 1.75, 1.875, 1.9375, 1.96875.) Ask students how we know that the series will never pass 2. Suggest to students that they write the partial sums as follows: 1 1 12 5 32 5 2 2 12 1 1 12 1 14 5 74 5 2 2 14 1 1 1 12 1 14 1 18 5 15 8 5 2 2 8 Students should be able to see that each partial sum is equal to the difference of 2 and a decreasing positive number. For example, the sum of the first three terms is 74 or 212 less than 1 so that the sum can 2. In general, Sn 5 2 2 2n21 never be greater than or equal to 2. Emphasize that even though we often call a1 1 2 r the sum of an infinite geometric series with r , 1, this is the limit that the sum approaches, not a sum that is actually reached by adding terms. A series that approaches a limit is called a convergent series. Note that while it is true that an infinite geometric series has a finite limit if r , 1, there are other types of series that converge. The sum of an infinite geometric series provides an alternate method for expressing a repeating decimal as a rational number in the form ba . Example: Write a fraction equivalent to 0.343434 . . . Solution: We can write this decimal as 0.34 1 0.0034 1 0.000034 1 c1 (0.34)(0.01) n21 1 c This is an infinite geometric series with a1 5 0.34 and r 5 0.01. Since r , 1, a S 5 1 21 r 0.34 0.34 34 512 0.01 5 0.99 5 99 The solution can be checked by using a calculator to divide 34 by 99. In calculus, students will learn that infinite series can be used to represent many numbers such as p, !2, sin 5p 2 , or ln 5. A major advantage of the series representation is that it allows us to approximate irrational numbers to any degree of accuracy. The symbol e was first used to represent the irrational number 2.718281828459045 . . . by the Swiss mathematician Leonhard Euler. It has come to occupy a special place in mathematics and its applications, and it will reappear in Chapter 8. EXTENDED TASK For the Teacher: In this exploration of the harmonic series, students see that the nth term converging to 0 is a necessary but not sufficient condition for the series to have a limiting sum. Students participate in the demonstration that the harmonic series is not bounded by working with specific partial sums and arranging terms in groups that are then replaced with smaller values that sum to 12. Since this process can be continued without end, the value of the sum can always be increased by another 12 and is, therefore, unbounded. Since the original sums were larger, the original series cannot have a limit. Although the harmonic series diverges, the partial sums do not grow rapidly. In fact, if j is very large, it has been shown that Sj is approximately equal to 0.577216 1 ln j. 14580TM_C06.pgs 3/26/09 12:12 PM Page 83 Name Class Date ENRICHMENT ACTIVITY 6-2 Arithmetic Sequences Suppose the set of numbers Sf9g 5 51, 2, 3, 4, 5, 6, 7, 8, 96 is separated into two smaller sets A 5 51, 4, 6, 96 and B 5 52, 3, 5, 7, 86 . Observe that in set B, the numbers 2, 5, 8 form an arithmetic sequence (difference 3) and the numbers 3, 5, 7 also form an arithmetic sequence (difference 2). Suppose S[9] is separated differently: A 5 52, 3, 86 and B 5 51, 4, 5, 6, 7, 96 Then, in set B, the arithmetic sequences 1, 4, 7 or 5, 6, 7 or 5, 7, 9 or 1, 5, 9 can be identified. Find ten different ways to separate the set S[9] into two sets A and B. Each set must have at least one element. Identify an arithmetic sequence of at least three terms in at least one of the sets and indicate the common difference. If you create a separation such that neither set contains a three-term arithmetic sequence, write NO SEQUENCE FOUND. Try sets of different sizes. Set A Set B Arithmetic Sequence (Common Difference) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Did you find any pair of sets A and B such that neither contained an arithmetic sequence? In fact, mathematicians have shown that whenever S[9] is separated into two sets, it is always true that at least one of the sets contains an arithmetic sequence of at least three terms. The proof uses cases and a combination of direct and indirect reasoning to show the existence of the required sequence in every case. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C06.pgs 3/26/09 12:12 PM Page 84 Name Next, consider the set Class Date Sf8g 5 51, 2, 3, 4, 5, 6, 7, 86 Does S[8] have the same property as S[9]? 11. Construct three different separations of S[8] such that at least one of the sets contains a three-term (or greater) arithmetic sequence. 12. Construct at least one separation such that neither set has an arithmetic sequence. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C06.pgs 3/26/09 12:12 PM Page 85 Name Class Date EXTENDED TASK The Harmonic Series The series ` 1 1 1 c an5112131 n51 is called the harmonic series. 1. What happens to the nth term of the harmonic series as n approaches infinity? 2. Find the first six partial sums for the series above. 3. As n becomes infinitely large, do you think the partial sums approach a limit? ` Do you think the infinite series a n1 has a sum? If so, what do you think the n51 value of the sum is? If not, explain why not. 4. Observe that S21 5 S2 5 1 1 12 1 2 S22 5 S4 5 1 1 12 1 13 1 14 . 1 1 12 1 14 1 14 5 1 1 22 since 13 . 14. Thus, S4 . 1 1 22. Use this result and a similar line of reasoning to show that S23 5 S8 . 1 1 32. 5. Use the result of the previous question to show that S24 5 S16 . 1 1 42. 6. Predict an inequality for the general case S2 j. What conclusion can be drawn ` about a n1 ? Explain. n51 Use a calculator for questions 7 and 8. j 7. Determine the smallest value of j such that a n1 $ 3. n51 j 8. Determine the smallest value of j such that a n1 $ 4. n51 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C06.pgs 3/26/09 12:12 PM Page 86 Name Class Date Algebra 2 and Trigonometry: Chapter Six Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–6: a. Write the next three terms of each sequence. b. Write an expression for the general term an. 1. 6, 12, 18, 24, . . . 2. 27, 25, 23, 21, . . . a. a. b. b. 3. 4, 512, 7, 812, c a. 4. 28, 4, 22, 1, . . . a. b. 5. 2 4 8 16 5, 5, 5, 5 , b. c 6. 21, 1, 21, 1, . . . a. a. b. b. In 7–12, find the indicated quantity of the arithmetic sequence. 7. a1 5 5, d 5 7; find a13. 9. a1 5 3, a10 5 75; find d. 11. a1 5 225, an 5 82, d 5 211; find n. 8. a1 5 26, d 5 24; find a17. 10. a1 5 4, an 5 148, d 5 9; find n. 12. a1 5 2.37, d 5 0.98; find a44. 13. Find two arithmetic means between 55 and 130. 14. Find four arithmetic means between 18 and 108. In 15 and 16, find the value of x that makes each sequence arithmetic. 15. 7, 15, 23, 5x 1 1, . . . Copyright © 2009 by Amsco School Publications, Inc. 16. x 1 12, 3x 1 7, 3x, . . . 14580TM_C06.pgs 3/26/09 12:12 PM Page 87 Name Class Date In 17 and 18: a. Write the terms of each series. b. Find the sum. 6 17. a (k2 1 1) a. b. a. b. k51 8 18. a (5k 2 3) k53 In 19–22, find the indicated partial sum for each arithmetic series described. 19. a1 5 1, a10 5 19, S10 20. a1 5 24, d 5 3, S16 21. a15 5 106, d 5 28, S15 225 22. a1 5 25 6 , a11 5 6 , S11 In 23–26, find the nth term of each geometric sequence. 23. a1 5 3, r 5 2, n 5 8 24. a1 5 4,096, r 5 14, n 5 7 25. a3 5 112, r 5 24, n 5 6 26. a1 5 34, a4 5 29, n 5 5 27. Find three geometric means between 9 and 144. In 28–31, find the indicated partial sum for each geometric sequence described. 28. a1 5 5, r 5 22, S6 29. 3 2 6 1 12 2 24 1 c, S9 30. 81 1 27 1 9 1 c, S8 31. a1 5 8, r 5 12, S7 32. A certain substance decomposes and loses 20% of its weight each hour. If there is originally 4,000 grams of this substance, how much remains after 5 hours? 33. Maria Lopez invests $3,000 at 4% interest compounded annually in a savings account. How much money is in her account at the end of 3 years? In 34–36: a. Find a1 and r for each geometric series. b. Find the sum if it exists; otherwise write “No sum.” 34. 6 1 3 1 32 1 34 1 38 1 c 35. 1.8 1 2.7 1 4.05 1 6.075 1 c 36. 5 2 2 1 4 2 8 1 16 2 c 5 25 125 Copyright © 2009 by Amsco School Publications, Inc. a. b. a. b. a. b. 14580TM_C06.pgs 3/26/09 12:12 PM Page 88 Name Bonus I: Class Date Each swing of a pendulum travels 90% of its previous distance. If the first swing is 8 feet long, determine the total distance traveled by the pendulum by the time it comes to rest. ` Bonus II: a. Write the series a n(n 11 1) as the sum of terms. n51 b. Is the series geometric? c. Find S1, S2, S3, S4, S5. d. Predict the values of S10 and S20. e. As n approaches infinity, what number does Sn appear to be approaching? 1 f. Observe that a n(n 11 1) can be written a A n1 2 n 1 1B. ` ` n51 n51 Write the terms for S5 without combining any terms. g. Using the results of part f, what conjecture can you make about an expression for Sn? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C06.pgs 3/26/09 12:12 PM Page 89 Name Class Date SAT Preparation Exercises (Chapter 6) I. MULTIPLE CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. Which of the following is not a geometric sequence? (A) 96, 48, 24, 12, . . . (B) 22, 6, 218, 54, . . . 1 1 1 1 (C) 243 , 81, 27, 9, c (D) 8, 4, 0, 24, . . . (E) !2, 2, 2 !2, 4, c 2. The 7th term of the geometric sequence 2,000, 200, 20, 2, . . . is (A) 0.2 (B) 0.02 (C) 0.002 (D) 0.0002 (E) 0.00002 3. Which number is a term of the sequence 3, 7, 11, 15, . . . ? (A) 45 (B) 63 (C) 85 (D) 110 (E) 120 4. An arithmetic sequence is represented by 3, 14, 3x 1 4, . . . . What is the value of x? (A) 7 (B) 10 (C) 14 (D) 21 (E) 25 5. The first term of an arithmetic sequence is 8 and each term is 13 more than the previous one. What is the tenth term in the sequence? (A) 117 (B) 125 (C) 130 (D) 138 (E) 142 6. What term of the sequence 212, 25, 2, 9, . . . is 65? (A) 10th (B) 11th (C) 12th (D) 13th (E) 14th 7. How many terms are there in the geometric sequence 58, 254, 52, c, 640? (A) 8 (D) 11 (B) 9 (E) 12 (C) 10 Copyright © 2009 by Amsco School Publications, Inc. 8. In a geometric series, S4 5 89 and a1 5 35. What is r? (A) 25 (B) 213 (C) 215 (D) 31 (E) 52 9. The first term of a geometric sequence is 5 and the third term is 7.2. Which of the following can be the fifth term? (A) 8.85 (B) 9.4 (C) 10.368 (D) 11.646 (E) 12.4416 10. If 219, 13, 21, . . . are the first three terms of a geometric sequence, which of the following must be true? I. The constant ratio r is 3. II. The next three terms are 3, 29, 27. III. The ninth term a9 is 729. (A) I only (B) II only (C) III only (D) I and II only (E) II and III only 11. Which arithmetic sequence does not have a term equal to 60? (A) 4, 11, 18, . . . (B) 23, 6, 15, . . . (C) 220, 25, 10, . . . (D) 230, 212, 6, . . . (E) 81, 79.5, 78, . . . 12. By how much does the arithmetic mean between 1 and 49 exceed the positive geometric mean between 1 and 49? (A) 0.5 (B) 7.5 (C) 12.6 (D) 17 (E) 18 13. What is the sum of the first eleven terms of the arithmetic sequence 3, 512, 8, . . . ? (A) 28 (D) 17012 (B) 14312 (E) 175 (C) 154 14580TM_C06.pgs 3/26/09 12:12 PM Page 90 Name Class 14. For the arithmetic sequence shown below w, 25, x, y, 4,z, c what is the value of w 1 x 1 y 1 z? (A) 22 (B) 0 (C) 8 (D) 10 (E) 15 15. The sum of the first n terms of an arithmetic series is 1,445, a1 5 13, and an 5 157. What is the third term of the series? (A) 23 (B) 31 (C) 33 (D) 40 (E) 41 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. If 3, 6, 12, 2y 1 6 is a geometric sequence, what is the value of y? 17. What is the 15th term of the sequence 213, 29, 25, 21, . . . ? Copyright © 2009 by Amsco School Publications, Inc. Date 18. What is the sum of the two arithmetic means between 3 and 60? 19. If the eighth term of a geometric 3 sequence is 38 and the ninth term is 16 , what is the third term of this sequence? 20. If a1 5 2 and an 5 2n 1 an–1, what is the sum of the first five terms? 21. The first term of a geometric sequence is 5 and the third term is 80. What is the quotient when the eighth term is divided by the fifth term? 22. What is the sum of the two geometric means between 40 and 135? 23. What is the sum of a19 1 a8 for the sequence 293, 282, 271, . . . ? 24. What is the sum of the infinite series 4,900 1 1,400 1 400 1 c? 14580TM_C07.pgs 3/26/09 12:12 PM Page 91 CHAPTER 7 EXPONENTIAL FUNCTIONS • To apply the rules of exponents with rational exponents. notational system; for example, the notation should be unambiguous, easy to use, and applicable in all instances. Present one or more polynomials and have students use the various methods shown in the chapter opener to express them. Ask students to create a system of their own. • To change from a radical to an exponential expression and from an exponential expression to a radical expression. 7-1 Aims • To review the laws of exponents. • To define zero and negative exponents. • To define and graph exponential functions. • To solve exponential equations. Students have used exponential notation in many of their previous mathematics courses. This chapter reviews the laws of exponents and uses these laws to motivate the definitions of zero, negative, and fractional exponents. From an exponential function and its graph, an intuitive understanding of the use of any real number as an exponent can be developed. The study of exponential equations begun in this chapter will be extended when logarithms are explored in the next chapter. Applications of exponential functions including compound interest and radioactive decay are examined. CHAPTER OPENER It is interesting for students to see how exponential notation evolved. Lead a brief discussion about the features that are desirable in a LAWS OF EXPONENTS Students may be familiar with the rules or laws of exponents presented in this section. A careful review, however, will reinforce their understanding and help them continue the study of exponents with confidence. Call attention to the changes in value that may result from the use of parentheses. For example, 262 (26)2 and (3x)3 3x3. Apply the rules to powers with numerical as well as literal bases. A common student error is to write the product 33 ? 34 as 97. Point out that the product is obtained not by multiplying 3 times 3, but rather by counting how many times 3 is to be used as a factor. 7-2 ZERO AND NEGATIVE EXPONENTS The question “What would happen if . . . ?” can motivate new ideas. When dividing powers with like bases, ask, “What happens if the exponent of the denominator is equal to or 91 14580TM_C07.pgs 92 3/26/09 12:12 PM Page 92 Chapter 7 larger than the exponent of the numerator?” In these cases, we find that the quotients have zero or negative exponents. Although zero and negative exponents do not allow us to define xa as “x used a times as a factor,” powers with zero and negative values are consistent with the rules for operations with powers. An example such as 1 x2 x5 1 2 5 xx ?? xx ? x ? x ? x 5 x13 and xx5 5 x225 5 x23 1 1 serves to establish that negative exponents are for x 2 0, another way of expressing reciprocals. The definition of a negative exponent can also be demonstrated as follows: x 2 0, xa ? x2a 5 x0 5 1 If the product of two numbers is 1, the numbers are reciprocals or multiplicative inverses. Therefore, x29 is the reciprocal of x9, and xa is the reciprocal of x2a (see Example 2). By demonstrating that powers with zero and negative exponents produce consistent results when the rules for multiplying or dividing powers of like bases are used, we show the validity of the new definitions. Emphasize to students that a factor can be moved from a numerator to a denominator or from a denominator to a numerator simply by changing the sign of the exponent. 7-3 FRACTIONAL EXPONENTS The definition of fractional exponents can be introduced by using the rules for multiplying powers with like bases as shown in the section. Since the nth root of x is defined to be one of the n equal factors whose product is x, and n 1 since the product of n factors of xn is xn or x1, 1 then xn satisfies the definition of the nth root of x. For example, 1 1 1 1 4 x4 ? x4 ? x4 ? x4 5 x4 5 x 1 Therefore, x4 is one of the four equal factors whose product is x and is a symbol for the 1 fourth root of x. The power x4 can be used as a n symbol for ! x because it satisfies the definition of the nth root of x. 1 n In the text, xn or ! x is defined only for x . 0. One unfortunate consequence is that 1 3 (28) 3 does not exist even though ! 28 is well1 defined (and equals 22), that is, xn is undefined for negative x when n is odd even though n the corresponding principal root ! x exists. 1 You may wish to define xn in the following way: • If n is a positive integer such that the prin1 n cipal nth root of x exists, then xn 5 ! x. To avoid the problems arising when working with negative numbers, also give the following definition: n m n • " x 5 A! x B m 5 x n only when m n is in lowest terms. m This results in an alternative approach that also satisfies the rules of exponents. As a practical matter, writing terms with fractional exponents allows you to perform operations on terms more easily when they have the same numerical or literal base. 4 3 3 2 Compare 7" x ? 9" x with the expression 3 2 7x4 ? 9x3 . Emphasize that when changing a radical expression to exponential form, the power is placed in the numerator, and the index or root is placed in the denominator of the fractional exponent. When the procedure is reversed to convert exponential expressions to radical expressions, the numerator of the fractional exponent is the power and the denominator is the index or root. Point out 2 3 2 3 that 63 may be written as either " 6 or A ! 6 B 2. To evaluate an expression such as 3 4 4 4 16 5 " 163 5 A ! 16 B 3, it is often easier to find the root first and then raise it to the power. Since powers and roots are on the same level in order of operations, this procedure is valid. Before students begin work on the section exercises, review the conditions for an expression to be simplified: 14580TM_C07.pgs 3/26/09 12:12 PM Page 93 Exponential Functions 1. It has no negative exponents. 2. The expression is not a complex fraction. 3. The index of any remaining radical is the smallest possible number. Use Enrichment Activity 7-3: Factoring Expressions with Rational and Negative Exponents to extend students’ skills. This topic is important for students going on to courses in calculus. 7-4 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Emphasize that the definition of an exponential function y 5 bx excludes 1 as a value for b because y 5 1x is not a one-to-one function. x Note that the graphs of y 5 2x and y 5 A 12 B are both one-to-one functions. In the figures, both graphs may appear to be horizontal near the x-axis, but in fact they are not. Exponential functions of the form y 5 bx will have a shape similar to y 5 2 x when x b . 1 and a shape similar to y 5 A 12 B when 0 , b , 1. When sketching the graph of y 5 2x , in addition to positive integers, we used 0, negative integers, and rational fractions as values of x. We evaluated y according to the definitions of these exponents and located points on the graph using the x- and y-values as coordinates. When we draw the smooth curve through these points, we are assuming that the expression 2x is defined for all values of x that are real numbers. Having made that assumption, we can read from the graph values of 2x for which x has irrational values, that is, values such as 2!3 and 2!5. More rigorous definitions of the values of such powers are derived in advanced mathematics courses. Students often fail to appreciate the difference between a linear and an exponential function in a practical situation. The following 93 example illustrates the difference between linear growth and exponential growth. • Exponential growth: If $1 is invested at 8% interest over a period of x years, the value y of the investment is y 5 (1.08)x. • Linear growth: If $1.08 is added to an initial investment of $1 each year over a period of x years, the value y of the investment is y 5 1 1 1.08x. Use a graphing calculator to sketch the graphs of these functions. First set the values of x from 0 to 100 with a scale of 10, and the values of y from 21 to 300 with a scale of 30. ENTER: WINDOW 0 ENTER 100 ENTER 10 ENTER 21 ENTER 300 ENTER 30 ENTER Now enter the equations. ENTER: Yⴝ 1.08 ^ X,T,⍜,n 1 ⴙ 1.08 X,T,⍜,n ENTER GRAPH DISPLAY: Notice that, throughout the period of 100 years shown on the graph, the y-values of the linear function rise steadily, and the y-values of the exponential function rise slowly at first but then more and more rapidly. For about the first 53 years, the y-values of the linear function are greater than the y-values of the exponential function. Thereafter, however, the y-values of the linear function are smaller than the y-values of the exponential function. 14580TM_C07.pgs 94 3/26/09 12:12 PM Page 94 Chapter 7 7-5 SOLVING EQUATIONS INVOLVING EXPONENTS To solve equations where the variable is the base of a power, the goal is to make the exponent equal to 1 so the variable can be isolated. If time allows, have students suggest how they might estimate the value of x if 5x 5 60. (Since 52 5 25 and 53 5 125, the value of x is between 2 and 3. The value of x should be much closer to 2 than 3. Alternatively, they could graph y 5 5x on a calculator and locate the approximate value of x for which y 5 60.) If A x3 B 5 x, 2 a 2 then x3a 5 x1 and 32a 5 1 or a 5 32, where x 0, 1. So A x3 B 2 5 x. 2 3 Therefore, to solve an equation such as 2 x3 5 4, it is necessary to raise both members of the equation to the 32 power in order to obtain an equivalent equation that has x as one side. 2 Note that x3 means that the cube root of x is squared. Therefore, we use the inverse operation of taking the cube root, which is raising the number to the third power, and the inverse operation of squaring the number, which is taking the square root of the number. A x3 B 2 5 42 2 3 3 7-7 APPLICATIONS OF EXPONENTIAL FUNCTIONS This section examines several interesting applications of exponential increase (growth) and decrease (decay). With regard to the discussion of compound interest, it is reasonable to assume that the more frequently the interest is compounded, the larger the compounded amount will be. However, the calculations show that although the compound amount increases, it does not increase dramatically. Continuing the calculations: Paid hourly for 1 year: 0.05 A 5 100 A 1 1 8,760 B < 105.1270947 or 105.13 8,760 3 x 5 42 x 5 A !4B 3 5 23 5 8 Remind students that variable expressions with fractional exponents are defined only for non-negative variables. 7-6 SOLVING EXPONENTIAL EQUATIONS This section demonstrates a relatively simple method of solving exponential equations. The equations solved here are special cases, since both sides of the equation can be expressed as powers of the same base. Their solutions provide additional practice with exponents. The general procedure for solving exponential equations is discussed in the next chapter, after the introduction of logarithms. Paid each minute for 1 year: 0.05 A 5 100 A 1 1 525,600 B < 105.1271107 or 105.13 525,600 Note that the expression A 1 1 k1 B can be interpreted as the compounding of an investment of $1 at a 100% interest rate compounded k times per year. Thus, no matter how often the interest is compounded, the compound amount will never exceed $2.72. In general, if Ao is invested for t years at a rate r compounded continuously, then the compound amount A is given by the exponential function k A 5 Aoert This function gives the general form for growth and decay. 14580TM_C07.pgs 3/26/09 12:12 PM Page 95 Exponential Functions EXTENDED TASK This activity is intended to introduce students to the exponential function through a handson activity. Students may work on this task in small groups or individually. Some students may see the patterns for the total number of holes before making all of the folds, punches, and counting the holes. They should be encouraged to extrapolate but should complete the activity to verify their predetermined patterns. 95 Some students may have difficulty expressing a number (like 24) as a constant times a power of 2. If so, they should be encouraged to look at the common factors of those numbers or to write the prime factorization of each number. This activity is a good example of connecting one topic in mathematics to another. Part IV of the extended task helps students relate pure mathematics (the exponential function) to real-world situations (examples of growth and decay). 14580TM_C07.pgs 3/26/09 12:12 PM Page 96 Name Class Date ENRICHMENT ACTIVITY 7-3 Factoring Expressions with Rational and Negative Exponents One reason that both rational and negative exponents are introduced in algebra is that you will need to understand and use them in more advanced courses such as calculus. Recall that you learned how to factor a monomial from a polynomial. Polynomials must have terms with nonnegative integer exponents. To factor x3 1 x7, you factor out the common variable with the smallest exponent. x3 1 x7 5 x3 (1 1 x4) The exponent on the variable term in parentheses was found by subtracting the lesser exponent from the greater exponent; 7 2 3 5 4. The same procedure is used to remove a common factor with rational or negative exponents from expressions that are not polynomials. Example 1 2 8 Factor x5 2 x5. Solution: The smaller exponent is 25. x5 2 x5 5 x5 A 1 2 x5 2 5 B 8 2 8 2 2 5 x5 A 1 2 x5 B 6 2 The answer can be checked using the distributive property and the laws of exponents. Example 2 Factor x23 1 x27 and then write the answer without negative exponents. Solution: Since 27 is less than 23, x27 is the term with the smaller exponent. x23 1 x27 5 x27 (x232(27) 1 1) 5 x27 (x4 1 1) 4 5 x x17 1 Simplify using positive exponents. Example 3 2 1 Factor x3 2 7x3 1 12. Solution: Observe that x3 5 Ax3 B 2. Let u 5 x3, so the expression becomes a trinomial. 2 1 1 2 1 x3 2 7x3 1 12 5 u2 2 7u 1 12 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C07.pgs 3/26/09 12:12 PM Page 97 Name Class Date Factor the trinomial. u2 2 7u 1 12 5 (u 2 4)(u 2 3) 1 Substitute x3 for each u. x3 5 7x3 1 12 5 A x3 2 4 B A x3 2 3 B 2 1 1 1 Multiply to check. In 1–10, factor each expression. Write the answer without negative exponents. 5 1 1 3 4 2 1. y2 1 y2 2. x3 2 x3 3. x25 1 x5 4. x2 2 x21 5. b21 2 b 6. 2w212 1 w27 5 1 3 7. x22 1 x22 1 x2 9 4 1 8. c25 2 3c25 1 c5 9. 2x24 2 6x25 10. 5x 2 10x21 In 11–16, factor each expression. 1 11. x 1 6x2 1 9 2 1 14. 2y7 1 y7 2 3 2 1 12. x5 1 4x5 2 5 1 15. 8b 1 2b2 2 1 Copyright © 2009 by Amsco School Publications, Inc. 1 1 13. x2 1 x4 2 20 1 1 16. 15x3 2 14x6 1 3 14580TM_C07.pgs 3/26/09 12:12 PM Page 98 Name Class Date EXTENDED TASK Holes, Holes, and More Holes: An Exponential Investigation In this extended task, you will investigate the pattern created when a folded piece of paper has holes punched in it. For this task you will need about 20 small squares of thin paper, such as tissue paper or origami paper, approximately 5 3 5, and a hole punch. In this activity, you will perform three similar tasks, tabulate data about the tasks, and examine the data for patterns. Part I Task 1: Punch a hole in one paper square. Take a second paper, fold it in half, and punch a hole. Fold a third paper in half twice and punch a hole. Fold a fourth paper in half three times and punch a hole. Fold a fifth paper in half four times and punch a hole. Fold a sixth paper in half five times and punch a hole. Open up your papers and record the results of the task in the table below, listing the number of holes and then expressing that number as a power of 2. The first two entries in the table have been done for you. # of folds 0 1 # of holes 1 2 # of holes expressed as a power of 2 20 21 2 3 4 5 a. What pattern do you observe? b. If you fold the paper n times, how many holes will you have? c. Express h, the total number of holes in the paper, as a function of n, the number of times you fold the paper. Task 2: Repeat Task 1, but punch two holes in the paper each time rather than just one hole. Record your data in the table below. # of folds 0 1 # of holes 2 4 # of holes expressed as a power of 2 21 22 2 3 4 5 a. Do you observe a similar pattern this time? b. Express the numbers in row 3 as the constant 2 times a power of 2; that is, 21 5 2 3 20 and 22 5 2 3 21. c. Express h, the total number of holes in the paper, as a function of n, the number of times you fold the paper. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C07.pgs 3/26/09 12:12 PM Page 99 Name Class Date Task 3: Repeat Task 1 again, but this time punch 3 holes in the paper each time. Record your data in the table below and express the number of holes in row three as 3 times a power of 2. # of folds 0 1 # of holes 3 6 # of holes expressed as a power of 2 2 3 4 5 3 3 20 3 3 21 Express h, the total number of holes that you will have, as a function of n, the number of times you fold the paper. Part II On graph paper, or using a graphing calculator, graph the functions written in the three tasks of Part I and discuss the nature of your graphs. Part III The type of function investigated in this extended task is referred to as the exponential function. Do you think this is an appropriate title? Explain your answer. Part IV Exponential functions are often associated with growth and decay. Research growth and decay functions and give at least two real-world examples of this type of function. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C07.pgs 3/26/09 12:12 PM Page 100 Name Class Date Algebra 2 and Trigonometry: Chapter Seven Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–8, evaluate each expression. 1 1 1. 273 3. A 15 B 2. 14(25) 2 23 2 4. (28) 3 1 5. 2120 6. 0.1622 8. A 814 B 2 3 3 7. 3225 In 9 and 10, evaluate each function for the given value. 9. g(x) 5 2x 1 222x; g(3) 10. f(x) 5 A xx9 B 6 22 ; f(4) In 11–16, simplify each expression and write the answers with positive exponents. All variables represent positive numbers. x3y21 1 11. xy23 12. 1 13. (25c22d4) 2 (5c2d) 22 15. a 1 x4y3 b 1 x2 y 2 9ab2 3 3a21b2 2(x3y) 14. 2(x23y) 21 16. a y4 23 2b y25 In 17 and 18, write each expression as a power with positive exponents and express the answer in simplest form. All variables represent positive numbers. 4 17. "4x2 Copyright © 2009 by Amsco School Publications, Inc. 3 18. "64x6y7 14580TM_C07.pgs 3/26/09 12:12 PM Page 101 Name Class Date In 19 and 20, write each power as a radical expression in simplest form. All variables represent positive numbers. 3 1 20. (25x10y) 2 19. (3x) 4 21. a. Sketch the graph of y 5 A 45 B for the interval 22 # x # 2. b. Sketch the graph of the reflection in the y-axis of the graph drawn in part a. x c. Write an equation for the function graphed in part b. In 22–27, solve each equation. 1 22. 2x2 5 8 23. x22 1 1 5 5 24. 4 2 3x21 5 x 25. 511x 5 125 26. 9x 5 27x21 27. 42x 5 8 28. Jackie invested $3,000 at 4% per year compounded monthly. Find the value of Jackie’s investment after 2 years. 29. A tractor that cost $28,000 depreciates in value by 15% after each year. Find the expected value of the tractor after 7 years. 30. A five-year CD pays 4.5% compounded continuously. Find the value of a $10,000 investment at the end of the five years. a11 a Bonus I: Simplify: 222 21202 3 3 Bonus II: Determine if " !x 5 " !x, x $ 0. Justify your answer. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C07.pgs 3/26/09 12:12 PM Page 102 Name Class Date SAT Preparation Exercises (Chapter 7) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. The product 3 ? 321 is not equal to (B) 1 (C) 33 (A) 9–1 (E) 90 (D) 30 2. If 4x 5 8y, then the ratio of y to x is (A) 232 (B) 223 (C) 31 (D) 23 (E) 32 3. If x 5 x2 1 x22, then 21 is equal to (A) 0 (B) 1 (D) 414 (E) 16 (C) 214 4. If 44x25 5 64, then what is the value of x? (A) 22 (B) 212 (C) 21 (D) 32 (E) 2 4 4 4 5. If 2 1 22x 1 2 5 6, then what is the value of x? (A) 22 (B) 2 (C) 3 (D) 4 (E) 7 x 6. If 9.9999 equals 10 2 10 , then x is (A) 21 (B) 22 (C) 23 (D) 24 (E) 25 7. If 10,000 1 6,000 5 2x1y ? 5x, then y is (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 8. If 2x13 5 10, then 2x equals (A) 7 (B) 2 (C) 35 (D) 10 (E) 54 7 Copyright © 2009 by Amsco School Publications, Inc. 9. The perimeter of a square with an area of 24x is (A) 22x11 (B) 22x12 (C) 24x1 1 4x12 8x (D) 2 (E) 2 10. If 4x1k is 25% of 4x, then k is (A) 21 (B) 212 (C) 41 (D) 12 (E) 1 11. The reciprocal of A 12 B 22 can be written as each of the following except 1 1 1 (A) (2) 22 (B) (2) 2 (C) A 12 B 2 1 (D) 1 1 (2) 2 (E) #12 x x x x 12. 2 1 2 2x11 12 1 2 5 (A) 0 (B) 1 (D) 3 (E) 4 (C) 2 13. What is the average of 3x, 3x11, 3x12, and 3x13? 4x 1 6 (A) 14x16 (B) 34x15 (C) 3 4 (D) 10(3x) (E) 30x 14. If " !x 1 !x 1 !x 1 !x equals !2, then x is (A) 2 (B) 12 (C) 14 1 (D) !2 15. 10116 2 10117 5 (A) 210917 (D) 10917 (E) 2!2 1 (B) 210 1 (E) 10 (C) 10117 14580TM_C07.pgs 3/26/09 12:12 PM Page 103 Name Class II. STUDENT-PRODUCED RESPONSE QUESTIONS In questions 16–24, you are to solve the problem. 16. If 3n 5 27, what is the value of 6n21? 17. If 5y2 5 25, then what is the value of 5y22? n 6 23 n 18. If 4 5 16 , then what is the value of n? n23 19. If 7 5 7 ? 7 of n? , then what is the value Copyright © 2009 by Amsco School Publications, Inc. Date 20. For what non-negative integer value of x will A 235 B 2x 1 1 be the smallest? 515 n 1 21. If 5 5 125 , what is the value of n? 22. If N* means NN23, then what is the value of (3*)*? 10 23. If 33x 1 3 5 27 3 , what is the value of x? 24. If f(x) 5 2x, what is the value of f(102) ? 2f(2100)? 14580TM_C08.pgs 3/26/09 12:12 PM Page 104 CHAPTER 8 LOGARITHMIC FUNCTIONS Aims • To define a logarithmic function as the inverse of an exponential function. • To apply the properties of logarithms to write equivalent logarithmic expressions. • To evaluate logarithmic expressions in any base. • To solve exponential and logarithmic equations. • To graph logarithmic functions. This chapter emphasizes the relationship between exponential and logarithmic forms and develops the skills students need to solve exponential and logarithmic equations. Students use their calculators to obtain the logarithmic values needed for their computations. CHAPTER OPENER Logarithms were first developed as an aid in computation and as such made possible the advancement of sciences such as astronomy that require long and tedious computations. Today, the development of high-speed computers has rendered obsolete the use of logarithms for computational purposes. Nevertheless, an understanding of logarithms 104 and the computational rules that govern their use is essential for the study of advanced mathematics. To impress upon students how their studies have been simplified by the availability of powerful hand calculators, you may wish to display an old table of logarithms. Explain that learning how to use this table required several lessons and involved terms such as mantissa and characteristic. 8-1 INVERSE OF AN EXPONENTIAL FUNCTION Under a reflection in the line whose equation is y 5 x, the image of the graph of an exponential function is the graph of the inverse of the exponential function. Since under this reflection the image of (x, y) is (y, x), the equation of the function that is the inverse of y 5 bx can be written as x 5 b y . However, the equation defining a function is usually written in a form that expresses y in terms of x. To solve x 5 by for y in terms of x, new notation is introduced. In the equation x 5 by, y is the value of the exponent and x is the value of the power. To solve this equation for y, we describe y in words and then replace the words with symbols and abbreviations that maintain the essential idea of the description. Encourage 14580TM_C08.pgs 3/26/09 12:12 PM Page 105 Logarithmic Functions 105 students to identify the core idea of the statement 3 5 log2 8 as “3 is the log” or “3 is the exponent.” Then the other parts can be added so that this statement is read as “3 is the exponent that must be used with the base 2 to give the power 8.” Call attention to the domains and ranges of the exponential and logarithmic functions. Since these are inverse functions, the domain of one is the range of the other. Since the base, b, of an exponential function is always positive, the function value, bx, is always positive. In other words, the range of the exponential function y 5 bx is the set of positive real numbers (y . 0), and the domain of the logarithmic function y 5 logb x is the set of positive real numbers (x . 0). Note that the domain of the exponential function and the range of the logarithmic function are unrestricted; both are the set of all real numbers. Since y 5 bx and y 5 logb x are inverses, their composites are the identity function. If f(x) 5 logb x and g(x) 5 bx, then f(g(x)) 5 logb bx 5 x and g(f(x)) 5 blogb x 5 x Explain to students that to x graph a logarithmic function such 1 as y 5 log3 x, they should change it 9 to exponential form, x 5 3y. Using 1 3 x 5 3y, they can construct a table 1 of values. The table is easier to develop if values are selected for y 3 and then corresponding x-values 9 are found. Alternatively, they can find the ordered pairs for y 5 3x and then interchange x and y in each ordered pair. y 22 8-2 LOGARITHMIC FORM OF AN EXPONENTIAL EQUATION This section continues to focus on recognizing the meaning of an expression such as log 5 25 5 2. After selecting the key idea (log 5 2), the student will more easily recognize the base as 5 and the value of the power as 25 in order to write the equivalent expression 52 5 25. Students usually consider exponential form to be easier than logarithmic form because exponents are more familiar to them, and will initially want to change the log equations into the more familiar exponential equations. Encourage them to become familiar, through repeated practice, with the log form as well. Some students may ask why they cannot use the LOG key on their calculators to do the exercises in this section. Explain that this key gives logarithms to the base 10 only. In Section 8-4 they will learn to use this key, and in the Hands-On Activity of Section 8-5 they will derive a formula that can be used to find logs to any base on a calculator. As students prepare to begin the exercises, remind them that they can find the value of a variable in a logarithmic equation logb x 5 y when values for two of the variables are known. 21 0 1 2 8-3 LOGARITHMIC RELATIONSHIPS The product rule, the quotient rule, and the power rule are derived in this section. Call attention to the fact that these rules are simply restatements of the laws of exponents. Compare the operations involved by making a chart such as the following: Operations Using Powers Operations Using Logs Multiplication Addition Division Subtraction Raising to a power Multiplication 14580TM_C08.pgs 3/26/09 12:12 PM Page 106 106 Chapter 8 Point out that the identity logb an 5 n logb a states that the log of a power is the product of two logs, logb a and n. Since n is an exponent, it is a log. For example, log2 323 5 3 log2 32 5 3(5) 5 15 Here it was necessary to recall that 25 5 32, but 3 is already a log since it is an exponent. Another helpful example is to have stu2 dents compare the expanded forms of log5 x3 2 and log5 A x3 B . 2 log5 x3 5 log5 x2 2 log5 3 Quotient rule 5 2 log5 x 2 log5 3 If time allows, have students find the log of 14 and the log of 140. log 14 < 1.1461 log 140 < 2.1461 They should notice that log 14 and log 140 have the same decimal part, 0.1461, called the mantissa, but different integer parts, 1 and 2. The integer part of a common logarithm of a number is called its characteristic and indicates the magnitude of the number. The characteristic is the exponent of 10 when the original number is expressed in scientific notation. So, 14 5 1.4 3 101 and 140 5 1.4 3 102. Also, log (140) 5 log (14 3 10) 5 log 14 1 log 10 5 1.1461 1 1 Power rule However, 2 log5 A x3 B 5 2 log5 A x3 B Power rule 5 2.1461 Have students predict values for log 0.014 and log 1,400,000. 5 2flog5 x 2 log5 3g Quotient rule 8-5 5 2 log5 x 2 2 log5 3 The examples and the exercises in this section give students practice in applying the rules for logarithms before they make use of calculators to evaluate logs. 8-4 COMMON LOGARITHMS This section introduces the student to the use of the calculator to find the common logarithm of a number and to find the number, or antilogarithm, when the common logarithm of that number is known. An alternative way of verifying the log of 8 that does not require storing the value is: ENTER: LOG 8 ENTER 2nd 10x 2nd ANS ENTER DISPLAY: NATURAL LOGARITHMS Since e is the base for natural logarithms, ln e 5 1. All properties of logarithms that students have learned apply to natural logarithms as well. The antilogarithm in base e is often denoted antiln x. Students should be aware that when they change a natural logarithm to exponential form, the base of the exponential expression must be e. If y 5 ln x, then ey 5 x. Since y 5 ln x is the inverse of y 5 ex, it follows that: ln ex 5 x eln x 5 x The general change of base formula is usually stated as follows: • For all a ., b . 0, and x . 0, a 1, b 1, log x loga x 5 logb a b log(8 .903089987 ^(Ans 8 14580TM_C08.pgs 3/26/09 12:12 PM Page 107 Logarithmic Functions 107 In the change of base formula, 10 is often used in place of base b because we know how to find common logarithms. Replacing base b with 10, we get log x log x loga x 5 log10 a or loga x 5 log a 10 Since e is a smaller base than 10, for x . 1 it follows that ln x . log x. The graph of y 5 ln x is above the graph of y 5 log x for x . 1. Enrichment Activity 8-5: Finding e presents a complex fraction that approximates e. Many other expressions have been used to represent this important number. 8-6 EXPONENTIAL EQUATIONS When both sides of an exponential equation cannot be written as a power of the same base, we often begin by taking the logarithm of both sides of the equation, as illustrated in the solution of 5x 5 32. Point out that while Example 1 could be solved using either common or natural logs, the solution to Example 3 should use natural logs because the formula involves e. Exercises 15–20 illustrate some of the many applications for exponential equations in business and the sciences. In Enrichment Activity 8-6: State Population Growth, students use researched data about their home state to create linear and exponential models. They use their models to predict population size in the future. 8-7 LOGARITHMIC EQUATIONS Emphasize that before the rule logb A 5 logb C S A 5 C can be used to solve a logarithmic equation, it is necessary to write each side of the equation as a single log with the same base. Two methods are shown for Example 1. A computational method is a third alternative. Evaluate ln 12 2 ln 3 using a calculator. Then use the result as the exponent for ex. ENTER: LN 12 ⴚ LN 3 2nd ANS ) ) ENTER 2nd ex ENTER DISPLAY: ln(12)-ln(3) 1.386294361 e^(Ans 4 The answer 4 will be displayed. Note that the parenthesis must be inserted after the 12. Example 2 shows that logarithmic equations may have extraneous solutions. When examining solutions, if you obtain the logarithm of a non-positive number, that answer is not a solution to the equation. EXTENDED TASK For the Teacher: This activity attempts to show students how a topic in pure mathematics was applied by a scientist (Richter) to a real-world phenomenon. Although the activity does not require students to use logarithms directly, it does demonstrate to students the relationship of mathematics (logarithms) to the real world. Students should be encouraged to research earthquakes and the Richter scale to better understand the theory behind the nomogram. There are many references to earthquakes in the library and on the Internet. Students might also research other realworld phenomena that apply logarithms. These include, but are not limited to: astronomy (brightness of a star), compound interest, sound (decibels), musical scales, f-stop on a camera, and atmospheric pressure. 14580TM_C08.pgs 3/26/09 12:12 PM Page 108 108 Chapter 8 Some of the major twentieth-century earthquakes that were recorded at 7.0 or above on the Richter scale are listed in the table below. Your students may find others. Year Location Size Year Location Size 1906 San Francisco 8.3 1992 Indonesia 7.5 1933 Japan 8.9 1993 Japan 7.7 1964 Alaska 8.5 1994 Sumatra 7.0 1976 China 8.0 1995 Russia 7.5 1978 Iran 7.7 1995 Mexico 7.6 1980 Italy 7.2 1996 China 7.0 1985 Mexico 8.1 1996 Indonesia 7.5 1989 San Francisco 7.1 1997 Pakistan 7.3 1990 Iran 7.7 1997 Iran 7.5 1990 Philippines 7.8 1998 Indonesia 7.8 1991 India 7.0 1999 Mexico 7.5 1992 California 7.5 For each 0.3 difference on the Richter scale, an earthquake is 100.3 or almost twice as powerful. For example, the 1997 earthquake in Pakistan had almost twice the magnitude of the 1996 earthquake in China, and the 1933 earthquake in Japan had 100.6 or almost four times the magnitude of the 1906 earthquake in San Francisco. 14580TM_C08.pgs 3/26/09 12:12 PM Page 109 Name Class Date ENRICHMENT ACTIVITY 8-5 Finding e One method of representing the irrational number e is with the complex fraction shown below. 1 21 1 11 1 21 1 11 1 11 1 41 1 11 1 11 1 61 11 1 1 11 811... 1. Evaluate the part of the fraction shown by entering it into a calculator as follows: 2 1 1 4 (1 1 1 4 (2 1 1 4 (1 1 1 4 (1 1 1 4 (4 1 1 4 (1 1 1 4 (1 1 1 4 (6 1 1 4 (1 1 1 4 (1 1 1 4 (8 1 1) 2. Write the number displayed. 3. Does this fractional expression appear to give a good approximation of e? 4. Research the history of mathematicians’ development of understanding of e. Include other representations, such as infinite series, equations, and formulas that have been used to approximate e. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 110 Name Class Date ENRICHMENT ACTIVITY 8-6 State Population Growth Use a reference book or Internet source to find the following information for the state in which you live. Name of state: Population in 1960: Population in 2000: 1. What was the change in population from 1960 to 2000? Be sure to indicate if the change was positive or negative. 2. What was the percent change from 1960 to 2000? 3. If the growth pattern from 1960 to 2000 was linear, what was the average yearly change in population for the state? 4. Write a linear equation to model the population change. 5. Use the linear model to predict the state population for: a. 2010 b. 2025 c. 2100 6. Suppose the population grew continuously during the period from 1960 to 2000. a. Write an exponential equation that can be used to find the yearly rate of growth. b. Find the growth rate to the nearest hundredth of a percent. 7. If the growth rate remains the same, predict the population for each year using the exponential model. a. 2010 b. 2025 c. 2100 8. Compare your predictions from the linear and exponential models. 9. If your state had a positive growth rate, how many years at that rate would it take for the 2000 population to double? If the growth rate was negative, how many years would it take for the 2000 population to be halved? 10. What rate of growth is needed for a population to double in 10 years? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 111 Name Class Date Extended Task Calculating the Magnitude of an Earthquake: A Mathematical Application Two measures are usually associated with earthquakes. The first of these, intensity, has to do with the degree of damage done to structures, with the amount of disturbance to the surface of the ground, and with the extent of animal reaction to the shaking. Hence, intensity depends upon the density of the population, the type of construction, the type of ground, and so on. In order to compare earthquakes worldwide, a measure is needed that does not depend upon the variables associated with intensity. The term magnitude is used for this measure. In California in 1936, Charles Richter developed a strictly quantitative scale to measure the magnitude that can be applied to earthquakes worldwide, whether the area is inhabited or uninhabited. The scheme was to use wave amplitude measured by a seismograph. Although there have been some modifications, the Richter scale, developed to measure the magnitude of an earthquake, is still in use today. Because the size of quakes varies enormously, the amplitudes of the waves differ by factors of thousands from earthquake to earthquake. It is therefore most convenient to compress the range of wave amplitudes by using some mathematical concept. Richter defined the magnitude of a local earthquake as the logarithm to the base 10 of the maximum seismic wave amplitude (in thousandths of a millimeter) recorded on a standard seismograph at a distance of 100 kilometers from the earthquake’s epicenter. This means that every time the magnitude goes up by 1 unit, the amplitude of the earthquake waves increases by more than 10 times. A nomogram can be used to find the magnitude of a quake on the Richter scale. The nomogram uses the difference in arrival times of the S- and P-waves and the amplitude to find the magnitude of the quake. The P-wave is the primary or fastest wave traveling away from a seismic event through the rocks. A P-wave compresses and expands the material in the direction it is traveling. The S-wave is the secondary seismic wave, traveling more slowly than the P-wave. An S-wave compresses and expands the material perpendicular to the direction it is traveling. The amplitude is the height of the maximum wave motion on the seismogram. To use the nomogram: 1. Find the difference in arrival times of the S- and P-waves by calculating the difference between the end of the P-wave and the end of the S-wave. 2. Measure the height of the maximum wave motion on the seismogram. 3. Place a straightedge between appropriate points on the distance and amplitude scales to obtain the magnitude. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 112 Name Class Date The diagram below illustrates this process for an amplitude of 23 millimeters and a 24-second difference in the arrival times of the S- and P-waves. P S 0 10 20 S – P = 24 sec 500 400 300 200 50 40 20 5 2 100 50 20 10 5 2 1 0.5 5 20 10 8 6 4 Amplitude = 23 mm 6 30 100 60 40 10 20 30 Nomogram 4 3 2 0.2 0.1 1 0 Magnitude 0 Distance S – P (km) (sec) Amplitude (mm) The magnitude of this earthquake is 5 on the Richter scale. The magnitude is obtained by drawing a line from the amplitude and the difference in arrival times on the scales. Notice also that this earthquake is about 200 kilometers away. Use the nomogram to find the magnitude of the earthquakes shown in the table below. The table gives the distance in kilometers and the amplitude in millimeters. Distance (kilometers) Amplitude (millimeters) 1. 5 50 2. 60 3. 100 100 4. 40 50 5. 300 1 0.5 Copyright © 2009 by Amsco School Publications, Inc. Magnitude 14580TM_C08.pgs 3/26/09 12:12 PM Page 113 Name Class Date Algebra 2 and Trigonometry: Chapter Eight Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–3, solve each equation for y in terms of x. 1. x 5 7y 2. x 5 82y 3. x 5 log6 y In 4–9, in each case: a. Write the expression in exponential form. b. Solve for x. 4. 0 5 log6 x a. b. a. b. 6. 4 5 logx 4 a. b. 7. x 5 log4 32 a. b. 1 8. log3 27 5x 9. logx 0.04 5 22 a. b. a. b. 3 2 5. log25 x 5 10. If f(x) 5 log4 x, find f(1,024). 11. If h(x) 5 log32 x, find h(16). 12. Write the expression 1022 5 0.01 in logarithmic form. In 13–18, for each equation, write log x in terms of log a, log b, and log c. 13. x 5 ab2 !a 14. x 5 bc 3 15. x 5 c#ba5 3 16. x 5 A ac b3 B 2 3 2 17. x 5 " ab 18. x 5 a2b !c Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 114 Name Class Date In 19–22, for each equation, write N in terms of a, b, and c. 19. log N 5 log a 1 2 log b 2 log c 20. log N 5 12 log a 2 2(log b 1 log c) 21. log N 5 12 (log a 1 2 log b) 2 log c 22. log N 5 2 log a 1 13 (log b 2 log c) 23. If log 463 5 b, write an expression for log 46.3 in terms of b. 24. Solve for x: log4 (x 2 2) 1 log4 (x 1 5) 5 2 log4 (x 1 1) 25. Solve for x: 2 log5 x 2 log5 (2x2 2 1) 5 2 26. a. Evaluate each expression to four decimal places. (1) 3 log 50 (2) log (3 3 50) (3) log 503 (4) log 3 1 log 50 b. Make two observations based on the evaluation of the expressions in part a. In 27–29, solve each equation for x to the nearest hundredth. 27. 27x 5 156 28. 3x11 2 12 5 24 x 29. 332 5 15 In 30–32, evaluate each expression to four decimal places. 30. ln 4.52 1 ln 0.085 31. ln 10,000 2 ln 999 32. (ln 0.1278)2 33. At the end of each year, employees of the Plumtree Software Company receive a pay increase of 2.5%. Devon had a starting salary of $24,000. For how many years has Devon worked for Plumtree Software if after his last wage increase his salary is over $58,000? 34. The Chans are saving money to go on a trip to Europe for their twenty-fifth wedding anniversary. They have five years to save $9,000 for the trip. If the five-year CD they buy now pays 514% interest compounded continuously, how much do they need to invest now in order to have $9,000 for the trip? (Use A 5 Pert.) 35. Find log9 22 to the nearest thousandth. 36. Find log8 77 to the nearest thousandth. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 115 Name Class Date In 37–39, find the value of each logarithm using the values log10 2 5 0.3010 and log10 5 5 0.6990. Show your work on a separate piece of paper. 37. log10 0.4 38. log10 2.5 39. log10 25 40. a. Sketch the graph of y 5 log5 x in the interval 0.04 # x # 25 on a separate piece of graph paper. b. On the same set of axes, sketch the graph of y 5 5x in the interval 22 # x # 2. c. Under what transformation is the graph drawn in part b the image of the graph drawn in part a? Bonus: Solve for x: logx 4 1 logx 9 5 2 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C08.pgs 3/26/09 12:12 PM Page 116 Name Class Date SAT Preparation Exercises (Chapter 8) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If V 5 pr 2h, then log V equals (A) 2p(log r 1 log h) (B) p log 2r 1 log h (C) 2 log pr 1 log h (D) log p 1 2 log r 1 log h (E) (p log r2)(log h) 8. The average of log 2 and log 18 is (A) log 6 (B) log 9 (C) log 10 (D) log 20 (E) log 36 9. If x 5 log2 5, then 4x is (A) 2 (B) !5 (D) 10 (E) 25 10. 2. If log 7 5 m, then log 490 equals (A) 2m 1 1 (B) m2 1 1 (C) 10m (D) 10m2 (E) 70m log2 8 If the area of the rectangle is 1, then the height is (A) log2 4 (B) log2 3 (C) log2 13 3. If log x 5 0.3721, then (A) 21 , x , 0 (B) 0 , x , 0.1 (C) 0.1 , x , 1 (D) 1 , x , 10 (E) 10 , x , 100 4. If 4x 5 8y, then the ratio of x to y is closest to (A) 2 (B) 32 (C) 34 (D) 32 (E) 12 5. If log x 1 log (x 1 3) 5 1, then which of the following could be a value of x? (D) log2 18 (E) log8 2 11. If log2 x is 25% of log2 y, then y, expressed in terms of x, is (A) 14x (B) 4x (C) x4 4 (E) !x (D) x4 12. c I. 25 (C) !10 b II. 2 III. 5 (A) I only (C) III only (E) II and III only 6. (B) II only (D) I and II only 2(2a) 2aq 2b In the division exercise above, b equals (A) 2a (B) 2a 1 1 (C) 2a 2 1 (D) 2a 1 2 (E) 2a 2 2 7. If 2 log x 5 log 2x, then x is (A) 0 (B) 1 (C) 2 (D) 4 (E) any number Copyright © 2009 by Amsco School Publications, Inc. a For a, b, and c shown above, 2 log c equals (A) log 2a 1 log 2b (B) 2 log a 1 2 log b (C) 2 log (a 1 b) (D) log (a2 1 b2) (E) log (2a 1 2b) 13. If the volume of a cube is 63a, then the surface area of the cube is (A) 62a+1 (B) 6(32a) (C) 6(26a) 3a 2a+1 (E) 3 (D) 6(2 ) 14580TM_C08.pgs 3/26/09 12:12 PM Page 117 Name Class 14. If loga b2 5 21, then a, expressed in terms of b, is (A) 2b (B) b2 (C) 22b (D) 22 (E) b2 b 1 15. log xy 5 (A) log x 1 log y (B) log x 2 log y (C) 1 2 log xy (D) 2(log x 1 log y) (E) 1 2 log x 2 log y Date 16. If y 5 log2 x, then what is the value of x when y 5 3? 17. Solve for z: log4 (log2 16) 5 z 18. Solve for y: log y 5 14 log 16 1 12 log 81 19. If log (0.1) ? log x 5 3, then what is the value of x? 20. What is the exact value of log5 25 !5? 21. Solve for w in the equation 2w(1,000) 5 2,048,000. 22. Solve for x: logx 27 1 logx 3 5 2. 23. Solve the equation for m: II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. Copyright © 2009 by Amsco School Publications, Inc. log5 32 2 4 log5 m 5 3 log5 12 24. If log x3 2 log y 5 4 and log x 1 log y 5 2, what is the value of log xy? 14580TM_C09.pgs 3/26/09 12:13 PM Page 118 CHAPTER 9 TRIGONOMETRIC FUNCTIONS Aims • To define the sine, cosine, and tangent functions. • To learn the reciprocal trigonometric functions: secant, cosecant, and cotangent. • To use a calculator to find trigonometric function values, when defined, for angles of any measure. CHAPTER OPENER Eratosthenes used the fact that at noon on the summer solstice, the longest day of the year, at Syene (the ancient name for Aswan), there were no shadows because the sun was directly overhead. However, at the same time on the same day in Alexandria, some 500 miles to the north, the angle formed by the top of a tower and the edge of its shadow was 7.2°. • To use a calculator to find the degree measure of an acute angle in Quadrant I when given a trigonometric function value. 7.2° Alexandria 7.2° • To use reference angles to determine trigonometric function values for angles in Quadrants II, III, and IV. • To derive basic trigonometric identities. In this chapter, another type of function, the trigonometric function, is defined in terms of the length of a line segment associated with the unit circle. The trigonometric functions relate measures associated with circular motion to measures associated with linear motion. This relationship is a common one in everyday life and is applied in many types of machinery. For example, in an automobile, the linear motion of the pistons is transformed into the circular motion of the wheels. The trigonometric functions describe mathematically this relationship of circular motion and linear motion. 118 Syene Center of Earth Assuming the sun’s rays are parallel, the angle at the center of Earth is congruent to the angle of the shadow (alternate interior angles). Then, using the fact that there are 360° in a full rotation, the following proportion can be written to find Earth’s circumference: 500 7.28 9-1 x 5 3608 and x 5 25,000 miles TRIGONOMETRY OF THE RIGHT TRIANGLE This section reviews the trigonometry of the right triangle, which most students will have studied in previous courses. 14580TM_C09.pgs 3/26/09 12:13 PM Page 119 Trigonometric Functions 119 If students are familiar with sine, cosine, and tangent expressed as ratios of the measures of the sides of a right triangle, it may be useful to review these ratios and to show how they are special cases as the more general definitions are developed. Exercise 1 focuses on the cofunction relationship sin u 5 cos (90 2 u). In Example 2 and Exercises 12 and 13, the special characteristics of the 45-45-degree right triangle and the 3060-degree right triangle are used to find trigonometric function values. For Exercises 18 and 19, remind students that an angle of depression or elevation is the angle formed by the line of sight and a horizontal line. 9-2 ANGLES AND ARCS AS ROTATIONS In previous courses, students have worked only with angles whose degree measure was limited to values between 0 and 180. This section, by considering an angle as a rotation, introduces the use of any rational number as the degree measure of an angle. Intuitively, it is clear that any real number can be the degree measure of an angle. However, this is not formally established until Section 10-4, after arc length has been introduced. Just as we find it convenient to associate linear distance with the real number line, so we find it convenient to associate angle measure with the coordinate plane. On the number line, any point to the right of 0 is associated with a positive number and any point to the left of 0 is associated with a negative number. Similarly, from the non-negative ray of the x-axis (the initial side of the angle), any angle measured counterclockwise has a positive measure and any angle measured clockwise has a negative measure. Coterminal angles are angles in standard position that have the same terminal side. Since the location of a point can be described by using a simpler coterminal angle measure, this concept will play a major role in simplifying and finding trigonometric function values. For example: sin 7508 5 sin (7508 2 3608) 5 sin 3908 5 sin (3908 2 3608) 5 sin 308 9-3 THE UNIT CIRCLE, SINE, AND COSINE This section uses the definitions of the sine and cosine as ratios of the measures of the sides of a right triangle in order to motivate the definitions of the sine and cosine functions. As the cases for the four quadrants are examined, emphasize that the ratios of the xcoordinate and y-coordinate of a point on the terminal side of an angle in standard position to the distance of the point to the origin do not depend upon the particular point but, rather, depend entirely on the angle. Thus, we can define functions from the set of angles in standard position to the sets of all such ratios. These ratios we call the sine and cosine of the angle, respectively. Since the distance d from the origin to the point (p, q) on the terminal side of any angle u is "p2 1 q2, we have q q or d "p2 1 q2 p p or d cos u 5 2 1 q2 p " sin u 5 For the unit circle "p2 1 q2 5 1, so q sin u 5 1 5 q p cos u 5 1 5 p Therefore, the point at which the terminal side of an angle in standard position with measure u intersects the unit circle has the coordinates (cos u, sin u). Encourage students to memorize the quadrants where sine and cosine are positive and where they are negative. Be sure students understand that the sign of the function value is determined by the quadrant where the terminal side lies, not by the sign of the measure of the angle. 9-4 THE TANGENT FUNCTION The tangent function assigns to every angle measure u a unique value t that is the y-coordinate of the point where the terminal side of 14580TM_C09.pgs 3/26/09 12:13 PM Page 120 120 Chapter 9 the angle intersects the line tangent to the unit circle at point (1, 0): tangent uht The illustrations for Case 5 show that the tangent is undefined for u 5 90° and u 5 270° because there is no intersection. The proof sin u that tan u 5 cos u is then given in Example 4. Since division by 0 is undefined, it therefore follows that the tangent is undefined for angles that have a sine of 0, that is, angles that have their terminal sides on the x-axis, namely 90° and 270°. The diagram below can help students remember the quadrants where the function values are positive: S Sine positive A All function values positive T Tangent positive C Cosine positive Students can use the mnemonic All Students Take Calculus. They may also enjoy making up other phrases having words with the relevant first letters and, in so doing, will remember the key ideas. 9-5 THE RECIPROCAL TRIGONOMETRIC FUNCTIONS Three new functions are defined in this section. For each function defined here, the function value is the reciprocal of the function value of a previously defined trigonometric function. For example: sec u 5 cos1 u (cos u 2 0) The domain of the cosine function is the set of all real numbers. The domain of the secant function is the set of all real numbers such that the cosine value for that real number is not 0. Therefore, the domain of the secant function is the set of all real numbers that are not 90 plus a multiple of 180 (90 1 180k, for any integer k). For every real number u that is in the domain of the secant function, if (u, c) is a pair of the cosine function, then A u, 1c B is a pair of the secant function. Notice that the second elements are reciprocals when the first elements are the same. Similar relationships exist between the domain and the range of the sine and cosecant functions and of the tangent and cotangent functions. Since sin u and tan u are 0 for any value of u that is a multiple of 180°, these values are excluded from the domain of the cosecant and cotangent functions. tan u is undefined for real numbers that differ from 90° by a multiple of 180°. By using the u definition cot u 5 cos sin u , we can define cot u as 0 for these values of u: cot p2 5 cos p 2 sin p 2 5 01 5 0 Notice that the secant, cosecant, and cotangent functions can be defined in terms of the sine, cosine, and tangent functions or in terms of the lengths of line segments associated with the unit circle. Finding values for reciprocal functions with a calculator is covered in Section 9-7. 9-6 FUNCTION VALUES OF SPECIAL ANGLES Although the derivations given in this section can be used whenever it is necessary to find an exact value of a trigonometric function that corresponds to an angle of 30°, 45°, or 60°, encourage students to memorize these function values, since they will be used frequently in the applications of the trigonometric functions. In this section, exercises are limited to trigonometric functions of angles with measures of 30°, 45°, and 60° and to function values of quadrantal angles. 14580TM_C09.pgs 3/26/09 12:13 PM Page 121 Trigonometric Functions 121 9-7 FUNCTION VALUES FROM THE CALCULATOR The calculator can be used to find the values of trigonometric functions for angles other than 30°, 45°, and 60° and to find the measure of an angle when one of its function values is known. Students should be made aware that, in some situations, a solution may require an exact function value. For this reason, it is important that students memorize function values for 30°, 45°, and 60° angles as presented in Section 9-6. In other situations and for angle measures other than those special values, we use the rational approximations displayed as function values by the calculator. Approximate function values are often rounded to the nearest ten-thousandth (four decimal places). Students should be taught how to determine function values of angles: first, with integral degree measures, such as 47° and 238°; second, with measures containing a portion of a decimal degree, such as 62.5°; third, with measures given in degrees and minutes, such as 75° 169. An angle measure of 25 degrees 9 minutes 27 seconds is written as 25° 99 270. Some calculators, such as the TI-831/841, can display angle measures in this format, called degree/minute/second (or DMS) mode. • To convert an angle in degrees to DMS mode, enter the angle, then 4 . press 2nd ANGLE • To write an angle in DMS mode, use ° (degree) and 9 (minutes) from the ANGLE menu, and 0 (seconds) or ⴖ . For example, to write ALPHA 25° 99 270, ENTER: 22 2nd 9 2nd ANGLE ANGLE 27 ALPHA 1 2 ⴖ Not all calculators have this feature, however, and students must learn how to enter a measure using degrees and minutes into any graphing calculator. Since there are 60 minutes of angle measure in 1 degree, and 60 seconds of angle measure in 1 minute, it follows that 9 27 258 9r 27s 5 A 25 1 60 1 3,600 B 5 25.15758 In the text, by disregarding seconds, we limit angle measures to degrees and minutes. Therefore, a measure such as 37° 459 is entered into a calculator as a mixed number of degrees, that is, as (37 1 45 4 60). For Example 2, sec 54° can also be found using the reciprocal key on the calculator: 8 ENTER: COS 54 ) x ⴚ1 ENTER Examples 4 and 5 show how to use the inverse capabilities of a calculator to find the measure of an angle when one of its function values is given. Students will learn more about inverses in Section 10-5. Positive function values, as shown in this section, lead to angle measures between 0° and 90°, or angles in Quadrant I. These acute angles can be regarded as reference angles that relate to all quadrants, as we will see in Section 9-8. For this reason, trigonometric function values are restricted here to positive numbers. If students attempt to enter negative function values on a calculator, they will discover that: • For sin21 and tan21, u is displayed as a negative angle between 0° and 290°, that is, as an angle in Quadrant IV. • For cos21, u is displayed as a positive angle between 90° and 180°, that is, as an angle in Quadrant II. Negative trigonometric function values will be studied in Chapter 10 as trigonometric graphs are drawn and special functions are defined. Enrichment Activity 9-7: Reflection and Refraction explains the physical laws that describe the behavior of light. The activity provides an application of finding trigonometric functions of a given angle and finding the angle when its sine is known. 14580TM_C09.pgs 3/26/09 12:13 PM Page 122 122 Chapter 9 9-8 REFERENCE ANGLES AND THE CALCULATOR The use of reflections in the axes or the origin simplifies the derivation of the reference angle formulas, helps to reinforce students’ understanding of transformations, and demonstrates a practical application of transformations. Again, emphasize that the sign of the function value is dependent on the quadrant in which the terminal side of the angle lies, not on the sign of the angle measure. Be sure students are aware of the differences in the two methods presented in Example 4. As discussed in Section 9-7, the calculator returns a negative angle measure between 0° and 290° when the arcsine of a negative value is requested. This negative angle is the reflection of a positive first-quadrant reference angle. This positive angle must be used to find the third-quadrant angle required in the example. EXTENDED TASK For the Teacher: This extended task helps students connect mathematics to the real world. You might wish to have students do some research on pilot rules and regulations for flying. The extended task provides an opportunity for some career education. If you live in the proximity of an airport, you may also wish to invite a pilot or air traffic controller into your classroom to discuss the role that mathematics plays in his or her job. 14580TM_C09.pgs 3/26/09 12:13 PM Page 123 Name Class Date ENRICHMENT ACTIVITY 9-7 Reflection and Refraction When light passes from one medium (such as air) into another medium (such as water), part of the incident light is reflected and part is refracted. Reflection of light at any interface follows a simple law—the angle of incidence (u1) of the incident ray equals the angle of reflection (u19) of the reflected ray. The refracted ray is the ray that enters the new medium. The angle relationship between the incident ray and the refracted ray is given by Snell’s law: incident ray normal u1 u19 air water u2 n1 sin u1 5 n2 sin u2 where n1 and n2 are the indices of refraction of the two media, u1 is the angle of incidence, and u2 is the angle of refraction. The index of refraction of a medium is the ratio of the speed of light c in a vacuum to the speed of light v in the indexed medium: Indices of Refraction Benzene 1.501 Diamond 2.419 Dry Air 1.000 Ethyl Alcohol 1.362 Plate Glass 1.523 Example 1 Quartz 1.458 A light ray passes from air to water with an angle of incidence equal to 30°. Find the angle of refraction (to the nearest degree). Solution: According to Snell’s law, Water 1.333 n 5 vc The table gives the index of refraction for some common materials. Since the index of refraction of air is very close to 1, it is usually treated as such in computations. n1 sin u1 5 n2 sin u2 In this case, n1 5 1.000, sin u1 5 sin 30° 5 0.5, and n2 5 1.333. (1.000)(0.5) 5 1.333 sin u2 0.5 1.333 5 sin u2 0.3751 5 sin u2 u2 5 228 Use a calculator to find the angle that has a sine equal to 0.3751. (nearest degree) Copyright © 2009 by Amsco School Publications, Inc. reflected ray refracted ray 14580TM_C09.pgs 3/26/09 12:13 PM Page 124 Name Class Date Exercises 1. The angle of incidence of a light ray passing from air to plate glass is 60°. Find the angle of refraction to the nearest degree. 2. The angle of incidence of a light ray passing from air to a diamond is 45°. Find the angle of refraction to the nearest degree. 3. The speed of light is about 186,000 miles per second. To the nearest mile per second, what is the speed of light in: a. Benzene b. Quartz c. Water 4. The angle of incidence of a light ray passing from air to a different medium is 30°. The angle of refraction is 17.6°. Find the index of refraction, to the nearest thousandth, of the medium. Total internal reflection occurs when there is no refracted light––all the incident light is reflected. In order for total internal reflection to take place, the following two conditions must be met: • The index of refraction of the medium where the light ray starts is greater than the index of the medium the light ray approaches. (The starting medium is more dense than the second medium.) normal n2 n1 • The angle of incidence is greater than the critical angle, uc. uc The critical angle occurs when the angle of refraction is 90° and depends on the medium. Different media have different critical angles. To find the critical angle for a medium with index n1 traveling into a less dense medium with index n2, substitute u2 5 908 into Snell’s law: n1 sin uc 5 n2 sin 908 n sin uc 5 n2 ? 1 1 n sin uc 5 n2 1 For any incident angles that are greater than the critical angle, all light will be reflected back into the first medium. Total internal reflection is the principle on which fiber-optic cable works. Light injected at the end of the cable will be internally reflected many times over its entire length because the cable is made of light-transmitting material with an index greater than 1. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C09.pgs 3/26/09 12:13 PM Page 125 Name Class Date Example 2 What is the critical angle for water with respect to air? Solution: The index of refraction for water is 1.333. The critical angle is the one where the angle of refraction is 90°. sin uc 5 1.000 1.333 sin uc 5 0.7502 (nearest ten-thousandth) uc 5 48.68 < 498 normal air 1.000 water 1.333 Exercises In 5–9, find the critical angle of each material with respect to air. Round to the nearest degree. 5. Benzene 6. Ethyl alcohol 7. Diamond 8. Plate glass 9. Quartz 10. Find the critical angle to the nearest degree for a fiber-optic material with index 1.800. 11. The critical angle for a material with respect to air is 58°. Find the index of refraction of the material to three decimal places. Copyright © 2009 by Amsco School Publications, Inc. uc 14580TM_C09.pgs 3/26/09 12:13 PM Page 126 Name Class Date EXTENDED TASK Trigonometry in Aviation Trigonometry is an area of mathematics very important to aviation and the airplane pilot. Most commercial planes are equipped with the necessary equipment to allow them to fly using Instrument Flight Rules when there is restricted visibility. On the other hand, many private planes do not have this capability. They are required to fly using Visual Flight Rules, which permit them to fly a plane only if the cloud ceiling (height of the cloud from the ground) is 1,000 feet or more and if the ground visibility is at least three miles. A trained eye is usually able to determine the cloud ceiling and the cloud ground visibility during daylight hours. At night, however, trigonometry is needed to help the pilot determine cloud height. This is accomplished by using a ground observer positioned 1,000 feet from a parabolic reflector light source posiCloud tioned at the same height as the height observer’s eye. The light is directed at Parabolic Observer's eye light the clouds at a constant angle of 70°. source The light from the parabolic light u 70° source hits the clouds and is reflected to the observer’s eye. The angle 1,000 ft formed by the reflected light and the Ground horizontal to the observer’s eye level is the angle of elevation, whose measure is u. 1. Express the height of the cloud as a function of the angle of elevation of the cloud from the observer’s eye such that u is the only independent variable in the equation. 2. Find the cloud height to the nearest ten feet for each angle of elevation. (Since the distance between the ground and the observer’s eye will vary from observer to observer, you can ignore it for these calculations. Since it will be approximately 5 feet, it would not be significant in determining the cloud ceiling.) a. 25° b. 45° c. 86° 3. For which of these angles would the cloud height be adequate to permit the pilot to fly the aircraft? 4. Find, to the nearest degree, the least possible angle of elevation that would provide a cloud cover that would allow the pilot to fly. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C09.pgs 3/26/09 12:13 PM Page 127 Name Class Date Algebra 2 and Trigonometry: Chapter Nine Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. 1. In PQR, R is a right angle, RP 5 7, and PQ 5 25. Give the value of each ratio as a fraction. a. cos P b. sin P c. tan P d. cot P e. csc P f. sec P g. sin Q h. tan Q P 25 7 Q R 2. A fisherman observes that the measure of the angle of elevation of the top of a 120-foot lighthouse to the boat is 6°. How far, in feet, is the boat from the lighthouse? Round to the nearest tenth. 1 3. In the diagram, QOPT intersects circle O at Q A 22 !2 2 , 23 B and h 1 P A 2 !2 3 , 3 B . RT y is tangent to circle O at R(1, 0). If P mROP 5 u, find: u a. sin u O b. cos u Q c. tan u d. cot u e. sec u f. csc u g. sin (180° 1 u) h. cos (180° 1 u) i. tan (180° 1 u) j. the coordinates of T 4. The terminal side of A is in Quadrant I and sin A 5 23. Find: a. cos A b. tan A c. csc A d. sec A e. cot A f. sin (180° 2 A) In 5–7, state the quadrant where the terminal side of u is located. 5. sin u , 0, cos u . 0 6. cos u , 0, cot u . 0 7. sec u , 0, csc u . 0 Copyright © 2009 by Amsco School Publications, Inc. T R x 14580TM_C09.pgs 3/26/09 12:13 PM Page 128 Name Class Date In 8–10, state a coterminal angle with a measure of u such that 0 # u , 360. 8. 495° 9. 670° 10. 2900° In 11–18, find the exact value of each expression. 11. sin 45° 12. tan 135° 13. cos 210° 14. cot (230°) 15. sin 270° 16. sec 240° 17. tan 30° 18. csc (2225°) In 19–26, find each function value to four decimal places. 19. sin 45° 20. cos 93° 21. sec 285° 22. tan 200.5° 23. cos 126° 529 24. csc 36° 25. tan (–128°) 26. cot 17.5° In 27–29, for each given function value, express u, where 0° # u , 360°: a. to the nearest degree b. to the nearest minute. 27. cos u 5 0.5619 a. b. 28. sin u 5 0.9200 a. b. 29. tan u 5 4.0511 a. b. 30. If sin u 5 0.9, find the value of sin (u 1 180°). 31. Find the degree measure of an angle in Quadrant III where sin u 5 cos u. In 32–35, write each expression as a function of a positive acute angle. 32. tan 320° 33. cos 210° 34. cos 700° 35. tan (2112.5°) In 36 and 37, solve for x. Round to the nearest hundredth. 36. 2x 1 sin 30° 5 cos 30° 37. 2x 1 sin 45° 5 sin 135° Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C09.pgs 3/26/09 12:13 PM Page 129 Name Class Date In 38 and 39, write each expression in terms of sin u, cos u, or both. Simplify the expression if possible. 38. sec u cot u u 39. tan sec u Bonus I: Find the image of the point (2, 0) under a rotation of 60° about the origin. Bonus II: In square WXYZ, the midpoint of side WZ is P. Find the degree measures of 1, 2, and 3 to the nearest tenth of a degree. W P Z Copyright © 2009 by Amsco School Publications, Inc. 2 X 3 1 Y 14580TM_C09.pgs 3/26/09 12:13 PM Page 130 Name Class Date SAT Preparation Exercises (Chapter 9) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 7. 1. In right triangle PQR, where R is 90°, the cosine of /P is 58. What is the sine of P? (A) 5 !39 39 (D) !39 (B) !39 8 (E) 8 !39 5 (A) 7 !3 2 (D) 14 (B) 7 (E) 14 !3 P (C) !39 5 2. If the length of an altitude of an equilateral triangle is 7 !3, then the length of a side is (C) 7 !3 8. P (A) 12 5 5 (B) 13 5 (D) 2 12 (E) 2 12 5 5 (C) 2 13 5. Which expression is equivalent to cos 310° 1 cos 190°? (A) 2cos 50° 2 cos 10° (B) 2cos 50° 1 cos 10° (C) cos 50° 1 cos 10° (D) cos 50° 1 cos (210°) (E) cos 50° 2 cos 10° 6. The expression sin 20° 2 cos 30° 1 sin 40° 2 cos 50° 1 sin 60° 2 cos 70° equals (A) 0 (B) 1 (C) sin 20° (D) 2cos 70° (E) 2cos 30° Copyright © 2009 by Amsco School Publications, Inc. Q T R S In rectangle PQRS, if tan /QPT 5 15 and tan /TSR 5 12, then tan PQS 5 (B) 80 sin 39° (D) 80 tan 39° 4. If P(25, 12) lies on the terminal side of u in standard position, then tan u is R S If QR 5 RS, then which of the following could be tan P? (A) 0 (B) 0.8311 (C) 1 (D) 1.312 (E) 2 3. When measured from a point on the ground that is 80 feet from the base of a lamppost, the angle of elevation is 39°. Which of the following equations represents the height of the lamppost? (A) 80 cos 39° (C) 80 sin 51° (E) 80 tan 51° Q 9 (A) 10 (B) 45 (D) 21 (E) 25 7 (C) 10 9. x° x° x° x° x° x° Given sin x° 5 12 and x is an acute angle. If all the rays pictured were extended indefinitely, how many intersection points would there be? (A) 3 (B) 4 (C) 6 (D) 8 (E) 9 14580TM_C09.pgs 3/26/09 12:13 PM Page 131 Name Class 10. 14. 12 13 x a Q y b 13 Date 12 In the figure above, which of the following must be true? II. tan x 5 tan a If mP is greater than mQ, which of the following is true? III. cos y 5 sin b (A) I only (C) I and II only (E) I, II, and III (B) III only (D) I and III only I. sin P . sin Q II. cos P . cos Q III. sin P . cos Q (A) I only (C) III only (E) I and III only Questions 11–13 refer to the following figure. R 15. 1 P r (A) r (B) pr (D) p1 (E) r (C) 1r p 1 sin R 13. The expression sin Pcos equals P (C) (E) p 1 r 4 Q P Q 12. If sin P 5 k(tan P), then k equals p 1 r r p 1 r p (B) II only (D) I and II only p 11. The expression (sin P)(sin P) is equivalent to (B) 1 2 p2 (A) 1 1 p2 2 (C) r (D) 1 2 r2 2 (E) 1 1 r (A) R P I. sin y 5 cos b (B) p 1 r (D) p 1 r 2 Copyright © 2009 by Amsco School Publications, Inc. x° y° R S If diameter PR measures 1 unit, then the perimeter of PQRS can be expressed as (A) (sin x°)(cos x°)(sin y°)(cos y°) (B) sin x° 1 cos x° 1 sin y° 1 cos y° (C) (tan x°)(tan y°) (D) tan x° 1 tan y° (E) sin(x 1 y)° 1 cos(x 1 y)° II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. In right triangle LMN, where M 5 90°, N 5 45°, and LN 5 20, what is the area of the triangle? 14580TM_C09.pgs 3/26/09 12:13 PM Page 132 Name Class 17. If equilateral triangle ABC has a side of 7.6 feet, what is the value, to the nearest tenth, of an altitude drawn to any side? Date 23. 8 18. In ABC, C is a right angle and tan /A 5 57. What is sin B to the nearest thousandth? In the figure above, if cos A 5 !3 2 , what is BC? 24. S 21. If cos u 5 0.600 and 0° , u , 180°, what is the value of tan u to the nearest thousandth? 22. If sin x 5 0.655, what is the value of tan x to the nearest thousandth? C A 19. In PQR, R is a right angle and cos /P 5 49. Find the degree measure of Q to the nearest tenth. 20. If cos 293° 5 cos B° and 0° # B # 90°, what is the value of B? B t R r s T In the figure above, if s is 20% less than t, and r is 25% less than s, what is the measure of sin R? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C10.pgs 3/26/09 12:14 PM Page 133 CHAPTER 10 MORE TRIGONOMETRIC FUNCTIONS For all x in S 2 p2 , p2 T , the expressions Aims • To define and use radian measure and to convert radians to degrees and degrees to radians. • To derive basic trigonometric identities. • To define the inverse trigonometric functions. • To know and apply the cofunction relationships between trigonometric ratios. In this chapter, students explore the domain and range of each trigonometric function. Since the trigonometric functions are not one-to-one functions, they do not have inverse functions. However, the domain of a trigonometric function can be restricted so that the inverse function exists. The chapter also covers the important Pythagorean identities and the very useful cofunction relationships. CHAPTER OPENER In calculus, it is shown that 3 5 7 sin x 5 x 2 x3! 1 x5! 2 x7! 1 c ` (21) n 5 a (2n 1 1)!x2n 1 1 n50 2 4 6 cos x 5 1 2 x2! 1 x4! 2 x6! 1 c ` (21) n 5 a (2n)! x2n 3 5 7 9 11 x sin x 5 x 2 x3! 1 x5! 2 x7! 1 x9! 2 11! 2 4 6 8 x10 cos x 5 1 2 x2! 1 x4! 2 x6! 1 x8! 2 10! are accurate to five digits. After students have learned about radian measure, suggest that they evaluate the expressions above for common angles such as p4 , p3 , p2 and compare the values calculated with the actual values of the sines and cosines. (The work is easily accomplished with a graphing calculator.) The accuracy is impressive. (See Section 10-2, Hands-On Activity 2.) 10-1 RADIAN MEASURE Degree measure of an angle is based on an arbitrary choice of 360° as the measure of one complete rotation. The 360° division may have been the invention of astronomers of antiquity, who divided the zodiac into 12 signs, each having 30 parts. The 360 subdivisions thus correspond roughly to the number of days in the year. Another conjecture is that the origin lies in the Mesopotamian use of numeration base 60, a number conveniently divisible by ten different factors. Radian measure of an angle is based on the ratio of arc length and measure of the radius of a circle with center at the vertex of the angle. An advantage of radians in working with trigonometric functions is that radian n50 133 14580TM_C10.pgs 3/26/09 12:14 PM Page 134 134 Chapter 10 measure, as a ratio, is independent of any unit of measure. Although a simple proportion enables students to determine the radian equivalent of any degree measure, they will find it convenient to know the radian equivalents of common angle measures such as 30°, 60°, 90°, and 180°. From these values, the radian measures of many other commonly used angles can be easily derived. Example: Since both 30° and p6 radians are measures of the same angle, an angle of 150° or 5(30°) has a radian measure of 5 A p6 B or 5p 6. 10-2 TRIGONOMETRIC FUNCTION VALUES AND RADIAN MEASURE This section provides practice with function notation and radian measure. Encourage students to become familiar with radian measure. Explain that, in advanced mathematics, radian measure is used almost exclusively. It is interesting to note that the sine of small angles can be approximated with the radian measure of the angles. Take the two sides of the triangle as equal to r. Hands-On Activity Instructions: 1. Draw a circle with a radius of at least 2 inches. 2. Cut a piece of string whose length is equal to the radius of the circle. Place the string along the circle and mark points at each end to locate the endpoints of an arc equal in length to the radius of the circle. 3. Draw rays from the center of the circle through the endpoints of the arc, forming a central angle. Using a protractor, find the degree measure of this central angle. 4. Repeat steps 1–3 two or three times, using a circle with a different radius each time. Discoveries: 1. The measure of the central angle whose intercepted arc is equal in length to the radius of the circle is 1 radian or about 57°. 2. The measure of the angle remains unchanged when the size of the circle changes. In Section 9–2, angular speed was briefly introduced in the homework exercises. Enrichment Activity 10-1: Angular Speed and Linear Speed continues its development with linear speed. The exercises present opportunities for students to see radian measure in real-world applications. h u s r For small angles, s h and this is nearly an isosceles triangle. The sine of the angle is sin u 5 hr and the angle in radians is u 5 sr, and so: sin u 5 sin sr 5 hr < sr That is, for small angles, the sine of the angle is approximately equal to the angle (in radians). p For example, the angle 30 is equal to 6° and sin 6° 5 0.1045. p 30 5 0.1047 The error in using the radian measure rather than the sine is 2 parts in 1,000 or 0.2%. Remind students that for the radian formula to work, lengths must be in the same unit. (See exercise 31.) Enrichment Activity 10-2: The Angle Between Two Lines presents a formula for the tangent of the angle between two lines. Students write lines in slope-intercept form, find the tangent, find the angle measure in degrees, and then convert to radians. 14580TM_C10.pgs 3/26/09 12:14 PM Page 135 More Trigonometric Functions 135 10-3 PYTHAGOREAN IDENTITIES Three new trigonometric identities are introduced in terms of an equation for the unit circle. Since this equation in turn derives from the Pythagorean relationship, the associated trigonometric equations are known as the Pythagorean identities. These, together with the reciprocal and quotient relationships previously defined, comprise the eight basic identities summarized in this section. It is often helpful to use simple algebraic equations to clarify for students the definition of an identity as an equation that is true for all values of the variable: x 1 x 5 2x, x ? x 5 x2, 2(x 1 1) 5 2x 1 2 The identities enable us to change a trigonometric expression into an equivalent one involving one or more different functions, and also to express each of the trigonometric functions in terms of any of the others. Therefore, if, for a given element of the domain, one trigonometric function value is known, the values of each of the other trigonometric functions can be found. 10-4 DOMAIN AND RANGE OF TRIGONOMETRIC FUNCTIONS Have students explore the domain and range for the sine, cosine, and tangent functions using their calculators. Have them select twenty values of u between 0 and 2p radians and evaluate the three trigonometric functions for each value. Ask them to describe their results including the least value they found for each function, the greatest value, and any angles for which a function was not defined. Some students may recognize patterns of how the values of the functions change as u increases from 0 to 2p. Ask them to predict the values for the functions from 2p to 4p and check their predictions. Then have them use the inverse function capability of the calculator to explore possible range values. The section carefully examines the domain and range of each of the six trigonometric functions, justifies the conclusions geometrically and algebraically, and summarizes the key results in a table. Understanding the domain and range for each function will help students in the next section when they learn about inverse functions and also in Chapter 11 where the functions are graphed. 10-5 INVERSE TRIGONOMETRIC FUNCTIONS The relation y 5 arcsin x is not a function, but by selecting some subset of this relation, we can obtain a function. To do this, we can select any subset of the domain where all values of the range occur exactly once (one-to-one) and the range changes from the minimum value (21) to the maximum value (1) (onto). Although there are infinitely many possible subsets, the one that is commonly selected includes the values of x smallest in absolute value, that is, the values closest to 0. The values in these restricted domains are sometimes called principal values. In some texts, capital letters are used to distinguish trigonometric functions with restricted domains from the usual trigonometric functions. So, for example, y 5 Sin x if and only if y 5 sin x and 2p2 # x # p2 In Chapter 11, the concept of inverse functions will again be considered when the trigonometric functions are graphed. Students will be able to apply the familiar vertical and horizontal line tests to determine how the domain of each trigonometric function must be restricted. The multiple representations of these functions contribute to the students’ increased understanding. Remind students that the composition of a function with its inverse must be the identity function; for example, (sin21 + sin)(x) 5 (sin + sin21)(x) 5 x 14580TM_C10.pgs 3/26/09 12:14 PM Page 136 136 Chapter 10 Students may wish to try Example 2 on their calculators. When correctly entered, the display will show the approximate value 0.8660254083. Since the solution to this example must be an exact value, student may recognize that the decimal is the approximation for !3 2 . 10-6 COFUNCTIONS The derivations of the identity sin u 5 cos (908 2 u) and the related identities that follow in this section state that u is the measure of an acute angle and (90° 2 u) is the measure of the complement. These are the conditions required for the definition of cofunctions. However, these identities are not restricted to first-quadrant angles but are true for all replacements of the variables where the function are defined. 14580TM_C10.pgs 3/26/09 12:14 PM Page 137 Name Class Date ENRICHMENT ACTIVITY 10-1 Angular Speed and Linear Speed The basic formula connecting rate R, distance d, and time t is R 5 dt. This rate is a speed, expressed as a measure of distance divided by a unit of time. As a point P on a circle moves to a new location P9 on the same circle, changes occur that result in two types of speed. • Linear speed (v) is the change in the arc length s divided by a period of time t. Just as R 5 dt, the formula for linear P9 P speed is v 5 st . R • Angular speed (v, the Greek letter omega) is the change in the angle measure u, expressed in radians, divided by a O period of time t. The formula for angular speed is v 5 ut . From Section 10-1, we know that the radian measure u of a central angle, the length s of the intercepted arc, and the radius r are related by the formula u 5 sr, or s 5 ru . By substitutions involving this formula, we see that v 5 st S v 5 rut S v 5 r A ut B S v 5 rv Linear speed and angular speed are used extensively in engineering, physics, and technological studies. A simple example will show how to use the formulas. Example A minute hand on a clock is 5 inches long. a. Find the angular speed of this minute hand in radians per hour. b. Find the linear speed of the endpoint of the minute hand. Solution: a. It takes 1 hour for the minute hand to make a full rotation. In a full rotation, u 5 2p radians. Thus, 5 2p radians>hr v 5 ut 5 2p radians 1 hr b. From part a, v 5 2p radians/hr. Thus, v 5 rv 5 5 in. A 12p hr B 5 10p in.>hr < 31.4 in.>hr Answers: a. 2p radians per hour b. 10p inches per hour Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C10.pgs 3/26/09 12:14 PM Page 138 Name Class Date Exercises Unless otherwise noted, express answers in terms of p to show an exact value. 1. a. On a clock, indicate the time required for each of the following hands to complete a full rotation: (1) an hour hand (2) a minute hand (3) a second hand b. Find the angular speed on a clock, expressed in radians per hour, for each of the hands listed in part a. c. The lengths of the hands are usually different on a clock. Find the linear speed, expressed in inches per hour, for: (1) a 6-inch hour hand (2) an 8-inch minute hand (3) a 10-inch second hand. For Exercises 2 and 3, assume that the radius of Earth is 6,400 kilometers. 2. Earth completes a full rotation on its axis every 24 hours or once per day. a. State the angular speed of Earth in: (1) radians per day (2) radians per hour b. State the linear speed of a point on Earth’s equator: (1) in kilometers per day (2) to the nearest 100 kilometers per hour c. State the linear speed of a point at Earth’s North Pole. Explain your answer. 3. A satellite travels 800 kilometers above Earth’s surface and circles the planet once every 6 hours. a. Find the angular speed of the satellite in radians per hour. b. Find the linear speed of the satellite in kilometers per hour. 4. A bicycle has a wheel with a 28-inch diameter. The bicycle is ridden at a steady pace, resulting in the wheel turning at 120 revolutions per minute. a. Find the angular speed of the bicycle wheel in radians per minute. b. Find the linear speed of a point on the wheel in: (1) inches traveled per minute (2) feet traveled per minute c. Is 880 feet per minute a good estimate of the linear speed found in part b(2)? Explain your answer. d. Express 880 feet per minute as a rate in miles per hour. 5. In a machine, a belt is connected to a wheel that turns at 240 revolutions per minute. The radius of the wheel is 3 inches. a. Express the angular speed of the wheel in: (1) radians per minute (2) radians per second b. Find the linear speed of the wheel in: (1) inches per minute (2) inches per second Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C10.pgs 3/26/09 12:14 PM Page 139 Name Class 6. A model plane, attached to a stake in the ground by a string, flies in circles, making one revolution every 5 seconds. The radius of the circle made by the plane is 15 feet. a. Express the angular speed of the plane in radians per minute. b. Express the linear speed of the plane in feet per minute. c. Find, to the nearest foot, the distance traveled by the plane in 1 minute. Copyright © 2009 by Amsco School Publications, Inc. Date 14580TM_C10.pgs 3/26/09 12:14 PM Page 140 Name Class Date ENRICHMENT ACTIVITY 10-2 The Angle Between Two Lines Suppose l1 and l2 are two intersecting lines. Then we define the angle from l1 to l2 to be the angle u through which l1 must be rotated counterclockwise about the point of intersection in order to coincide with l2. Thus, 0 # u , p. There is a relationship between u and the slopes of the lines. If l1 and l2 are two non-vertical lines that are not perpendicular and l1 has slope m1 and l2 has slope m2, then the tangent of the angle u from l1 to l2 is given by l1 u l2 m 2 m 2 1 tan u 5 1 1 m1m2 Exercises In 1–5, find: a. the tangent of the angle u from l1 to l2. b. the angle u to the nearest degree. c. the radian equivalent of the answer to b. 1. l1: y 2 2x 5 2 l2: 2y 1 5x 5 17 4. l1: x 1 y 5 6 l2: x 1 2y 5 8 2. l1: y 5 4x 2 2 l2: 3y 5 22x 1 7 3. l1: 2y 5 x l2: y 2 3x 5 7 5. l1: y 5 2x 2 2 l2: y 5 25x 1 5 6. The equation of line A is x 1 y 5 6 and the equation of line B is 3x 2 4y 5 4. Find, to the nearest degree: a. the angle from line A to line B. b. the angle from line B to line A. (Make sure to express each angle as a positive angle.) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C10.pgs 3/26/09 12:14 PM Page 141 Name Class Date Algebra 2 and Trigonometry: Chapter Ten Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–4, express each given degree measure in radians. 1. 75° 2. 18° 3. 2240° 4. 510° In 5–8, express each given radian measure in degrees. 5. 3p 5 7. 6. 23p 7p 4 8. 4p 9 9. Find the radian measure of a central angle that intercepts an arc of 12 inches in a circle whose radius has a length of 5 inches. 10. Circle O has a radius of 1.4 inches. What is the length, in inches, of an arc intercepted by a central angle whose measure is 3 radians? In 11 and 12, find, to the nearest ten-thousandth, the radian measure u of a first-quadrant angle with the given function value. 11. sin u 5 0.4658 13. If f(x) 5 sin 4x 2 cos 2x, find f A p8 B . 12. tan u 5 8.2794 14. Find the value of (sin2 u 1 cos2 u 2 5)2. 1 15. Find the value of (3 sec2 u 2 3 tan2 u 1 3) 2. In 16–18, write each expression in terms of sin u, cos u, or both. Simplify the expression if possible. 16. tan2 u 1 1 17. sin u (cot2 u 1 1) u csc u 18. costan u In 19–24, find the exact value of u in radians. 19. u 5 arcsin !2 2 21. u 5 arctan (21) 23. u 5 arctan !3 Copyright © 2009 by Amsco School Publications, Inc. 20. u 5 arccos 0 22. u 5 arcsin (20.5) 24. u 5 arccos A 212 B 14580TM_C10.pgs 3/26/09 12:14 PM Page 142 Name Class Date In 25–30, find the exact radian measure u of an angle with the smallest absolute value that satisfies the equation. 25. u 5 arccot !3 26. u 5 arccsc (22) 29. cot u 5 0 30. csc u 5 22 27. sec u 5 !2 28. u 5 arcsec (21) In 31–34, find the value of each expression. 7 31. cos A arcsin 25 B 5 32. tan A arccos 13 B 33. sec A arccos 25 B 34. sin (arctan A2!3B B 35. What is the greatest negative integer in the range of the secant function? 36. If cos u 5 sin (u 1 20°) and u is the measure of an acute angle, find the value of u. 37. If tan 2u 5 cot (2u 1 15) and u is the measure of an acute angle, find the value of u. 38. A box contains 181 chips. Each chip is marked with one of the integers from 290 to 90. Each integer occurs exactly once. What is the probability that if a chip is randomly selected, the tangent function will be undefined at the value indicated by the chip? u sec u Bonus: If tan u 5 45, find sincot u . Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C10.pgs 3/26/09 12:14 PM Page 143 Name Class Date SAT Preparation Exercises (Chapter 10) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. The length of the arc intercepted by a central angle of 4 radians in a circle with a radius of 2.5 centimeters is (A) 0.625 cm (B) 1.6 cm (C) 4 cm (D) 5 cm (E) 10 cm 2. In a circle whose radius measures 1.8 inches, a central angle intercepts an arc of length 4.5 inches. The radian measure of the central angle is (A) 0.4 (B) 1.25 (C) 2.5 (D) 2.7 (E) 8.1 3. Angle M measures 7p 10 and angle N measures 5p 12 . The difference in their measures M 2 N expressed in degrees is (A) 12° (B) 20° (C) 24° (D) 51° (E) 75° 4. Which is the radian measure of an angle that measures 495°? (A) 7p 2 (B) 9p 3 (D) 13p 6 (E) 15p 12 5. 2 csc p3 1 8 tan p6 5 (A) 4 !3 6. A tan (A) 1 9 (D) 2 (B) 1 3 (B) (D) !3 5p 2 6 B (E) 10 !3 3 !3 2 (C) 11p 4 7. To four decimal places, 2 5p sin2 5p 9 1 cos 9 5 (A) 0.8116 (B) 0.9698 (D) 1.0744 (E) 1.7814 8. 1 1 1cos x 1 1 2 1cos x is equivalent to (A) cos22 x (B) 2 sin2 x 2 (D) tan2 x (C) 2 sec2 x (E) 2 csc2 x 9. Evaluate A 3 sec2 p4 2 3 tan2 p4 1 5 B . 2 (A) 64 (D) 8 (B) 36 (E) 4 (C) 16 10. cos 320° 5 (A) sin 40° (C) sin 50° (E) 2cos 40° (B) cos 50° (D) cos 60° 11. tan 190° 5 (A) 2tan 10° (C) tan1108 (E) 2cot 10° (B) tan 80° (D) cot 80° 12. Which is not in the domain of y 5 sec x2? (A) 0 (B) p2 (C) p (D) 3p 2 (C) 2 !3 (C) 1.0000 (E) 2p 13. Which is not in the range of y 5 csc x? (A) 217 (B) 23 (C) 0 (D) 1 (E) 4 14. If u 5 arcsin (cos (2p)), then a coterminal angle with u has measure 5 (C) 1 (E) 3 (A) p2 (D) 3p 2 (B) 3p 4 (E) 2p 15. tan (cos21 (20.5)) 5 (A) 2!3 (D) !3 Copyright © 2009 by Amsco School Publications, Inc. (C) p (B) 2 !3 2 (E) 3 !3 2 (C) !3 2 14580TM_C10.pgs 3/26/09 12:14 PM Page 144 Name Class II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. A sector has a radius of 8 centimeters and a central angle of 13p 36 . To the nearest hundredth, find its arc length. Date 19. Find the exact value of A tan p3 2 sec p3 B A tan p3 1 sec p3 B 1 3. 20. If x 5 cos21 A !3 2 B , what is the degree measure of x? 21. If sin (2x 1 5) 5 cos (7x 2 5), find x. 5 17. Evaluate sin A tan21 12 B as a fraction in lowest terms. 22. If tan 215° 5 cot u and u is an angle in Quadrant I, what is the degree measure of u? 18. Find the exact value of (5 2 4 sin2 x 2 4 cos2 x)3 when x 5 10100. 24. If cos u 5 34, find tan u (cot u 1 tan u). Copyright © 2009 by Amsco School Publications, Inc. 23. If f(x) 5 22 sec 2x, evaluate f A p3 B . 14580TM_C11.pgs 3/26/09 12:14 PM Page 145 CHAPTER 11 GRAPHS OF TRIGONOMETRIC FUNCTIONS Aims • To graph the sine, cosine, and tangent functions. • To define amplitude and period and to study the effects of amplitude and period on the trigonometric graph. • To graph the reciprocal functions. • To write the equation of the trigonometric function that is represented by a given periodic graph. • To graph the inverse trigonometric functions. In this chapter, students will study the graphs of the trigonometric functions and relate characteristics of the graphs to the equations of the functions. Since the trigonometric functions are periodic, their graphs are symmetric with respect to a translation in the horizontal direction. CHAPTER OPENER The hertz (Hz) is the unit of measure for frequency; for example, 440 waves per second is equal to 440 Hz. A healthy human ear can hear sound frequencies from about 20 Hz to 20,000 Hz. As people get older, they often lose the ability to detect sounds at high frequencies. Frequencies above 20,000 Hz are called ultrasonic waves. Most humans cannot hear ultrasonic frequencies, but bats can detect frequencies as high as 100,000 Hz. Sonar, or sound navigation, uses ultrasonic sound waves to estimate the size, shape, and depth of submerged objects. Ultrasound is also used in many medical diagnostic tools and treatments. Infrasonic waves have frequencies that are less than 20 Hz. These waves may be produced by sources such as thunder or heavy machinery. Their effects are sensed as a rumble rather than actually heard. 11-1 GRAPH OF THE SINE FUNCTION In previous chapters, students have found the values of trigonometric functions for angles measured in degrees and radians. They have found exact values for special angles and approximate values using a calculator. As is often the case, a graph is an effective way of presenting the data; by studying the graph, key patterns and characteristics can be recognized. Graph paper that is divided into 1-centimeter blocks, each of which is further divided into five 2-millimeter sections, is well suited for student work. A convenient scale uses two large blocks to represent 1 along the y-axis and one large block to represent p6 along the x-axis. See, for example, the graph of y 5 sin x on page 435 of the text. (Note that the scale used in the text is slightly smaller than 1 centimeter.) 145 14580TM_C11.pgs 3/26/09 12:14 PM Page 146 146 Chapter 11 Call attention to various symmetries of the graph. Because the function is periodic, the graph is its own image under a translation T 2kp,0 for any integer, k. When k 5 1, 2kp assumes its smallest positive value, 2p, which is the period of the function. Point out that this translational symmetry makes it possible to sketch the graph of y 5 sin x for all values of x by simply repeating the curve drawn for the values from 0 to 2p. The graph has other line and point symmetries. For example, the graph has point symmetry since it is its own image under a reflection in the origin or in any point on the x-axis whose x-coordinate is an integral multiple of p. Also, the graph has line symmetry since it is its own image under a reflection in the line whose equation is x 5 p2 or any other vertical line whose x-intercept differs from p2 by an integral multiple of p, for example, x 5 3p 2 or x 5 5p . 2 The Hands-On Activity is a very dynamic demonstration of the relationship between the unit circle and the sine and cosine functions. Because they are both defined using a unit circle, the sine and cosine are often called circular functions. Amplitude, period, and frequency will be discussed in greater depth in Section 11-3. 11-2 GRAPH OF THE COSINE FUNCTION The graph of y 5 cos x is presented in a manner similar to that used for y 5 sin x. The graph of y 5 cos x has the same translational symmetry as the sine graph since it has the same period, 2p. Like the sine graph, the cosine graph has point and line symmetries. It is symmetric with respect to the y-axis or with respect to any vertical line whose x-intercept is a multiple of p. The symmetry with respect to the y-axis means that, for all x, cos x 5 cos (2x). This identity will be proven algebraically in Chapter 12. The cosine graph has point symmetry with respect to the point A p2 , 0 B or any other point on the x-axis whose x-coordinate differs from p2 by an integral multiple of p, for example A 3p 2 , 0 B or A 2p2 , 0 B . Encourage students to describe ways in which the functions y 5 sin x and y 5 cos x are the same and ways in which they are different. Ask students the following question: Suppose you graphed y 5 sin u and then decided you wanted to describe the graph using the cosine function instead. How could you do this? A If y 5 sin u, then y 5 cos A u 1 p2 B . B 11-3 AMPLITUDE, PERIOD, AND PHASE SHIFT Call attention to the careful definition of amplitude. Students may ask whether amplitude may be alternatively defined as the maximum value of the function. This alternate definition is correct only for functions where the minimum and maximum values have the same absolute value, such as the function defined here. Notice that the definition is not correct for a function such as y 5 3 1 2 sin x that has a maximum value of 5, a minimum 1 value of 1, and an amplitude of 5 2 2 or 2. The generalization that the function y 5 a sin x has an amplitude of a can be extended to y 5 c 1 a sin x, which also has an amplitude of a. For the discussion of period, be sure that students understand the order of operations. To find values for the graph of y 5 sin 2x, it is necessary to choose a value of x, multiply that value by 2, and then find the sine values. Call attention to the fact that, no matter what value is assumed by 2x, the sine cannot be greater than 1 or less than 21. For the interval 0 # x # 2p, the function y 5 sin x attains its maximum value once and its minimum value once, y 5 sin 2x attains its maximum value twice and its minimum value twice, and y 5 sin 12x attains its maximum value once but does not attain its minimum value. In the case of sound and light, waves that are in phase (crests of the two waves are 14580TM_C11.pgs 3/26/09 12:14 PM Page 147 Graphs of Trigonometric Functions 147 together and troughs are together) interfere constructively and the amplitude of the resulting wave is the sum of the individual amplitudes. Waves that are out of phase interfere destructively; if two waves of the same amplitude and same frequency are 180° (p) out of phase, the resultant amplitude is zero and no sound would be heard. Sometimes a problem is written so that it appears as if the period and phase shift are inside the trigonometric function; for example y 5 sin (2x 1 p) makes it look like the period is twice as fast and the horizontal shift is p. However, all phase shifts must be factored out to be analyzed, so y 5 sin 2 A x 1 p2 B shows that the period is twice as fast but the horizontal shift is p2 . Note that the frequency is the number of cycles per unit time and is measured in hertz. The angular frequency, on the other hand, is the number of cycles in the 2p interval and is measured in radians per second. If the |b| frequency is 2p , then the angular frequency is b. In Enrichment Activity 11-3: Graphing Combined Functions, students use tables of values and a graphing calculator to draw and analyze the graphs of sums and differences of sine and cosine functions. 11-4 WRITING THE EQUATION OF A SINE OR COSINE GRAPH The steps for writing an equation of a sine or cosine graph are the same. For an equation using sine, the basic cycle is found by identifying a portion of the graph that begins at 0, increases to its maximum, decreases to 0, continues to decrease to the minimum, then increases to 0. For an equation using cosine, the basic cycle begins at the maximum value, decreases to 0, continues to decrease to the minimum, increases to 0, then increases to the maximum. Be sure students understand Step 3 of the solution to Examples 1 and 2. The interval for one cycle is found in the previous step by the visual process described above. The difference between the endpoints of the interval is the period. Since the period of the function was defined as P 2p b P , we can find b by equating the two expressions for the period. Since the basic sine and cosine curves begin at 0, the phase shift c is positive if the cycle on the given graph begins to the left of 0 and negative if the cycle begins to the right of 0. Have students compare this relationship to the equation of the basic absolute value equation y 5 x or the parabola y 5 x2 that is translated left or right. If students have difficulty answering Exercise 2, suggest that they graph the two functions with their graphing calculator. They may also verify their equations for Exercises 3–14 by graphing. It is convenient to use the trigonometric window, found by pressing ZOOM 7, for these graphs. In this window, each tickmark on the x-axis represents p2 and each tickmark on the y-axis represents 1 unit. Enrichment Activity 11-4: Polar Coordinates is the first of two activities focusing on this topic. Students have learned how to associate a pair of coordinates on the unit circle with the sine and cosine functions, and polar coordinates are an extension of this idea. The activity allows students to practice converting from rectangular to polar coordinates and vice versa. In the next activity, they explore some of the interesting polar graphs. Both activities can be used at any time during the chapter. 11-5 GRAPH OF THE TANGENT FUNCTION The graph of the tangent function is similar to that of the sine or of the cosine function in several respects: 1. It is the graph of a periodic function with a period of p. Therefore, it has translational symmetry. 2. It has point symmetry since it is its own image under a reflection in the origin or in any point on the x-axis whose x-coordinate is an integral multiple of p2 . 14580TM_C11.pgs 3/26/09 12:14 PM Page 148 148 Chapter 11 The graph of the tangent function differs from that of the sine or of the cosine function in several respects: interval; in this case the vertical line test fails, which indicates that the inverse relation is not a function for the interval. Have them perform 1. It is discontinuous at p2 and at any value of x that differs from p2 by an integral multiple of p. a similar analysis for the interval 2p2 to p2 to reinforce the reason why the domain of the sine function is restricted to this interval. The interval includes all of the values of sin x from 21 to 1 and the sine is a one-to-one function on this interval, so the inverse is also a function. 2. It is always increasing. 3. It has no maximum or minimum values and, therefore, no amplitude. 4. It has no line symmetry. 11-8 SKETCHING TRIGONOMETRIC GRAPHS 11-6 GRAPHS OF THE RECIPROCAL FUNCTIONS To draw the graph of y 5 csc x, use the table that was made to graph y 5 sin x and determine the value of csc x by finding the reciprocal of the given value of sin x. Whenever sin x 5 0, csc x is undefined. Students may find it helpful to find values of csc x for values of x close to those for which csc x is undefined, p 14p 16p , 15 , 15 , and 29p such as 15 15 . Remind them that the calculator must be in radian mode and p 1 that csc 15 must be evaluated as p . sin 15 ENTER: 1 ⴜ ⴜ DISPLAY: 15 SIN 2nd ) ENTER p 1/sin(π/15) 4.809734345 The right parenthesis after 15 is optional. 11-7 GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS Have students use the horizontal line test with the graph of y 5 sin x from 0 to p. Since the horizontal line test fails, they know the function is not one-to-one on this interval. Have them identify the portion of the graph of the inverse sine function that corresponds to this After students have learned how changing the values of a and b affect the graphs of y 5 a sin bx and y 5 a cos bx, it should no longer be necessary for them to sketch a graph by making a table of values and plotting points. Emphasize, however, the importance of locating the points of intersection with the x-axis, and the maximum and minimum points, before sketching the curve. Encourage students to determine the period of the curve, to mark off intervals equal to the period on the x-axis, and then to divide these intervals into four sections. The maximum point(s), minimum point(s), and points of intersection with the x-axis will occur at the endpoints of these four sections. An accurate sketch can be made after these critical points are located. If students have been comparing their graphs with those displayed on a graphing calculator, be sure that they understand that a graph shown on the calculator varies with the viewing window. For example, the graph of y 5 sin x when WINDOW is from Xmin 5 22p to Xmax 5 2p Ymin 5 22 to Ymax 5 2 will look exactly like the graph of y 5 2 sin 2x when WINDOW is from Xmin 5 2p to Xmax 5 p Ymin 5 24 to Ymax 5 4 14580TM_C11.pgs 3/26/09 12:14 PM Page 149 Graphs of Trigonometric Functions 149 Students should have ample practice drawing the graphs of trigonometric functions on paper and comparing these graphs with those displayed on a calculator. In Exercises 21–23, students may realize they are solving trigonometric equations. For example, Exercise 21 can be used to solve 2 sin x 5 cos x graphically. Each side of the equation is graphed and the points of intersection of the two graphs are the solutions on the specified interval of the equation. Trigonometric equations are discussed in Chapter 13. Enrichment Activity 11-8: Graphing Polar Equations continues the work with polar coordinates begun in the previous enrichment activity. Students use their graphing calculators in polar mode to graph the polar rose family r 5 a sin bu and the cardioid family r 5 a(1 6 sin u) and r 5 a(1 6 cos u). They identify how changing values alter the appearance of the graphs. Students also graph the Archimedean spiral r 5 au. EXTENDED TASK For the Teacher: The purpose of this activity is to help students see that there are phenomena in the real world that are modeled by trigonometric curves. The activity will help reinforce the concepts of the period and amplitude of trigonometric curves. If desired, you could have the students research data for their own city rather than Buffalo, New York. You might also have them research data for their own city and draw the corresponding graph and compare it with the data and graph for Buffalo, New York. A comparison of the Buffalo graph and equation with similar results for a city in the southern hemisphere would also provide interesting discussion material. An extension of this task is to have them find another phenomenon besides weather that can be modeled by a trigonometric graph. Possibilities include tides, the number of daylight hours, certain predator-prey relationships, or the motion of a Ferris wheel. 14580TM_C11.pgs 3/26/09 12:14 PM Page 150 Name Class Date ENRICHMENT ACTIVITY 11-3 Graphing Combined Functions Trigonometric functions can be combined by addition or subtraction, and the resulting functions can be graphed. The table below shows values of sin x and cos x for 0 # x # 2p. Note that to graph combined functions, it is often useful to show many function values. x 0 p 6 p 4 p 3 p 2 2p 3 3p 4 5p 6 p sin x 0 0.5 0.71 0.87 1 0.87 0.71 0.5 0 cos x 1 0.87 0.71 0.5 0 20.5 20.71 20.87 21 7p 6 5p 4 4p 3 3p 2 5p 3 7p 4 11p 6 2p sin x 20.5 20.71 20.87 21 20.87 20.71 20.5 0 cos x 20.87 20.71 20.5 0 0.5 0.71 0.87 1 sin x 1 cos x sin x 2 cos x x sin x 1 cos x sin x 2 cos x 1. Complete the two rows of values for y 5 sin x 1 cos x. 2. Sketch the graph of y 5 sin x 1 cos x. 3. a. What is the maximum value of y 5 sin x 1 cos x? Where does this value occur? b. What is the minimum value of y 5 sin x 1 cos x? Where does this value occur? c. What is the period of y 5 sin x 1 cos x? 4. Complete the two rows of values for y 5 sin x 2 cos x. 5. Sketch the graph of y 5 sin x 2 cos x. 6. Answer questions 3a–c for y 5 sin x 2 cos x. 7. For what values of x is sin x 1 cos x 5 sin x 2 cos x? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 151 Name Class Date A graphing calculator is a useful tool for displaying and analyzing the graphs of combined functions. To see how the combined function was derived, you can graph the component functions and the combined function on the same screen. Make sure the calculator is in RADIAN mode and begin by selecting the trigonometric window. For the examples below, adjust Ymin 5 22 and Ymax 5 2. 8. Enter Y1 5 sin x, Y2 5 0.5 sin 2x, and Y3 5 Y1 1 Y2. Graph Y1, Y2, and Y3 in the interval 0 # x # 2p. (To view only Y3, deselect Y1 and Y2.) 9. Answer questions 3a–c for y 5 sin x 1 0.5 sin 2x. 10. a. Graph Y1 5 sin x and Y2 5 sin (x 1 p). b. Predict what you will see on the screen if you graph y 5 sin x 1 sin (x 1 p). Give reasons for your answer. c. Verify your prediction by graphing the combined function. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 152 Name Class Date ENRICHMENT ACTIVITY 11-4 Polar Coordinates In your study of algebra, you have learned to identify each point in the plane with its coordinates in a Cartesian coordinate system. However, there are other coordinate systems, one of the most important being the polar coordinate system. Any point P in the plane can be identified with two numbers r and u P(r, u) y where r r 5 distance from the origin to the point u 5 angle between the positive x-axis and the line drawn from the origin to the point x The pair (r, u) is called a set of polar coordinates for the point P. For point P there are infinitely many possible choices of u, any two differing by a multiple of 2p. However, there is only one set of polar coordinates (r, u) such that r . 0 and 0 # u , 2p. The figures below show several points in the plane along with their polar coordinates. y (2, p4 ) (2, 3p4 ) y x y x y x x (1, 3p2 ) (1, 5p4 ) Every point in the plane has both Cartesian and polar coordinates, and it is possible to convert from one type of coordinates to the other. Suppose a point P in the plane has polar coordinates (r, u) and Cartesian coordinates (x, y). From the definition of sine and cosine: y r P y 5 r sin u x 5 r cos u u x 5 r cos u y 5 r sin u x If the Cartesian coordinates of P are (x, y), then y r2 5 x2 1 y2 and tan u 5 x for x 2 0 Copyright © 2009 by Amsco School Publications, Inc. u 14580TM_C11.pgs 3/26/09 12:14 PM Page 153 Name Class Date Example 1 Find the Cartesian coordinates of the point P having polar coordinates A 3, p4 B . 3 !2 Solution: x 5 3 cos p4 5 3Q !2 2 R 5 2 3 !2 y 5 3 sin p4 5 3Q !2 2 R 5 2 3 !2 The Cartesian coordinates of P are Q 3 !2 2 , 2 R. Example 2 Find the polar coordinates of the point P having Cartesian coordinates A25, 5 !3B . Solution: r2 5 (25) 2 1 A5 !3B 2 5 25 1 75 5 100 r 5 !100 5 10 Also, !3 5 2!3 tan u 5 525 Since A25, 5!3B lies in Quadrant II, tan21 A2!3B 5 2p 3 . (Note that we must be sure that u lies in the correct quadrant.) The polar coordinates of P are A 10, 2p 3 B. In 1–8, find the Cartesian coordinates for each point with the given polar coordinates. 1. A 3, p3 B 5. (1, 3p 2 B 2. A 2, p2 B 3. A 4, 3p 4 B 7. A 2, 3p 2 B 6. (5, 0) 4. 8. A 1, 11p 6 B A 3, 7p 6 B In 9–14, find the polar coordinates for each point with the given Cartesian coordinates. 9. (3, 3) 12. (24, 0) 10. (4, 24) 13. A3, 3 !3B 11. (0, 5) 14. A22 !3, 2B An equation written in Cartesian coordinates can be converted to polar form by substituting x 5 r cos u and y 5 r sin u in the equation. For instance, in polar form, the equation y 5 2x 1 2 becomes y 5 2x 1 2 r sin u 5 2r cos u 1 2 15. Find the polar equation of the circle x2 1 y2 5 a2, where a . 0. 16. Write the equation of the line 2x 1 3y 5 4 in polar coordinates. Express the answer in the form r 5 f(u). Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 154 Name Class Date ENRICHMENT ACTIVITY 11-8 Graphing Polar Equations Polar coordinates allow for the graphing of some very interesting figures. To graph polar equations, you must select Pol graphing mode before you enter values for window variables or any polar equations. Radians must also be used for angles. ZTrig 7 ) followed by ZSquare ( ZOOM 5 ) works well for most of the ( ZOOM graphs below; adjust the window variables as needed. Part I 1. The polar equations r 5 a sin bu and r 5 a cos bu for a . 0 and b , 0 are the equations of a rose. To explore the appearance of the rose for different values of b, graph each equation below and make a sketch of the graph for reference. a. r 5 2 sin 3u b. r 5 2 sin 2u c. r 5 2 cos 5u d. r 5 2 cos 4u 2. Try other values of a and b in the equations r 5 a sin bu and r 5 a cos bu, and observe how the new values affect the graph. 3. How many petals does the graph have if b is odd? How many if b is even? 4. What type of symmetry do graphs involving the sine have? What type of symmetry do graphs involving the cosine have? 5. How does the value of a affect the appearance of the rose? Part II 6. The polar equations r 5 a(1 6 sin u) and r 5 a(1 6 cos u) for a . 0 both graph a cardiod. To explore the appearance of the cardiod for different equations, graph each equation below and make a sketch of the graph for reference. a. r 5 2(1 1 sin u) b. r 5 2(1 2 sin u) c. r 5 2(1 1 cos u) d. r 5 2(1 2 cos u) 7. How are the graphs the same? How are they different? 8. What type of symmetry do graphs involving sine have? What type of symmetry do graphs involving cosine have? 9. Try different values for a in the equations r 5 a(1 1 sin u) and r 5 a(1 1 cos u), and explain the effect on the graphs. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 155 Name Class Date Part III 10. The polar equation r 5 au graphs an Archimedean spiral. For this graph, set the window variables umin 5 0 and umax 5 6p. Graph each equation below and make a sketch of the graph for reference. a. r 5 0.2u b. r 5 20.2u 11. How do the two spirals differ? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 156 Name Class Date EXTENDED TASK Temperature Trigonometric graphs are periodic, that is, they repeat themselves over an interval, or period. For example, the sine function produces a curve that is periodic. The graph of the equation y 5 sin x has a period of 2p radians and an amplitude of 1. Many familiar phenomena are periodic: sound waves, electrical currents, business cycles, air pollution levels, and so on. In this extended task, you will graph real-world data and compare the resulting graph to a trigonometric function that models the physical situation. For this task, we will use data familiar to all: monthly normal temperatures The table below gives the normal monthly temperatures, in degrees F, for Buffalo, New York. Normal Temperature Month Normal Temperature January 24° July 71° February 25° August 69° March 34° September 62° April 45° October 51° May 57° November 41° June 66° December 29° Month Graph, on a set of axes, the normal monthly temperatures for Buffalo. Place the months January through December on the horizontal axis and the temperatures in °F on the vertical axis. Use an appropriate scale. Use only half the horizontal axis for the months January through December. Observe the resulting curve and make any statements about it that you can. Now extend the labeling on the horizontal axis to include another January through December and graph the data a second time. This will give you a repeat curve congruent to the first curve drawn. 1. What is the range for the monthly normal temperatures for Buffalo, New York? 2. Find one-half of this range and add this value to the lowest normal monthly temperature. What is this temperature? 3. Draw a horizontal line on your graph at this temperature. 4. Superimpose a coordinate grid on your graph. Place the x-axis along the horizontal line drawn in Exercise 3 above and the y-axis at the first point where this x-axis intersects the temperature curve. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 157 Name Class Date 5. Using a pen of a different color, highlight one cycle of your curve, beginning at the origin. 6. Based on the placement of the x-axis and y-axis, which trigonometric curve models the normal monthly temperatures for Buffalo, New York? 7. How many days are there in the period of your curve? 8. What is the amplitude of your curve? 9. Find another set of weather data and graph those data in a similar fashion. 10. Discuss your graph, answering questions such as: a. Is the graph periodic? b. Does the graph model one of the trigonometric curves? c. How does it compare to the normal monthly temperature graph for the Buffalo data? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 158 Name Class Date Algebra 2 and Trigonometry: Chapter Eleven Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–3, for each function: a. Find the amplitude. b. Find the period. c. Sketch the graph in the interval 0 # x # 2p. Clearly label the points where the graph crosses the x-axis and where the minimum and maximum values occur. 1. y 5 cos x a. b. 2. y 5 3 sin 2x a. b. 3. y 5 2 sin 12x a. b. 4. a. Sketch the graph of y 5 cos 2x in the interval 2p # x # p. b. On the same set of axes, sketch the graph of y 5 2sin x in the interval 2p # x # p. c. Which of the graphs drawn in parts a and b is symmetric with respect to the y-axis? d. Which of the graphs drawn in parts a and b is symmetric with respect to the origin? e. How many points do the graphs of y 5 cos 2x and y 5 2sin x have in common in the interval 2p # x # p? In 5–8, for each function, find: a. the amplitude. b. the period. c. the phase shift. 5. y 5 cos A x 1 p4 B 6. y 5 2 sin A x 2 p3 B 7. y 5 cos 2 A x 1 p2 B 8. y 5 12 sin 4 A x 2 p6 B a. b. c. a. b. c. a. b. c. a. b. c. In 9 and 10, write the equation of the graph as a: a. sine function b. cosine function. In each case, choose one cycle with the lower endpoint closest to zero to find the phase shift. 9. 2 10. y 1 1 23p 2 2p 21 y x p 2 p 3p 2 x 25p 24p 3 3 22p 2p 3 3 p 3 21 22 a. 2p b. Copyright © 2009 by Amsco School Publications, Inc. a. b. 2p 3 p 4p 3 5p 3 14580TM_C11.pgs 3/26/09 12:14 PM Page 159 Name Class Date 11. What is the domain of the secant function? 12. What is the range of the cotangent function? 13. In the interval p # x # 3p 2 , how are the values of the function y 5 csc x changing? 14. What is the value of y 5 cot u when tan u 5 !3 3 ? In 15–18, find each exact value in degrees. 15. y 5 arcsin !3 2 16. y 5 arccos 12 17. y 5 arctan Q2 !3 3 R 18. As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the starting point. A particular wheel has a diameter of 46 feet and travels at a rate of 3 revolutions per minute. a. Sketch a graph in which the horizontal axis represents time and the vertical axis represents height in relation to the starting point. b. What is the period? c. What is the amplitude? 19. Is there an interval where the graphs of y 5 cos x and y 5 sec x are both increasing? Give the interval or explain why no interval satisfies the condition. Bonus: a. Sketch the graph of y 5 sin x for the interval 22p # x # 2p. b. What is the period of the function? c. Under what transformation(s) is the graph of y 5 sin x its own image? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C11.pgs 3/26/09 12:14 PM Page 160 Name Class Date SAT Preparation Exercises (Chapter 11) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 4. P( p6 , 1. If, for some real number x, x 1 cos x 5 7, then x 1 2p 1 cos (x 1 2p) is (A) 7 1 2p (B) 9p (C) 7 2 2p (D) 5p (E) Not determinable (C) A p3 , 1 B (E) A p6 , 0.6 B x 1 (B) A 5p 6 , 22 B 1 (C) A 5p 6 , 2B 1 (D) A 7p 6 , 22 B 1 (E) A 11p 6 , 22 B y 2 p x The shaded area lies between the x-axis and the graph of y 5 sin x, 0 # x # p. Which point lies in the shaded area? (A) (1, 1) Q(p, 0) (A) A 12, p6 B 5. O ) If point R (not shown) is the reflection of P in Q, then the coordinates of R are y 1 1 2 O 2. The graph of y 5 0.5 intersects the graph of y 5 sin x, 0 # x # 2p, at how many points? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 3. y (B) A p2 , 2 B (D) A p4 , 12 B 1 x O p 2p The equation of the graph shown is (A) y 5 sin (x 1 1) (B) y 5 1 1 sin x (C) y 5 sin 2x (D) y 5 2 1 sin x (E) y 5 2 sin x 6. For what value of k is y 5 cos 2x the image of y 5 sin 2x under the translation Tk,0? (A) p4 (B) p2 (D) p (E) 5p 4 (C) 3p 4 7. Which is not in the domain of y 5 sec x2? (A) 0 (B) p2 (C) p (D) 3p 2 Copyright © 2009 by Amsco School Publications, Inc. (E) 2p 14580TM_C11.pgs 3/26/09 12:14 PM Page 161 Name Class 8. At which values of x can cot x be equal to tan x? (A) 0 and p (B) 0 and 2p (C) p4 and p2 (D) p4 and 3p 4 Date 15. The maximum value of y 5 sin x 1 cos x in the interval 0 # x # p2 is (A) 0 (D) 2 (E) p and 2p 9. If sin x 5 cos (2p), then x can be equal to (A) p2 (B) 3p (C) p 4 (D) 3p 2 (E) 2p (B) 1 (E) 2 !2 (C) !2 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 10. The graph y 5 arcsin x is obtained by reflecting the graph of y 5 sin x in the line (A) y 5 0 (B) y 5 x (C) y 5 2x (D) x 5 0 (E) x 5 p 16. What is the amplitude of the function y 5 3 cos 12x 2 2? 11. The period of y 5 12 sin 3x is (A) 12 (B) p2 (C) 2p 3 18. What is the period of the curve whose equation is y 5 13 sin px? (D) 3 (E) 3p 12. For how many values of x in the interval 0 # x # 2p does cos x 5 2cos x? (A) 0 (B) 1 (C) 2 (D) 4 (E) An infinite number 13. For how many values of x in the interval 0 # x # 2p does 2 sin 4x 5 1? (A) 0 (B) 2 (C) 4 (D) 8 (E) 16 14. Which is an increasing function in the interval p # x # 3p 2? I. y 5 csc x II. y 5 cos x III. y 5 tan x (A) I only (C) III only (E) I, II, and III (B) II only (D) I and III only Copyright © 2009 by Amsco School Publications, Inc. 17. If x 5 cos21 !3 2 , what is the degree measure of x? 19. If f(x) 5 22 cos 2x, evaluate f A 3p 4 B. 20. To the nearest hundredth, what is the maximum value of y 5 arcsin x? 21. To the nearest hundredth, what is the maximum value of y 5 sin x 2 cos x for 0 # x # 2p? 22. What is the exact value of cot A arcsin A 56 B B ? 23. For how many values of x in the interval 2p # x # p does cos 2x 5 2 sin x? 24. For how many values of x does the graph y 5 cos nx have a minimum value in the interval 0 , x , 2p if n is a positive integer? 14580TM_C12.pgs 3/26/09 12:14 PM Page 162 CHAPTER 12 TRIGONOMETRIC IDENTITIES Aims • To distinguish between conditional equations and identities. • To demonstrate methods of proving identities. • To prove basic identities that use the sum or difference of two angle measures. • To prove basic identities that use twice or half the angle measure. In previous chapters, students learned eight basic identities resulting from the definitions of the trigonometric functions. In this chapter, these basic identities are reviewed and then used to prove other identities and to derive relationships that make it possible to find function values of a given angle using function values of other angles. CHAPTER OPENER After students examine the illustration of the parking angle, ask them what value of u corresponds to the maximum road space used. (0°) For u . 0, the length of the space is the hypotenuse of a right triangle; if that length is fixed at, say, 10 feet, then the length of the road space required is 10 cos u and the width of the space is 10 sin u. As u increases from 0° to 90°, the length of the road space decreases and the width increases. 162 12-1 BASIC IDENTITIES Since the solution set of a conditional equation is a proper subset of the domain, this is equivalent to saying that there is at least one element in the domain where the equation is false. An identity, however, is an equation that is true for all values of the domain. To show that an algebraic equation is an identity, we usually use the properties of the field of real numbers, such as the commutative property or the distributive property, and arithmetic substitutions to transform one side into the other. To show that a trigonometric equation is an identity, we usually substitute one or more of the basic identities in the chart on page 483 and use the properties of the field of real numbers. In Example 1, call attention to the question mark above the equal sign. The question mark is used to emphasize that we do not know whether the two expressions on either side of the equation are equal. Thus, proving an identity is similar to checking the solution of an equation. 12-2 PROVING AN IDENTITY Some students look upon proving identities as similar to solving puzzles and enjoy the work, while other students may find the work tedious and unrewarding. Explain to students 14580TM_C12.pgs 3/26/09 12:14 PM Page 163 Trigonometric Identities 163 that the real importance of trigonometric identities becomes clear in advanced courses such as calculus where trigonometric identities facilitate the derivation and application of key formulas and processes. Emphasize that the proof of an identity is different from the solution of a conditional equation. In solving an equation, we work on both sides of the equation simultaneously by performing the same operation on both sides. In proving an identity, we transform an expression either on one side of the equality only, or on both sides independently. There is often more than one series of steps to establish an identity. Either side of the equality can be transformed to prove an identity, although it is usually most efficient to work with the more complicated side, if there is one. Examples 1 through 4 show solutions obtained by working with only one side of the equality. It is also a valid method to work with each side of the equality, transforming each expression separately into the same form. You may wish to present an example. Example: Prove the identity tan u 1 cot u 5 sec u csc u. Solution: ? tan u 1 cot u 5 sec u csc u sin u cos u sin2 u sin u cos u u ? 1 1 1 cos sin u 5 cos u ? sin u 2 u 1 1 sincos u cos u 5 cos u sin u ? sin2 u 1 cos2 u ? 1 sin u cos u 5 cos u sin u 1 sin u cos u 5 cos u1sin u ✔ By showing that both sides of the equality reduce to a common form, we have proved the identity. The comment following Example 1 can be further explained by pointing out that when we prove a trigonometric identity, we are really working backward. The last step of the proof is really the first step in the reasoning process. Since that step is true, we may conclude that the next-to-last step is also true, and so on continuing all the way back to the original equation. 12-3 COSINE (A 2 B) Before presenting the proof of the identity cos (A 2 B) 5 cos A cos B 1 sin A sin B, you may wish to review the distance formula d2 5 (x2 2 x1)2 1 (y2 2 y1)2. The proof places two angles A and B in standard position. The terminal sides of the angles intersect the unit circle at points P and Q, whose coordinates can be written in terms of the sine and cosine values of A and B. With these coordinates, we can find the distance PQ by using the distance formula. By rotating OQP through an angle of 2B, we obtain an angle in standard position whose measure is (A 2 B) and a distance P9Q9 that is equal to the distance PQ. Equating these two expressions for distance enables us to write cos (A 2 B) in terms of the sine and cosine values of A and B. Note that the identities given in this and the following sections have other possible derivations. The derivations given here were chosen because they are valid for all values of A and B and because they make use of relationships previously established in this course. In Example 1, the exact function value for cos 15° is found by writing 15° as (60° 2 45°) and using the cosine rule for cos (60° 2 45°). Ask students whether the same exact function value can be found by using cos (45° 2 30°); they should see this approach is valid and will be applied in Exercise 18. Students should also recognize that the cosine rule can be used for angles in degree or radian measure. 12-4 COSINE (A 1 B) The proof of the identity cos (2u) 5 cos u is relatively straightforward. However, the proof of the identity sin (2u) 5 2sin u may be less obvious to students. We begin by replacing A in the identity sin A 5 cos (90° 2 A) with 2u to obtain the identity sin (2u) 5 cos (90° 2 (2u)). 14580TM_C12.pgs 3/26/09 12:14 PM Page 164 164 Chapter 12 If, at this point, we were to rewrite the right side of this identity using the formula for cos (A 2 B) , we would obtain sin (2u) 5 sin (2u). Therefore, we first express (90° 2 (2u)) as The symbol 6 means plus or minus. Note that in the cosine formula, when the sign on the left side of the equation is plus, the sign on the right side is minus; when the sign on the left side is minus, the sign on the right side is plus. The signs match each other in the sine formula. (u 2 (290°)) to derive the desired relationship sin (2u) 5 2sin u. To derive an identity for cos (A 1 B), we write A 1 B as the difference of two measures and then use the identity for the cosine of the difference of two angle measures. In Example 1, the exact value of cos 105° is !6 found to be !2 2 . Have students check 4 this result by using rational approximations found with a calculator; first the approxima!2 2 !6 4 tion for < 20.2588, and second, the approximation for cos 105° 20.2588. Ask students if there is another way to find the exact value of cos 105° using known values. (Possible answer: cos (150° 2 45°)) 12-5 SIN (A 2 B) AND SIN (A 1 B) Ask students to provide examples that show sin (A 2 B) sin A 2 sin B sin (A 1 B) sin A 1 sin B (Possible answers: sin 60° sin 90° 2 sin 30° and sin 90° sin 60° 1 sin 30°) In this section, the identity sin u 5 cos (90° 2 u) is again used to derive new identities. By replacing u with A 2 B and regrouping the variables in the expression (90° 2 (A 1 B)), we are able to use the formula for the cosine of the sum of two angle measures to obtain an expression equal to sin (A 2 B). The sum and difference of angle formulas can be summarized as follows. For all values of A and B, cos (A 6 B) 5 cos A cos B 6 sin A sin B sin (A 6 B) 5 sin A cos B 6 cos A sin B 12-6 TANGENT (A 2 B) AND TANGENT (A 1 B) Call attention to the fact that, when dividing A cos B by the ratio cos cos A cos B, we are dividing by a ratio that is equal to 1 if cos A 0 and cos B 0. If cos A 5 0, then tan A is undefined; if cos B 5 0, then tan B is undefined. Therefore, the equation A 1 tan B tan (A 1 B) 5 1tan 2 tan A tan B is true for all values of A and B where both tan A and tan B are defined. If 1 2 tan A tan B 5 0, then tan (A 1 B) is undefined. Similarly, the equation A 2 tan B tan (A 2 B) 5 1tan 1 tan A tan B is true for all values of A and B where tan A and tan B are defined. If 1 1 tan A tan B 5 0, then tan (A 2 B) is undefined. In Exercise 18, this identity is derived in Section 9-8 for the special case when u is an acute angle. Here, by applying the sum formula for tangent, the identity is shown to be true for all values of u where tan u is defined. 12-7 FUNCTIONS OF 2A Of all the identities presented in this chapter, the identities for sin 2A and cos 2A are, perhaps, the ones most frequently used in advanced mathematics courses. It is important that students memorize them or be able to derive them from the formulas for cos (A 1 B) and sin (A 1 B). Point out to students that the form of the identity chosen for cos 2A depends on what you are given in a problem and what you are asked to find. In Exercise 23, the identity can 14580TM_C12.pgs 3/26/09 12:14 PM Page 165 Trigonometric Identities 165 be proved by substituting 1 2 sin2 u for cos2 u in cos2 u 2 sin2 u 5 cos 2u. 12-8 FUNCTIONS OF 1 2A The identities for the sine, cosine, and tangent of 12A are unique in that they are the only common identities we have derived so far where it is necessary to choose the correct sign for the result. The sign of sin 12A, cos 12A, and tan 12A must be determined from the quadrant where 1 2A lies. When students use their calculators to explore graphs of identities, remind them that cot x must be entered as 1 4 tan x. Ask if they can think of other ways to check an identity graphically. Some students may suggest that if the difference of the two sides of the identity is treated as a function, then the screen should be blank. For example, if sin2 x 1 cos2 x 2 1 is entered as Y1, no points will be graphed (other than y 5 0). See Enrichment Activity 12-8: Forming Identities where students form identities by matching trigonometric expressions in one column with those in a second column. One approach to find matches is to have students simplify both columns to simpler expressions. EXTENDED TASK In this activity, students are given values for sin 1° and cos 1° and are asked to construct a table of sines and cosines for angles measuring 1° to 10° using the formulas for sin (A 1 B) and cos (A 1 B). They are also asked to find the values with a calculator and compare the two sets of values. Students will find that the values compare very favorably. The activity is then extended using the sine and cosine values to obtain tangent values. Again, the values from the formula computation are close to the values from a calculator. This activity will reinforce students’ memorization of the sum identities and give practice with computation and rounding skills. Although students have seen the algebraic proofs of the identities, the work in this activity will provide additional verification of the formulas. Note that the activity may be assigned at any time following Section 12-5. 14580TM_C12.pgs 3/26/09 12:14 PM Page 166 Name Class Date ENRICHMENT ACTIVITY 12-8 Forming Identities For each expression in Column I, there is one and only one equivalent expression in Column II. Match the equivalent expressions so that, for all possible replacements of the variable whereby each side of the equality is defined, an identity is formed. Column I Column II 2u 1. sec2 u 2 cos cos u a. sin u cot u 2. sin (908 1 u) b. cot u 1 3. csc u 2 cot u cos u c. sec2 u 2 tan2 u 4. cos (2708 2 u) d. sec2 u csc2 u cos2 u 1 sin2 u 5. 1 22tan u 2 sin 2u e. 2cos u tan u sin u 6. sin u A cos u 1 cot uB f. tan u csc u u 1 sin 2u 7. sinsec u 1 2 g. cos u tan u 8. (sin u 1 cos u) 2 2 (sin u 2 cos u) 2 h. cos2 u 3 u 2 2 cos u 9. (sincos u 2 1) 2 1 2 sin u i. sin u cos u 2 cos2 u 2 1 10. cos 2 u 2 sin2 u j. 2 csc u cot u 2u 11. tan u cot u 2 2sin cot u k. 2sin u cot u 2 12. cos u 1 sin u cos u cot u 2 1 l. sin u sec u 13. cos u 2sincos2uu cos 2u m. csc u 14. sec2 u 1 csc2 u n. 2 sin 2u 15. 1 2 1cos u 2 1 1 1cos u o. cos 2u Answers: 1. 2. 3. 4. 5. 6. 7. 9. 10. 11. 12. 13. 14. 15. Bonus: On separate paper, find the exact value of each expression in Column I when u 5 30°. Copyright © 2009 by Amsco School Publications, Inc. 8. 14580TM_C12.pgs 3/26/09 12:14 PM Page 167 Name Class Date EXTENDED TASK You are given the following information: sin 1° 5 0.01746 cos 1° 5 0.99985 1. Construct a table of sine and cosine values using the identities for the sine and cosine of the sum of two angles. For each computation, use 1° as one of the angles in the sum. Then compare the values you calculated using the formulas with the values obtained directly from a calculator. Complete the table below rounded to four decimal places. By Formula Angle Sin From Calculator Cos Sin Cos 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° 2. How did the values for sine and cosine found using the formulas compare with the values from the calculator? sin x 3. Use the identity tan x 5 cos x to find the tangent values for the angles from 1° to 10°. Compare with the values from a calculator. Complete the table below rounded to four decimal places. Angle Tan (by formula) 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° Copyright © 2009 by Amsco School Publications, Inc. Tan (calculator) 14580TM_C12.pgs 3/26/09 12:14 PM Page 168 Name Class Date Algebra 2 and Trigonometry: Chapter Twelve Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1 and 2, write each expression as a single function. 1. cos2 u sec u csc u 2. sin u (1 1 cot2 u) In 3–6, write true or false if A 5 90°. 3. sin 2A 5 2 sin A cos A 4. cos 2A 5 2 sin A 213 A B A 5. tan 2A 5 1 22tan tan2 A 6. cos 12A 5 sin 12A 7. If tan A 5 2 !5 2 and A is in the second quadrant, find: a. cos 2A b. the quadrant where 2A lies 8. If x is the measure of an obtuse angle and cos x 5 218, find sin 12x. 9. If B 5 arcsin Q2 !7 4 R , find sin 2B. In 10–13, use a sum or difference formula to find the exact value of each expression. 10. cos 285° 11. sin 255° 12. cos (2210°) 13. cos 195° In 14–19, sin x 5 0.28, cos y 5 0.6, and x and y are measures of acute angles. Find the exact value of each expression. 14. cos x 16. sin 1 2x 18. sin (x 1 y) Copyright © 2009 by Amsco School Publications, Inc. 15. sin y 17. cos 2y 19. tan (y 2 x) 14580TM_C12.pgs 3/26/09 12:14 PM Page 169 Name Class Date 20. From the top of a lighthouse 220 feet above a river, the keeper spots a boat sailing directly toward the lighthouse. The keeper observes that the angle of depression of the boat is 6° and then, sometime later, observes the angle to be 14°. To the nearest tenth of a foot, find the distance the boat has sailed between the times the two observations were made. In 21–24, in each case: a. Prove that the given statement is an identity. Show your work on a separate piece of paper. b. Find the domain where the identity is defined. 21. csc x ? tan x 5 sec x 2 2 2 b. 2 22. cos x 1 cos x cot x 5 cot x b. sin 2x 2 sin x sin x 23. cos 2x 1 cos x 5 cos x 1 1 cos x 24. sec x1 1 1 1 sec x1 2 1 5 2sin 2 x b. b. Bonus: In the diagram, BG and CF divide rectangle ADEH into three congruent squares. The degree measures of DAE, DBE, and DCE are x, y, and z, respectively. Prove that x 1 y 1 z 5 90°. (Hint: Find tan (x 1 y).) H G y x A Copyright © 2009 by Amsco School Publications, Inc. F B E z C D 14580TM_C12.pgs 3/26/09 12:14 PM Page 170 Name Class Date SAT Preparation Exercises (Chapter 12) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If sin u 5 0.438, what is the value of sin u cos u tan u to the nearest hundredth? (A) 1.00 (B) 0.71 (C) 0.44 (D) 0.19 (E) Cannot be determined 9. sin2 x 1 3 cos2 x 5 (A) 22 1 cos 2x (C) 2 1 cos 2x (E) 2 2 cos 2x 10. 2. cos4 u 2 sin4 u is equal to (A) 2 (B) 4 (C) cos 2u (D) sin 2u (E) cos2 u 1 sin2 u 3. sin A u 1 p6 B 1 cos A u 1 p3 B 5 (A) sin u (B) cos u p (C) sin A u 1 3 B (D) cos A u 1 p6 B (E) 1 4. If sin u 5 0.9 and 1 2 cos2 u 5 k, what is k? (A) 0.81 (B) 0.405 (C) 0.19 (D) 0.1 (E) Cannot be determined 5. Where defined, tan x sin x cos x csc2 x is equivalent to (A) tan2 x (B) sec x (C) sin x (D) cot2 x (E) 1 6. If sin x cos x 5 52 and tan x 5 12, then sin2 x 5 (A) 15 (D) 2 !5 5 (B) 52 (C) !5 5 (E) 53 7. The value of (sin x 1 cos x)2 2 sin 2x is (A) 22 (B) 21 (C) 0 (D) 1 (E) 2 8. If cos 2x 5 45 and cos x 1 sin x 5 k, then cos x 2 sin x is (A) 45k (B) 35k (D) 35k 4 (E) 5k (C) 54k Copyright © 2009 by Amsco School Publications, Inc. (B) 22 1 sin 2x (D) 2 1 sin 2x y 1 Q O P R S p x Rectangle PQRS is inscribed in the graph of y 5 sin x, 0 # x # p. If PQ 5 sin k, then PS equals (A) k (B) p2 2 k (C) p 2 k (D) 2p 2 k (E) p 2 2k 11. x° x° If each edge of the square has a measure of 1, then the shaded region has an area of (A) 12 (B) 1 2 sin x° (C) 1 2 sin 2x° (D) 1 2 12 sin x8 (E) 1 2 12 sin 2x8 14580TM_C12.pgs 3/26/09 12:14 PM Page 171 Name Class 12. II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. S 16. In PQR, R is a right angle. What is the value of sin2 P 1 sin2 Q? R x° x° P 17. What is the exact value of (4 sin 2u)(5 sin 2u) 1 (2 cos 2u)(10 cos 2u)? Q If PQ 5 2 and RQ 5 1, then PS equals (A) 2 !5 3 (B) 35 (E) 2 !5 (D) 10 3 (C) 2 !10 3 (A) 12 cos 558 1 !3 2 sin 558 (B) 12 cos 558 2 !3 2 sin 558 22. Angle A is in the second quadrant and cos A 5 235. Angle B is in the third (E) !2 2 (cos 408 1 sin 408) 2 14. Where defined, (sin x 1 cos x) 2 (tan2 x 1 cot2 x) 1 sec2 x 1 csc2 x 5 (A) 23 (B) 21 (C) 0 (D) 1 (E) 3 P S 1 1 R If PQ 5 2 and QR 5 SR 5 1, then 2!2 cos PQS equals (A) 1 1 !3 (B) 1 2 !3 (C) !3 2 1 (E) !2 1 1 7 quadrant and sin B 5 225 . What is the exact value of cos (A 1 B)? 1 23. If tan (A 1 B) 5 217 6 and tan A 5 2 , what is the exact value of tan B? 24. If cos 2u 5 !5 4 , what is the exact value of 2 cos x? Q 2 19. If 6(cos 80° cos 20° 1 sin 80° sin 20°) 5 x, what is x? 21. If sin u 5 sin 37° cos 12° 2 cos 37° sin 12° and u is an angle in the second quadrant, what is the degree measure of 3u? (C) 12 cos 258 2 !3 2 sin 258 !3 (D) 2 cos 258 1 12 sin 258 15. 18. What is the exact value of sin4 u 2 cos4 u cos2 u 2 sin2 u 1 7? 20. If cos u 5 15 and u is an angle in the fourth quadrant, find the exact value of sin 2u . 13. cos 85° equals 2 Date (D) !2 2 1 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C13.pgs 3/26/09 12:14 PM Page 172 CHAPTER 13 TRIGONOMETRIC EQUATIONS Aims • To solve first-degree trigonometric equations algebraically. • To solve higher-order trigonometric equations using factoring and the quadratic formula. • To develop methods of solving trigonometric equations using substitution of identities. • To solve trigonometric equations graphically. In this chapter, categories of trigonometric equations are examined and the appropriate solution methods for each are developed. While the same procedures used for solving algebraic equations can be used to solve trigonometric equations, students need to understand how the value of the unknown leads to all solutions of the equation. As usual, there is often more than one way to solve a trigonometric equation. Although some methods may be more efficient, students should use the approach that makes sense to them and that they can apply with confidence. 172 13-1 FIRST-DEGREE TRIGONOMETRIC EQUATIONS Students should be aware that trigonometric equations are not identities. An identity is true for any angle in the domain of the function involved. A trigonometric equation is true for some specific angles, if at all. For example, the equation sin x 5 22 has no solution since 22 is outside the range of the sine function. Emphasize Step 3 of the procedure for solving a linear trigonometric equation illustrated in the example for 5 cos u 1 7 5 3. Students may not recognize the correct method for working backward from the value of 143° to the reference angle and then using this reference angle to find the second solution in the third quadrant. Trigonometric equations are usually solved for values of the variable in the interval f08, 3608) or f0, 2p) . There are other solutions outside that interval. These other solutions differ by integral multiples of the period of the function. In this section, the period of each function is 360° or 2p radians. The use of a graphing calculator greatly simplifies the solving of trigonometric equa- 14580TM_C13.pgs 3/26/09 12:14 PM Page 173 Trigonometric Equations 173 tions. The calculator is especially helpful when the equation involves several different functions, involves multiples of angles, has fractional or decimal coefficients, or has constants that do not easily lend themselves to standard algebraic methods. When 3(sin A 1 2) 5 3 2 sin A is solved by graphing Y1 5 3(sin X 1 2) Y2 5 3 2 sin X we are forming a system of equations and finding their intersection points. Alternatively, the equation can be solved by simplifying as follows: 3(sin A 1 2) 5 3 2 sin A 3 sin A 1 6 5 3 2 sin A 4 sin A 1 6 5 3 4 sin A 1 3 5 0 If Y1 5 4 sin X 1 3 is then graphed, we look for the zeros of that function. Both methods will produce the same solutions. 13-2 USING FACTORING TO SOLVE TRIGONOMETRIC EQUATIONS Review factoring techniques (greatest common factor, difference of squares, factoring trinomials) and the solution of second-degree equations before beginning this section. It is important that students realize that the equation 3x2 2 5x 2 4 5 0 is a quadratic equation in x, since x is the variable that is squared, but that 2 3 cos x 2 5 cos x 2 4 5 0 is a quadratic equation in cos x, since cos x, not x, is the variable that is squared. If students have difficulty with this concept, they may find it helpful to rewrite equations in a nontrigonometric form before attempting to solve them. Example: Find all the values of u in the interval 0° # u , 360° that satisfy the equation 2 cos2 u 1 cos u 2 1 5 0. Solution: Let x 5 cos u. The equation can now be written as 2x2 1 x 2 1 5 0 (2x 2 1)(x 1 1) 5 0 2x 2 1 5 0 x1150 x 5 12 Replace x with cos u. cos u 5 12 u 5 608 or x 5 21 cos u 5 21 u 5 1808 or u 5 3008 Answer: {60°, 180°, 300°} Emphasize that factoring can be used only if one side of the equation is 0. Example 2 illustrates two solution methods: factoring using the difference of two squares and the square root method, an approach used with quadratics of the form ax2 1 c 5 0 where c is negative or 0. Any factoring method may be applied to trigonometric equations. Factoring by grouping works in some special cases where there are an even number of terms and common factors can be found for groups of them. For example, 4 sin x cos x 2 2 sin x 2 2 cos x 1 1 5 0 2 sin x(2 cos x 2 1) 2 1(2 cos x 2 1) 5 0 (2 sin x 2 1)(2 cos x 2 1) 5 0 The solution can then be completed by setting each factor equal to 0 as usual. 13-3 USING THE QUADRATIC FORMULA TO SOLVE TRIGONOMETRIC EQUATIONS Factoring can be applied only to cases where the quadratic equation has rational roots. The quadratic formula can be used for any seconddegree equation. 14580TM_C13.pgs 3/26/09 12:15 PM Page 174 174 Chapter 13 As suggested in Section 13-2, students who have difficulty recognizing the values for a, b, and c in the quadratic formula may find it helpful to change equations to non-trigonometric form. After some practice, they should be able to forgo this intermediate step and work directly with the given equation. cot2 u 5 csc u 1 1 cos2 u sin2 u 5 sin1 u 1 1 cos2 u 5 sin u 1 sin2 u One more substitution is needed, that is, cos2 u 5 1 2 sin2 u. 1 2 sin2 u 5 sin u 1 sin2 u 13-4 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING MORE THAN ONE FUNCTION Although various suggestions are given in this section for solutions of trigonometric equations, here, too, students should realize that there is often more than one way to proceed. Careful observation may suggest special techniques for some equations. For instance, some algebraic operations, such as squaring, may produce answers that are not solutions of the original equation. Therefore, it is necessary to check solutions in the given equation. In general, if an equation involves two or more different functions, try replacing each function using either a ratio identity or a reciprocal identity. In the new expression: • simplification may be possible, resulting in an equation that is either factorable or solvable by the quadratic formula. • the substitutions may create fractions that can be combined using an LCD. The products that result from the multiplications used to create equivalent fractions may be parts of identities that can be substituted to simplify the expression. As students work the exercises, suggest they first think whether there is an obvious substitution that will express one function in terms of the other. If not, as in the case of Exercise 13, they should try writing all functions in terms of sine and cosine: Simplifying results in the equation 2 sin2 u 1 sin u 2 1 5 0 which can be solved by factoring. In Enrichment Activity 13-4: Solving Trigonometric Inequalities, students use graphing to identify the intervals where inequalities involving constants or two functions are true. 13-5 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING DIFFERENT ANGLE MEASURES This section extends the concept of using substitution to solve trigonometric equations by focusing on identities for multiple angle measures. Where an equation containing a function of a multiple angle can be solved directly, students must know how to obtain all solutions of x between 0° and 360°. For example, 2 sin 3x 5 1 sin 3x 5 12 3x 5 308 S x 5 108 3x 5 3908 S x 5 1308 3x 5 7508 S x 5 2508 3x 5 1508 S x 5 508 3x 5 5108 S x 5 1708 3x 5 8708 S x 5 2908 14580TM_C13.pgs 3/26/09 12:15 PM Page 175 Trigonometric Equations 175 To obtain all solutions of x between 0° and 360°, it is necessary to solve for all solutions of 3x between 0° and (3)360° 5 1,080°. To do this, add 360° to each of the original solutions two times and divide each result by 3. In general, if a solution is found for nu, add 360° or 2p to each of the original solutions (n 2 1) times and divide each result by n. EXTENDED TASK For the Teacher: This extended task is designed to give students practice in working with the skills pre- sented in this chapter as well as those skills that are prerequisite to working with trigonometric identities and equations. Students could be encouraged to make up their own grids with equations, using a variety of expressions. As well as being very useful, students should see mathematics as an avenue for recreation. Hence, this extended task was designed as a mathematical puzzle. 14580TM_C13.pgs 3/26/09 12:15 PM Page 176 Name Class Date ENRICHMENT ACTIVITY 13-4 Solving Trigonometric Inequalities In Section 5-9, you learned how to use graphs of quadratic inequalities in two variables to solve quadratic inequalities in one variable. Trigonometric inequalities can be solved in the same way. Example Solve the trigonometric inequality sin x . 212 for the interval 0 # x , 2p. Solution: First, find the solutions to the corresponding equation: sin x 5 212 11p The roots of the equation are 7p 6 and 6 and separate the number line into three intervals: 0 # x , 7p 6, 7p 6 , x , 11p 6 , 11p 6 , x # 2p The solutions to the inequality sin x . 212 correspond to the interval(s) where the graph of y 5 sin x is above the graph of y 5 212. 1 y 0.5 20.5 21 x p 6 p 3 p 2 2p 5p 3 6 p 7p 4p 3p 5p 11p 6 3 2 3 6 2p From the graph, we see that this relationship is true in the intervals 0 # x , 7p 6 and 11p . , x # 2p 6 11p Answer: 0 # x , 7p 6 or 6 , x # 2p In 1–4, solve each inequality for 0 # x , 2p. 1. tan x $ 1 3. cos x # 0 2. sec x , 21 4. cot x , !3 5. a. If 0 # x , 2p, for what exact values of x does sin x 5 cos x? b. Explain how you would solve sin x , cos x by graphing. What are the solutions? c. For what values of x is the inequality sin x $ cos x true? 6. Solve sin x $ cos x for 0 # x , 2p. 7. Solve cos x , !3 2 for 0 # x ,2p. 8. Solve cot x $ tan x for 0 # x , 2p. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C13.pgs 3/26/09 12:15 PM Page 177 Name Class Date EXTENDED TASK Find the Letter: A Trigonometric Puzzle 1. Solve each of the problems a–k and shade in the correct answer in the chart below. 45°, 135°, 225°, 315° 30° 2 !5 5 7p 12 csc x sin 2x 63 65 12 7 p cot x 1 sin x cos x 135° and 315° 135° cos2 x 240° sin2 x 1 2 cos x sin x 100° !5 5 sec x p 3 1 2 sin x cos x 45° and 225° cos 2x 33 65 210° 260° a. The expression 1 2 cos2 x is equal to . b. How many degrees are there in 3p 4 radians? c. Express sec x 2 tan x as an expression in sin x and cos x. x d. The expression cos sin x is equal to . e. What is the smallest positive angle which satisfies the equation 2 cos2 x 2 5 cos x 1 2 5 0? f. If 2 sin x 1 1 5 0 and x is in Quadrant III, what is a possible value for x? g. The expression sin1 x 5 . h. If cos A 5 35 and angle A is acute, find the exact value of sin 12A. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C13.pgs 3/26/09 12:15 PM Page 178 Name Class Date i. In the interval 0° # x # 360°, sin x 5 cos x. Find all values of x. 5 j. If sin A 5 35, sin B 5 13 , and angles A and B are acute angles, what is the value of cos (A 2 B)? k. How many radians are in 105°? 2. In the chart below, find the letter that corresponds to each answer that you shaded on the previous page. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 3. In the blanks below, use the letters you have shaded in Exercise 2 to write an expression. (Note: Two extra vowels have been supplied to aid you in your task.) O I ? Copyright © 2009 by Amsco School Publications, Inc. . 14580TM_C13.pgs 3/26/09 12:15 PM Page 179 Name Class Date Algebra 2 and Trigonometry: Chapter Thirteen Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–12, find the exact solution set of each equation if 0° # u , 360°. 1. 8 sin u 1 1 5 23 2. 3 tan u 2 2 5 tan u 3. 2 sin2 u 2 1 5 0 4. 3 tan2 u 2 1 5 0 5. 2 cos2 u 2 7 cos u 5 4 6. 2 sin2 u 1 3 sin u 1 1 5 0 7. sec u 5 sec1 u 8. 2 cos u sin u 2 cos u 5 0 9. sec2 u 2 tan u 2 1 5 0 11. cos 2u 1 cos u 1 1 5 0 10. 3 cos 2u 2 5 cos u 5 1 12. tan u 5 3 cot u In 13–20, find to the nearest degree all values of u in the interval 0° # u , 360° that satisfy each equation. 13. 10(cos u 1 1) 5 6 14. 2 cos2 u 2 5 cos u 2 1 5 0 15. tan2 u 5 8 tan u 2 5 16. cos2 u 2 3 sin u 1 2 5 0 17. 6 cos u 1 1 5 5 sec u 18. tan2 u 5 tan u 1 1 19. 3 cos 2u 1 5 cos u 1 2 5 0 20. 2 cos 2u 1 cos u 5 0 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C13.pgs 3/26/09 12:15 PM Page 180 Name Class Date 21. Express, in radians, the exact values of u in the interval 0 # u , 2p that are roots of the equation 4 sin3 u 2 sin u 5 0. 22. Express, in degrees, the exact values of u in the interval 0 # u # p that are roots of the equation 4 sin2 3u 5 3. Bonus: Solve the equation sin x 5 cos x for 0 # x , 360°. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C13.pgs 3/26/09 12:15 PM Page 181 Name Class Date SAT Preparation Exercises (Chapter 13) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If 0° # u , 360°, which of the following are solutions of I. 30° 3 tan u 1 !3 5 2 !3? II. 150° III. 210° (A) I only (C) I and II only (E) I, II, and III (B) II only (D) I and III only (B) p2 (D) 3p 4 (E) 5p 3 8. Which is not a solution of sin 2u 5 tan u if 0° # u , 360°? (A) 45° (B) 105° (C) 180° (D) 225° (E) 315° 9. In the interval 0 # u , 2p, what is the solution set of 2 sin 2u cos u 2. One root of the equation 3 cos u 2 1 5 1 2 cos u is (A) p4 7. Which equation is equivalent to cos 2u 2 2 sin u 1 2 5 0? (A) 22 sin2 u 2 2 sin u 1 3 5 0 (B) 22 sin2 u 2 2 sin u 2 1 5 0 (C) 2 cos2 u 1 cos u 1 2 5 0 (D) 2 cos2 u 2 2 sin2 u 1 1 5 0 (E) 2 cos u 2 sin2 u 2 3 5 0 (C) 2p 3 3. To the nearest degree, the measure of the acute angle that satisfies the equation csc u 5 3(csc u 2 5) is (A) 4° (B) 8° (C) 15° (D) 82° (E) 172° 4. In the interval 0° # u , 360°, what is the solution set of cos2 u 2 cos u 2 2 5 0? (A) {0°} (B) {90°} (C) {180°} (D) {0°, 180°} (E) {0°, 90°, 180°} 5. The smallest non-negative measure for which 2 sin2 u 2 3 sin u 1 1 5 0 is (A) 0° (B) 30° (C) 45° (D) 60° (E) 120° 6. If 0° # x , 360°, how many real roots does the equation 16 sin3 x 2 sin x 5 0 have? (A) 0 (B) 2 (C) 3 (D) 4 (E) 6 Copyright © 2009 by Amsco School Publications, Inc. 2 sin1 u 5 0? (A) { } (B) U p3 , 2p 3 V (C) U p3 , 5p 3 V (D) Up6 , 5p 6 V 7p 11p (E) U p6 , 5p 6, 6, 6 V 10. If 0° # u , 360°, how many solutions does the equation cos u 2 cos u sin2 u 5 0 have? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 11. If cos (u 2 40°) 5 sin 60°, then the degree measure of u can be (A) 20° (B) 30° (C) 60° (D) 70° (E) 90° 12. How many solutions to cos2 u 2 5 cos u 1 6 5 0 are there in the interval 0° # u , 360°? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 13. Which is not a solution of tan x 2 sin x tan x 1 sin x x 2 1 5 sec sec x 1 1 ? (A) p8 (B) 3p 5 (D) 7p 2 (E) 13p 4 (C) 5p 6 14580TM_C13.pgs 3/26/09 12:15 PM Page 182 Name Class 14. If 0° # x , 360°, which of the following are solutions to sin x 5 1 1 cos x? I. 908 18. What is the sum of all the degree values of u in the interval 0° # u , 360° that satisfy the equation 3 tan2 u 2 1 5 0? 19. If A and B are acute angles and II. 1808 III. 2708 (A) I only (C) I and II only (E) I, II, and III Date sin (A 1 B) 5 (sin A 1 cos A) sin B, (B) II only (D) I and III only 11p 15. If 7p 6 and 6 are all the solutions to 2 sin2 u 2 3 sin u 1 p 5 0 for 0 # u , 2p, then the value of k is (A) 23 (B) 22 (C) 21 (D) 2 (E) 4 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. In the interval 0 # u , 2p, what is the radian measure of the greatest angle that is a solution to 2 sin u 5 tan u? 17. Find, to the nearest degree, the least positive value of u in the interval 0° # u , 360° that is a solution to 3 cos2 u 1 2 sin u 2 1 5 0. Copyright © 2009 by Amsco School Publications, Inc. then what is the degree measure of B? 20. If cos 2x 5 cos2 2x and 0° , x , 90°, what is the degree measure of x? 21. If twice the product of sin x and cos x equals the product of tan x and cot x, what is the least positive value for x, in degrees? 22. Find x to the nearest tenth of a degree if 3 csc x 2 6 5 csc x and 90° , x , 180°. 23. Find to the nearest tenth of a degree the smallest positive value of x if 2 sin x 5 4 sin x 1 1. 24. In the interval 2p # x # p, how many solutions are there to the equation sin 2x 5 x? 14580TM_C14.pgs 3/26/09 12:15 PM Page 183 CHAPTER 14 TRIGONOMETRIC APPLICATIONS Aims • To be able to express any point in the coordinate plane in terms of its distance from the origin and the measure of an angle. • To derive and apply the Law of Cosines. • To derive and apply the Law of Sines. • To determine the area of a triangle or a parallelogram, given the measures of two sides and the included angle. • To recognize the ambiguous case and to be able to determine the number of possible solutions. • To be able to find the missing measures of the sides and angles of a triangle when sufficient information to determine the triangle is provided. In the study of triangle congruence, students learned what information is sufficient to determine the triangle, that is, to establish that one and only one triangle can be constructed using the given information. When it has been decided that a triangle can be constructed using the given data, the missing measures of the sides and angles, as well as the area of the triangle, can be found. The formulas derived in this chapter make it possible to compute the desired measures. CHAPTER OPENER If a distress signal is received at two coast guard stations, the distance between the stations and the direction of the ship’s signal at each station can be used to determine the ship’s exact location, which coast guard station is closer, and from which station help can arrive more quickly. In this situation, the measure of two angles and the included side of a triangle are known. The exact position of the ship may be determined by drawing the triangle with the two coast guard stations and the ship at the vertices and then using the Law of Sines to find the distance of the ship from each coast guard station. The problem is similar to Exercise 22 in Section 14-5. Analagous problems can be described to apply to aviation (the distance between two airports and an aircraft) or forestry (the distance between two ranger stations and a fire). 14-1 SIMILAR TRIANGLES This section reviews the conditions that must be met for two triangles to be similar and then applies the similarity relationship to derive the formulas used to write rectangular coordinates in trigonometric form.Any point can be located using its distance from the origin and the measure of the angle in standard position. The ideas in this section are prerequisite for students’ understanding of the derivation of the Law of Cosines that follows in the next section. 183 14580TM_C14.pgs 3/26/09 12:15 PM Page 184 184 Chapter 14 In Example 2, students must be aware of the correct steps to use once the calculator has returned a value for the inverse cosine or inverse sine. The calculator does not give the reference angle, nor does it give the required third-quadrant angle. For Exercise 1, students should explain that R 5 arctan 28.48 25.3 5 588, so it is not necessary to find OA first. To summarize the Law of Cosines, explain that the measure of one side is defined in terms of the measures of the other two sides and the angle opposite the first side. Emphasize that the Law of Cosines can be used with the following given information: • the measures of two sides and the included angle (SAS), or • the measures of three sides (SSS) 14-2 LAW OF COSINES For the derivation of the Law of Cosines given in the text, the vertex A of the angle whose measure is known is placed at the origin of the coordinate plane and one side whose measure is known is placed along the x-axis so that a second vertex, B, is on the x-axis. The derivation would have been essentially the same if point C had been placed on the x-axis. To repeat the derivation expressing b in terms of a, c, and cosine B, position ABC on the coordinate plane so that the coordinates of B are (0, 0) the coordinates of A are (c, 0) and the coordinates of C are (a cos B, a sin B). An alternate derivation that does not involve coordinates can be shown using the diagram below. C b A h x D a c2x B c Consider ABC with height h and sides with measures a, b, and c. Let AD 5 x. Then DB 5 c 2 x. Using the Pythagorean Theorem for DCB, we have a2 5 (c 2 x) 2 1 h2 a2 5 c2 2 2cx 1 x2 1 h2 In ADC: b2 5 x2 1 h2 so: a2 5 c2 2 2cx 1 b2 Since cos A 5 xb: x 5 b cos A a2 5 c2 2 2c(b cos A) 1 b2 Then: a2 5 b2 1 c2 2 2bc cos A Problems involving the SAS situation are examined in this section and the SSS case is considered in the next section. 14-3 USING THE LAW OF COSINES TO FIND ANGLE MEASURE A triangle is determined by the measures of its three sides; that is, if the measures of three sides are given, the triangle can have exactly one size and exactly one shape. The measures of the three angles can be found using the Law of Cosines. Recall that in order to construct a triangle using three given line segments as sides, the measure of each side must be less than the sum of the measures of the other two. For example, 10, 3, and 4 cannot be the measures of the sides of a triangle because 10 . 3 1 4. If we try find A using 10, 3, and 4 as the values of a, b, and c in the Law of Cosines, the result is a value for cos A that is less than 21 and therefore not acceptable: 2 c2 2 a2 cos A 5 b 1 2bc 2 42 2 102 5 3 12(3)(4) 5 9 1 16242 100 5 23.125 Since 21 # cos A # 1, there is no angle whose cosine is 23.125. This example verifies our original observation that 10, 3, and 4 cannot be the measures of the sides of a triangle. Students are asked to consider a similar situation in Exercise 1. For Exercises 13–18, encourage students to anticipate the location of the angle before using 14580TM_C14.pgs 3/26/09 12:15 PM Page 185 Trigonometric Applications 185 the calculator to find the measure. If cos A is between 0 and 1, A is an acute angle; if cos A is between 21 and 0, A is an obtuse angle; if cos A 5 0, A is a right angle.Although for any given value of cos A there are infinitely many values of A, the calculator will always return a value for A between 0° and 180°, a value that can be the measure of an angle of a triangle. 14-4 AREA OF A TRIANGLE Note that in this section the use of A to represent area is avoided since in the discussion A denotes the vertex of an angle. When presenting the derivation in class, you may wish to use a letter such as K to denote the area. The letter b also has two different meanings in this section: (1) the base of a triangle in the formula for area and (2) a particular length of a side in the formula using the sine. The formula derived in this section uses the measures of two sides and the included angle (SAS). The derivation of a formula for the area of a triangle in terms of two angles and an included side (ASA) is suggested as a bonus question in the Chapter Test for this chapter. In Exercise 15, students are asked to find the area of a particular equilateral triangle. They can easily derive the general formula for the area of an equilateral triangle with side s. Area of equilateral nABC 5 12ab sin /C 5 12 (s)(s) sin 608 5 12s2 !3 2 2 5 s !3 4 The formula for the area of a triangle in terms of the measures of the three sides of the triangle is called Heron’s formula. Enrichment Activity 14-4: Heron’s Formula leads students through the steps needed to derive this formula. 14-5 LAW OF SINES The Law of Sines can be applied to a triangle when the given information is: • the measures of two angles and the measure of any side (ASA and AAS), or • the measures of two sides and the angle opposite one of the sides (SSA) Note that in the first case, the sum of the measures of the two angles must be less than 180°. In the second case, a unique solution does not always exist, and sometimes a solution does not exist at all. The ambiguous case of SSA is examined in Section 14-6. The use of the area formula derived in Section 14-4 simplifies the derivation of the Law of Sines. This identity indicates that, for a given triangle, the ratio of the measure of any side to the sine of the angle opposite is constant. The largest side is opposite the angle with the largest sine value and, therefore, the largest angle of the triangle; the smallest side is opposite the angle with the smallest sine value and, therefore, the smallest angle of the triangle. If C is a right angle, then ABC is a right triangle where: • c is the measure of the hypotenuse • a is the measure of the leg opposite A • b is the measure of the leg opposite B Since sin C 5 sin 90° 5 1, the Law of Sines can be used to prove that sin A 5 ac and sin B 5 bc (see Exercise 18). 14-6 THE AMBIGUOUS CASE Emphasize that when students solve problems involving triangles, they must analyze the data to determine whether or not a solution exists and, if a solution does exist, whether it includes one or two triangles. The measures of two sides and the angle opposite one of them (SSA) are not sufficient to determine a triangle. From such information, one triangle, two triangles, or no triangle may be constructed. This section derives a set of rules to determine which of these possibilities applies for a given set of data. Although the rules are summarized at the end of the section, students should be encouraged to use reasoning rather than memorization as they solve problems. As shown in 14580TM_C14.pgs 3/26/09 12:15 PM Page 186 186 Chapter 14 Example 1, the number of solutions can be found by applying the Law of Sines and then interpreting the results. In that example, two possible values of B in ABC are found and tested with the measure of A, the given value. Since the sum of the measures of two angles of a triangle must be less than 180, only one of the measures of B is a solution and only one triangle is possible. If the value of sin B had been greater than 1, no triangle would have been possible. It is interesting to examine how the ambiguous case is reflected when the Law of Cosines is used to find the possible solutions for SSA. After substituting in the Law of Cosines, a quadratic equation is obtained. If the discriminant of the equation is negative, no triangle is possible since there are no real roots. If the discriminant is positive, there may be zero, one, or two positive values for the measure of the third side, giving zero, one, or two possible triangles. An example is given below. Example: Use the Law of Cosines to find two possible measures of c in ABC when a 5 7, b 5 8, and mA 5 60. Then use the results to find, to the nearest degree, two possible measures of B in ABC. Solution: To find c, use the Law of Cosines with a on the left-hand side. Use this form because after known values are substituted, it is the only equation that will have just one variable to solve for. a2 5 b2 1 c2 2 2bc cos A Substitute a 5 7, b 5 8, m/A 5 60: 72 5 82 1 c2 2 2(8)c cos 608 49 5 64 1 c2 2 2(8)c(0.5) 49 5 64 1 c2 2 8c 0 5 c2 2 8c 1 15 The quadratic is factorable: 0 5 (c 2 3)(c 2 5) So c 5 3 or c 5 5. Next, let c 5 3. Either the Law of Cosines or the Law of Sines can be used to find the measure of B. Using the Law of Cosines: b2 5 a2 1 c2 2 2ac cos B 64 5 49 1 9 2 2(7)(3) cos B 64 5 58 2 42 cos B 6 5 242 cos B 6 cos B 5 242 6 cos21 A 242 B < 98.218 < 988 Note that mC is easily found using this result. m/C 5 1808 2 (m/A 1 m/B) 5 1808 2 (608 1 988) 5 228 A similar calculation using c 5 5 gives mB 5 82° and mC 5 38°. In this example, the quadratic equation was solved by factoring and the two solutions were obtained. Often, however, the computation required to solve the quadratic equation makes this a much lengthier problem than the approach that begins with the Law of Sines. Hands-On Activity Instructions: Choose any acute A and any line segment of length c as an angle and an adjacent side of a triangle. In this activity, line segments of different lengths will be used for possible sides opposite A. The activity can be done using either a compass, straightedge, and pencil, or geometry software. 1. For convenience, draw A with one ray horizontal. Find point B on the oblique (non-horizontal) ray of A such that AB 5 c. 2. From B, construct the perpendicular to the horizontal ray of A. Call the foot of the perpendicular D. Let BD 5 h. One right triangle has been drawn, ABD, with sides of lengths c and h and A opposite side h. Note that sin A 5 hc or h 5 c sin A. 14580TM_C14.pgs 3/26/09 12:15 PM Page 187 Trigonometric Applications 187 3. Draw a line segment of length a such that c . a . h. Open your compass to length a. With the point of the compass at B, h draw two arcs that intersect AD. Label one point of intersection C and the other C9. Draw BC and BCr. Two triangles have been drawn, ABC and ABC9, with sides of length c and a and A opposite side a. 4. Draw a line segment of length e such that e . c. Open your compass to length e. With the point of the compass at B, draw h an arc that intersects AD. Label the point of intersection E. Draw BE. One triangle has been drawn, ABE, with sides of length c and e and A opposite side e. 5. Draw a line segment of length f such that f , h. Open your compass to length f. With the point of the compass at B, show h that no arc can be drawn to intersect AD. No triangle can be drawn with sides of lengths c and f and A opposite the side of length f. Discoveries: Given the lengths of two sides, a and c, and A opposite side a: 1. One right triangle can be drawn if a 5 c sin A. 2. Two triangles can be drawn if c . a . c sin A. 3. One triangle can be drawn if a . c. 4. No triangle can be drawn if a # c sin A. What If . . . The given angle is obtuse? (One triangle can be drawn if a . c. No triangle can be drawn if a # c.) 14-7 SOLVING TRIANGLES This section summarizes the work of the preceding sections and provides some suggestions for approaching problems where the measures that are given and those to be determined are related as parts of a triangle. The importance of a well-drawn figure should be constantly stressed. Although it is incorrect to draw conclusions based on the diagram, the figure can help students to visualize the relationships and verify the results. At each step in the solution process, students should be encouraged to question the reasonableness of the answers they obtain. Students should be aware that there are many other identities that can take the measures of the sides of a triangle and relate them to an expression involving trigonometric functions. A well-known set of equations is named for Karl Mollweide, who was an astronomer and a teacher. Mollweide’s equations involve all six parts of a triangle: a 1 b c 5 a 2 b c 5 cos 12 (A 2 B) sin 12 C sin 12 (A 2 B) cos 12 C In Enrichment Activity 14-7: The Law of Tangents, students will be introduced to this less commonly used law and will be guided through the steps for applying the relationship to problem situations. EXTENDED TASK For the Teacher: This extended task is intended to show students how a person uses trigonometry to do his or her job. In this case, a land developer (or surveyor) uses trigonometry to find the number of square feet in the area of a building lot subdivision. The task also helps students connect to other parts of mathematics, particularly geometry and arithmetic. It might also be seen as a task involving consumerism. Students should recognize that they have two options available to complete the task. Since ABCD is a trapezoid, they could use the formula for the area of the trapezoid. They must first, however, find the length of the altitude. The other option is to draw one of the two diagonals and find the areas of the two triangles formed using the trigonometric formula for area of a triangle, A 5 12ab sin C . Whichever option the student chooses, he or 14580TM_C14.pgs 3/26/09 12:15 PM Page 188 188 Chapter 14 she must first use the Law of Sines and/or Law of Cosines to find the measures of needed sides and/or angles. In addition to the trigonometric skills needed to complete the task, the following skills are necessary: ability to round numbers to a stated precision, conversion skills, equivalences (1 acre 5 43,560 square feet), arithmetic skills, and finding percent. The task requires that students draw upon mathematics they have learned in their previous schooling. As an extension to this task, you might give students the option of taking the 18-acre tract of land that Mr. Kronau purchased and designing it as a development. Lots could be of varying sizes, and the infrastructure such as roads, a park, a pond, and so on could be included. Other variables can be introduced into the task to create extensions. 14580TM_C14.pgs 3/26/09 12:15 PM Page 189 Name Class Date ENRICHMENT ACTIVITY 14-4 Heron’s Formula If we know the measures of the three sides of a triangle, the size and the shape of the triangle are determined. We should be able to find the area of the triangle in terms of these measures. The formula for this purpose is found in the works of Heron of Alexandria, although some historians believe that the formula was originally developed by Archimedes. To derive Heron’s formula, we will need the following algebraic relationships: a2 1 2ab 1 b2 5 (a 1 b) 2 a2 2 2ab 1 b2 5 (a 2 b) 2 (a 1 b) 2 2 c2 5 f (a 1 b) 1 cg ? f (a 1 b) 2 cg The perimeter P of ABC, whose sides have measures a, b, and c, is P 5 a 1 b 1 c. The semi-perimeter s is one-half the perimeter, or s 5 a 1 2b 1 c. 2 b2 2 c2 1. Use the Law of Cosines, cos C 5 a 1 2ab , to show that 1 1 cos C 5 (a 1 b) 2 2 c2 2ab 2 a 1 b 1 c a 1 b 2 c 5 ab ? ? 2 2 2. Show that a 1 2b 2 c 5 s 2 c. 2 ? s(s 2 c) . 3. Using steps 1 and 2, show that 1 1 cos C 5 ab 2 2 2 b 2 c 4. Use the Law of Cosines, cos C 5 a 1 2ab , to show that 1 2 cos C 5 c2 2 (a 2 b) 2 2ab 2 c 1 a 2 b c 2 a 1 b 5 ab ? ? 2 2 5. Show that c 1 a2 2 b 5 s 2 b. 6. Show that c 2 a2 1 b 5 s 2 a. 2 ? (s 2 b)(s 2 a) . 7. Using steps 4, 5, and 6, show that 1 2 cos C 5 ab 8. Using steps 3 and 7, show that 9. Show that 2 sin C 5 ab !s(s 2 a)(s 2 b)(s 2 c). Area 5 !s(s 2 a)(s 2 b)(s 2 c) (Heron’s formula) In 10–13, in each case, find the area of the triangle by using the methods specified. A 10. In right ABC, mC 5 90, a 5 5, b 5 12, and c 5 13. 13 Find the area of ABC by using: a. Area 5 1 2 (base)(height) 12 b. Heron’s formula B Copyright © 2009 by Amsco School Publications, Inc. 5 C 14580TM_C14.pgs 3/26/09 12:15 PM Page 190 Name Class Date 11. In BCD, b 5 21, c 5 13, d 5 20, and h 5 12, where h is the altitude drawn to base b. Find the area of BCD by using: a. Area 5 12bh b. Heron’s formula B c 5 13 D d 5 20 12 C b 5 21 12. In equilateral RST, the length of each side is 4. Express, in simplest radical form, the area of RST by using: S a. Area 5 12bh (First find the height, h.) b. Area 5 12rs sin T c. Heron’s formula 13. In ABC, a 5 12 meters, b 5 20 meters, and c 5 28 meters. Express, in simplest radical form, the area of ABC by using: a. Area 5 12ab sin C (First use the Law of Cosines to find mC.) b. Heron’s formula Copyright © 2009 by Amsco School Publications, Inc. 4 R 4 4 T 14580TM_C14.pgs 3/26/09 12:15 PM Page 191 Name Class Date ENRICHMENT ACTIVITY 14-7 The Law of Tangents Although less commonly used than the Law of Cosines or the Law of Sines, the Law of Tangents can be applied when two sides of a triangle and the angle between those sides are known: • In ABC with sides a, b, c opposite angles A, B, C respectively: a2b a 1 b 5 tan 12 (A 2 B) tan 12 (A 1 B) or b2c b 1 c 5 tan 12 (B 2 C) tan 12 (B 1 C) or c2a c 1 a 5 tan 12 (C 2 A) tan 12 (C 1 A) The problem below illustrates the use of the Law of Tangents in solving a triangle. Complete each step as indicated. Problem A In ABC, a 5 26, b 5 14, and C 5 80°. Find the remaining sides and angles of ABC. 1. Use the measures of sides a and b to write an equation relating the tangents of the sum and difference of unknown angles A and B. 2. Since mC 5 80, what must the sum of the measures of A and B equal? 14 80° C 26 3. Substitute the value you found for the sum of the angles into the equation derived in Step 1 and simplify the result. This will give a numerical value for tan 12 (A 2 B) . Round to four decimal places. 4. Use the inverse tangent function on your calculator to find 12 (A 2 B) and then A 2 B. Round to three decimal places. 5. Write and solve a system of equations using the values found for A 1 B (Step 2) and A 2 B (Step 4). Round to the nearest tenth. 6. Use the values found in Step 5 and the Law of Sines to solve for side c. Round to the nearest tenth. 7. Summarize the results of ABC. a5 A5 b5 B5 c5 C5 8. If you did not know the Law of Tangents, how could you have solved ABC? Describe the steps you would use. Copyright © 2009 by Amsco School Publications, Inc. B 14580TM_C14.pgs 3/26/09 12:15 PM Page 192 Name Class Date 9. In ABC, a 5 6, b 5 2.5, and C is a right angle. a. What is the sum of A and B? b. What is the value of tan 12 (A 1 B) ? c. Use the Law of Tangents to find tan 12 (A 2 B) and then A 2 B to the nearest tenth. d. Complete the solution of ABC using the Law of Sines. Round to the nearest tenth. e. Verify the values for a, b, and c using the Pythagorean Theorem. f. Verify the values for A and B using the tangent ratio for a right triangle. 10. Solve ABC by using the Law of Tangents. Round your answers to the nearest tenth. A 14 120° C 21 B Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C14.pgs 3/26/09 12:15 PM Page 193 Name Class Date EXTENDED TASK Land for Sale: A Trigonometric Investigation Mr. Kronau is a land developer. He has just purchased an 18-acre tract of land, which he is subdividing into building lots. One of the lots that lie along the main road is shown below and labeled ABCD. The measures of the sides are rounded to the nearest 5 feet and marked on the diagram. The measure of ABC, to the nearest degree is 95°. The lot is in the shape of a trapezoid with AB CD. (Note: The diagram is not drawn to scale.) B 135 ft 95° A 270 ft 190 ft C 345 ft D Road 1. Find the area of the subdivision to the nearest 100 square feet. 2. What is the approximate size of the lot in acres? 3. If Mr. Kronau makes all of the subdivisions about the same size, how many lots will he have available to sell? 4. If Mr. Kronau sells the lots for $5,500 per acre, about how much is the cost of one lot? 5. If Mr. Kronau sells all the available lots, about how much will he receive in all? 6. If Mr. Kronau bought the tract of land for $65,000, what would be his percent of profit to the nearest whole percent? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C14.pgs 3/26/09 12:15 PM Page 194 Name Class Date Algebra 2 and Trigonometry: Chapter Fourteen Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may only receive partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–3, write in simplest radical form the coordinates of each point A if A is on the terminal side of angle u in standard position. 1. OA 5 3, u 5 45° 2. OA 5 12, u 5 150° 3. OA 5 0.75, u 5 300° 4. The coordinates of point A are (8, 6). a. Find OA, the distance of point A to the origin. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains point A. 5. In ABC, b 5 9, c 5 15, and sin B 5 0.3. Find the measure of acute C. 6. In PQR, q 5 10, r 5 15, and cos P 5 0.23. Find p. 7. In DEF, d 5 12, e 5 3, and f 5 10. Find mD to the nearest degree. 8. In ABC, b 5 10, B 5 32°, and A 5 40°. Find a and c to the nearest tenth. 9. In ABC, b 5 12, c 5 23, and A 5 100°. Find the area of ABC to the nearest tenth. 10. Find to the nearest integer the area of a rhombus if the measure of a side is 27 centimeters and the measure of one angle is 65°. 11. A ship sails on a course forming an angle measuring 23.7° with a straight coastline. After sailing 74.3 nautical miles on this course, the ship changes heading and sails back toward the coast. After sailing 98.7 nautical miles on the new course, the ship arrives back at the coast. To the nearest tenth, how far from its starting point is the ship when it again meets the coast? 12. A sign is placed on the roof of a building that is 50 feet high. From a point on the ground, the angle of elevation of the bottom of the sign is 35° and of the top of the sign is 48°. Find the height of the sign to the nearest foot. 13. In ABC, angle A measures 40° and angle B measures 70°. The length of side a is 10 inches. Find the perimeter of ABC to the nearest tenth. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C14.pgs 3/26/09 12:15 PM Page 195 Name Class Date 14. The measure of diameter BC is 30 centimeters, and the measure of chord AB of the circle is 12 centimeters. Find to the nearest degree the measure of minor arc AB. X 15. To avoid a marshy stretch of a straight path, a hiker walks 200 feet in a direction that is at an angle of 20° with the path, then turns and walks 300 feet back to the original path. a. Find, to the nearest degree, the measure of the acute angle where the path followed by the hiker returns to meet the original path. b. Find to the nearest 10 feet the distance between the points where the hiker left and returned to the path. 16. In ABC, A 5 40°, b 5 10, and a 5 8. a. How many distinct solutions are there for ABC? b. Show the solution(s) indicated in part a rounded to the nearest tenth. Bonus: If in ABC we are given the measures of A, B, and c (ASA), use the Law of Sines and the formula for the area of a triangle (area 5 12bc sin A) , to show that: 2 sin A sin B Area 5 2csin (A 1 B) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C14.pgs 3/26/09 12:15 PM Page 196 Name Class Date SAT Preparation Exercises (Chapter 14) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. In ABC, mC 5 100 and cos A , cos B. Which of the following is true? I. m/A , m/B 7. II. m/A . m/B III. sin A . sin B (A) I only (C) III only (E) II and III only 6. The sides of a parallelogram measure 6 inches and 8 inches and the smaller angle is 60°. The length of the longer diagonal of the parallelogram is (A) 2 !13 inches (B) 10 inches (C) 2 !37 inches (D) 13 inches (E) 15 inches R (B) II only (D) I and III only 2. The angles of a triangle are in a ratio of 1 : 3 : 8. The ratio of the longest side of the triangle to the next longest side is (A) 8 : 3 (B) !2 : 1 (C) 2 : !3 (D) !6 : 2 (E) !8 : !3 If sin P is 125% of sin Q, then PR is what percent of RQ? 5. The area of ABC is 35!3. If a 5 7 and b 5 20, the measure of C could be (A) 30° only (B) 30° or 150° (C) 45° or 135° (D) 60° only (E) 60° or 120° (A) 6623% (B) 75% (D) 125% (E) 150% 4 3 A If sin A 5 k(cos A), then k is (A) !7 3 (D) 3 !5 5 (B) 3 !7 7 (E) !21 3 (C) 53 9. R P Q If the radius of circle P is 1, then the radius of circle Q is (A) 2 sin P (B) 2 cos P 1 (C) 2 sin P (E) 2 tan P Copyright © 2009 by Amsco School Publications, Inc. (C) 80% 8. 3. In ABC, A 5 30°, b 5 16, and a 5 8 !2. Which could be the degree measure of C? (A) 135° (B) 105° (C) 90° (D) 75° (E) 60° 4. For which set of measures can more than one triangle be formed? (A) p 5 12, q 5 10, mp 5 100 (B) p 5 12, q 5 10, mP 5 30 (C) p 5 10, q 5 14, mP 5 150 (D) p 5 6, q 5 12, mP 5 30 (E) p 5 6, q 5 10, mP 5 30 Q P 1 (D) 2 cos P 14580TM_C14.pgs 3/26/09 12:15 PM Page 197 Name 10. Class 12 P 15. In isosceles triangle ABC, AB 5 AC 5 4 and the area of the triangle is 4. Find the degree measure of the base angles. (A) 30° (B) 60° (C) 75° (D) 77° (E) 81° R 4 3 Q In the rectangular box pictured, sin RPQ equals (A) 14 (B) 31 5 (D) 13 (E) 21 5 (C) 12 11. If sin A 5 0.5 and A is an obtuse angle, what is the value of tan A rounded to three decimal places? (A) 21.733 (B) 20.577 (C) 0.500 (D) 0.577 (E) 1.733 12. In triangle ABC, the measure of C is 90° and the measure of A is 40°. If AB 5 7, what is the value of CB to the nearest tenth? (A) 4.5 (B) 5.4 (C) 7.0 (D) 9.0 (E) 10.9 13. In a circle with center O, the length of radius OA is 13 units. A chord BC that is 24 units long is drawn. If mOBC 5 u, then sin u is equal to (A) 5 13 13 (D) 12 (B) 5 12 Date (C) 12 13 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. The coordinates of the vertices of PQR are P(6, 8), Q(28, 0), and R(0, 0). Find the degree measure of R to the nearest tenth. 17. The sides of a triangle are in the ratio of 4 : 5 : 6. Find the measure of the smallest angle to the nearest tenth. 18. In isosceles ABC, mC 5 30 and BC 5 14. What is the least possible area, in square units, of ABC? 19. Two small airplanes depart from the same airport at the same time on courses forming an angle measuring 57°. If the planes fly at speeds of 150 and 180 miles per hour, respectively, how far apart are the planes, to the nearest tenth of a mile, at the end of 30 minutes? 20. B 110° (E) 13 5 14. Find the degree measure, to the nearest tenth of a degree, of the largest angle of a triangle if the sides measure 10, 12, and 13 units. (A) 38.7° (B) 43.1° (C) 46.9° (D) 71.8° (E) 72.1° Copyright © 2009 by Amsco School Publications, Inc. 20 130° O A C Triangle ABC above is inscribed in circle O, mAB 5 110, mBC 5 130, and BC 5 20. Find the area of ABC to the nearest integer. X X 14580TM_C14.pgs 3/26/09 12:15 PM Page 198 Name Class 21. Triangle ABC is drawn in the coordinate plane with point A at (22, 0) and vertex B on the positive ray of the x-axis. The measure of angle A is 65° and BC 5 20. Find the x-coordinate of point C to the nearest hundredth. 22. A triangular plot of land has sides of lengths 120 feet, 90 feet, and 70 feet. Find, to the nearest degree, the measure of the largest angle. 23. In an isosceles triangle, the measure of the vertex angle is 64° and the base is 12 units. Find the length of the altitude to the base of the triangle to the nearest tenth. Copyright © 2009 by Amsco School Publications, Inc. Date 24. R 10 O T S In the figure above, two lines are drawn from point T tangent to circle O at points R and S. Chord RS is drawn, separating circle O into two arcs whose measures are in the ratio of 3 : 1. If RS 5 10, find the area of RST. 14580TM_C15.pgs 3/26/09 12:15 PM Page 199 CHAPTER 15 STATISTICS Aims • To understand the differences among various methods for collecting data and to determine factors that may affect the outcome of a survey. • To review different methods of organizing and displaying data. • To review the measures of central tendency. • To define various measures of dispersion and their applications. • To define and apply the normal distribution. • To construct scatter plots and determine the regression model that is most appropriate. • To interpret the value of the correlation coefficient as a measure of the strength of the relationship for a linear regression model. • To use the regression model to interpolate and extrapolate from data. Modern technology has the capability of making available vast quantities of information. Intelligent use of information requires an understanding of its applicability to our lives. Statistics provides methods of organizing and analyzing data to enable us to identify trends and their implications for the future. Understanding statistical methods helps us to be critical consumers of information and to avoid being misled by invalid conclusions. The material on statistics in this chapter is introductory but is intended to help students to begin to understand some of the statistical measures used to analyze data and to lay a foundation for more comprehensive work with statistics in their advanced studies and career preparation. CHAPTER OPENER Students who are preparing for college have already taken or will soon take one of the standardized tests (SAT or ACT) that most schools require of their prospective students. The companies that administer these tests provide information about the distribution of scores on their tests. Discuss with students the information about test scores given in the booklets prepared by the test companies. Explain that companies field-test the questions for standardized tests and use the feedback to determine the level of difficulty of each question and the percent of students who are expected to answer each question correctly. Information about standardized tests can also be obtained from the Internet. 199 14580TM_C15.pgs 3/26/09 12:15 PM Page 200 200 Chapter 15 15-1 UNIVARIATE STATISTICS Univariate data (one-variable data) describe a single characteristic of a population or sample. For example, for the set of students in a class, univariate data would include each student’s height or age. Quantitative data take on numerical values. Qualitative or categorical data describe some characteristic of the population, such as eye color or favorite ice cream flavor. A census is a very complex, time-consuming, and expensive project. These factors have motivated a second strategy for data collection: the sample survey. The process of drawing conclusions about the nature of the entire population based on data from a sample is called statistical inference and is the basis for the usefulness of statistics in daily life. The question of how accurately the sample statistics will represent the actual population begins with the method used to select the population sample. The selection process is called sample design. Some sampling techniques are flawed because they may be biased. Bias is a systematic error that favors a particular subset of the population or that tends to encourage only certain outcomes in the data. Voluntary response sampling is used by radio shows that invite listeners to call in their opinions about a controversial issues. The fact that people with strong opinions respond works well for the show, since the discussions may become quite emotional, but the opinions are not necessarily representative of the whole population. In Example 2, the method of choosing every 4th person is called systematic sampling. For systematic sampling to be valid, the researcher must be careful that the ordering principle is not connected to the nature of the population. In addition to the wording of the question, survey answers can also be biased by the order of choices (the first answer tends to receive a greater number of selections), the appearance and/or demeanor of the interviewer, and the honesty of the answers given by the participants (guaranteeing confidentiality is often used to motivate honest responses). Newspapers and magazines often provide details of the surveys and studies that are being reported. Ask students to find examples of surveys, observational studies, and controlled experiments and critique the methods used. Review the construction of stem-and-leaf diagrams, frequency distribution tables, and histograms, as these data displays will be used throughout the chapter. The Hands-On Activity that begins in this section will be analyzed using the methods described in subsequent sections. 15-2 MEASURES OF CENTRAL TENDENCY This section reviews the measures of central tendency (mean, median, and mode), first and third quartiles, and the box-and-whisker plot. Graphing calculators and some scientific calculators can be used to find these measures. To understand what these measures represent, however, it is important that students have some initial practice in finding statistical measures with calculator use limited to basic computations. Discuss the circumstances under which one of these measures may be more appropriate than the others. The mode is the easiest average to find but is the one that tells us the least about the other data entries. Call attention to the fact that although, for the sake of simplicity of computation, sets of data with few entries are often used as exercises, the mode has significance only when the number of occurrences of the modal value is significantly higher than the number of occurrences for the other values of the data. The median is the middle value. The number of data entries larger than or equal to the median is equal to the number of data entries smaller than or equal to the median. The median is often considered to be a better measure of central tendency than the mean when there are outliers. The mean is the measure of central tendency that is most frequently used. The mean is commonly called the average. The sum of 14580TM_C15.pgs 3/26/09 12:15 PM Page 201 Statistics 201 the absolute values of the differences, 2 a (x 2 x) , for data values smaller than the mean is always equal to the sum of these differences for data values greater than the mean. The mean is the average used to find other statistical values such as the variance and the standard deviation. The first quartile is often denoted by Q1, the second by Q2 and the third by Q3. Emphasize that if the data set contains an odd number of values, then the median is the middle data value and is not included in the calculation of either the first or third quartiles. If the number of data values is even, then the median is the average of the two middle values and all data values are used in the calculation of the first and third quartiles. Be sure students understand the method described in Example 2b for locating the median and first and third quartiles on a stemand-leaf diagram and the note below the example. The interquartile range as a measure of dispersion is discussed in Section 15-4. As the Hands-On Activity continues, students use the data collected about estimates of a minute to prepare the five statistical summary. 15-3 MEASURES OF CENTRAL TENDENCY FOR GROUPED DATA In most serious statistical studies, the data is usually large, and many values occur more than once. In such situations, the data is organized into a table where xi represents a value within the range of the set of data, fi represents the number of times that value occurs, and xi fi is the total value of the fi entries having the value of xi. The mean is found by summing all the xi fi entries and then dividing by the sum of the fi entries. When data is grouped into intervals, it is convenient to use one number to represent each interval. For the purpose of finding the mean of interval data, it is assumed that all values within an interval are represented by the midpoint of the interval. In the example involving weights, the actual values can be any number in an interval (length, weight, temperature are examples of continuous variables). Since the intervals are recorded using whole numbers of pounds, the weights that would fall in the interval 240–249 extend from 239.5 to 249.5. Percentiles are useful in providing positional information in large data sets. For example, a score of 510 on a standardized test does not tell us the position of this score relative to the entire set of scores. If we are told that the score of 510 is in the 56th percentile, then we know the position of the score relative to the entire set. In some discussions, percentile is defined as the percentage of data values below the specified value. With that definition, the percentile rank of 7 misspelled words would be based only on the number of essays with fewer than 7 misspelled words: 87 100 5 87% The method for approximating the median for grouped data assumes that the number of values in an interval are evenly spaced in the interval. Drawing number lines as shown will help students determine the position of the median. The Hands-On Activity is continued by having students group their data into intervals of 5 seconds and then finding the mean using the methods of this section. 15-4 MEASURES OF DISPERSION Although the range is the easiest measure of dispersion to calculate, it tells us nothing about how the data is distributed within the limits of the range. The range is based only on the extremes and does not account for intermediate values. The interquartile range (IQR) specifies the length of the interval that contains approximately the middle 50% of the data. The interquartile range corresponds to the “box” of a box-and-whisker plot. Similarly, each “whisker” of a box-and-whisker plot represents about 25% of the data. 14580TM_C15.pgs 3/26/09 12:15 PM Page 202 202 Chapter 15 The interquartile range also provides criteria for identifying outliers in the data. If a data value is greater than 1.5 3 (interquartile range) plus the upper quartile or less than the lower quartile minus 1.5 3 (interquartile range), that data value is an outlier. Note that when an outlier is present, the whiskers of a box-and-whisker plot extend to the most extreme non-outlier value on either side of the box as appropriate. Some graphing calculators allow the option of a box plot identifying no outliers or a modified box plot indicating outliers. These options are identified by different icons on the STAT PLOT menu of the TI-831/841 graphing calculators. The concepts of this section are applied as the Hands-On Activity continues. estimates are better when the formulas use division by n 2 1 rather than n. • If the data represents a population: n Variance 5 s2 5 n1 a (xi 2 2 x )2 i51 n Standard deviation 5 s 5 Ç 1 n a (xi 22 x) 2 i51 Strictly speaking, the greek letter m should be used to denote the mean of a population. Greek letters are generally used to indicate population measures, while English letters are generally used to indicate sample measures. • If the data represents a sample: n 1 22 Variance 5 s2 5 n 2 1 a (xi 2 x) i51 15-5 VARIANCE AND STANDARD DEVIATION Today, the standard deviation is the most frequently used measure of dispersion in statistical studies. Standard deviation is particularly important because of its relationship to the normal curve. This relationship is explained in detail in Section 15-6. Since standard deviation is the square root of another statistical measure called variance, we begin by defining variance. Variance measures the deviation of each data value from the mean. The variance is the mean of the squares of the differences of each data value from the mean of the data values. We square each difference so that positive and negative differences do not cancel each other out. The standard deviation is the square root of the variance. There are two variances and standard deviations that can be calculated for any data set. If the data represents a population, then the exact values of the variance and standard deviation can be found. However, if the data represents a sample of a population, then we can find only estimates for the variance and standard deviation. It has been found that these Standard deviation 5 s n 5 Ç 1 n 2 1 a (xi 22 x) 2 i51 Note that s 5 0 only when all the data values are the same; otherwise, s . 0. Moreover, standard deviation is strongly influenced by extreme values. Most graphing calculators can determine standard deviation. However, students will have a better understanding of the measure if they carry out the necessary steps using a calculator at first for basic computation only. They may then check their results using the statistical capabilities of the calculator. 15-6 NORMAL DISTRIBUTION Distributions of physical measurements (such as heights and weights) or test scores for very large data sets often have symmetric, bellshaped graphs. Normal distribution curves have an infinite base—the long, flat-looking tails actually extend to infinity. The normal curve is symmetric about a vertical line through its highest point. The x-coordinate of the highest point represents the mean, the median, and the mode, which coincide in a 14580TM_C15.pgs 3/26/09 12:15 PM Page 203 Statistics 203 normal distribution. The y-coordinate of the highest point represents the frequency of the mean, the median, and the mode. The area under the standard normal curve is 1, and the area under a portion of the curve between two vertical lines represents the probability that values that fall between the values that are the x-coordinates of those lines will occur. (This will be explored in Chapter 16.) There is a point on each side of a normal curve where the slope is steepest. These two points are called points of inflection and the distance from the mean to either point is equal to one standard deviation. The greater the standard deviation, the less steep the sides of the curve are. before computers and graphing calculators, this conversion allowed us to use a single standard normal table to determine probabilities for any normal distribution. Conversion to z-scores is a combination of two transformations: 1. By subtracting the mean from each x, we transform the mean to 0. 2. By dividing each difference x 2 2 x by the standard deviation, we transform the standard deviation to 1. The set of z-scores for the data set (assuming it represents the population) {2, 4, 6, 8, 10, 12, 14, 16} is point of inflection point of inflection x2s x x1s {–1.528, 21.091, 20.655, 20.218, 0.218, 0.655, 1.091, 1.528}. The mean of the transformed data is 0, the standard deviation is 1, and each data value has the same relative position in the set. Note that given a z-score, we can reverse the procedure that was used and find the original x-value: x 5 2 x 1 zs. 15-7 BIVARIATE STATISTICS Normal curves with the same mean but different standard deviations. Note that while the standard deviation can be calculated for any distribution, the rule for percentages of data in the “standard deviation regions” is reliable only for normal curves. The z-score allows us to compare the position of a data value relative to the mean and standard deviation of the data. When you convert a set of data to a set of z-scores, the transformed data has a mean of 0 and a standard deviation of 1, regardless of the mean and standard deviation of the original data. In the days When we explore bivariate or two-variable data, we are interested in determining if an association exists between the two variables observed, how this association can be measured, and how it can be expressed mathematically and used to make predictions. A scatter plot is a technique for visually displaying a relationship between two quantitative variables. Individual observations are written in (x, y) form and then plotted. The resulting scatter plot is examined to see if the points show a distinct pattern. Once the strength and direction (positive or negative) of the association have been determined, algebraic models can be fit to the data for predictive purposes. The term linear regression refers to the process of finding an equation of the form y 5 ax 1 b for a set of bivariate data. This 14580TM_C15.pgs 3/26/09 12:15 PM Page 204 204 Chapter 15 linear model is called the regression line. One purpose of this linear equation is to enable us to predict a typical value of a dependent (or response) variable from a given value of the independent (explanatory) variable. If yi is the actual or observed value of the dependent variable and ŷi is the predicted value (called the “fit”) of the dependent variable, then we wish to choose a “line of best fit” that minimizes the residual (yi 2 ŷi) at xi for each data value. The least squares regression line is one such line of best fit that minimizes the sum of the square of the residuals or n 2 a (yi 2 ŷi) . The “squares” in the name i51 refers to the squares of the residuals. For example, the linear regression equation given in the text for the SAT scores is y 5 0.693x 1 151.03 Using this equation, the predicted reading score for a math score of 530 is ŷ 5 0.693(530) 1 151.03 ŷ 5 518.32 < 518 Since the actual reading score for a 530 math score is 530, the residual at this point is 530 2 518 5 12. Similarly, for a math score of 566, the predicted reading score is ŷ 5 0.693(566) 1 151.03 ŷ 5 543.268 < 543 For this point, the residual is 543 2 543 5 0. Two important characteristics of the regression line are: 1. It always includes the point (2 x, 2 y) . 2. The sum of the residuals is 0. On the TI-83+/841 calculators, when a linear regression is executed, the residuals are automatically stored in the list RESID. This list can then be selected from the LIST NAMES menu. 15-8 CORRELATION COEFFICIENT The correlation coefficient r is a number between 21 and 1 that can be used to determine the strength and direction of the linear association between the variables of a set of bivariate data. A negative r implies an inverse relationship (as one variable increases, the other decreases) and a positive r implies direct variation (as one variable increases, so does the other). When r 5 1, the data pairs, when plotted on a scatter plot, are collinear. Emphasize that although a value of the correlation coefficient can always be found, the data should be plotted to determine if it makes sense to look for a linear relationship. This idea will be discussed in more detail in Section 15-9. In addition to the value of r, many graphing calculators provide a value of r2. The value r2 is called the coefficient of determination and provides information regarding the percent of variation in the dependent variable that is accounted for by the regression line. Alternatively, we can say that r2 gives the percentage of variation in y that is predictable from a knowledge of x. So, in Example 3, the value of r2 5 0.33 indicates that 33% of the change in the number of pounds purchased is accounted for by the cost per pound. More importantly, we can conclude that 67% of the change in pounds purchased must be associated with other variables that are not included in the linear model. In Enrichment Activity 15-8: Calculating the Correlation Coefficient, students are given the formula for the correlation coefficient and they are guided through the steps for completing the calculations. 15-9 NON-LINEAR REGRESSION Often, a straight line is not the best model for representing a relationship between two variables. Such cases are clearly indicated when the scatter plot shows a distinctive curved pattern. Emphasize that students should not make a judgment about the appropriateness of a 14580TM_C15.pgs 3/26/09 12:15 PM Page 205 Statistics 205 model based on a comparison of the correlation coefficient of different models. For logarithmic, exponential, and power models, the correlation coefficient describes the goodness of fit for transformed data that has been linearized, not the original data. A larger correlation coefficient does not necessarily mean a better fit. The following describe the transformations used to obtain each model: Logarithmic Model y 5 a 1 b ln x Replace ln x by x9 Linear equation: y 5 a 1 bx9 Power Model y 5 axb ln y 5 ln a 1 b ln x Replace ln y by y9, ln a by a9, ln x by x9 Linear equation: y9 5 a9 1 bx9 Exponential Model y 5 abx ln y 5 ln a 1 x ln b Replace ln y by y9, ln a by a9, ln b by b9 Linear equation: y9 5 a9 1 b9x Students have an opportunity to perform some of these transformations in the Exploration at the end of the chapter. In that activity, they will work with the data from Examples 1 and 2. Note that when the TI-831/841 perform any of the polynomial regressions, it returns only the value R2. The absence of a value for r and the fact that an upper-case R2 is shown indicates that some technique other than linearization was used to carry out the regression. However, as the coefficient of determination, R2 still indicates the percentage of variation in y that is accounted for by the polynomial equation. 15-10 INTERPOLATION AND EXTRAPOLATION When we use the regression equation to make predictions, it is often worthwhile to replace the x and y variables with the names of the variables in the context of the problem. So, in Example 1, we could write miles 5 32.365 gal 2 2.076 In this example, the coefficients are interpreted as follows: slope: the number of miles driven is predicted to increase by 32.365 for each increase of 1 gallon of gas y-intercept: the number of miles driven is predicted to be 22.076 when 0 gallons of gas are needed. As this example shows, sometimes the y-intercept makes no sense in the context of the data set. A negative number of miles driven is meaningless here. While extrapolation must be applied with great care, it is often the technique for making predictions about population growth, product sales, and other variables. EXTENDED TASK For the Teacher: This activity is intended to provide students with an opportunity to make use of their knowledge of statistics by designing and conducting a survey on their own. It allows them to see that statistics is a valuable tool in the real world. You may want to work cooperatively with the language arts teachers and have the students present an oral report of their survey and findings. A four-point scoring guide is given below as a sample that might be used: [4] The student independently completes the task. He/she has a well-defined investigation and has developed three to five questions that will adequately aid in the investigation. The population selected for the survey is appropriate and unbiased. The data are organized and displayed in an orderly manner. The analysis of the data is thorough. Appropriate measures of central tendencies and dispersions are accurately computed. The student conducts the survey with the required number of participants. The written report is neat and thorough. It contains conclusions, inferences, and or predictions that are appropriate from the collected data. [3] The student needs some assistance to complete the task. He/she has a well-defined 14580TM_C15.pgs 3/26/09 12:15 PM Page 206 206 Chapter 15 investigation and has developed three to five questions that will adequately aid in the investigation. The questions, however, are not well worded. The population selected for the survey is appropriate and unbiased. The data are organized and displayed, but are not particularly neat or well presented. The analysis of the data is somewhat difficult to follow. Appropriate measures of central tendencies and dispersions are computed but contain some minor errors. The student conducts the survey with the required number of participants. The written report is fairly neat but somewhat sketchy. The conclusions, inferences, and or predictions from the collected data are somewhat vague. [2] The student needs considerable assistance from the teacher to complete the task. His/her investigation is not well defined but does contain the minimum of three questions. The population selected for the survey is either inappropriate or biased. The data are poorly organized and displayed, making the analysis difficult and vague. Appropriate measures of central tendencies and dispersions are either lacking or contain serious errors. The student fails to conduct the survey with the required number of participants. The written report is sketchy, containing few meaningful conclusions, inferences, and or predictions from the collected data. [1] Even with considerable assistance from the teacher, the student is unable to complete the task. The investigation is poorly defined and flawed. It contains fewer than three good questions. The student demonstrates a lack of ability to select an appropriate and unbiased population. The data lack any organization. They are either not displayed at all or so poorly displayed that they are meaningless. There is little, if any, analysis of the data. Few, if any, appropriate measures of central tendencies or dispersions are included. The student fails to obtain the required number of participants. The student summary is omitted or meaningless, and he/she makes no attempt to state any significance of the survey. 14580TM_C15.pgs 3/26/09 12:15 PM Page 207 Name Class Date ENRICHMENT ACTIVITY 15-8 Calculating the Correlation Coefficient The correlation coefficient is a number that measures the strength and direction of linear association of the variables in a set of bivariate data. For a sample data set, the correlation coefficient, denoted r, can be found using the formula: n xi 2 2 x yi 2 2 y 1 r5n2 1 a A sx B A sy B i51 The formula is actually the sum of the products of the z-scores of the two variables divided by 1 less than the sample size. To calculate the correlation coefficient for this sample data set {(1, 3), (2, 6), (4, 20), (7, 22)} follow the steps below. Round your answers to three decimal places. x ). 1. Find the sample mean of the x-values (2 y) . 2. Find the sample mean of the y-values (2 3. Find the sample standard deviation of the x-values (sx). 4. Find the sample standard deviation of the y-values (sy). 5. Use the values you found in steps 1–4 to complete the table. xi yi 1 3 2 6 4 20 7 22 xi 2 2 x sx Q yi 2 2 y sy xi 2 2 x yi 2 2 y sx R Q sy R Total: 6. Divide the total from the table by n 2 1. 7. Compare the value of r calculated using the steps above with the value given by a calculator. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 208 Name Class Date 8. Complete the table and calculate the correlation coefficient. xi yi 0 4 2 7 5 16 9 14 12 8 17 25 23 19 30 53 xi 2 2 x sx Q yi 2 2 y sy Total: Copyright © 2009 by Amsco School Publications, Inc. xi 2 2 x yi 2 2 y sx R Q sy R 14580TM_C15.pgs 3/26/09 12:15 PM Page 209 Name Class Date EXTENDED TASK Taking a Survey: Designing a Statistical Study Many surveys are conducted in today’s world. There are surveys to find out what products people prefer, surveys to obtain opinions of political ideas, surveys to determine the effectiveness of a new medicine, and so on. Many economic, political, and social decisions are made based on the results of surveys. Surveys are conducted by statisticians who, after they collect data, organize, display, and analyze those data. In this extended task, you will play the role of a statistician. Your task is to decide upon some topic that you would like to investigate and design a study as part of your inquiry. Your investigation must include a survey questionnaire consisting of a minimum of 3 questions and a maximum of 5 questions. The questions should be in either a multiple-choice format, a Likert scale format, or an open-ended format. Examples Multiple-Choice Format: How many times have you attended a movie at a movie theater during the past 6 months? (A) 4 or more (B) 3 (C) 2 (D) 1 (E) none Likert Scale Format: How much do you agree/disagree with the following statement: “All students should be required to take 4 years of mathematics during their high school career.” (1) Strongly agree (2) Slightly agree (3) Slightly disagree (4) Strongly disagree Open-Ended Format: Approximately how many miles do you drive per year? Select an appropriate and unbiased population that you will use for your survey. Using the questionnaire that you have designed, survey a total of fifty people from your selected population. Organize and display your data in an orderly manner that will make it possible for you to effectively analyze it. Analyze your data using appropriate measures of central tendencies and dispersion. Analyze the data for each question in your survey. Write a thorough description of your survey, including conclusions, inferences, predictions, and so on, that you can make as a result of the analysis of your data. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 210 Name Class Date Algebra 2 and Trigonometry: Chapter Fifteen Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. 1. Each employee in a corporation fills out a questionnaire for a survey about commuting times. What type of data collection strategy does this represent? 2. The owner of an excursion boat recorded the number of passengers on each of the 12 trips that the boat made one weekend as follows: 20, 19, 22, 17, 22, 23, 23, 22, 21, 20, 21, 22 Find: a. the mean b. the median c. the mode d. the first quartile e. the third quartile f. the range g. the variance to the nearest hundredth h. the standard deviation to the nearest hundredth 3. Use the data set below 27, 39, 51, 69, 46, 60, 81, 23, 53, 55, 54, 46 to find: a. the interquartile range b. any outliers in the set 4. The records kept by two teachers of the numbers of students absent from their classes in the past month are shown below. Mrs. Alvarez: 4, 2, 1, 1, 3, 2, 2, 0, 1, 2, 4, 3, 1, 2, 6, 3, 1, 0, 0, 2 Mr. Kazin: 0, 0, 1, 4, 4, 7, 1, 0, 2, 6, 0, 0, 0, 1, 2, 6, 2, 3, 1, 0 a. Using Mrs. Alvarez’s record, find: (1) the mean (2) the median (3) the mode (4) the standard deviation to the nearest hundredth Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 211 Name Class Date b. Using Mr. Kazin’s record, find: (1) the mean (2) the median (3) the mode (4) the standard deviation to the nearest hundredth c. Which set of data more closely resembles a normal distribution? Explain your answer. 5. The table shows the recorded high temperatures for the month of September in New York City. For this set of data, find: Temperature (°F) Number of days 90–94 1 85–89 3 80–84 6 75–79 12 70–74 7 65–69 1 a. the mean b. the standard deviation to the nearest hundredth 6. The mean of the ages of the employees of a large department store is 42 years with a standard deviation of 8. Find the z-score of the age of an employee who is a. 30 b. 62 7. The mean of the math scores on an SAT test is 511 with a standard deviation of 112. In the same year, the mean of the math scores on the ACT test is 20.6 with a standard deviation of 5.0. Carol scored 650 on the SAT and 28 on the ACT. a. On which test did Carol have the higher z-score? b. On which test did Carol do better? Explain. 8. In a set of data that approximates a normal distribution, the mean is 5 and the standard deviation is 2.1. a. What percent of the values are expected to be between 2.9 and 5? Round your answer to the nearest whole percent. b. What percent of the values are expected to be above 9.2? Round your answer to the nearest tenth. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 212 Name Class Date 9. The heights of 800 students have a normal distribution with a mean of 65 inches and a standard deviation of 5 inches. To the nearest tenth, what percent of the students are between 60 and 72 inches? 10. The mean score on a standardized reading test is 487 and the standard deviation is 93. If the scores are normally distributed and 9,000 students took the exam, about how many students had scores from 394 to 580? Round your answer to the nearest hundred. In 11 and 12, for each of the given scatter plots, determine whether the correlation coefficient would be close to 21, 0, or 1. 11. 60 12. 300 50 250 40 30 200 150 20 10 100 50 0 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 13. The table below lists the twelve sales representatives employed by BestTech, Inc., each person’s annual sales for 2008, and the number of years the person has worked for the company. Sales Representative Years with Company Annual Sales ($1,000s) Arnold 25 650 Bass 1 80 Chan 5 120 Dern 4 180 Etlis 12 360 4 140 Gold 20 450 Hardy 30 550 7 280 Jackson 17 300 Kendall 18 350 Lamont 8 220 Ferrara Ives a. Find an equation for the line of best fit for the data. Let years with company be the independent variable. b. Using the equation found in part a, predict the sales for a person who worked for the company for 10 years. Round to the nearest thousand. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 213 Name Class Date c. If Hardy stays with the company for another 5 years, to the nearest thousand, what sales does the model predict for the final year? What method is used to make this prediction? 14. Several years ago, Janine Lucas opened a sandwich shop called Between the Bread. As the store became popular, she opened more locations. The table below shows the years since she opened her first store and the number of stores she had each year. Year of Operation 1 2 3 4 5 6 7 Number of Stores 1 2 4 7 15 27 52 a. Make a scatter plot of the data. b. Identify the type of model that would best fit the data and find the equation. c. Using the equation found in part a, how many stores would Janine expect to have in her ninth year of business? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C15.pgs 3/26/09 12:15 PM Page 214 Name Class Date SAT Preparation Exercises (Chapter 15) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 1. If x 1 y 5 14, y 1 z 5 11, and z 1 x 5 9, what is the mean of x, y, and z? (A) 17 3 (D) 17 (B) 17 2 (E) 34 (C) 34 3 2. What is the arithmetic mean of 39, 315, and 324? (A) 48 (B) 316 (C) 33 1 35 1 38 (D) 38 1 314 1 323 (E) 347 3. On Tuesday, 20 of the 25 students in an algebra class took a test and their average was 86. On Wednesday, the other 5 students took the test, and their average was 81. What was the average for the entire class? (A) 81.8 (B) 82 (C) 83.5 (D) 85 (E) 87 4. If the mode, mean, and median for the values x, 1, 1, 2, 2, 3, 3, 4, 4, 8, and 8 are integers and mode . mean . median, then x is (A) 1 (B) 2 (C) 3 (D) 4 (E) 8 5. Let M be the median and D be the mode of the following set of numbers: 12, 60, 90, 20, 60, 40 What is the mean of M and D? (A) 22 (B) 40 (C) 50 (D) 55 (E) 60 Copyright © 2009 by Amsco School Publications, Inc. 6. The number of boys and girls in a tenthgrade class is the same. The average weight of the boys is 146 pounds. The average weight of the girls is 122 pounds. What is the average weight of all the students in this class? (A) 178 pounds (B) 168 pounds (C) 134 pounds (D) 124 pounds (E) Cannot be determined 7. If x 5 3 and y 5 4, what is the value of the median of the following set? 5x 1 2y, 3x 2 y, 3(x 1 y), x 1 4y6 (A) 8 (D) 14 (B) 11 (E) 15 (C) 13 8. If the mean of four different negative integers is 29, what is the least possible value for any of the four integers? (A) 236 (B) 233 (C) 230 (D) 218 (E) 23 9. On a standardized test with a normal distribution of scores, the mean is 40 and the standard deviation is 6.6. Approximately what percent of the scores fall in the range 26.8 to 46.6? (A) 34% (B) 68% (C) 81.5% (D) 95% (E) 99% 10. Measure (xi) Frequency (fi) 27 400 29 350 33 150 35 600 41 300 What is the median of the data shown in the table above? (A) 33 (B) 34 (C) 35 (D) 150 (E) 900 14580TM_C15.pgs 3/26/09 12:15 PM Page 215 Name Class Date 11. The scores of an exam have a normal distribution with a mean of 60.1 and a standard deviation of 7.3. The percentage of scores that are between 74.7 and 82 is closest to (A) 2% (B) 20% (C) 40% (D) 84% (E) 97.5% III. There is a very strong association between daily calorie intake and blood pressure. (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III 12. The scores on a test are normally distributed with a mean of 79. If the interval 71 to 87 contains approximately 68% of scores, which value can be the standard deviation? (A) 2 (B) 4 (C) 8 (D) 12 (E) 16 II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 13. In a large survey of teenagers regarding bottled water, the mean number of bottles consumed per week was 20 with a standard deviation of 3.5. If a normal distribution is assumed, which interval represents the total number of bottles per week that approximately 95% of this group will drink? (A) 6–34 (B) 13–20 (C) 13–27 (D) 20–27 (E) 20–34 14. Which of the following would you expect to be true about the correlation between number of family members and weekly food budget? (A) strong and positive (B) weak and positive (C) strong and negative (D) weak and negative (E) zero 15. Suppose a study finds that the correlation coefficient relating daily calorie intake with blood pressure is r 5 0.95. Which of the following are proper conclusions? I. A high-calorie diet causes high blood pressure. II. Low blood pressure is the result of a low-calorie diet. Copyright © 2009 by Amsco School Publications, Inc. 16. Let :x; 5 the greatest integer that is less than or equal to x. For example, :4.23; 5 4 and :8; 5 8. What is the average of :10.75; and :23.68; ? 17. To the nearest hundredth, how much smaller is the population standard deviation of the data set than the sample standard deviation of the same data? 57, 3, 23, 156 18. A packing machine fills cardboard boxes with oatmeal. Suppose the amount of oatmeal per box forms a normal distribution with a mean of 16.1 ounces and a standard deviation of 0.08 ounce. If 2,000 boxes are filled with oatmeal, how many can be expected to have between 15.94 ounces and 16.26 ounces of oatmeal? Round your answer to the nearest hundred. 19. Consider the set of points {(2, 7), (3, 10), (4, 13), (5, s), (8, t), (11, 34)}. What is the value of s 1 t when there is a perfect linear correlation between the x- and y-values? 20. If the mean of p, q, and 14 is 114, what is the average of p and q? 21. The ages of the members of a women’s bridge club are 49, 58, 62, 68, 74, 76, 76, and 89. If Winifred joins the club, the median will not change. What is Winifred’s age? 14580TM_C15.pgs 3/26/09 12:15 PM Page 216 Name Class 22. What is the mean of the integers 249,000 to 50,000 inclusive? 23. The mean of 7 positive integers is 405. Two of the integers are 103 and 199 and the other numbers are greater than 199. If all 7 integers are different, what is the greatest possible value for any of the 7 integers? Copyright © 2009 by Amsco School Publications, Inc. Date 24. The mean of the test scores of a class of n students is 68 and the mean of the test scores of a class of m students is 94. The mean score, when both classes are combined, is 88. What is the n value of m ? 14580TM_C16.pgs 3/26/09 12:15 PM Page 217 CHAPTER 16 PROBABILITY AND THE BINOMIAL THEOREM Aims • To review probability concepts learned in previous courses. • To use permutations, combinations, and the Counting Principle to determine the number of elements in a sample space. • To study the probability associated with events where there are two outcomes. • To know and apply the binomial probability formula to events involving the terms exactly, at least, and at most. CHAPTER OPENER Have students suggest questions involving probability that might be of interest to medical researchers. For example, what is the probability that a person will contract a specific disease? What is the probability that a person’s condition will improve if a certain medicine is used? What is the probability that a piece of diagnostic equipment will fail to detect a physical problem or indicate a problem is present when the patient is actually healthy (that is, a false positive result)? • To use the normal distribution as an approximation for binomial probabilities. 16-1 THE COUNTING PRINCIPLE • To expand powers of binomials using the Binomial Theorem. This section reviews the counting principle and independent and dependent events. For experiments with a large number of outcomes, it is inefficient to use a list or tree diagram to find the total number of outcomes. The counting principle can be applied to decisions or events that take place in succession. Emphasize that in using the counting principle, we determine the number of choices available for the second activity by assuming that the first activity has taken place. So if there 5 different fruits in a bowl and you eat 2, you have 5 choices for your first selection but only 3 choices left for your second selection (the selections are dependent; the first selection affects the second). Many of the statistical concepts used to evaluate data in business, medicine, government, technology, and population growth were first developed to predict results in games of chance. The study of statistics has, from its beginnings, been linked with the study of probability. In this chapter, basic concepts of probability and combinatorics will be reviewed and then extended to finding the probability that a given outcome will occur a specified number of times in repeated trials of an experiment. These probability concepts will then be used to derive the Binomial Theorem. 217 14580TM_C16.pgs 3/26/09 12:15 PM Page 218 218 Chapter 16 Point out that the counting principle gives only the number of outcomes. For some problems, if we want more information about the nature of each outcome, then a list or tree diagram is more helpful. Note that many of the exercises preview the application of the counting principle for permutations that will be discussed in the next section. important. The formula for the number of combinations of n things taken r at a time (r # n) is nCr P n 5 Q r R 5 nr!r A variation of this formula is obtained by substituting the rule for nPr: 5 (n 2n!r)!r! By examining the formulas above, students can see that the number of combinations is always less than the corresponding number of permutations by a factor of r!. Note that nCn 5 1 and nC1 5 n. nCr 16-2 PERMUTATIONS AND COMBINATIONS Review the counting principle and its application to the formula for permutations. The formula for the number of permutations of n things taken r at a time (r # n) P 5 n(n 2 1)(n 2 2) c(n 2 r 1 1) n r n! 5 (n 2 r)! can be considered to be the result of repeated applications of the counting principle. The elements to be arranged are selected from the same set without repetition, as that the number of possible selections decreases by 1 each time. Point out that the question, “How many different four-digit numbers can be made using each of the digits 2, 3, 7, 9 exactly once?” can be answered either by using the counting principle or permutations. However, the question, “How many different four-digit numbers can be made using the digits 2, 3, 7, 9 if digits can repeat?” can be answered only by using the counting principle. Note that for n objects taken n at a time, n! n! 5 (n 2 n)! 5 0! 5 n! (since 0! 5 1) Sometimes the set of elements to be arranged contains a subset whose elements cannot be distinguished from each other. In such a case, we first determine the number of permutations as if the elements are all different, and then we divide by the number of arrangements of the identical elements among themselves. The key difference between permutations and combinations is that permutations are used when the order of the objects matters, and combinations are used when order is not n Pn 16-3 PROBABILITY This section reviews the terminology and principles of probability. Combinatorics can be applied to answer probability questions as long as we keep in mind that for equally likely outcomes, the probability of an event E is the ratio of the number of outcomes in the event, n(E), to the number of outcomes in the sample space, n(s). n(E) P(E) 5 n(S) Students may find different solution methods for the problems in the text. Consider the situation involving Jacob and Emily. Two approaches are given, but others are possible. The probability that Jacob is picked is the probability of two disjoint events: Jacob is chosen on the first pick or Jacob is chosen on the second pick. We find the probabilities and then add them. P(Jacob chosen first) 5 18 ? 77 5 18 P(Jacob chosen second) 5 78 ? 17 5 18 P(Jacob chosen) 5 18 1 18 5 28 5 14 Similarly, 1 11 1 P(Emily chosen first) 5 12 ? 11 5 12 1 1 P(Emily chosen second) 5 11 12 ? 11 5 12 1 1 2 1 12 5 12 5 16 P(Emily chosen) 5 12 14580TM_C16.pgs 3/26/09 12:15 PM Page 219 Probability and the Binomial Theorem 219 Then, P(Jacob and Emily chosen) 5 P(Jacob chosen) ? P(Emily chosen) 1 5 14 ? 16 5 24 For Example 2, the probability that both students have fewer than two absences is the product of the probabilities that the first and second students chosen have fewer than two absences. Using the probability form of the counting principle for dependent events: P(first , 2) ? P(second , 2 given first , 2) 1,176 9,730 49 48 84 5 140 ? 139 5 5 695 < 0.12 Enrichment Activity 16-3: Chi-Square (x2) Test for Goodness of Fit presents a simplified introduction to this topic. The problem of deciding whether or not a set of data fits a given distribution is discussed and the method of calculating the chi-square statistic is explained. The concepts of degrees of freedom for each distribution and the significance level are avoided; instead, students are given the critical value and told that the chi-square statistic must be compared with the given critical value. 16-4 PROBABILITY WITH TWO OUTCOMES The conditions for a binomial probability experiment are: • The n trials or observations are independent of each other. • There are only two outcomes on any trial, which can be considered success or failure. • The probability of a success is the same for each trial or observation. • The number of trials, n, is fixed in advance. Every experiment can be considered to have two outcomes, a favorable outcome and a non-favorable one that consists of all other outcomes. When a question asks for the probability of exactly 3 heads on 5 tosses of a coin, it is asking for the probability of 3 heads and 2 tails. If a question asks for the probability of exactly 7 juniors on a committee of 10 chosen at random from the student body, it is asking for the probability of randomly choosing 7 juniors and 3 students who are not juniors. Students should be encouraged to reword a problem in this way before writing the solution since, in the second case, it is necessary to consider the probability of all 10 outcomes (7 juniors and 3 not juniors) to obtain the correct probability. Hands-On Activity Instructions: A calculator can be used to simulate an activity with two outcomes when the probabilities of the component parts are known. The steps given below describe how to use calculator-generated random numbers to solve the following problem. The probability that a customer in a dress shop will make a purchase is 15 . What is the probability that if there are 4 customers in the shop, exactly 2 will make purchases? 1. Seed the random number generator of your calculator using the rand command. Randomly press a few digits. Then press 你 STO佡 MATH ENTER ENTER . A sample screenshot is shown below: 7898923547 >rand 7898923547 2. Use the randBin command to simulate a trial of 4 customers walking into the store. 你 MATH 7 4 , 1 ⴜ ENTER: 5 ENTER DISPLAY: randBin(4, 1/5 0 The randBin command takes as input the number of trials (4) and the probability of success A 15 B and returns the simulated number of successes observed in the 4 trials. Note that your answers will vary based on the value used for the random seed in step 1. 14580TM_C16.pgs 3/26/09 12:15 PM Page 220 220 Chapter 16 3. Repeat the previous step for a total of 20 times. 4. Of the 20 simulations, count the number of times there were exactly 2 successes. Write the ratio of this number to the total number of simulations, 20. This ratio is the empirical probability that exactly 2 of 4 customers will make a purchase. How does this value compare with the theoretical probability? 5. Now combine the results of all the students in the class. Write the ratio of the total number of times there were 2 successes to the total number of simulations. How does this value compare with the theoretical probability? Discoveries: 1. A calculator can be used to simulate a situation under consideration in a probability study. 2. The greater the number of simulations, the more closely the empirical probability matches the theoretical probability. Enrichment Activity 16-4: Geometric Probability Distribution explains the difference between a binomial distribution and a geometric distribution. In a geometric experiment, we are interested in the number of trials that are needed to achieve the first success. (Note this is not the geometric probability discussed in Section 16-3.) The general formula is derived and students practice applying the formula to solve problems. A method for simulating this type of situation is demonstrated using a calculator. 16-5 BINOMIAL PROBABILITY AND THE NORMAL CURVE Direct students’ attention to the chart preceding Example 1. Ask them to suggest any other phrases that mean the same as at least or at most; for example “up to 8 successes” or “a minimum of 8 successes.” Call attention to when the endpoint is included. As shown in Example 1, problems involving at least or at most have two methods of solution. Method 1: P(5 on at most seven dice) 5 P(one 5) 1 ? ? ? 1 P(seven 5s) Method 2: P(5 on at most seven dice) 5 1 2 P(5 on more than seven dice) Students should understand both methods and should be encouraged to use the shorter solution when possible. The table for the number of heads when 10 coins are tossed shows that the binomial distribution when p 5 12 is symmetric. For other values of p, binomial distributions will be only somewhat symmetric. Before technology became readily available, a normal approximation to the binomial was the preferred method for finding binomial probabilities when n was large or when many cases needed to be considered. For example, the probability of tossing more than 65 heads in 100 coin tosses would require a very long calculation. The normal approximation method works best when n is large and when the probability of success, p, is not close to 0 or 1. Alternatively, some mathematicians recommend the criteria that np $ 10 and n(1 2 p) $ 10. The reason for these conditions is that, when they are met, the binomial distribution is relatively symmetric and the normal curve provides a fairly good fit. If either condition is not met, the binomial distribution will be skewed significantly and the approximation by a normal curve will not be accurate. Students should also be aware that the binomial probability distribution is a discrete distribution and, by applying the normal curve approximation, we are finding a discrete probability using a continuous distribution. By adding or subtracting 0.5 to the endpoints of the region of interest, we are making a continuity correction to make up for gaps between the discrete values that are being approximated by a continuous function. 14580TM_C16.pgs 3/26/09 12:15 PM Page 221 Probability and the Binomial Theorem 221 So that students may understand how a distribution may be skewed, have them find the probability of n successes for n 5 0, 1, 2, . . . , 10 when p 5 .1. Ask them to make a probability plot similar to the one in the text. (See Exercise 2.) 16-6 THE BINOMIAL THEOREM Students should note the following patterns in the expansion of (x 1 y)n: and 10x2y3 and another two terms are 5x4y1 and 5x1y4. • The number of terms in the expansion of (x 1 y)n is n 1 1. In the sigma notation form of the binomial theorem, i 5 0 for the first term, i 5 1 for the second term, and so on. In general, the value of i is always one less than the number of the term you want to find. For example, the eighth term of (x 1 y)12 is given by 5 7 12C7x y n • The exponent of (x 1 y) , n, is the exponent of x in the first term and the exponent of y in the last term. • In successive terms, the exponent of x decreases by 1 and the exponent of y increases by 1. • The sum of the exponents in each term is n. • The coefficients are symmetric. The coefficients increase and then decrease from left to right. Terms where exponents are reversed have the same coefficients. For example, in (x 1 y)5, two terms are 10x3y2 5 792x5y7 When the binomial theorem is used to compute probability, students must think carefully to choose the correct term. For example, if Sara’s probability of making a basket on any attempt is p 5 34, then her probability of missing is m 5 14. To find the probability that Sara will make exactly 5 baskets in 8 attempts, compute the term of (m 1 b)8 where the exponent of b is 5. Since i is always one less than the number of the term, find the term where i 5 5 or the sixth term. 3 5 8C5m b 5 56 A 14 B 3 A 34 B 5 1,701 5 8,192 < .2076 14580TM_C16.pgs 3/26/09 12:15 PM Page 222 Name Class Date ENRICHMENT ACTIVITY 16-3 Chi-Square (x2) Test for Goodness of Fit An important question is whether or not a set of observed data values are compatible with a set of expected values. For example, you may want to know if public opinion on marriage has changed in the past 40 years. (In this case, the expected values would be the results of a 40-year-old survey, and the observed values would be the results of the same survey done in the present.) A Chi-square test for goodness of fit allows us to answer such questions. Chi-square statistic: x2 5 a (observed 2 expected) 2 expected The chi-square statistic takes into account the size of the difference between each observed and expected value relative to the size of the expected value. So an observed difference of 10 is more significant if it comes from an expected value of 100 than from an expected value of 10,000. The bigger the chi-square value, the more likely the observed values do not conform to the expected values. Problem 1 A manufacturer packs a rebate coupon in each box of its cereal and claims that the probabilities of getting various amounts are as follows: Coupon Amount $0.50 $1.00 $2.00 $3.00 .40 .30 .20 .10 Probability To see if the company’s claim is accurate, a consumer group randomly selects 500 boxes of this cereal and obtains the following data: Coupon Amount $0.50 $1.00 $2.00 $3.00 224 162 82 33 Number Do these data provide evidence that the company is lying? That is, can any differences be due to chance (random variation), or are the differences significant enough to doubt the company’s assertion? 1. Complete the table. The first row is done for you. Expected Sample Expected Observed Amount Probability Size Frequency Frequency $0.50 .40 500 .40(500) 5 200 224 $1.00 .30 500 162 $2.00 .20 500 82 $3.00 .10 500 33 Copyright © 2009 by Amsco School Publications, Inc. (Observed 2 Expected) 2 Expected (224 2 200) 2 200 5 2.88 14580TM_C16.pgs 3/26/09 12:15 PM Page 223 Statistics Name Class Date 2. Calculate the chi-square statistic by finding the sum of the numbers in the last column of the table. 3. To decide how large a calculated chi-square value must be to be significant, we must compare it with a critical value. You will learn more about finding the critical value in advanced courses. For this problem, the critical value is 7.81. If the calculated chi-square value is greater than 7.81, then we must reject the company’s claim about the probability of getting various coupon amounts. (Otherwise, the only conclusion we can make is that we have insufficient evidence to reject the company’s claim.) Moreover, this critical value is chosen so that our chances of incorrectly rejecting the company’s claim is only 5%. Based on your calculations, should the claim be rejected? Problem 2 A die is rolled 600 times and the following results are recorded. Face Frequency 1 2 3 4 5 6 113 88 103 117 75 84 Do these data cast doubt on the fairness of the die? To decide, calculate the chi-square statistic and compare it with the critical value of 11.07. 1. Complete the table below. Expected Sample Expected Observed Face Probability Size Frequency Frequency 1 1 6 600 2 1 6 600 3 1 6 600 4 1 6 600 5 1 6 600 6 1 6 600 2. Calculate the chi-square statistic. 3. Should the claim that the die is fair be rejected? Explain. Copyright © 2009 by Amsco School Publications, Inc. (Observed 2 Expected) 2 Expected 14580TM_C16.pgs 3/26/09 12:15 PM Page 224 Name Class Date Problem 3 1. Conduct your own experiment. Toss a die 120 times and record the frequency for each face. 2. Compute the chi-square statistic for your data. 3. Compare your statistic with the same critical value as above (11.07). What conclusion can you draw based on your experiment? 4. Make a list of the chi-square statistics calculated by all members of the class who performed the experiment. What percent of the class had values that were greater than the given critical value? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 225 Name Class Date ENRICHMENT ACTIVITY 16-4 Geometric Probability Distribution Recall the conditions for a binomial experiment: • The n trials are independent of each other. • There are only two outcomes on any trial, which can be considered success or failure. • The probability of a success is the same for each trial. • The number of trials, n, is fixed in advance. The conditions for a geometric experiment are similar with one important difference: • The n trials are independent of each other. • There are only two outcomes on any trial, which can be considered success or failure. • The probability of success is the same for each trial. • The number of trials is not fixed. A geometric distribution shows the number of trials needed until the first success is achieved. Example Suppose the probability that a box of cereal has a free movie ticket is .2. Find the probability that the first free ticket is found in: a. the 1st box purchased b. the 2nd box purchased c. the 3rd box purchased Solution: Since P(ticket) 5 .2, P(no ticket) 5 1 2 .2 5 .8. a. P(ticket in 1st box) 5 .2 b. P(first ticket in 2nd box) 5 P(no ticket, ticket) 5 P(no ticket) 3 P(ticket) 5 .8 3 .2 5 .16 c. P(first ticket in 3rd box) 5 P(no ticket, no ticket, ticket) 5 P(no ticket) 3 P(no ticket) 3 P(ticket) 5 .8 3 .8 3 .2 5 .128 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 226 Name Class Date It is interesting to note that the greatest probability that a ticket will be found occurs for the first box! The results of the example can be generalized. • If p is the probability of success, then P(first success occurs on the rth trial) 5 (1 2 p)r21 3 p In 1–3, find all probabilities to three decimal places. 1. A battery manufacturer determines that 5 out of every 100 batteries are defective. Find the probability that a quality control engineer finds the first defective battery in the a. 4th battery tested b. 7th battery tested 2. Let a sum of 8 when two dice are rolled be considered a success. a. Find P(8) in fractional form. b. Find the probability that the first sum of 8 occurs on the 5th roll. 3. At Rosedale High School, the probability that a student studies a foreign language is .6. If students at Rosedale High are randomly chosen and asked if they study a foreign language, find the probability that a. the first student who studies a foreign language is the 6th student asked b. the first student who does not study a foreign language is the 10th student asked A simulation can be used with regard to a geometric distribution. This type of simulation is sometimes called a wait-time simulation since we continue the simulation until a condition occurs. Consider the example that involved buying cereal boxes until a free movie ticket was found. Since the tickets are placed in 20% of the boxes, assign the digits 0 and 1 to a free ticket. Using a random-number table or a calculator, begin each simulation by noting the number of digits necessary until a 0 or 1 occurs. Record the number of digits including the first success. Repeat the simulation for a total of 20 times. For example, using the T1-83+/84+, execute the command randInt(0, 9) repeatedly until a 0 or 1 appears. (randInt is found in the MATH PROB menu.) Count the number of executions including the success and record the number. Then begin the next simulation. The results of 20 simulations are recorded in the table below. Simulation First success 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 15 4 1 1 1 3 5 1 2 18 2 3 3 2 2 3 6 5 1 1 Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 227 Name Class Date In the table, the ordered pair (1, 15) tells us that in the 1st simulation, the first success occurred in the 15th try. The ordered pair (2, 4) tells us that in the 2nd simulation, the first success occurred in the 4th try, and so on. The empirical probability that the first ticket is in the third box is then number of simulations where success occurs in 3rd execution total number of simulations 4 5 20 5 .2 Note that these particular simulations gave a result greater than the theoretical value. Also, since the sum of the digits of all the trials is 79, the average number of boxes needed to obtain a ticket is estimated as 79 20 5 3.95 so we expect to purchase a minimum of 4 boxes before finding a prize. 4. The probability that a construction company will be awarded a project is .4. a. Conduct 10 simulations using a calculator to estimate the probability that the company will land its first project on its 4th bid. (1) Describe the method of assignment of digits to the situation. (2) Describe how each simulation is carried out. (3) Show your work to find the empirical probability. b. Compare the calculated probability with the theoretical value. Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 228 Name Class Date Algebra 2 and Trigonometry: Chapter Sixteen Test Write your answers legibly in the space provided below. Show any work on scratch paper. An incorrect answer with sufficient work may receive partial credit. A correct answer with insufficient work may receive only partial credit. All scratch paper must be turned in at the conclusion of this test. In 1–8, evaluate each expression. 1. 6P6 2. 7P3 3. 20C18 4. 13C0 5. Q 10 R 3 6. 1099 P1 7. (12C4)(8C5) 8. 11P5 1 9 P4 9. A fast-food store offers the following choices: Hamburger: Single / Double / Triple Roll: White / Whole Wheat / Sesame Topping: Ketchup / Cheese / Onion / Chili / Relish / Mayonnaise How many different orders are possible for a hamburger on a roll with one topping? 10. In how many different ways can the letters of the word CIVILIZATION be arranged? (Answer may be left in factorial notation.) 11. What is the probability that a random arrangement of the letters of the word CIVILIZATION begins and ends with I? 12. Every week, Sharon works 4 days and does not work the other 3 days. In how many different ways can her work schedule be arranged? 13. The weather channel gives the probability of rain on Saturday as 30% and the probability of rain on Sunday as 40%. What is the probability that it will rain on one or both days? 14. An equilateral triangle is drawn on a square target such that the base of the triangle is one side of the square. If a dart hits the target, what is the probability that it lands in the triangle? 15. A summer reading list includes 8 novels, 10 nonfiction books, and 5 plays. How many different selections of 2 novels, 3 nonfiction books, and 1 play can be chosen from the list? 16. A 20-member club has an equal number of freshmen, sophomores, juniors, and seniors. If 5 members are selected at random, what is the probability that at least 4 will be seniors? Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 229 Name Class Date 17. The owner of Katonah Krafts estimates that 1 out of every 3 persons who come into the store makes a purchase. Today, 4 people came into the store the first hour it was open. What is the probability that at least one person made a purchase? 18. Of the students at Eastchester High School, 2 out of 3 have jobs for the summer. If 10 students are selected at random, what is the probability that exactly 4 will have jobs? Round your answer to four decimal places. 19. On a multiple-choice test, the probability of selecting the correct answer by guessing is 14. What is the probability that a student who guesses every answer will have exactly 13 correct answers on a test with 20 questions? Round your answer to four decimal places. 20. The buyer for a shoe store estimates that 10% of the shoes sold are size 6. If the store sold 25 pairs of shoes yesterday, to four decimal places, what is the probability that at most 2 pairs of size 6 shoes were sold? In 21 and 22, use the normal approximation to estimate each probability. Round answers to four decimal places. 21. A fair coin is tossed 100 times. What is the probability of getting at least 45 heads? 22. The probability that the Cougars will win any game is .4. What is the probability that the team will win between 10 and 20 of the 50 games it plays this season? 23. Expand the binomial (x 1 2)6. 24. Expand the binomial (3x 1 2y)4. 25. What is the fourth term of (a 2 2b)7? 26. What is the fifth term of A 3 1 3t B ? 27. Use the binomial theorem to write an expression for the volume of a cube with sides measuring (2n 2 3) inches. 5 28. What is the sum of the numerical coefficients of the expansion of (x 1 y)20? (Hint: Use (1 1 1)20.) Bonus: In a small town, a study showed that 10% of registered voters had lived at the same address all their lives. A sample group of registered voters is to be selected at random from the town. How large of a group is needed to have the probability be greater than 40% that at least one person has always lived at the same address? (Use a calculator and trial and error.) Copyright © 2009 by Amsco School Publications, Inc. 14580TM_C16.pgs 3/26/09 12:15 PM Page 230 Name Class Date SAT Preparation Exercises (Chapter 16) I. MULTIPLE-CHOICE QUESTIONS In 1–15, select the letter of the correct answer. 6. 4 3 1. How many distinct arrangements of the letters of the word TENNESSEE can be made? (A) 3,780 (B) 7,560 (C) 10,480 (D) 15,120 (E) 30,240 2 1 5 2. How many ways can a row of four girls followed by four boys be arranged if the girls must sit together and the boys must sit together? (A) 16 (B) 256 (C) 576 (D) 10,080 (E) 40,320 The probability of getting each number on the spinner is proportional to the number. For example, P(4) 5 2 ? P(2). What is the probability of getting a sum of 5 from two spins? 2 (A) 45 4 (B) 45 3. A building has two small elevators. Each elevator can safely carry 3 people. There are 5 people waiting to take the elevators to the second floor. If both elevators arrive at the first floor at the same time, how many ways can the people group themselves in the elevators? (A) 4 (B) 6 (C) 10 (D) 12 (E) 20 8 (D) 225 2 (E) 75 4. Mrs. Grasso’s eleventh-grade class has 12 boys and 14 girls. In how many ways can at least 2 boys be selected for a committee of 3 students? (A) 66 (B) 286 (C) 924 (D) 1,144 (E) 4,004 5. A spinner is divided into red, white, and blue sectors. The area of the white sector is twice the area of the red, and the area of the blue sector is one and a half times the area of the white. The spinner is spun twice. What is the probability that one spin lands on red and one spin lands on blue? 1 (A) 12 (B) 16 (D) 21 (E) 23 3 (C) 12 Copyright © 2009 by Amsco School Publications, Inc. 4 (C) 75 7. a b A target consists of one square inside another. If a dart has an equally likely chance of striking any point on the target, what is the probability that it will strike the shaded region? 2 (A) ba2 2 1 2 (D) ba2 1 1 2 (B) 1 2 ba2 (E) ba 2 (C) ba2 8. The math department has 8 teachers. What size subcommittee should be selected to provide the largest number of combinations of teachers on the committee? (A) 8 (B) 7 (C) 6 (D) 5 (E) 4 14580TM_C16.pgs 3/26/09 12:15 PM Page 231 Name Class 9. If 10 coins are flipped simultaneously and P(k) is the probability of getting k heads, 10 what is the value of a P(k) ? k50 1 10 (A) 0 (B) (D) 1 (E) 10 (C) 1 210 10. If 10 coins are flipped simultaneously and P(k) is the probability of not getting 10 k heads, what is a P(k) ? k50 1 10 (A) 0 (B) (D) 1 (E) 10 11. If xCx–2 5 15, then x is (A) 3 (B) 4 (D) 6 (E) 7 (C) 1 210 (C) 5 12. Q 25 R 2 Q 25 R 1 Q 25 R 2 c 1 Q 25 R 0 2 24 1 25 2 Q R equals 25 (A) 0 (B) 1 (C) 2 (D) 4 (E) 25 x x 13. If Q 1 R 2 Q 0 R 5 24, then x is (A) 4 (D) 24 (B) 6 (E) 25 Date II. STUDENT-PRODUCED RESPONSE QUESTIONS In 16–24, you are to solve the problem. 16. The probability of the Jaguars winning any one game is 30%. To four decimal places, what is the probability that the Jaguars will win exactly 2 games in a 5-game series? 17. How many different ways can a group of 3 dogs and 2 cats be chosen if a pet store has 10 dogs and 8 cats? 18. A committee of 7 students is to be chosen from a group of 20 seniors and 14 juniors. Find, to four decimal places, the probability that the committee will include 4 seniors and 3 juniors. 19. The probability of Slugger Sam getting a hit in any one at bat is .75. To four decimal places, what is the probability that Sam will get at most 5 hits in his next 6 atbats? 20. A gardener will plant 3 yellow tulips, 2 red tulips, and 4 purple tulips along a straight path. How many different arrangements can she make? 21. How many diagonals are there in a regular hexagon? (C) 12 14. What is the sum of the numerical coefficients of (x 2 y)10? (A) 2252 (B) 25 (C) 0 (D) 5 (E) 252 15. The expression (2 2 i)5 where i 5 !21 is equivalent to (A) 238 2 41i (B) 38 2 39i (C) 42 1 40i (D) 258 1 56i (E) 142 1 120i Copyright © 2009 by Amsco School Publications, Inc. 22. What is the coefficient of the x5 term of the product of (x3 1 3x2 1 3x 1 1) and (x5 1 5x4 1 10x3 1 10x2 1 5x 1 1)? 23. If n is an integer between 0 and 100, then 100 Cn ? (100 2 n)! is equal to 100 P100 2 n for what values of n? 24. A fair die is tossed three times. What is the probability that, on at least two of the throws, the face that appears on top is greater than 4? 14580BM.pgs 3/26/09 12:11 PM Page 232 Summary of Formulas and Trigonometric Relationships Functions and Relations • To determine if a relation is a function, use the vertical line test. • To determine if a function is one-to-one, use the horizontal line test. • If f(x) is a function, then: The graph of f(x) 1 a is the graph of f(x) moved a units up or down. The graph of f(x 1 a) is the graph of f(x) moved a units to the left when a is positive and |a| units to the right when a is negative. The graph of 2f(x) is the graph of f(x) reflected in the x-axis. The graph of f(2x) is the graph of f(x) reflected in the y-axis. When a . 1, the graph of af(x) is the graph of f(x) stretched vertically by a factor of a (or, alternatively, af(x) is the graph of f(x) compressed horizontally by a factor of a). When 0 , a , 1, the graph of af(x) is the graph of f(x) compressed vertically by a factor of a (or, alternatively, af(x) is the graph of f(x) stretched horizontally by a factor of a). Absolute Value Equations and Inequalities • If x 5 k for positive k, then x 5 2k or x 5 k. • If x , k for positive k, then 2k , x , k. • If x . k for positive k, then x . k or x , 2k. • If x 5 k for negative k, then the solution set is { }. • If x , k for negative k, then the solution set is { }. • If x . k for negative k, then the solution set is the real numbers. Quadratic Inequalities in One Variable • To solve a quadratic inequality: 1. Find the roots of the corresponding equality. 2. The roots of the equality separate the number line into two or more intervals. 3. Test a number from each interval. An interval is part of the solution if the test number makes the inequality true. • Alternatively, to solve 0 . ax2 1 bx 1 c or 0 , ax2 1 bx 1 c, graph y . ax2 1 bx 1 c or y , ax2 1 bx 1 c, respectively. The solution is the set of x-coordinates of the points common to the graph of y . ax2 1 bx + c or y , ax2 1 bx 1 c and the x-axis. 232 14580BM.pgs 3/26/09 12:11 PM Page 233 Summary of Formulas and Trigonometric Relationships 233 Quadratic Inequalities in Two Variables • The solution set of y . ax2 1 bx 1 c is the set of coordinates of the points above the graph of y 5 ax2 1 bx 1 c. • The solution set of y , ax2 1 bx 1 c is the set of coordinates of the points below the graph of y 5 ax2 1 bx 1 c. Quadratic Equations b b , c 2 4a • The vertex of the parabola y 5 ax2 1 bx 1 c is A 22a B and the axis of symmetry is b x 5 22a. 2 • To complete the square in ax2 1 bx 1 c 5 0: 1. Divide by a if a 1. 2. Isolate the terms in x on one side of the equation. 2 3. Add the square of one-half the coefficient of x or A 12b B to both sides of the equation. 4. Write the square root of both sides of the resulting equation and solve for x. • The quadratic formula gives the roots of ax2 1 bx 1 c 5 0 as 2 x 5 2b 6 "2ab 2 4ac • In ax2 1 bx 1 c 5 0, the value of b2 2 4ac is the discriminant. The number of x-intercepts of the function is When the discriminant b2 2 4ac is The roots of the equation are . 0 and a perfect square real, rational, and unequal 2 . 0 and not a perfect square real, irrational, and unequal 2 50 real, rational, and equal 1 ,0 imaginary numbers 0 • If A and B are two roots of a quadratic equation, then a quadratic equation with these roots is x2 2 (A 1 B)x 1 A ? B 5 0 • In ax2 1 bx 1 c 5 0: 2ba is equal to the sum of the roots. ac is equal to the product of the roots. Complex Numbers • For any integer n: i 4n 5 1 i 4n 1 2 5 21 i 4n 1 1 5 i i 4n 1 3 5 2i 14580BM.pgs 3/26/09 12:11 PM Page 234 234 Summary of Formulas and Trigonometric Relationships • The complex conjugate of a 1 bi is a 2 bi. Also, (a 1 bi)(a 2 bi) 5 a2 1 b2. • The multiplicative inverse of any non-zero complex number a 1 bi is 1 a 1 bi a b 5 a2 1 b2 2 a2 1 b2 i Sequences and Series • For an arithmetic sequence: an 5 a1 1 (n 2 1)d 5 an–1 1 d Sn 5 n2 (a1 1 an) 5 n2 f2a1 1 (n 2 1)dg • For a geometric sequence: an 5 a1rn–1 5 an21r a1 (1 2 rn) 1 2 r Sn 5 • For a geometric series, if r , 1: ` a n21 5 1 21 r a a1r n51 Exponential Growth and Decay • If a quantity A0 is increased or decreased by a rate r per interval of time, compounded n times per interval, its value A after t intervals of time is A 5 A0 A 1 1 nr B nt • If the increase or decrease is continuous for t units of time, the formula becomes A 5 A0ert Exponents and Logarithms Powers Logarithms Exponent of Zero b0 5 1 logb 1 5 0 Exponent of One b1 5 b logb b 5 1 Products bx1y 5 bxby bx y Quotients bx2y 5 b Powers (bx) y 5 bxy Power of a Product (bc) x 5 bxcx Power of a Quotient A bc B x x 5 bcx logb cd 5 logb c 1 logb d logb dc 5 logb c 2 logb d logb cn 5 n logb c logb (cd) n 5 n logb (cd) 5 n(log b c 1 log b d) n logb A dc B 5 n log b A dc B 5 n(log b c 2 log b d) • To change the base of a logarithm to any base b: log y logb y 5 log b ln y logb y 5 ln b 14580BM.pgs 3/26/09 12:11 PM Page 235 Summary of Formulas and Trigonometric Relationships 235 Trigonometry of the Right Triangle In right ABC: hyp sin A 5 hyp opp csc A 5 sin1 A 5 opp adj sec A 5 cos1 A 5 adj hyp cos A 5 hyp opp adj sin A tan A 5 cos A 5 adj cot A 5 tan1 A 5 opp Trigonometry of the Unit Circle • If P(x, y) is a point on the unit circle and ROP is an angle in standard position with measure u, then x 5 cos u y 5 sin u • If P(x, y) is any point in the coordinate plane, then the formulas become where r 5 "x2 1 y2. x 5 r cos u y 5 r sin u Trigonometric Function Values of Special Angles u (degrees) 0° 30° 45° 60° 90° u (radians) 0 p 6 p 4 p 3 p 2 sin u 0 1 2 !2 2 !3 2 1 cos u 1 !3 2 !2 2 1 2 0 tan u 0 !3 3 1 !3 undefined Reference Angles If u is a second-, third-, or fourth-quadrant angle: Second Quadrant Reference Angle Third Quadrant 180 2 u u 2 180 Fourth Quadrant 360 2 u sin u sin (180 2 u) 2sin (u 2 180) 2sin (360 2 u) cos u –cos (180 2 u) 2cos (u 2 180) cos (360 2 u) tan u –tan (180 2 u) tan (u 2 180) 2tan (360 2 u) Radian Measure • To convert x in radians to degrees, multiply x by 180 p . p • To convert x in degrees to radians, multiply x by 180 . 14580BM.pgs 3/26/09 12:11 PM Page 236 236 Summary of Formulas and Trigonometric Relationships Trigonometric Functions For y 5 a sin b(x 1 c) or y 5 a cos b(x 1 c): • The amplitude, a, is the maximum value of the function and 2a is the minimum value. • b is the number of cycles in the 2p interval (also called the angular frequency), the period 2p is b |b| the length of the interval for one cycle, and the frequency 2p is the reciprocal of the period. • The phase shift is 2c. If c is positive, the graph is shifted c units to the left. If c is negative, the graph is shifted c units to the right. Function Domain (n is an integer) Range Sine All real numbers [21, 1] Cosine All real numbers [21, 1] p 2 1 np Tangent All real numbers except Cotangent All real numbers except np p 2 All real numbers 1 np Secant All real numbers except Cosecant All real numbers except np All real numbers (2`, 21] < [1, `) (2`, 21] < [1, `) Sine Function with a Restricted Domain Inverse Sine Function y 5 sin x y 5 arcsin x or y 5 sin21 x Domain 5 U x : 2p2 # x # p2 V Range 5 {y : 21 # y # 1} Domain 5 {x : 21 # x # 1} Cosine Function with a Restricted Domain Inverse Cosine Function y 5 cos x y 5 arccos x or y 5 cos21 x Domain 5 {x : 0 # x # p} Domain 5 {x : 21 # x # 1} Range 5 {y : 21 # y # 1} Range 5 {y : 0 # y # p} Tangent Function with a Restricted Domain Inverse Tangent Function y 5 tan x y 5 arctan x or y 5 tan21 x Domain 5 Ux : 2p2 , x , p2 V Range 5 {y : y is a real number} Domain 5 {x : x is a real number} Range 5 U y : 2p2 # y # p2 V Range 5 Uy : 2p2 , y , p2 V 14580BM.pgs 3/26/09 12:11 PM Page 237 Summary of Formulas and Trigonometric Relationships 237 Trigonometric Identities Cofunction Identities cos u 5 sin (90° 2 u) sin u 5 cos (90° 2 u) tan u 5 cot (90° 2 u) cot u 5 tan (90° 2 u) sec u 5 csc (90° 2 u) csc u 5 sec (90° 2 u) Reciprocal Identities Quotient Identities Pythagorean Identities sec u 5 cos1 u sin u tan u 5 cos u cos2 u 1 sin2 u 5 1 csc u 5 sin1 u u cot u 5 cos sin u 1 1 tan2 u 5 sec2 u cot u 5 tan1 u Sums of Angle Measures cos (A 1 B) 5 cos A cos B 2 sin A sin B cot2 u 1 1 5 csc2 u Differences of Angle Measures cos (A – B) 5 cos A cos B 1 sin A sin B sin (A 1 B) 5 sin A cos B 1 cos A sin B sin (A – B) 5 sin A cos B 2 cos A sin B A 1 tan B tan (A + B) 5 1tan 2 tan A tan B A 2 tan B tan (A – B) 5 1tan 1 tan A tan B Double-Angle Identities sin (2A) 5 2 sin A cos A cos (2A) 5 cos2 A 2 sin2 A cos (2A) 5 2 cos2 A 2 1 cos (2A) 5 1 2 2 sin2 A Half-Angle Identities sin 12A 5 6#1 2 2cos A cos 12A 5 6#1 1 2cos A 2 cos A tan 12A 5 6#11 1 cos A A tan (2A) 5 1 22tan tan2 A Trigonometric Relationships • The Law of Cosines gives a relationship between an angle, the opposite side, and the two adjacent sides in ABC: a2 5 b2 1 c2 2 2bc cos A 2 c2 2 a2 cos A 5 b 1 2bc b2 5 a2 1 c2 2 2ac cos B 2 c2 2 b2 cos B 5 a 1 2ac c2 5 a2 1 b2 2 2ab cos C 2 b2 2 c2 cos C 5 a 1 2ab 14580BM.pgs 3/26/09 12:11 PM Page 238 238 Summary of Formulas and Trigonometric Relationships • The Law of Sines gives a relationship between all the sides and angles of ABC: a sin A 5 sinb B 5 sinc C • Given an angle and two adjacent sides of a triangle, the area of ABC can be calculated using the following formula: Area of ABC 5 12bc sin A 5 12ac sin B 5 12ab sin C • In ABC, to determine the number of possible triangles given a, b, and mA (ambiguous case), let h 5 b sin A: C C C C b h b A A A is: and: Possible triangles: b ha B b a h a B A B9 C C a B h a A h B a b B a h b A A B Acute a,h Acute h5a Acute h,a,b Acute a.b Obtuse a#b Obtuse a.b None One, right Two One None One Statistics Formulas 1 x , of a set of N numbers is equal to the sum of the numbers divided by N or N • The mean, 2 a x. • The median of a set of N numbers arranged in numerical order is the middle number. a. If N is odd, the median is the number in the N 21 1 position. N b. If N is even, the median is the average of the numbers in the N 2 and 2 1 1 positions. • The mode is the number that occurs most often in a set of data. a. If two or more numbers occur more often than all other data values and these two or more numbers have the same frequency, then each of these numbers is a mode. b. If each data value occurs with the same frequency, then the data set has no mode. • Quartiles are determined when a set of data is arranged in ascending numerical order, usually from left to right. a. The second quartile, Q2, is the median of the data, separating the data into two halves. b. The first quartile, Q1, is the median of the lower half of the data, not counting the median. c. The third quartile, Q3, is the median of the upper half of the data, not counting the median. • The range is the difference between the highest and lowest numbers in a data set. • The interquartile range is the difference between the upper and lower quartiles. • The five statistical summary is the minimum, first quartile, median, third quartile, and maximum of a data set. 14580BM.pgs 3/26/09 12:11 PM Page 239 Summary of Formulas and Trigonometric Relationships 239 • An outlier is any data value that is more than third quartile 1 1.5 3 (interquartile range) or less than first quartile 2 1.5 3 (interquartile range). x) 2. • The population variance, s2, of a set of data is n1 a (x 2 2 1 22 • The sample variance, s2, of a set of data is n 2 1 a (x 2 x) . • The population standard deviation, s, is the square root of the population variance or 1 x) 2. #n a (x 2 2 • The sample standard deviation, s, is the square root of the sample variance or 1 x) 2. #n 2 1 a (x 2 2 Statistics Formulas for Grouped Data If a set of data is grouped in terms of the frequency f of a given value x: xf x , is a f . • The mean, 2 a f(x 2 2 x )2 • The population variance, s2, is a . • The sample variance, s2, is af 2 2 a f(x 2 x ) . A afB 2 1 • The population standard deviation, s, is • The sample standard deviation, s, is É 2 a f(x 2 x ) . f a É 22 a f(x 2 x ) . A afB 2 1 Probability Formulas • If A and B are independent events, then the probability of both occurring is P(A) 3 P(B) . • In permutations (or arrangements), order is important. The permutation of n things, taken n at a time is nPn 5 n! 5 n(n 2 1)(n 2 2) c3 ? 2 ? 1 The permutation of n things, taken r at a time, where r # n is n! n(n 2 1)(n 2 2) c or nPr 5 (n 2 nPr 5 8 r)! r factors The permutation of n things where there are a1, a2, c, ar repetitions is n! a1! ? a2! ? c ? ar! 14580BM.pgs 3/26/09 12:11 PM Page 240 240 Summary of Formulas and Trigonometric Relationships • In combinations (or sets), order is not important. The combination of n things, taken n at a time, where r # n is nCr nCn P 5 nr!r or nCr 5 (n 2n!r)!r! 5 1 and nC0 5 1. For whole numbers n and r where r # n: nCr 5 nCn – r . • In a Bernoulli experiment or binomial experiment, if the probability of success is p and the probability of failure is q 5 1 2 p, then the probability of r successes in n trials is r n2r nCrp q r The probability of at most r successes in n trials is a nCipiqn2i. i50 n The probability of at least r successes in n trials is a nCipiqn2i. i5r Normal and Binomial Distributions • For a normal distribution, the following relationships exist. The mean and the median of the data values lie on the line of symmetry of the curve. Approximately 68% of the data values lie within one standard deviation from the mean. Approximately 95% of the data values lie within two standard deviations from the mean. Approximately 99.7% of the data values lie within three standard deviations from the mean. 99.7% of data values 95% of data values 68% of data values 13.5% 34% 34% 13.5% x–23s x–22s x–2s x– x–1s x–12s x–13s Normal distribution • If x is a data value of a normal distribution, its z-score is x 2 2 x x 2 2 x z-score 5 standard s deviation 5 • A binomial distribution of n trials with probability of success of p can be approximated by a normal distribution with mean np and standard deviation !np(1 2 p) . • When n is a positive integer, (x 1 y)n can be expressed as a polynomial called the binomial expansion: n (x 1 y)n 5 a nCixn 2 iyi i50 14580BM.pgs 3/26/09 12:11 PM Page 241 Summary of Formulas and Trigonometric Relationships 241 Linear Regression • The regression line is appropriate only for data that appears to be linearly related. • A high correlation coefficient, r, does not necessarily mean that one variable causes the other. • When r 5 1 or 21, there is a perfect linear relationship between the data values. • When r 5 0, no linear relationship exists between the data values. • When r is close to 1, the data has a strong linear relationship. Values between 0 and 1 indicate various degrees of moderate correlation. • The sign of r matches the sign of the slope of the regression line. 14580BM.pgs 3/26/09 12:11 PM Page 242 Table of Measures Length English System 1 foot (ft) 12 inches (in.) 1 yard (yd) 3 feet 1 mile (mi) 1,760 yards 1 mile (mi) 5,280 feet Metric System 1 centimeter (cm) 10 millimeters (mm) 1 decimeter (dm) 10 centimeters 1 meter (m) 10 decimeters 1 meter 100 centimeters 1 kilometer (km) 1,000 meters Weight (Mass) English System 1 pound (lb) 16 ounces (oz) 1 ton (T) 2,000 pounds Metric System 1 gram (g) 1,000 milligrams (mg) 1 kilogram (kg) 1,000 grams 1 metric ton (MT) 1,000 kilograms Liquid Measure English System 1 pint (pt) 16 fluid ounces (fl oz) 1 quart (qt) 2 pints 1 gallon (gal) 4 quarts Metric System 1 liter (L) 1,000 milliliters (mL) 1 kiloliter (kL) 1,000 liters 1 cubic centimeter (cm3) 1 milliliter 1 liter 1,000 cubic centimeters Time English System 1 minute (min) 60 seconds (sec) 1 hour (hr) 60 minutes 1 day 24 hours 1 week 7 days 1 year 52 weeks 1 year 12 months Metric System Same as the English system Conversion factors from the English system to the metric system: Length 1 inch 2.54 centimeters 1 foot 0.305 meter 1 yard 0.914 meter 1 mile 1.61 kilometers Holistic Scoring Rubric 242 14580BM.pgs 3/26/09 12:11 PM Page 243 Holistic Scoring Rubric To be used with Cumulative Reviews (Chapters 2–16) Part II (2 credits) Score General Description 2 Credits Complete and correct • • • • • Demonstrates thorough understanding of the concepts involved. Employs appropriate strategies. Performs all calculations correctly. Graphs and diagrams are accurate and appropriate. Deductive arguments are used to justify conclusions. Response Criteria 1 Credit Partially correct • • • • • Demonstrates limited understanding of the concepts involved. Employs appropriate strategies. Performs most calculations correctly. Graphs and diagrams are accurate and appropriate. Arguments are constructed with some mathematical basis. 0 Credits Completely incorrect • • • • • Demonstrates insufficient understanding of the concepts involved. Employs inappropriate strategies. Performs most calculations incorrectly. Graphs and diagrams are neither accurate nor appropriate. Arguments are constructed with no mathematical basis. Part III (4 credits) Score General Description Response Criteria 4 Credits Complete and correct • • • • • 3 Credits Fundamentally correct • Demonstrates satisfactory understanding of the concepts involved. • Employs appropriate strategies. • Performs most calculations correctly; flaws reflect inattentive execution of mathematical procedures. • Graphs and diagrams are accurate and appropriate. • Arguments are constructed with some mathematical basis; flaws reflect inattentive execution of written arguments. 2 Credits Partially correct • • • • • Demonstrates adequate understanding of the concepts involved. Employs appropriate strategies. Performs most calculations correctly. Graphs and diagrams are accurate and appropriate. Arguments are constructed with some mathematical basis. 1 Credit Incomplete and flawed, but not completely incorrect • • • • • Demonstrates limited understanding of the concepts involved. Employs inappropriate strategies. Performs most calculations correctly. Some graphs and diagrams are accurate or appropriate. Some correct reasoning or justification is used. 0 Credits Completely incorrect • • • • • Demonstrates insufficient understanding of the concepts involved. Employs inappropriate strategies. Performs calculations with major errors. Graphs and diagrams are neither accurate nor appropriate. Arguments are constructed with no mathematical basis. Demonstrates thorough understanding of the concepts involved. Employs appropriate strategies. Performs all calculations correctly. Graphs and diagrams are accurate and appropriate. Deductive arguments are used to justify conclusions. 243 14580BM.pgs 3/26/09 12:11 PM Page 244 244 Summary of Formulas and Trigonometric Relationships Part IV (6 credits) Score General Description 6 Credits Complete and correct • • • • • Response Criteria 5 Credits Essentially correct • Demonstrates satisfactory understanding of the concepts involved. • Employs appropriate strategies. • Performs most calculations correctly; flaws reflect inattentive execution of mathematical procedures. • Graphs and diagrams are accurate and appropriate. • Deductive arguments are used to justify conclusions. 4 Credits Fundamentally correct • Demonstrates satisfactory understanding of the concepts involved. • Employs appropriate strategies. • Performs most calculations correctly; flaws reflect inattentive execution of mathematical procedures. • Graphs and diagrams are accurate and appropriate. • Arguments are constructed with some mathematical basis; flaws reflect inattentive execution of written arguments. 3 Credits Partially correct • • • • • Demonstrates adequate understanding of the concepts involved. Employs appropriate strategies. Performs most calculations correctly. Graphs and diagrams are accurate and appropriate. Arguments are constructed with some mathematical basis. 2 Credits Partially incorrect • • • • • Demonstrates limited understanding of the concepts involved. Employs appropriate strategies. Performs most calculations correctly. Graphs and diagrams are accurate and appropriate. Arguments are constructed with some mathematical basis. 1 Credit Incomplete and flawed, but not completely incorrect • • • • • Demonstrates limited understanding of the concepts involved. Employs inappropriate strategies. Performs most calculations correctly. Some graphs and diagrams are accurate or appropriate. Some correct reasoning or justification is used. 0 Credits Completely incorrect • • • • • Demonstrates insufficient understanding of the concepts involved. Employs inappropriate strategies. Performs calculations with major errors. Graphs and diagrams are neither accurate nor appropriate. Arguments are constructed with no mathematical basis. Demonstrates thorough understanding of the concepts involved. Employs appropriate strategies. Performs all calculations correctly. Graphs and diagrams are accurate and appropriate. Deductive arguments are used to justify conclusions. 14580BM.pgs 3/26/09 12:11 PM Page 245 Conversion Chart To be used with Cumulative Reviews (Chapters 2–16) Chart for Converting Total Cumulative Review Raw Scores to Scaled Cumulative Review Scores Raw Score Scaled Score Raw Score Scaled Score Raw Score Scaled Score 44 100 29 75 14 45 43 98 28 73 13 42 42 96 27 71 12 40 41 95 26 70 11 37 40 93 25 68 10 34 39 91 24 66 9 31 38 90 23 64 8 28 37 88 22 62 7 25 36 87 21 60 6 22 35 85 20 58 5 19 34 84 19 56 4 15 33 82 18 54 3 11 32 80 17 51 2 8 31 79 16 49 1 4 30 77 15 47 0 0 To determine the student’s cumulative review score, find the student’s total raw score in the column labeled “Raw Score” and then locate the scaled score that corresponds to that raw score. This scaled score is analogous to the scaled score given for comparable work on the Regents examination. A passing score is any scaled score of 65 or higher. Cumulative reviews that earn a scaled score between 60 and 64 may be scored a second time using the Holistic Scoring Rubric to ensure that responses to questions in parts II, III, and IV received all merited credits. 245