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メインページ Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis

Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis

Richard G. Brown

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Richard G.  T L i -  athematics  Pfecuiculu with Discrete and Data Analysis  thematics  THIS BOOK  IS  THE PROPERT Y OF:  STATE Book No.  PROVINCE  COUNTY  Enter information  PARISH  in  SCHOOL  spaces  to the left as  DISTRICT  instructed  OTHER  CONDITION  Year  ISSUED TO  RETURNED  ISSUED  Used  4 ..U.(Lr&M..)>yb^ iXo2ucl<&4  PUPILS to or  whom  Lk&sa  this  mark any part of  is issued must not write on any page any way, consumable textbooks excepted.  textbook it  in  1.  Teachers should see that the pupil's every book issued.  2.  The following terms should be used Poor; Bod.  in  name  is  clearly  written  in  ink  in  the  recording the condition of the book:  spaces above  New; Good;  in  Fair;  FEATURES AND BENEFITS  Advanced Mathematics Precalculus with Discrete Mathematics and Data Analysis  To  lay the groundwork for further study of mathematics at the college level, all standard precalculus topics are presented, as well as substantial new material on discrete mathematics and data analysis, pp. iii - xiii (contents).  Numerous applications  lessons, examples,  mathematics 698 - 700.  life  to  everyday  and a variety of  and exercises establish the importance of scientific and technical fields, pp. 43 - 48,  Integration of technology throughout the lesson presentations, examples, and exercises fosters effective learning  and prepares students for participation  in  a technological  society, pp. 75, 131,690.  variety of types and multiple levels of exercises meet many different teaching and learning needs. New communication exercises, including Reading, Writing, Discussion, and Visual Thinking exercises, are labeled for easy reference, as are  The wide  application exercises, pp. 107, 129, 154.  Numerous worked-out examples, important  results in tinted boxes, Class Exercises,  Chapter Summaries, Chapter Tests, and Cumulative Reviews make the text accessible and easy  to use, pp. 126, 172, '176.  Activities in lesson presentations, Investigation  and Research exercises, and e; nd-of-  chapter Projects are especially useful in promoting active learning, pp. 19, 156, 210.  The  flexible outline, incorporation of technology, applications orientation, and  provisions for active student learning meet contemporary standards (see  and Approaches, pp. xiv -  Key Topics  xvii).  Careers in Mathematics pages, Biographies, and other text references, establish the  multicultural nature of mathematics and  its  importance to people of varied interests  and background today, pp. xxvi, 366, 537.  A  Entrance Examinations gives students an excellent opportunity these important examinations, pp. 790 - 799.  section on College  to prepare for  dvanced kTi  athematics  Precalculus with Discrete Mathematics and Data Analysis Richard G.  Brown  Editorial  Adviser Andrew M. Gleason  Teacher Consultants Martha  Dane Maria  R. F.  A.  Brown  Camp G. Fierro  Wallis Green  Linda Hunter  Carolyn Kennedy  Littell Inc.  A  Houghton  Evanston,  Illinois  Mifflin  Company  Boston  Dallas  AUTHOR Richard G. Brown, Mathematics Teacher, Phillips Exeter Academy, Exeter, New Hampshire. A teacher and author, Mr. Brown has taught a wide range of mathematics courses (or both students and teachers at several schools and universities. His affiliations have included the Newton (Massachusetts) High School, the University of New Hampshire, Arizona State University, and the North Carolina School for Science and Mathematics during the school year beginning in 983. Currently a member of the COMAP Consortium Council, he is an active participant in professional mathematics organizations and the author of mathematics texts and journal articles. 1  EDITORIAL ADVISER  Andrew M. Gleason,  Hollis Professor of  Philosophy, Harvard University.  mathematician and a  member  Professor  Mathematics and Natural  Gleason  a well-known research  is  of the National Academy of Sciences.  He  has  served as President of the American Mathematical Society.  TEACHER CONSULTANTS Martha A. Brown, Prince George's  Dane  Camp,  R.  Supervisor of Mathematics,  County Public Schools, Prince George's County, Maryland Mathematics Teacher,  Downer's Grove High School, Downer's Grove,  Maria  F.  Cerritos  High School,  Illinois  G. Fierro, Mathematics Department Chairperson, Cerritos, California  Wallis Green, Mathematics Teacher, C.  E.  Jordan High School, Durham, North Carolina  Linda Hunter, Mathematics Department Chairperson, Douglas MacArthur High School, San Antonio, Texas  Carolyn Kennedy, Mathematics West High  Teacher,  Ohio  School, Columbus,  TECHNOLOGY CONSULTANT Wade Ellis, Jr., Mathematics Instructor, West  Valley College, Saratoga, California  ACKNOWLEDGEMENTS The author wishes  to thank  Jonathan Choate,  Mathematics Teacher, Groton  School, Groton, Massachusetts, for contributing the material on dynamical sys-  tems  in  Chapter 19. The author also wishes  Professor of Mathematics, Bentley College, tributing the projects that follow  Copyright  No  part of  ©  Chapters 5, 9,  1997, 1994 by Houghton  this  Mifflin  work may be reproduced or  IL  Donna DiFranco,  1  2,  1  8,  transmitted  and  in  and  1  All rights  any form  Manager,  Rights  is  con-  or  reserved.  by any means,  recording, or by any informaLittell  Inc.  expressly permitted by federal copyright law. Address inquiries  and  Permissions,  McDougal  Littell  Inc.,  P.O. Box 1667, Evanston,  60204.  ISBN: 0-395-77114-5  for  9.  storage or retrieval system without prior written permission of AAcDougal  unless such copying to  thank  Company.  electronic or mechanical, including photocopying tion  to  Waltham, Massachusetts,  23456789-000 99 98 97  \  ntroduction  05>  %  Dear Student,  Welcome to this course you  will  have studied I  in  Advanced Mathematical This year well as extend those you  many new topics as  explore in  previous courses.  have written this book with the goal of making mathematics  clear, interesting,  and  relevant.  As  a result, you  will  see many  real-world applications of the topics you study. For example: •  Exponential functions model population growth, decline of natural resources, and cost of a college education.  •  Logarithmic functions measure the intensity of  earthquakes, the loudness of music, and the brightness of stars. •  Trigonometric functions describe  AM/FM  radio waves, the  pattern of the tides, and the daily change  in  the time  of sunset. •  Discrete mathematics provides techniques for calculating  the return on a financial investment, deciding which  mix of products to manufacture, and predicting the  course of a •  flu  epidemic.  Probability theory predicts potential gains  and losses  with car insurance, business decisions, and even multiple choice tests. •  Statistics helps organize and analyze  many types  of  numerical information, such as percentile ranks, sports  data, and advertising claims, that bombard us  daily.  As you see the wide range of fields that use mathematics, mathematics will play a role in your own  think about how future.  Take  full  advantage of this course to prepare for the  many opportunities ahead. I'd like  to offer you the advice that  Math  is  I  offer  my own  5*T^  students:  not a spectator sport!  Don't just watch other people do mathematics! Stay actively involved by doing the activities, participating  in  classroom dis-  cussions and group work, reading the text and examples, and working on the exercises and projects. I  hope you  find  '  the course useful, stimulating, and enjoyable.  Sincerely,  •  s  !L^  j_*  ontents • Functions,  CHAPTERS 1-6  Graphs, and Applications  alcul  pplications Calculate a swimmer's rate of speed or see how the State Conservation Department uses estimation skills to approximate the number of deer in a mountainous area.  roblem Solving Take a closer look at the motion of the through trigonometric modeling.  tide  easoning Create your  something  own argument  proving  either right or  wrong.  is  ommunication Increase your ability to explain a math concept clearly and concisely.  echnology Move  into the wonderful world of mathematics through calculators and computers.  \ Functions, Graphs,  m  m  &**  l&fe Chapter  1  Linear and Quadratic Functions  Linear Functions 1-1 1  -2  3 1  -4  Points  and  Lines  1  7  Slopes of Lines  14  Finding Equations of Lines Linear Functions  and Models  19  Quadratic Functions 1  -5  The Compl 2x Numbers  25  1  -6  Solving Quadratic Equations  30  1  -7  Quadratic Functions and Their  37  Graphs 1  -8  43  Quadratic Models  10, 19  Activities |  Technology '  HH  Graphing Calcu ator/Software  3, 5,  36,  40,41,42 Calculator Exercises  13,29  III/ Computer Exercises  29,36  II  Tests  and Review Chapter Summary  48  Chapter Test  50  ^\  Chapter 2  Polynomial Functions  Zeros and Factors of Polynomial Functions 53  2-1  Polynomials  2^2  Synthetic Division; The  Remainder  and Factor Theorems  Graphs;  58  Maximums and Minimums  2-3  Graphing Polynomial Functions  2-4  Finding  62  Maximums and Minimums 68  of Polynomial Functions  Polynomial Equations 2-5  Using Technology  to  Approximate  75  Roots of Polynomial Equations  2-6  Solving Polynomial Equations by  80  Factoring  2-7  General  Results for Polynomial  85  Equations Activities  62, 63,  v  64  Technology  J  Graphing Calculator/Software  Chapter 3  62, 63, 64,  Inequalities  65, 67, 69, 72, 74, 75, 78, 93 Calculator Exercises  74  //// Computer Exercises  58, 79  !  Historical Development Tests  Inequalities in  3-2  88  Chapter Test  Variable  Polynomial Inequalities  in  Inequalities in  91  3-3  93  100  Variables  Polynomial Inequalities  Two 3-4  Two  Linear  95  One  Variable  and Review Chapter Summary  One  Linear Inequalities; Absolute Value  in  104  Variables  Programming  108 101  Technology  J  Graphing Calculator/Software  Tests  i  1  102, 104  and Review Chapter Summary  114  Chapter Test  115  Cumulative Review 1 -3  116  Careers  in  Planning  Architecture  and Urban  117  Chapter  \  4  Properties of Functions 4-1  Functions  119  4-2  Operations on Functions  1  24  Graphs ond Inverses of Functions 4-3  Reflecting  4-4  Periodic Functions; Stretching  4-5  + *++  Graphs; Symmetry  131  and  Graphs  Translating  *  Inverse Functions  1  38  1  46  Applications of Functions  Two  4-6  Functions of  4-7  Forming Functions from Verbal  Variables  151  ^^^^^^^bbbb^i^^hi  157  Descriptions  Chapter Analytic Geometry  131  Activities  Technology I  Graphing Calculator/Software  1  27,  130, 131, 138, 145, 150, 158, 162, 164, 167 1 1  Calculator Exercises  Tests  1  50  and Review Chapter Summary  1  Chapter Test  Introduction  213  6-1  Coordinate Proofs  214  6-2  Equations of Circles  219  6-3  Ellipses  225  6-4  Hyperbolas  231  6-5  Parabolas  238  6-6  Systems of Second-Degree  6-7  A New Look at Conic Sections  29,  1  1  6  65 66  242  Equations  247 226  Activities  Technology  Chapter 5  Calculator/Software J Graphing  Exponents and Logarithms  220, 223,  1  224, 228, 229, 230, 235, 241  Exponents 5-1  Growth and Decay:  5-2  Growth and Decay: Rational  //// Computer Exercises  Integral  Exponents  Communication  175  Biographical Note Tests  Reading  Exponents  1  169  5-3  Exponential Functions  180  5-4  The Number e and the Function e*  186  .ogarithms 5-5  Logarithmic Functions  191  5-6  Laws of Logarithms  197  5-7  Exponentia Equations; Changing 1  203  Bases  180 ,187  Activities  Technology  Graphing Calcu ator/Software  1  85, 187,  191 3,195,201,205,206,207 ,210 1  //// Computer Exercises Tests  190  and Review Chapter Summary  208  Chapter Test  209  Newton's Law of Cooling  210  Project  ,  242, 245  231  246  237  and Review Chapter Summary  252  Chapter Test  253  Cumulative Review 4-6  254  Trigonometry  •3*  m/./l  Chapter 7  Trigonometric Functions i  Angles, Arcs, 7-  1  7-2 1  and  Sectors  Measurement  257  of Angles  y  263  Sectors of Circles  rhe Trigonometric Functions 7-3  The Sine and Cosine Function:  7-4  Evaluating  7-5  The Other Trigonometric Fund ions  7-6  The  and Graphing  Sine  268 i  275  and Cosine  282  Inverse Trigonometric '  Functions Activities  286 269  Technology  J Graphing Calcu ator/ Software //// Computer Exercises Tests  &£, 280, 285, 290  <  i  274  and Review Chapter Summary  291  Chapter Test  293  *»*».  *. Chapter 8  Trigonometric Equations and Applications  Equations and Applicat ions of Sine  Waves  8-1  Simple Trigonometric Equations  295  8-2  Sine and Cosine Curves  301  8-3  Modeling Periodic Behavior  308  Identities  > V  and Equations  8-4  Relationships  8-5  Solving  Among  More  the Functions  317  Chapter 10  Difficult  Trigonometric Addition Formulas  323  Trigonometric Equations  1  0-1  Formulas for cos (a ±  /3)  1  0-2  Formulas for tan (a ±  /?)  and  sin  (a±0 369  318,319  Activities  375  Technology 10-3  1 Graphing Calculator/Software 306, 31 0, 31 4, 31  5, 31 9,  321  Double- Angle and Half-Angle  298, 300, 303, ,  325, 326, 327  //// Computer Exercises  380  Formulas 1  0-4  Solving Trigonometric Equations  Activities  Biographical Note Tests  386  322  380  323 Technology  and Review Chapter Summary  328  Chapter Test  328  Graphing Calculator/Software  ~]  374, 378,  384, 386, 388, 38 9, 390, 391  //// Computer Exercises  391  Communication  379  Discussion  Chapter  9  Triangle Trigonometry 9-1  9-2  The Area of a Triangle  339  9-3  The Law of Sines  345  9-4  The Law of Cosines  350  9-5  391  Chapter Test  393  359  Chapter  346  Polar Coordinates  1  1  1  -3  Powers of Complex Numbers  1  1  -4  Roots of  1  1  Graphing Calcu lator/Software  334, 338,  Tests  338, 355  and Review Chapter Summary  Complex Numbers  407 412  Technology  364 Graphing Calcu ator/Software  3 97, 400, 401  365 > ?///  Project  Computer  Exerci ses  Biographical Note  The People of Mathematics  403  409  Activities  "1  Chapter Test  395  Geometric Representation of  Complex Numbers  34C ), 343, 344  //// Computer Exercises  and Graphs  -2  Technology  L  1  Polar Coordinates and Complex Numbers 11-1  Activities  Chapter Summary  355  Exercises  Applications of Trigonometry to  Navigat on and Surveying  and Review  331  Solving Right Triangles  Mixed Trigonometry  375  Biographical Note Tests  366  Tests  402 411  and Review Chapter Sl mmary Chapter Te St Cumulative Review 7-1  414 415 416  Discrete Mathematics  and Data Analysis Chapter 12  Vectors and Determinants Properties 12-1  and  Basic Operations  Geometric Representation of  419  Vectors 1  2-2  Algebraic Representation of  426  Vectors 1  2-3  Vector and Parametric Equations:  Motion 1  2-4  432  a Plane  in  and Perpendicular  Parallel  441  Vectors; Dot Product  rhree Dimensions 1  2-5  Vectors  in  1  2-6  Vectors  and Planes  Three Dimensions  446 452  Determinants and Their Application! 1  2-7  Determinants  1  2-8  Applications of Determinants  1  2-9  Determinants and Vectors  Three Dimensions Activities  458 461  in  465 434, 459  Technology  Graphing Calcu ator/Software  '•  434, 437,  1  438, 439 '///  Computer Exerc  Tests  ses  440, 467  and Review Chapter  Si jmmary  468  Chapter  Te St  469  Project  Uniform C ircular Motion  470  Chapter 13 8  Sequences and Series Finite  Sequences and Series  13-1  Arithmetic  and Geometric  473  Sequences 13-2 1  3-3  and Geometric Sums  Arithmetic Their  Infinite  and  Series  486  Sequences and Series  1  3-4  Limits of Infinite  1  3-5  Sums  1  3-6  Sigma Notation  13-7  479  Recursive Definitions  493  Sequences  500  of Infinite Series  506  510  Mathematical Induction  ES1BH  474 480 <  Technology  Graphing Calculator/Software  j  480, 483, 485 486, 489  Calculator Exercises  498, 506  I'll Computer Exercises  499,510  I  499  Biographical Note Tests  and Review Chapter Summary  514  Chapter Test  515  Chapter 14  Matrices  _  Matrix Operations i 1  4-  1  4-2  Matrix Multiplication  1  4-3  Applying Matrices  1  Matrix Addition and Scalar  517  Multiplication  523  to Linear  Systems  530  Applications of Matrices  i  m mm  1  4-4  Communication Matrices  537  1  4-5  Transition Matrices  543  1  4-6  Transformation Matrices  551  Activities  553  Technology  Graphing Calculator/Software  536,541, 542, 547, 550  Biographical Note Tests  ^^:>%m  537  and Review Chapter Summary  560  Chapter Test  560  Cumulative Review 12-14  562  O.RTH CAROLINA  " thS-  ALOHA STATE  Combinatorics 15-1 1  5-2  Venn Diagrams The Multiplication, Addition, and  Complement  Principles  1  5-3  Permutations and Combinations  1  5-4  Permutations with Repetition; Circular Permutations  Mixed Combinatorics 1  5-5  Exercises  The Binomial Theorem; Pascal's Triangle  Activities  Technology  //// Computer Exercises Tests  and Review Chapter Summary Chapter Test  J  Graphing Calculator/Software  //// Computer Exercises  618 618  Communication Reading Tests  623  and Review Chapter Summary  636  Chapter Test  636  9  nrm  Chapter 17 Statistics  Descriptive 1  7-  1  7-2  1  Statistics  Tables, Graphs,  and Averages  Box-and-Whisker  Plots  17-3  Variability  17-4  The Normal Distribution  Inferential Statistics 17-5 1  7-6  669  Sampling Confidence  and  Intervals for Surveys  Polls  Activities  Technology  //// Computer Exercises  Note and Review  Biographical Tests  Chapter Summary  680  Chapter Test  681  Curve Fitting and Models 1  8-1  Introduction to Curve Fitting;  The  Least- Squares Line  ns  1  8-2  1  1  Fitting  Exponential Curves  8-3  Fitting  Power Curves  8-4  Choosing the Best Model  700  Activities  Technology J  Graphing Calculator/Software  684, 689,  690, 693, 700, 705 Tests  and Review Chapter Summary  709  Chapter Test  710  Cumulative Review Careers  in  1  5-1  Genetics and Statistics  71 71  Project  Preference Testing  71  Limits and Introduction to Calculus »  • •  ^  •  •  Chapter 19 ' •  Limits, Series,  and Iterated Functions  •  •  •  Limits Limits of Functions  717  Graphs  726  19-1 1  9-2  of Rational Functions  Series 1  9-3  Using Technology to Approximate the  Area under a Curve  729  1  9-4  Power Series  733  Iterated Functions Analyzing Orbits  737  Applications of Iterated Functions  744  19-5 1  9-6  726, 730, 737, 739  Activities  Technology j  Graphing Calculator/Software 726, 728, 729, 731 Biographical  Tests  ,  722, 724,  745, 746, 748, 749, 750  736  Note  and Review Chapter Summary  748  Chapter Test  749  Project  Chaos  in  the  Complex Plane  750  .:  •  #  <&  Chapter  An Introduction  20  to Calculus  20- 1  The Slope of a Curve  20-2  Using Derivatives  757  Curve  in  764  Sketching  20-3  Extreme Value Problems  20-4  Velocity  and Acceleration  769  774  769  Activities  Communication  763  Visual Thinking Tests  and Review Chapter Summary  781  Chapter Test  782  Cumulative Review  1  9-20  783  784  Geometry Review Properties of the Real  Number System Trigonometry Review College Entrance Examinations Tables  Appendix  788 789 790 800  1  822  Graph Theory Appendix 2  830  The Median-Median Line Appendix 3 Descartes' Rule of Signs  and Bounds  for Real Roots  Appendix  Partial Fractions  Appendix  835  4  838  5  Tangents and Normals to Conic Section: ,842  6 Graphing Calculator Applications  Appendix  Refresher Exercises List  of Symbols  Visual Glossary of Functions  Glossary Index  Answers to  Selected Exercises  846  854 864 866 868 874  pplications Discussion Pic  t.  "When  mam  am  ever  I  10.  variables  m  in  Dis>  It  b.  Skclch some con  ih.n  person pays lor auto insurance  ,i  whai some  s  = N/T 2  /(.. v>  a.  Communications Consumer Economics Economics  .Consumer Economics vested  4  a func-  38.  The  Consumer Economics  39.  Social  CM  What  1  the  ai  are the  About how many years does double your money est' at  '  intcrc-M  \ti'i  Consumer Economics  2.  The graph  various speeds tor 2400  ll  gtven  is  ?  ciency cling  h.ishee  -is  ViOO  shows  ihe tucl efficiency  can. Tne fuel efficiency  is  when  has  ir  speeds  trav  does  as  a  have approxi-  lb car  mately the same fuel Ciency  a  effi-  25 Hli/h?  ai  what  At  C.  lb  have approxi-  lb car  mately the same fuel  '  and 3600  .  what other speed will  3000  lb.  lunc  I  _?_ and  lion of b. At  the right below  at  3000  lb.  miles per gallon (mi/gall  in  Fuel efficiency  a.  interest  1291 inter-  at  '  lake to  it  X'ft  at  compounded annually  2400  a  effi-  car  lb  I  r.Hc uf increase  Research Look Then determine  in  (U  ihe ncarcsl tenth of a percent)  from 1984  |Q  traveling at 55 mi/h?  1988?  ihe average annual rale of increase  do  what  d. Regardless qj  an almanac to find Ihe current consumer once index  weighs, ihc nearest tenth ol a  iched"  from 114.001.  ai i  h>  universiiy  state  to  $18,500  in the  What has been  in  the  average annual growth rale  4  expenses  '  this  ll  what  continues,  penses he 4  L-jrs  \  Research Find Mining  Psychology Transportation  you and Us  25 30 35 40 45 50 55 60 65  maximum  is  luei effici  Yearly expenses  caJ  ..  approximately  ai  what speed  percent! for this index since l°R4. (See E'  41. Education  Polling  shown  the rightmost  = 100(1 OB)  \  equations of the other two curves' C.  Economics The rMUtlflW price inde\. (CPh is a measure of the average cost of goods and services The United Slales government sei ihe indci ji HKi lot the period 1982 1984 In 1988, ihe index was 18.3. What w.is ihc average annual  40.  Science Archaeology Education  is  (or several  i  di  i  The equation of  right  curses  price of firewood lour >c  (he annual rate of increase in ihe  versus  1  consnVni values ol  Statistics  cord.Tmijv acnirfnf wood costs $182 To (he nearest  i'  i  of  Ihe graphs of  b.  ihc nearest percent,  the value of  in-  is  compounded  *  fa later is \ = 100(1 1 The formula shows that A l  Manufacturing Operations Research in 1980 was sok what was the annual ntt ol ap ihc house"  moo  it  rate  amount  Linear Programming  house bought tor 150,000  interest  at  annually, then the accumulated  a.  finance Industrial Design  A  —  =  WRITTEN EXERCISES  Business  To  hi  Duelneee  going to uee this?"  37. Finance  5).  )  domain of Ha,  h) noi in the  Q  R-4,3), and/(0.  curves ol the function  ni  HI  ol these variables  find ,<3.4i.  "v-  COS)  growth will  in  rale  the  ex-  from HOW the  present  that  interests  several  yean  ago Find the average jnnual growth rate Solve. >'  43. a. (4,  45.  £-  Z^  4"  V Factor.  = 9*  =  -  : I  =  15.  Dally Life  81  49. a.  i  1 + ir' - 4<i>  factor out a  ~b  .]  Forestry Horticulture  landscaping Music Plumbing  |.r  19.  1  -4|<3  16.  |..  a3  18.  '  =  20.  |. |.i  y 7|  -  1  23. [4j  Q  K|  4  -4|s 5 + 8|<9  21. |2t  Carpentry n F.xercise 49iai  17.  25. HotltCuHure  > <  J  2| 3|  =  7  -  71  + •  -  22. |3j |34  |6-  24.  -I "  "| 2:  <  lr|  12  Plant experts advise that gar-  denias kept indoor, musi have high humidplenty ol sunlight during the day. and  ity,  4  temperatures  cool  degrees Fahrenheit  C = ^{F  that  sponding  -  The recom-  night.  at  mended nighttime  range  temperature is  60°  express  .12).  temperature  in  F s 65° Given  =s  corre-  ihe  range  degrees  in  Celsius 2*. a. Writing  Recreation  Use  the geometric definition of  and  |.i|  the definition of  ;  bined inequality to wnte a sentence that gives the meaning of the e  Sports  b.  Solve the inequality 2  c.  Solve the inequality  I  < £  \\\  <  |ij  <5  4 and graph  and graph  In Exercises 27-32, solve Ihe given inequality 27.  |  £ \x - 4) £ 3  29.  S  31.  —<  solution  its  solution.  and graph  28. 2  - 7j < 2  (jr  its  2 {Hint Consider two case  I se the following definition of  |.i  t  |  .)  <  solution  its  \x  -  6|  s  1  +  3|  <  30.  0<  32.  7^"7>4  1  5 1  o complete E,ercwes 33-37:  w-L  It  .  if  1  2 <  0. 0.  Solve.  Q33. 35.  |.|  +  Show 1  3)  36. a.  a  -2| =  2  that \ah\  =  |..  >  and h  34. \a\ • \h\  <  f>.  when  and  Give ihrce examples  (4)  investigation plex  number  Use the a. to  -  Chapter  if  >  + 1  h\ 1  £  we  |«|  |«|  and h  and h  mangle  +  + |.-2|>5  >  0: (2)  <  a  and h  > 0.  < inequality  \h\  will define the absolute value of a  com-  V i + v- Decide whether the triangle v + w| a and h are complex numbers.  as follows:  inequah'y holds 37.  In  <  illustrating the \a  b.  a  (I)  a  |  =  2  triangle inequality stated in Exercise _36(a) to prove h\  -  \a\  +  |n|  b.  M  -  \h\  S  Iff  -  h\  tneaualiues  *  Earth Science Conservation  Q  Geography w  52. Geology The Richicr scale  proposed  s  1935 by Charles Richier  in  was  It  Geology Meteorology Oceanography  re-  The Richier magnitude. R.  fined in 1979  of an earthquake  given by.  is  R = 0.67 log <0.37£) + 1.46. £ is the energy in kW h released  where  •  by the earthquake. a.  b.  Show Show  E = 2-7 10'* " if R increases by •  thai thai  *6 "067  '  .  £  unit,  1  increases by a factor of about 31.  53.  If  9  ,v "'  strip  is  multiplied out and  typed on a  is  of paper. 3 digits per centimeter,  about how many kilometers long would  The common logatell you how many  the paper be? [Hint-  WRITTEN EXERCISES  rithm of a number can digits the  number  The San Andieas  has.)  you know about positive numbers a and h  54. Suppose that  a.  Find /<!  fault  In Exercises 1—4. state any ei  =/<oM  /is that /(oft)  a functic  all  b. Prove that fla  |  2 )  =  If la) and/tV)  What  3/(o)  1.  generaliz  A  senator explain-  2.  Prove does  that  this  f(\a) = \j\a) and fC&a)  What  -j-/(a).  Prove  that  fU)  ~f(b).  g.  If/(IOi =  A  (  4.  that satisfies the original equation.  which /To = 2 and /<*)  find the values of r for  =  i  by stating that 60<V of (he mail  bill  bill.  form on which they tan choose one  to mail in a  is  the  choice of the majority of the citizens.  first  radio talk show host invites listeners to telephone the station and talk about  lechngs on a proposed highway to be  iheir  m  Try to find a function / I.  3.  =/(«)  f.  you think might  >  newspaper asks readers  \ liis  charge  =  that  of three ways to finance improvements to the zoo. Based on the responses, the newspaper reports that financing the improvements b> raising the admission  generalization  suggest?  d. Prove that/(^)  B vote  received favored the  suggest' c.  s  situation discussed.  I.  built in their  A new spaper  reporter randomly slops people going "Docs your family use newspaper coupons?"  in a  count)  grocery store am!  .isks  Farmingion High School, there are 360 ninth- or tenth-grade students and 320 eleventh- or twelfth-grade students. A poll shows thai 2 out of 40 ninth-  5. In  3.  1  or tenth-grade students intend to vote for Lahey as student council president  5-7 Exponential Equations; Changing Bases  jImi for  Objective  To  cuuatum\ and  e xponcnliat  sol\  fromo An exponential equation  base is  to  change logarithms  to  6.  another in the  you  In this section,  solve exponential equations involving a  \  Twenty-five percent of a city's employees  who  exponent.  how  will see  live in the city  |  7.  as lime.  employees preferring an increase  Estimate the  Manufacturing  I.  suburbs  if  and  75%  thc>  live in the  employees would prefer  II)  number of years of  light  =  in  in  pay  in pay.  Estimate the percent of  to an increase in bench's  service ol 200 (actors w orkers based  5 where the years of serv ice reported are  on  15. 8. 20, 5. 12.  one hour, a  Life  packaged 150 boxes of  factory  ami Logarithm  s  total  a sample of size n  to use logarithms to  Exr>»  live in the  It  to vote  increases in pay or increases in benefits. Eight of the to city dwellers and four  city  8.  amble such  and 10 who  live in the city  of the 10 suburbanites preferred an increase  These exponential equations are special because both sides of each equation can easily be expressed as powers of the same number Usually exponential equations  my  that  surrounding suburbs The mayor conducts a small sursey asking  an equation that contains a variable  Here are exponential equations you can solve from Section 5-2. -3 2' 9^ = 3^3 = 8  cannot be solved this  24 out of 40 eleventh- or twelfth-grade students intend Lahey. Estimate the percent of the student body favoring Lahey  -hows  Estimate  bulbs.  number of defective packaged  sample of 8  the light  total  bulbs  Science  hour based on a  that  bOXS  containing the  following numbers of defective  9.  .  3a .  Draw two vectors a and  Investigation  -  What do you nonce'  3b.  A  Navigaticn  before  km  -  3(a  b,  km  west from port and then 240  -  must take from port  find the course that a rescue ship  due  in  3.  Ecology  The  1,2,9,  5.  4, 3  to estimate the  number of deer  a mountainous  It  in  On  6  N  west acting on a body  approximate direction  On  Aviation  strength  its  (as a  .  12 arc lagged. Estimate the number of deer  order to reach  at  illustrates a force  Illustrate the resultant  in the area.  number of degrees west of  400 knots  blowing toward the northeast  N  nc  sum of  tb  of 8  Nutrition *  of an  wind velocity of 50 knots  Illustrate a  If the airplane  north) of this force.  that illustrates the velocity  encounters  wind,  this  illustrate its  Estimate the resultant speed and direction of the airplane.  resultant velocity  Physiology  Then, using trigonometry, determine the  graph paper, make a diagram  airplane heading east  Physical Science  iThe direction i> the angle the resultant makes with due north measured clockwise from north.) .  A  a. Sports 2  swimmer  km/h The  river  is  4  leaves point  km w ide  A swimming south across  and flows east  at the rate  of  1  a  mer  ji  km/h. Make a  vector diagram showing her resultant velocity  How  long will  it  take her to  swim  across the  river'*  How  far east  does she  On graph paper, make a vector diagram showing an airplane heading southwest at 600 knots and encountenng a wind blowing from the v Show the plane's resultant velocnv when the wind blows at lai 30 knots. (b) 60 knots, and (c) 90 knots  13. Aviation  14. Navigation  On  graph paper, make a  Astronomy Aviation  b. Calculate her resultant speed. c.  Chemistry Civil  Engineering  Electronics Engineering  vector diagram showing a motorboat  heading east  31  10 knots.  Add  diagram a vector representing rent  moving  (c)  to your  if  the  the current  at (al 2 knots, lb) 3 knots,  and  4 knots.  15. a. Navigation  Make  a  diagram show-  mi 040° followed by  ing the result of sailing a ship 3  on  a course of  sailing b.  it  8 mi on a course of 100".  From your diagram,  estimate the  distance of the ship from  its start-  ing point. c.  Find the exact distance Of the ship  from its starting point by using the law of cosines.  424  Chapter 7VW,,  Mechanics  a cur-  Show  southeast  boat's resultant velocity  moves  Biology  Ecology Medicine  s  Statistic:  graph paper, make a diagram that  a force of  two forces and estimate  .  Animal Science  captures and tags 80 deer  and then releases them. Later n captures 156 deer and finds that  disabled ship.  and  Agriculture  State Conservation  Department wants area.  b>.  Use trigonometry  disabled. Illustrate this in a vector diagram  is  it  Sketch a  7  ?00  ship travels  b.  bulbs: 5,  navigation Optics Physics Space Science Surveying Telecommunications  Thermodynamics  XVIII  roblem Solving "Did you ever try to eolve a problem but  know where to begin?"  didn't  1-7 Quadratic Functions and Their Graphs To define and graph amadrnhc functions  The graph  of the quadratic function ft*)  o( points (i. i) thai satisfy the equation > niu. a in the  curve that can be teen, for example,  =  *hcrr  i  ai~ + hi +  in the  * u  ,j  This graph  i  iv  If  vertical  the axis  and the  yon  include mathematical modeling to help you describe real-world situations.  Chapter 6  rwwwm then » hen you fold the graph aJong this two halves of the graph coincide. The graph of a quadratic function has a nil Dl \vmmetry. or axis. The vertex of the parabola is the pomi where of symmetry intersects the parabola. If a > 0. the parabola opens upward, function has a minimum value. If a < 0. the parabola opens downward,  a graph has an <iuj «y  axis, the  Sections  the vei  a jsu/oh-  is  cables of a suspension bndgc and  path of a thrown ball. Parabolas can also be defined geometrical h.  wifl sec in  and the function has is_  = «JT + ltr +  a  .  maximum  In the figure at the far tight  value  The bigger  the narrower the parabola  \a\ is.  below, the graph of  =  i  Sx*  narrower than the  is  v  i  graph of  ends of a string to a piece of card-  with  board  Make  thumbtacks  sure  the  N  some slack Keeping the vtnnj: draw a curve on the cardboard as shown Describe the curve traced by the string has  taut,  P Repeat  pencil point  moving  the lacks  the experiment by  involved F,ie.O) and  If  set  of  all  points  PF The  .-intercept of  .  parabola with equation  ax  ;  +  bx-i  i\  f2 <-r.0t  m the plane and a  then the  2  an ellipse This  ]  is  P  in the  c  < a,  understanding.  plane such ihjt  the geometric definition of  A hands-on  an  are called the foci of ihc  L-ks  10-4 Solving Trigonometric Equations  two fixed  <  ui.\ In  activity where  On  are the foci and the  mi (or the ellipse with foci F,t<  .  hi.  To use  identities  toward  Earth's  the  onrfnee  FF2  physics, g  '+  is  ally constant  we have  - Or)) 2 + 2 = in V(x + c) 2 + y 2 - U - V(7^ J = 4u - 4oVU - c)' + y 2 + [U V(jt  In theoretical  g.  is  varies  but  A  with latitude  :  =  usually considered  is  However, g  constant.  in  I  called acceleration due to gravity,  often denoted by  not actu-  j.  4a - 4aV(x 2  4a  -AaVfx  c)2  + y2 +  e) 2  +  y  j  z  -  =  For .in  9.78049(  I  example, » - 066V1  0As you  *-  terms of  -= - a \7T  Gl.it/ acting on itt) creates  you  .ind  two methods  8  N,n  live  -  0.000006  Chicago,  in  sui  2  this waterfall  2»)  which  has  a  42  latitude  N  Therefore, g = 9.803* mis' example, some problems involve trigonometric have multiples of angles or numbers. The following suggests  that  this  may  be helpful in solving such equations The  first  method  gives a graphical method using a graphing calculator or computer, and the  Sections present alternative  methods  second method gives  waw  (or solving the equation algebraically  mm  <  MM.  Use a graphing calculator or computet to graph v = f(x\ and y = giti on the same set of axes. Use the zoom or trace feature to find the x<oordinales of any intersection points of the two graphs.  of  Use  the following guidelines.  may be  solving  j  problems.  b. If the  It  helpful to  see roughly  draw a quick sketch of  where the solutions  v  =  /(  *  i  and  are.  equation involves functions of 2x and  i.  transform the  functions of 2t into functions of i by using identities. c.  If  the equation involves functions of 2j only,  solve for 2x directly and then solve for d.  Be  careful not to lose n»ois  when you  il  is  usually better to  t  divide both sides of an  equation by a function of the variable. Review the discussion about losing roots on pages J2  386  Chapter Ten  •V-V)  on 20 - 0.9945.  can see from  equations that  t  '+y 2  0.  :  0.005288  if  .  rwiginf terms:  the latitude in degrees  g  2<  2  by using the following formula. in  +  c) 2  slightly  good approxima-  tion to the value of g can be lound  which expresses g  F 2 t-i.0t. we  0) and  the ellipse, then express PF, and  acceleration of a body falling  J3  .  substitute b~  and  help build  2a  of fin  :  are  a constant.  is  ~PF 2 = "*y,  may have two, one, b — 4ac. we have shown in the diagra  The  get you  apart or closer  farther  together  poind  Activities  Vi  y  2  2 c  r  2 »  lei  you construct an ellipse.  _.-;^  ;.'»:  Q  23. If 4  a zero of fix)  is  24. If 2f  is  a zero of /(«)  = Si3 + fct - 2. = i* + x' + a.  A quadratic  26.  The leading  coefficieni of a cubic polynomial  linear term  is  no  P  polynomial  What  27. a.  -5.  If  PtO) = 7 and  PQ) =  P  and  0. 2.  ±  Find a quartic function with zeros  =  7*  30. If gix)  =  3  31. If /<  = mx +  29. If fix)  i)  + -  depend on  not 32. lf/(  r)  =  r  a. /(9.2)  2, find:  k.  i  show  or h  and h  *  -  /in = 3(l +  -  IK-r  2)?  * 4 b. fix  b. g{x  «<4.25)  value of  that the  - 4)?  in  and -4.  3 and  -/(8.2)  8i. find: a. git.25)  I  3.  3.  are the zeros of the quadratic function  b. Find a quadratic function with zeros 3  and the coefficient of the  /HI = 2n -  arc the zeros of the quadratic function  Find a quadratic function with zeros 2 and  Find a cubic function with zeros  What  is 2,  21. find P(3).  b.  c.  a constant term of 6. and  of P.  c.  28. a.  - 2.  has leading coefficient  linear term. Find the zeros  V %  find the value of *. find ihe value of a.  25.  fi *  +  h)  W  *  + +  \)  2)  3*  - fix) - gix) *  where h  docs  0.  Investigations  Inlerprei this result graphically 0. find the  value of  fjx*h)-/U)  expression independeni of the values of z and  the value of this  Is  help you discover relationships and connections In  for the function of  h.  Exercise 317 33. Investigation  Multiply several pairs of nonzero polynomials  What  the  is  relationship between the degree of the product and the degrees of the factors?  How  can you use this relationship to justify saying that the polynomial no degree, even though all other constant polynomials have degree 0? 34. a. Consider fix)  =  2  x  the  +  2x  following  +  table  of  values  for  the  quadratic  mathematics.  has  function  3.  JY  b.  What pattern do \ou observe in Make a difference table similar What pan f(x) = 2-«" - 3i -  c.  Do  1  the differences in the values  like (hose  you observed  35. Investigation  If  Experiment vUhdifl  (See Exercise 34.)  Chaos  Can you  Complex Plane  the  in  in parts  difference tabic  detect a  you've explored some of the  systems  dynamical  Section 19-5 and  presented  Projects  in  19-6, then you're  probably well aware that iterating a  function— no matter how simple rule  —can  produce  its  provide you with opportunities to explore interest-  unexpectedly  complicated orbits for certain seeds  The  results are  and exciting  even more unusual v,  hen  function  a  domain and range, which were  of  sets  allowed to be ber,  sets  instead.  In  s  ing topics through experiments.  until nov.  numbers,  real  are  of complex numfact,  iterating  function with a complex  a  Discover how patterns can produce exceptionally Intricate and  domain  and range can produce exceptionally intricate  images, like  and beautiful graphic the one shown.  $»  FiltaHn Julia  ol tiz)  = z2 -  Oin  *  0.861'  Materials: calculator with  a computer or programmable  graphics  mode  Complex Orbits Suppose we have a  --:=/!--,  I  :  pnnter that  : is  gives the orbit of  a  c  beautiful graphic images.  a graphics s  (preferablv color graphics)  function /ui where  some complex seed  i  :omplex number. Iterating/!:) using  :n  =/ 2 l--o> For some parts of the following exercises, you may wish  For example, the orbit of /(:>  =  :-  :,, .-,  .-,  :,  :  =  7. a.  is:  =1+2, = 11 + 2il ! = -3 + 4i = (-3 + 4.|-*= -7-24i = (-7 - 24iV = -527 +  lor.  b.  |  calculation of such orbils is besl  At the lop of the neM page  /CI =  2  for  any seed  computer  A sends messages to Ship B. Ship B sends and receives messages from Ships C and E. Ship D sends and receives messages from Ship C. Ship E receives messages from Ships A and C. Draw this network. Ship  M that models this communication network, labeling row*  Write the matrix  and columns alphabetically. 2 Explain what the element Find  M  in the fifth  row, -second column means.  number  ways messages can be  .  d. Find the matrix that represents the  The  to use  software or a calculator that performs matrix operations. 1  e.  is  i  ro-  8. a.  Yog do  not need to calculate  M  rangers,  |  row 4 of  M  sent  are used b> forest rangers to  is.  Make  a  communi-  diagram showing five pairs having  two-way communication, some  pairs having  Current technology  one-was communication, and some having no oonunliniattiOfl Write a matrix to model the communication between the forest rangers.  M  M  Find 9.  some  in  .)  Suppose radios of varying quality  cate with each other in a large national park  LJ  ot  from one ship to another using at most one relay. Reason from the network diagram what the last element  +  W.  What information does  this  network  The diagram below models  matrix give you?  is used to solve complex  real-world problems.  If a  person has only outgoing arrows,  then that person  a transmitter of the  is  rumor A perNon having only incoming  A  i  the  I  r  person having both outgoing and  incoming arrows  is  a rela> point for  the spreading of the rumor. a. Identify the transmitters, receivers,  and relays for b.  How  this  tify  10. Let  network.  can you identify a transmitter  by looking at rumor matrix?  the  corresponding  How  can you iden-  a receiver?  M be a matrix that illustrates com-  munication operators  among If  M~  several  Ham  radio  contains no zeros.  what can you conclude 1  easomng What do you  haulm; company needs  can be moved along  traitor  highway  a  to the top  the trailer  If  of the  9  ts  w>de and  in  n  traileri. will  a large house  passes under a bridge with an  m wide at (he  2  1  w nether  determine  10  thai  opening in (he shape of a parabolic arch. the center  Vrove it to me!" You need to decide!  A  42. Trarnaorrjtioi  think?"  base and  ft  mult measured from  3-2  m high  in  the ground  I  under the bridge  fit  9  8 Quadratic Models Wnbng  To model real-»or1d uruatiois usint quadratic functions  and 16 you made a prediction Which of these predictions do you think has a better chance of being correct? Write a few leniences explaining y a  I?.  In each of Exercises 15  The table snows statistic* for the ten players of a college basketball team opening same of the season. I se the data lo complete Exercises |»->0.  "  HMMtx pined 1  Hi  III  fa*  1  II  1  1  25  21  II  16  13  ;:  10  15  II  1  6  4  1  HI  4  6  .'  5  2  3  2  :  1  21  musts are fascinated by the ability of birds to travel great distances without ipinc L sing wind tunnels to monitor parakeets' oxygen consumption and cardiosidc production in flight, they have investigated the rale at which parakeets  The data in the tabic below show the number of number of calories needed to move each gram  end energy  in level flight.  met burned »dj weight  per  gram hour  (the  for  one houn  for three different (light speeds.  ii  1  we want lo predict the bird's energy some other flight speed, we will want  If I  I  Lse compuler software to  make  a scatter plot,  at  draw  ind a good mathematical model baaed on  the least-squares line, find  The information  kc data.  equation, and find the  its  bed on  each  ended decreases arty  flight  is  Decision  making  19.  minutes  1  <  can find many kinds of  ^  138"  through these data points  £  12ft"  ;  H°  We is  One  would not be  more  possible curve  21. Writing  Refer no the results of Exercise 20. Docs the positive correlation  between fouls committed and points scored suggest a cause -and -effect relationship between fouls and points* As a coach, would you adsisc playing more defensively and thus committing ers' potnis vcorecP  Q  22.  Wnte  a short  more  fouls as a  way of  a. b.  suita-  which  called cur\e fining,  and  n the graph into  c.  '00  Chapter  detail in  "*'_  a parabola with  is  * pes of  ISO  increases.  ISO  thai pass  process  then  -j  will discuss in  fouls  and  a linear model  to use.  m  r<<  speed increases, the energy  the  set of data.  in the tabic is  the graph in figure (ai. Note that  correlation coefficient for  To  c  find the  l(  ;  substitute the data  shown  this equation, as  the next page.  increasing your play-  paragraph explaining your reasoning.  The diagram at the right shows a waiter plot of a set of data and a line with slope  m  that contains the  pomi  For each point point  and the  **,.»,». fci d, represent the vertical distance  line. tHu  Skom  random-number  13. In the  between the  0-9. occurs  that:  42M3  02431  69414 89353  18519 17889 33584 70391  96960 19620 Tan* 67893  29|H D5063 23218 72452  92900  go)  ;..,=  (0333  For example, to use Monte Carlo simulations  "heads"  in 3 tosses  of a coin,  random-number  a.  Of  the  ft  I  heads' '  =  )  b.  Begin with the  m j al c.  How many  1  Iff  d.  ulaunns  of a  comes  lies  the digits  using  In order  in  fi  nd the probabitily of getting  Monte Carlo  c  t  Bod  i  1  fae  0-9 be  assigned to the possible out-  far each exercise  some  1  iWutr  may be  It  Cross off  digits  all  before  in the table i  Perform Monte Carlo simulations 2 and Exercise 7  you  si  pttibmfA  how should  Exercises 2-12.  actually doing a simulation  Compare  for Exercise  the probabilities  find to the theoretical probabilities found  using the binomial probability theorem .  4  Sports  A  cent  750.  is  basketball player's free-throw per-  What  is  the probability thai she  scores on exactly 4 of her next 5 free .  45JS69J I44J54  "heads " Thus, based on 16 Simula-  1  occurrences of those digits  math  l|7imj9  4J704I II5JI9  the probability of getting  sec Exercise  i  asked far  helpful to eliminate  .  1  "he:  '  '  'heads.  Based on these  i  1 always improve the accuracy of the approximation' Explain  Discussion  validity  conjecture  "tails  16 simulations of 3 tosses of a coin.  Discusston When you use Monte Carlo simulations to approximate the theoretical probability of an event, will increasing the number of simulations  .  37260  digits of the first  30 si rnulations.  is  13751  0—1 with the outcome " Then the first row of  simulations can be done using the whole table  simulations, what  Diecuee the  2  last  45869 14454 64906 15021 54268 13433 21709  25 Compare this probability to the u^eoretical "heads" in 3 tosses of a coin (see Exercise l(d» row and perform 14 more simula-  0.3  probability of getting  22551 23214  to find the probability of getting  outcome  8|7S2|2  simulations. 5 produce  P10  lions,  the  table represents  2«|66?| 620(58  5J280J2  each digit  associate the digits  first  5—9 with  "'heads" and the digits the  494}*?  that  can a-' fiV table to perform  87822 I47C1 86056 62812 86864 55808  25252 97738  Chapter Etfhteei  the probability  We  0.1  is  order to find certain probabilities empirically  in  20667 ansa 4631? 14(03 11106  494S7 52101 29480 91539  690  shown below,  table  any given position  in  Monte Carlo simulations  Discussion theorem  to  complete part  (ai  assumes  probability of a successful free throw I  t  Using the binomial probability  assumption  s  slid  I  is  thai the  always  Follow a  argument  \oq\ca\  Suppose  thai  a  small  TV  manufacturing  company produces console and  portable  TV's  using three different machines. A. B. and C. The  shows how many hours are required on each machine per day in order to produce a table below  console  TV  or a portable  TV  These requirements for the different machines can be described by the following inequalities, where x represents the number of console TV's  and y represents the number of portable TV's:  { The number '  Machine  A  of TV's cannot be negativi  needs  hour for each console  I  each portable Thus,  Machine for at  (The ,  4x +  ;  i  24  A  needs x  +  for  i  most 16 hours a day.  last  chines  two  TV  x  +  2y  s  2  hou  have  solutions to any  to satisfy all five inequalities.  the region that  procedure  is  is  problem subject  The  =  log*  b.  easiest  way  to these constraints  would  to find these values is to find  inequalities.  the  graph of >  =  log„  related  (  graph of  the  to  x.  Check your prediction by graphing y  A  step-by-step  =  nd y  I  =  log  (  —  )  on a single  set of axes.  r  34. a. Predict  formed by the graphs of the system of  shown below.  v  one above,  are available for at most 9 hours and 24  respectively  The possible  how  35. a. Predict  is av,  16.  inequalities are similar to the  B and C  and  console TV's and y portable  2y hours. Since this machine  b.  how  the graph of y  =  log* i~  Check your prediction by gr-phing  .  is  =  related to the graph of y log  i  and y  =  =  Make and  log,, *.  2  log x on a single set  teat conjectures  ommunication "Have you ever had trouble explaining something because you didn't fully understand it?  CUSS 1.  EXERCISES  As  Example  in  If  suppose  1.  drawn  a card is  queen of hearts?  a. the  d. a red card?  e.  are rolled, what  two dice  are rolled, find (he prob-  If  b. a c.  a  If  tomorrow  one  least 7.  Discussion  In the solution of  5 =  why  A  Does  the set 5  "beads"  9.  A  pan  (al  card  is  5 =  ability to in  10.  medical research, and the methods of data analysis play an important role  1  sums  Example 3. another possible sample space for the two droG  3.4. 5.6. 7.8.9. 10.  (2.  possibilities  1  i*  11.  is  12}  correspond to a sum of 9 or 10. some people  to say that the probability  reasoning  tht*  of a  sum of 9  or 10  is -t  Discus*  incorrect.  -  1.2.  (0.  come  could  thai  3).  which gives  the four different  thai since  one of the four sample points  corresponds to 2 "heads." then P(2 "heads") picked  at  random from  a standard deck.  {club, spade, red card, face card)  Comment on  numbers of  up. satisfy the definition of a sample space?  wrong with reasoning  b. V. hat is  our  is  m  penny, a nickel, and a dime are each tossed.  a.  and Statistics  show the same number  manu-  the probability ot  is  Since (wo of these  in  in a  one month  the set of the 11 possible  8.  f. a red face card  probability that both  accident''  might be lempted  of stud> contribute  the  prob-  If the  of no accident*  0.82. what  field*  is  the probability of  facturing plant during  Mam  whai  cards,  a queen  c.  is 40%. what is the no nun tomorrow ?  OcOMattonal Safety ability  4  me  is  1  sum ol 3 sum of*4 sum of 3 or 4  a. a  •w  understand, control, and curt disease  Pincgauvc number)  c.  of getting:  ability  Discussion  The twentieth century has seen unprecedented progms  each probability  60)  a face card?  two dice  4.  probability of  in Genetics  b. 1 heart'  3. If  rain  Careers  b. /'(factor of  random from a standard deck of 52  ai  probability of getting  5. MtttorolOfly  Further your understanding of math by discussing it with others.  a die is rolled. Find  square)  a. /'(perfect 2.  is  in  = —1  Explain why  the  Id  nor a sample space.  the following reasoning  whose names begin with the letter C (California. Colorado, call them "C-suies " Likewise, there are 3 "O-states" if a person is chosen at random from the U.S. population, that person has the same probability of being from a "C-  There are 3  states  and Connecticut),  (Ohio. Oklahoma, and Oregon) Thus, Genetics, the study of heredity, encompasses a  _Jf »i of research. Examples ^Pucb uses rfffTrrnnf analysts  number  ot  staie" as from an  "O-tMe  "  include population genetics. to find  -4tnd (he occurrence of  m gene  partem*  di-.tr*-  genetx diseases: microbial  which explores the process of mutation by study  Reading  multiply ing rnsowjrgaoisms: and cytogenetics. fcok*. at hereditary activity at (he cellular level. I  Stroud-Lee a University of Chicago Ph.D..  ne research  Learn about a variety of careere that uee  in  cytogenetics at the Los  Alamos Sa-  Laboraiory. B> observing the effects ot radiation and  aenucals on chroniosotnes. she sought due* to ablitic* in genetK material Her results have improved iJerstanding of birth defects  "  s,  she say s.  tier  their pursuit  and cancer  '  '1  have made  Find two angles, one poMtite and on* negatne. thai are culermlnal «ilh each  other scientists can use my work of science " Ms. Stroud-Lee retired  mathematics.  max experts •  ey*.  in the  and questionnaires used  ful data,  and  in the  design of experiments, *ur-  to generate accurate  Q  and use-  meaningful analysis of data. Their  tant  Statistic* i*  an impor-  d  «*  d  T  c  19. a  23.  180*20'  Give an expression are coterminal with  24.  Joe Fred Gonzalez. Jr.. a mathernaticaJ statistician whose work includes developing estimation pro-  sire to  b.  116.3*  b.  -WW  b.  -270*30'  r  -60.4'  d  -315.3°  c.  -50.8*  d  -320.7*  c.  3*21  d.  115*15'  c.  11*44'  d  in  rmr  terms of (he integer n for the measure of  terms of the integer n for the measure of are coterrrunal with an angle of - 1 16'10  Gi*e an expression  all  angle*  in,  all  angle  th,  an angle of 29.7". in  Each of Exercises 25-30 gives the speed of a revolving gear. Find <ai th number of degrees per minute through which each gear turns and <b) lb number of radians per minute. Give answers to the nearest hundredth.  intellectually stimulating."  His de-  understand application* of mathematics led him to an  M.S. degree ersity.  *•«  -100°  NCHS  math "challenging and in statistics  from the George Washington Uni-  Believing in the value of education. Mr. Gonzalez  irnc university  -60*  b.  22. a.  cedures and national sample survey designs, always found  Fnd Gambia A  b.  1000*  source of information for researchers and policymaker*  care used, nutrition, exposure to nsks. and age.  -  500*  18. a.  21. a. 360*30'  N'CHS analysis of data reveals connections between such v aned health factors as socioeconomic status, type of health  for  angle.  17. a.  2». •. 38.4'  skills  are indispensable to medical and health research.  The National Center for Health  gnen  is  25. 35  also a pan-  31.  Coreers  in Genetics  and  Statistics  rpm  713  rpm  2*. 27  rpm  28. 6.5  and college professor.  29.  14.6  27. Z 5  rpm  34).  rpm  19.8  rpm  Reading On page 257. you were told that when a car with wheels of radius 14  in  turn  i*  driven at 35 mi/h. the wheels  approximate  an  at  420 rpm. Show how  of  rate  to obtain (hi* rate  of turn.  Suppose you can  32. Recreation  Research  ride  a  bicycle a distance of 5 mi in 15 nun.  If  you nde  al  a constant speed and  if  bicycles wheels have diameter 2 7  the in  .  find the wheels' approximate rate of  turn tin rpm).  Investigate interesting facts  4  and app\\ca~t\or\3 of mathematics.  33.  Research Con*ull an encyclopedia or an atlas map are located by using latitude and longitude  to see  how  points on a world  coordinates given in degrees.  minutes, and seconds. a.  If  you  from a given point on Earth, about how many miles do 9 go to traverse an angle of I ? if you travel west  travel south  you have b.  to  Explain why your answer io pan (ai might be different instead of south.  34.  262  Research Consult a book of astronomy or a star atlas to sec how stars on celestial map are located by using angles of right ascension and d Describe how each of these angles is measured, and give examples  Chapter  Snen  a  XXIII  22.  Geography  The mamed.  ever  Aje  table  below  pvw  givi  20-24  25-29  30-34  35-44  45-54  39  71  84  92  95  2  22  57  75  89  94  ever married  At the right a  populatic  6  mimed  Percent of males  a.  US  15-19  Potent of females ever  data on the percent of the  I  pan of  is  comparative histo-  two  showing  gram  ban  side-by-side  each  of  for  age  the  Complete  groups.  the histogram.  Writing  b.  Write  paragraph  Writing  a  explain-  what  ing  20-24  15-19  data  the  show.  The  23. Nutrition  below gives data on the recommended energy  table  kilocalories (Call, for average females  A*eero«p  (in  Cal  for  i  Draw  math  19-22  23-50  5,-75  276  2200  2100  2100  20O0  1800  1600  2700  2800  2900  2700  2400  2050  males  for  each of the age groups. ( See Exer-  22  for an  example of such  Express,  skills.  in  your own words, the meaning and understanding of concepts.  histogram  showing two side-by-side bars cise  and your  skills  writing  females  comparative  a  Bring together your  intake,  age groups  15-18  Energy needs  a.  in various  ,,-H  Energy needs (in Call for  and males  a  histogram.) b.  Write a paragraph ex-  Writing  show  plaining what the data 24. For a group of  10 teenagers, the  mean age is 17.1, the median is 16.5. and the mode is 16. If a 21-year-old joins the group, give the mean, median, and mode for the ages of the 1  people.  WRITTEN EXERCISES  Exercises  For Exercises 1-4, suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of the following.  [J  1. a.  a black card  b. a  2. a. a black face card a.  a red  4.  a.  a jack  5.  Mr and  tickets. If  4280  winner  = x"\  find/'U).  b.  For what values of  c.  Sketch the graph of /(.*).  may need  a.  x is f'{x) (If  undefined?  you use  7.  computer or graphing  a  to enter the rule for the function as ix')  how  graph.) Explain  the function fix)  =  x  ifi  to obtain the  b.  of the integers between  1  even?  complete  fix)  =  4x>  32. fix)  =  x''  35. H\x)  Q  = 5-r 4 = lVx  Bronx,  37. a. Use the binomial theorem (page 591) to wri  term  last  b.  Use pan  expansion of  in the  (a)  U  +  i  =  36. h'(x)  = 3G  the first three terms  a.  and the  and the definition of fix) to prove Theorem 2 on  b.  39. Prove  Theorem 4 on page  Use  760. iHini  the definition of /'( the definition  what  is  the probability that  and 20. inclusive,  1  is  picked  c at  to explain  none of the S  random What  Interesting  is  is:  by 3?  c.  problems.  a prime?'  di-  is  and  Brooklyn,  Suppose York City  number  Explain  why  ability  is  that  the probis  it  a  telephone  number is not ~. What do you need to know in order to find in  8.  Theorem 3 on page 760. (Win* Use  in all,  the correci probability i  2.)  38. Prove  and other subjects  Island.  New  a  Manhattan  positive integral values of n.  Theorem  knowledge of math  it  randomly chosen.  3X  n)".  (Note The binomial theorem can be generalized to apply to any number n. This form of the binomial theorem can then be used general proof of  City  telephone 5  33. g*(jr>  jack  Combine your  Manhattan. Queens, the  Staten  .  In Exercises 31-36, find a function thai has the given derivative.  34, g'ix)  c. neither  one of the 5 Smiths?  b. divisible  New York  that  31.  not a black ja  not a black di  vided into five boroughs:  the graph of fix) supports the result of part (b).  29 using  30. Repeat Exercise  you  calculator,  were sold  tickets  the probability that the integer  29. a. If fix)  not a spade  c  c  is:  Mr. or Mrs. Smith?  One  diamond  b. a jack or king  the grand prize  6.  b. a black  Mrs. Smith each bought 10 raffle tickets. Each of their three children  bought 4  a.  c  spade  b. a black jack  diamond  3.  pan  Suppose  (a)?  that a  member  of the U.S. Senate and a  member of the  U.S. House of  Representatives are randomly chosen to be photographed with the PresidenL  i).)  Explain  of fix).)  why  ~  is  the probability thai the senator  not the probability thai the representative  is  from Iowa, and why  ^  is  from Iowa.  is  In Exercises 9-12, use the table on page 601. which gives the 36 equally likely s  The saying "a picture is worth t thousand words" is often true inn maocs One well-drawn diagram may be a very convincing a  9. a  10. a.  id own. without being a formal proof Diagrams and sketches c may be true, and they can offer clues d  evidence that a conjecture  Far example, conasdet the diagram at the right where the lengths a and b arc  1.  bow  Explain  areas  the  of  (« 2.  +-  you 3.  b)  Explain  By  2  -  (a  how  -  the  to conclude that  b)  1  +  from  4ab.  the  diagram  a  greater than  that (a  +  1  b)  >  its parts'*  allows  4or>.  taking tie square root of each side of the inequality above, what  relationship  is  b.  even.  11.  The two dice show  1?  The  red die  b.  Sum Sum  c Sum  is 7. is  c.  12.  Sum  is 8. is less  than 12.  different numbers,  shows a greater number than  t  k  white die.  Probability  the laying "the whole  to conclude  dice are rolled. Find the probability of each event  is 6.  Visual Thinking  squares and rectangles in the dia-  gram allow you  when two  Sum Sum  do you discover between m. Vab. of a and  the anthrnetK  W  mean,  —-—  ,  and  ••*•  Learn to  communicate through a diagram or picture.  visually  603  XXIV  echnology  Polynomial Equations 2-5 Using Technology to Approximate Roots  "Technology enhances understanding, encourages exploration, and opens the door to many new and fascinating applications!' Technology  MtMtKtonug A  I  long  shown a.  its  width  to  is  You  already  know how  will  dtacUH  boil lo solve a  lo solse linear  approximate the real roots of a poi\-  Eiample  1  Solution  and quadratic equations  some polynomial equations can be solved Solvf  Draw  r  i  - S* + 2 = Obj  hy factoring:  the graph of Ihc polynomial function  3  r  -jTa  the roots of the equal ion the graph  II  -ir + 2  estimating (he  not labeled, you could use the  i-  i  -intercepts ol the graph arc  intercepts  is  I  volume  Hmr Volume =  /iM>M  I  of the cutter as a function of  one of the equal  feature,  if  available.  i.  From  sides.  (he  i  tolumc of  the gutter a  the graphs of y  the roots of i'  =  x~  — x — Sx +  - x 2 - Sx + 2 =  2 shown above, we estimate  are x  = -2.  maximurr example, we  out drawing a graph.  Part (hi of Exercises 25- 2V requires the use of a computer  From a.  If  a raft .SO  bar 100  m offshore, m down the  the lifeguard  mmg  technology suggested to get the full benefit of studying the lesson  We  will solve the  equation  will instead use the lot ation  .i  -  prim  «*  -  5r  x  = 04. and  +  2  =  iplr. stated at the  swims  a lifeguard wants lo  beach, as  at  and running time  I  /  shown  m/s and runs  swim  to shore  at the left  at 3  below  m/s. express th  as a function of the distance  diagram  | Uae  a computer or graphing calculator to find the  minimum  or exercise. Ex.2*  Ex. 25  A  24. Engineering  60  m  power station and a factory are shown at the nghl above A cable  wide, as  station to the factory  li  costs  $25 per meter  to  r  opposite sides of a river ist  be run from the power  the cable in the nver and  $20 per meter on land a.  indicates specific exercises for which  technology necessary.  total cost C as a function of i. the distance downstream from power station to the point where the cable touches the land a computer or graphing calculator to find the minimum cost  Express the the  | Use  27. landscaping sides, as  A  rectangular area of  shown The  made of a.  is A  Use  The  m  2  has a wall as one of  its  made of  side parallel to the wall  is  decorative fencing that costs $8 per meter.  Express the length  | 28.  60  sides perpendicular to the wall arc  fencing (hat costs J6 per meter.  advanced  a  i  total  com C  of the fencing as a function of the  of a side perpendicular to the wall.  computer or graphing calculator to find  cylinder  is  inscribed  in  a sphere of radius  the  minimum  I.  J  a.  Express the volume  | Use  Functions  I  of the cylinder as a function of the base radius  r.  a computer or graphing calculator to find the radius and height of the  cylinder having  164  because  m  wide and 10  area of trapezoid X length of gutter)  For what value of  mack  difficult  TRACE feature, if available  ft uprose the accuracy of the approximations, you could rescale or use the is  calculator.  indicates advanced  we  using a computer or graphing calculator  with a computer or graphing calculator. The  if  In this section  polynomial equation of higher degree, such as  a graphing or programmable calculator In ihc nest section  jt I  In the next  ~1  u>  nomiat equation  form a gutter with a cross  the length in centimeters of  b.  on  To uif icihnnti'iiv  ]  at the right  fcxprcss the  I  60  Polynomial Equations  [flB/Wtftf  an fftfttfJ trapezoid with I2tf angles, as  section that  Logos  lheet of metal  hem along  li  of ^J  maximum  volume.  Chapter Four  Statistics  Volar Graph.5  with-  top of  .  -  that  2  h  "^  XXV  1  *  dcuU* 5  Frecalculue ?\otter Plus The diagram on placed  ai the  ihe previous  measured  can be used as formulas  Many  scientific  Example 2  page  b.  posmvi  counterclockwise direction. The gil  in the  converting from one coordin  ("or  and graphing calculators have a.  buil(-u  Give polar coordinates for the point (3.4). Give the rectangular coordinates for the point 3?  B= * =  tan  (3.4) ray.  4  Thus.  Quadrant  is in  and so  r  =  5.  I  + 42 = *  If  =  !  possible  same  B  = rcos30 = 3cos30 andy = rsin30°  mode. Many calculaiors use  If  (5.  point  • Use data to create histograms, scatter plots, and regression  | i  = (  lines of  graph of a polar equation can be drawn by a graphing calculaiorl  described below, provided the calculator has a polar  If the  1  Therefore, polar coordinates are  o  1  instead of  mode  data.  the calculator has parametric  r  =/(/)  x  = rcosr =/(')l = r sin, =/(,)?  v  The two equations in greater detail in  for  1  and  Chapter  1  are called  v  2,  parometru equations  where ; often represents time  = 2 tin  Example 3  Sketch the polar graph of r  Solution  With a graphing calculator In polar  We  will study  them  • Test statistical  1  in a practical probler  hypotheses through sampling experiments.  26.  mode, enter  &££«*.>, In parametric  mode, enter  5  • Explore sequences and series.  v=:*,n2l»n. The  spreadsheet  or a paramefl  ft  mode, enter the equation: mode, enter the equations:  calculator has polar  and  plot the graphs.  pomi isl  53.1°. then the  Thus, rectangular coordinates of the point are  BThe  • Enter equations 1  (3.  5  l*53.l 8  possible pair of polar coordinates of the b. i  Graphing software that lets you  illustrates the c  origin with the polar axis coinciding with the  positive angles  following  values  were used for the graph of r  =  2 sin  s ,  2.1  •  shewn:  2tt. step:  f  2. scale:  I  2. scale:  I  Ferform matrix operations.  t  -2<x< -2sy<  • Check your  Polar Coordinate* ana" Complex Numbfi  understanding by determining equations for displayed graphs.  Inequalities  Matrices  Trigonometry  These pages  will introduce you to the basic features of most graphing calculators. Because of the variety of graphing calculators available, specific keystrokes are not given. Refer to your calculator's instruction manual for details.  Setting the When  RANGE  Variables  using a graphing calculator to display graphs, think of the screen as a  On many calcuwindow uses values from -10 to 10 on both axes. You adjust the viewing window by entering values for the RANGE variables, which appear on the screen when you press the RANGE key. "viewing window" that shows a portion of the coordinate plane. lators, the  "standard" viewing  RANGE The x-axis for  -2 < x <  The /-axis for  will  4.  will  -3 < y <  fXmin = -2 IXmax = 4 Xscl = 1 <r fYmin = -3 lYmax = 3 Yscl = 1  be shown  be shown 3.  With scale variables set to equal 1 tick marks will be 1 unit apart on both axes. ,  Graphing a Function To graph a function, enter its equation and set the RANGE variables for an appropriate viewing window. (You may need to experiment to find the best viewing window.) Here, the graph of the cubic equa3 tion / = x 3x 2 + 2 is shown using the viewing window described above. Note that the scale labels shown here, and on similar diagrams throughout the book, do not actually appear on a calculator display.  " l\  Function form equation to be graphed must be entered in the form y = ..., that is, / must be expressed as a function of x. For example, before graphing x 2y + y = 4,  An  first  solve the equation for /. Enter the equivalent equation  Using parentheses Be careful when you enter an equation Enter  The second equation  XXVI  Y = 4 + will  (X2  +  like  1),  the  —  .  xl +  1  one discussed above.  not  be interpreted as / =  / =  Y = 4 * X2 +  —  +  1 ,  not as  1.  / =  x2 +  1  Try This and graph each equa fion, using an app rop Mate viewing window. You may need to solve for/ first.  1. Enter  -/  = -8  calculator often look distorted. Adjusting  some  a. y = x 2 + x - 5  b. 6 + 2/ = x  c.  Ixl  Appearance of Graphs Graphs displayed on a graphing of the settings  on your calculator may improve a graph's appearance.  Squaring the screen  A  square screen  is  a viewing window with equal  Standard viewing window  unit  spacing on the two axes.  Square screen window  \  f \  Here, the circle x 2 +  y  2  = 25  appears stretched horizontally.  J  Displayed on a square screen, the  same graph  is  undistorted.  On many about 2  make  to 3.  graphing calculators, the ratio of the screen's height to its width is For a square screen, choose values for the RANGE variables that  the "length" of the /-axis about two-thirds the "length" of the x-axis:  (Ymax - Ymin) ~ ^-(Xmax - Xmin)  Connected mode; graphs with asymptotes your calculator is in connected mode, the individual plotted points on a graph are joined by line segments. As a result, graphs often look jagged. Also, the separate pieces of a graph with vertical asymptotes may appear to be connected. If you take your calculator out of connected mode, only points on the graph will be plotted. This may give you a better sense of the true shape of the graph, although there can be iarge gaps between the points.  When  XXVII  M  H ZOOM How many  graph of / = x 3 - 6.2x 2 + 9.6x + 0.05 (shown below) have between x = 2 and x = 4? To answer this question, use the calculator's ZOOM feature to enlarge the section of the graph near the point (3, 0). On many calculators, you can do this by creating a "ZOOM BOX" around the point of interest. The contents of this box can then be drawn at full-screen size. x-intercepts does the  Standard viewing window  ZOOM-BOX window  m ZOOM BOX  -\J 4  3  Y = -.47619  \ = 4.31579  Now you  The coordinates of one corner of the  ZOOM BOX are displayed. On  can see  that the  graph  has no x-intercepts near x = 3.  most graphing calculators, the  ZOOM  window. Many calculators  feature offers several  ways  to  "zoom-in" (show a smaller portion of the coordinate plane), or "zoom-out" (show a larger portion of the coordinate plane) on a point you select, changing the RANGE variables by facadjust the viewing  tors that  details  on  you  specify.  ZOOM  will  Consult your calculator's instruction manual for specific  procedures.  Try This 2. Use the fourth-degree equation y = 30x 4 +  1  22x 3 - 3x 2 - 492x.  a. Graph the equation using a viewing window with -4 < x < 4 and  -500 b. Use  <y< 1000. a ZOOM BOX  to  enlarge  the graph's "flat" section. For  an even more detailed view of this portion of the graph, set Ymin = 450 and Ymax = 500. Describe the shape of the "flat" section of the graph.  XXVIII  X  = -1.22105  Y = 404.761  U^ing a Graphing Calculator  TRACE TRACE feature. When you appears on the graph. The x- and /-coordinates of the cursor's location are shown at the bottom of the screen. Press the  After a graph  press the  left-  is  TRACE  displayed, you can use the calculator's key, a flashing cursor  and right-arrow keys  to  move  the  Finding a point of interest You can use the TRACE and ZOOM  TRACE  cursor along the graph.  features to find the coordinates of a point of  interest on a graph, such as an x-intercept or a high or low point. Consider the graph shown in Exercise 2, which has an x-intercept between and 2. To find the coordinates of this x-intercept, begin by pressing the TRACE key. 1  Move  TRACE  the  cursor to a  point just below the x-axis.  The /-coordinate of is  X=  point  Y = -57.660  1.642  Now move  the  a point  above  just  TRACE  is  cursor to  the x-axis.  The /-coordinate of  X=  this  negative.  this  point  positive.  35.808  1.726  Somewhere between these two points, the graph must cross the x-axis where / = 0). Therefore, the x-intercept is between .64 and .73. 1  "Zoom  (at  a point  1  on a point near the graph's  in"  Move  x-intercept.  the  TRACE  cursor along  100  below the x-axis the graph until when x ~ .695 and then just above the x-axis when x ~ 1 .697. it  is  just  1  X=  When  rounded  to the nearest  Y=  1.27425  hundredth, the two x-va!ues mentioned above are .70. If you wish, is x = by repeating this process.  the same. Thus, to the nearest hundredth, the x-intercept  you  1.69684  can increase the accuracy of this approximation  1  XXIX  Finding an intersection point of two graphs You can use a similar process to find the coordinates  of an intersection point of on one graph near the intersection point, note the value of the x-coordinate. Then press the up-arrow key. This moves the TRACE cursor to the point on the other graph that has the same x-coordinate. Compare the /-coordinates of the two points. You can "zoom-in" and repeat this process until the /-coordinates are the same to the desired degree of accuracy.  two graphs.  When  the  TRACE  cursor  is  Try This 3. a. Graph / = 2~ x - 5, as shown at the right. Find the x-intercept of the  b.  graph  to the nearest tenth.  Add  the  the  same  graph of / = 2x  to  set of axes. Find the coordi-  nates of the intersection point of the  two graphs  to the nearest hundredth.  Solving Equations with a Graphing Calculator Finding approximate solutions to equations that are algebraically  is  difficult  a powerful and important use of a graphing  or impossible to solve calculator.  Using an x-intercept to solve an equation To solve the fourth-degree equation x 4 - 6x + 4 = 0, use the graph of the 4 related fourth-degree function / = x 6x + 4.  For any point on the x-axis,  / =  0.  Thus, each x-intercept of the function  / = x 4 - 6x + 4 is  a  solution of the equation  x 4 - 6x + 4 = To the nearest  You can use the x - 6x + 4 =  XXX  tenth,  one  solution of  0.  x 4 - 6x + 4 =  is  x =  1.5.  ZOOM and TRACE features to find both real solutions of the equation to  any desired degree  of accuracy.  i  sing a Graphing Calculator  Using an intersection point to solve an equation To solve the radical equation vx + 3 = 7 - x, use the graphs functions  y =  v  +3 and y  x  The coordinates of the (a, b)  must  satisfy  — 7 -  drawn on  x,  the  same  of the two  set of axes.  intersection point  both equations:  b = Vc7 + 3 and b = 7 - a  Va~ + 3 = 7 -  Th us,  a.  Therefore, x = a equation Vx  Using  is a solution of the = + 3 7 - x.  ZOOM and TRACE, you can  find that the solution  is  x = 2.44.  Try This 4. For each equation,  find  all  real solutions to the nearest tenth,  = x 5 - 3x 2 + 3  a.  5. Use a graph  to  b.  Vx  + 5  =  |  x  l  determine the number of real solutions of  x + 4 = -x  4  + 3x 3 -  1  equation.  this  .5x + 5  Graphing Parametric Equations In  the seconds after a baseball  hit,  zontally in  is  moved both horiand vertically, as shown  the ball has  — baseball, after  the diagram.  Instead of using  one equartical  tion to describe the path of the  you can use two equations, one to express x in terms of t (the time in seconds) and one to  after  t  t  being  seconds hit  position  seconds  ball,  express  /  in  O  .  x = horizontal position after  /  seconds  terms of t  x = lOOf  and  / = -16f 2 + 40f + 3  These two equations, used to express two variables (x and variable (f) are called parametric equations. The variable f to define the variables  x and  /) in  terms of a third  the  parameter used  is  /.  XXXI  Parametric  mode  If your graphing calculator has a built-in parametric mode, you can enter and graph parametric equations that express the variables x and y in terms of the variable The RANGE-variables screen will have three additional quantities for you to specify Tmin, Tmax, and Tstep (called "pitch" on some calculators). r.  —  RANGE  The calculator  f=0  \i^  Tmin = Tmax = 3 J Tstep = .05 < Xmin = Xmax = 300 Ymin = Ymax = 70 Each f-value the point  is  will  f =  plotted  learn  will  use t-values from  3.  The difference between successive f-values will be 0.05. Note:  The  x-  omitted here,  substituted in both the equation for  (x, y) is  You  to  and will  y-scale variables, also appear.  x and the equation  for y; then  on the calculator screen.  more about parametric equations  in  Sections 11-1  and  1  2-3.  Try This Graph ball.  RANGE  variables as  Then use the TRACE feature ball to  on page xxxi for the path of the shown on the calculator, screen above.  the parametric equations given  Set the  reach  its  maximum  to  determine  how many seconds  it  takes the  height.  Other Capabilities of a Graphing Calculator In  addition to  may have  its  ability to display the  graphs of functions, your graphing calculator  other capabilities that will be useful to you  Statistical  in this  course.  graphs  Many graphing  calculators can display his-  line graphs, or scatter plots of data you have entered. For example, the histogram at the right displays the data from the list that appears on page 639 of fifty scores on a standardized  tograms,  5 mathematics achievement test. The histogram 100 shows the number of students whose scores fell in each 100-point interval. You can obtain statistics about the data, such as the mean and standard deviation (see Chapter a graphing calculator just as you would from a scientific calculator.  n  XXXII  1  7),  from  Using a Graphing  Curve Curve  a  fitting  (see Chapter 1 8) is the process of finding an equation that describes ordered pairs. Often, the first step is to graph the paired data in a scatter For example, the chart shown below gives the winning times in the men's fitting  set of  plot.  Olympic 400 m freestyle swimming race. The data can be entered in a graphing calculator and then displayed as plotted points (/, s), where y is the number of years since 1 900 and s is the time in seconds.  Year  Winning time (seconds)  1972  240.27  1976  231.93  1980  231.31  1984  231.23  1988  226.95  1992  225.00  240  225  72  92  The relationship between y and s can be approximated by a line. A graphing tor will give an equation of the "line of best fit," s = -0.653y + 285, and a tion coefficient that reflects how well the equation models the data.  calculacorrela-  Matrices  Many  graphing calculators allow you to enter numerical information in matrix 1 4) and can then perform a variety of matrix operations. For example, the matrices A and 8 shown below display the quiz and homework averages of 3 precalculus students for each of two units. form (see Chapter  Unit  1  Mary Jose  Kasha  Quiz  HW  68 89 92  75 77 95  =  A  Quiz  Mary  76  80  Jose  81  81  Kasha  85  90  r  Unit 1 will be worth 40% and Unit 2 will be worth 60% of each student's midterm quiz and homework grades. A graphing calculator with matrix capabilities can calculate 0.4a + 0.6b for  each pair of corresponding elements in matrices A and 8, and display the results in a new matrix that gives each student's quiz  HW  Unit 2  =  B  '  [72.8  78]  [84.2  79.4]  [87.8  92]  and homework averages.  And more Throughout the year, you are sure to find other topics that can be explored with a graphing calculator, and you will discover new methods and uses for this valuable tool. Be sure to share your discoveries with your classmates and your tc cher. XXXIII  Careers in Mathematics anil Science Today's society  is  changing so rapidly  that  you cannot foresee the career  may be available to you within a few years. You can be however, that many of the most exciting careers will involve the use  opportunities that sure,  of mathematics, science, and technology. Recent studies have shown that  even now there  is a shortage of scientists and engineers. By choosing to complete four years of high school mathematics you have already made one decision that will help you keep your options open. Continuing to  study mathematics and related fields in college will prepare you to take  advantage of a wide variety of career choices, including jobs sciences and the  arts,  as well as engineering, medicine,  in the social  and  scientific  research.  Now  meet seven people whose careers use the tools and methods of mathematics, the language of science and technology.  IMMUNOLOGY  An  immunologist investigates the body's immune response,  the process  by which the body  identifies, reacts to,  fights off toxic or disease-causing agents.  munology  Research  in  and im-  contributes to the medical profession's ability to  enhance the body's capacity  to fight cancer  and other  life-  threatening diseases.  Martha C. Zuniga. who  holds a Ph.D. in biology  from Yale University, conducts research on the immune (  system's ability to distinguish alien virus and tumor cells  from the body's healthy  tissues. In  ence Foundation honored her with  1989 the National Sciits  Presidential  Young  Investigator Award, a grant to provide her with funding to  continue her research.  Martha  C.  As  Zuniga  a recipient of the  an active role  mote  in  NSF award,  Dr. Zuniga maintains  undergraduate education, helping to pro-  the importance of studying science.  At University of California,  Santa Cruz, where she teaches immunology, Dr. Zuniga encourages her students to enjoy the challenges of difficult academic work. "People get  excited about heroism, and  I  think there can be heroism in intellectual  who  works with students through the Society for the Advancement of Chicanos and Native Americans in  pursuits," says Dr. Zuniga,  Science.  XXXIV  Careers  in  Mathematics and Science  also  COMPUTER GRAPHIC ARTS Computer graphics have expanded the world of many artists. Computers help artists who work in traditional media to make decisions about composition, design, and color. For other  artists,  image stored duced on  the  end product of the  in the  computer's  an  is  then repro-  film, printers, or video.  Midori Kitagawa De Leon, Visualization Laboratory at Texas  came  process  artistic  memory and  interested in  painting at the  She went on  a Ph.D. student at the  M  A &  University, be-  computer graphics while majoring  Women's College  to earn an  M.A.  in  in oil  of Fine Arts, in Tokyo.  Computer Graphics and  Animation from Ohio State University. There she wrote programs  to generate  three-dimensional  "branching ob-  jects," such as trees and other plant-like forms, and to simulate the  Midori Kitagawa  De Leon  growth of the plants using genetically determined  growth patterns as well as information about  Ms. Kitagawa De Leon's tions of surrealistic plant  shows and magazines. landscape architects,  life,  artistic  environment.  work, which includes film anima-  has been featured in computer graphics  In addition to  who  their  its artistic  programs  will use the  work  value, her  art  will help  to project the future  appearance of their designs.  SPACE EXPLORATION  Any United  States citizen holding a degree in mathematics,  science, or engineering can apply to be an astronaut.  Once  selected, an astronaut candidate goes through a one-year  may spend many more  years  working on the ground before getting an opportunity  to fly.  training program,  and then  Guion Bluford, Jr. was a teenager when the space in 1957. He was fascinated by flying objects, from the model airplanes he built to the newspapers he age began  tossed each day on his paper route, and he dreamed of entering the relatively  new  field of  aerospace engineering.  He  graduated from Pennsylvania State University and the Air  Force Institute of Technology, earning his Ph.D. space engineering in 1978. In the same year, Guion Bluford.  Jr.  cepted him into  its  in aero-  NASA  ac-  astronaut training program.  August of 1983 Colonel Bluford flew on NASA's eighth shuttle mission, becoming the first African- American to travel into space. Two years later, his second space flight saw one Dutch, two West German, and five American astronauts fly together on a mission run by In  West Germany.  In 1991 Colonel Bluford completed a third mission.  Careers  in  Mathematics and Science  XXXV  ENGINEERING Aerospace, biomedical, chemical, environmental, industrial  —  these are a few of the branches of the vast field of  engineering. Engineers apply mathematics and science to  They design  the solution of practical problems.  airplanes,  buildings, highways, artificial limbs, lasers, and computers,  among  other things.  Robert K. Whitman, deputy director of The American Indian Science and Engineering Society, remembers his -  early interest in engineering. radio, it  much  to their  dismay.  pick up sounds. ...  lot  I  read  *I I .  .  .  .  took apart  wanted .  .  to see  .  .  my .  parents'  what made  that engineering requires a  of training in math and science. In high school.  I  took  all  math and science courses I could get." Bob Whitman went on to study electrical engineering at the University of New Mexico and at Colorado State University. He received a scholarship from NASA and an Outstanding Achievement Award from the Navajo Nation (1978). He has worked for IBM, where his projects the  Robert K. Whitman  included developing printed circuit boards used in electronic equipment,  and designing computer software.  COMPUTER PROGRAMMING Computers help doctors make diagnoses, assist architects with their designs, and regulate the functioning of all kinds of machines from heart pacemakers to rocket engines. To earn out these and other instructions; thus, the  improve our tion  lives  tasks,  HHL  ^-jteraSj »  'imf*  power of computers  to  change and  can be realized only through the imagina-  and ingenuity of the people who program them.  microcomputer.  vated  Installed  on  a  wheelchair,  the  Katalavox gives quadriplegics unprecedented control of their lives. In addition, the  ^ll  1  Katalavox has become indispen-  sable in the operating room,  where surgeons' hands are  Martine  free  work while surgical microscopes are guided by voice. Ms. Kempf, a native of France, was studying astronomy  at the  Uni-  Bonn in 1982 when she first wrote the Katalavox program. She moved to the United States to run her own business marketing her  versity of later  -  computers require explicit  Martine Kempf was still in her twenties when she succeeded in programming a computer to recognize and respond to the human voice. Using her program she invented the Katalavox, a small black box containing a voice-acti-  to  H  invention.  XXXvi  Careers  in  Mathematics and Science  Kempf  PSYCHOLOGY Psychologists study  human and animal  on the mental functions involved  behavior, focusing  in the emotional, intellec-  and physical development of individuals. Trained in the methods of scientific research and mathematical analytual,  sis  of data, they formulate and  predict behavior.  They  and body on each  test theories to  explain and  also study the influence of the  mind  other.  Patricia Cowings, a psychophysiologist for the Space Life Sciences Division of  NASA,  studies the body's re-  sponse to the weightlessness experienced  She teaches astronauts  to control,  space travel.  in  through mental tech-  niques, such physical functions as blood pressure and heart rate,  allowing them to counteract some of the negative ef-  fects of weightlessness.  Patricia  Cowings first studied psychology at the State Uniof New York at Stony Brook, and then pursued graduate degrees  Cowings  Dr. versity  the University of California at Davis.  began working for  NASA  at the  While  Ames  still  at  a graduate student she  Research Center. She earned a  Ph.D. in psychology in 1973.  PHYSICS Physicists seek to understand  all  aspects of matter and  They  energy, the fundamental components of our world.  pursue knowledge about subjects as varied as electromagnetism, optics, thermodynamics, acoustics, and  quantum  theory. All branches of physics require a thorough back-  ground  in  higher mathematics.  Samuel Chao Chung Ting,  a researcher in the field  of high-energy particle physics, studies the composition and  behavior of subatomic particles. In 1974, while working  Brookhaven  at  National Laboratory, Dr. Ting demonstrated  the existence of the J particle, a discovery heralded as an  important breakthrough in twentieth-century understanding of atomic structure. Also  known  as psi, the particle  was  independently observed by Dr. Burton Richter of the Stan-  Samuel Chao Chung Ting  ford Linear Accelerator Center. In 1976 Dr. Ting and Dr.  Richter jointly received the Nobel Prize for their discovery.  American by  birth, Dr.  Ting grew up  in China, returning to the  United States to attend the University of Michigan, where he earned a Ph.D.  in physics. In  1967 he joined the faculty of the Massachusetts  Insti-  tute of Technology.  Careers  in  Mathematics and Science  XXXvii  unctions  Linear Functions and Lines  1-1 Points Objective  Each point  To find the intersection of two lines and to find the length and the coordinates of the midpoint of a segment. can be associated with an ordered pair of numbers, called  in the plane  the coordinates of the point. Also, each ordered pair of  numbers can be associated with a point in the plane. The association of points and ordered pairs is the basis of  coordinate geometry, a branch of mathematics that connects geometric and algebraic ideas.  To  up a coordinate system, we can choose two  set  Second Quadrant  perpendicular lines, one horizontal and the other vertical, as the x-axis and the j>-axis and designate their point of  measure,  we mark  number  off the axes as  The axes divide  located at the origin.  3  Using a convenient unit of  intersection as the origin.  4- First Quadrant R(0,2)  2  lines with zero  G(-3,0)  1  -h- -  the plane into four  -9-10 -4-3-2-1  quadrants.  The diagram shows and j-coordinate -2.  We  P  that  has x-coordinate 4  -2f  write P(4, -2). Points with  x-coordinate 0, such as R,  lie  on the  v-axis.  /  >  3  4  (4,-2)«  -3-  Third Quadrant  Points  Q, lie on the x-axis. The coordinates of the origin, O, are (0, 0). with y-coordinate 0,  12  1  such as  -4 -\- Fourth Quadrant  Linear Equations  A  solution of the equation 2x  —  3y  the equation true. For example, (0,  Several solutions are  of  all  in the  12  —4)  is  is  an ordered pair of numbers that makes  a solution because 2(0)  — 3(— 4) =  12.  diagram. The set  points in the plane corresponding to solutions  of an equation  The graph diagram.  is  called the  of 2x —  We  call  3v  —4  = the  12  graph of is  the line  the equation.  shown  in the  y -intercept of the graph —4).  We  6 the x-intercept of the graph because the  line  because the call  shown  =  line intersects the v-axis at (0,  intersects the x-axis at (6, 0).  Any  equation  of the  form  Ax + By =  C,  where A and B are not both 0, is called a linear equation because its graph is a line. Conversely, any line of a linear equation. The graph is often referred to as "the call  Ax + By = C  ^  In this  the general  view of Chicago from the  suggests a coordinate  form of  air,  in the plane is the  line  graph  Ax + By = C." We  a linear equation.  the regular pattern of streets intersecting at right angles  grid.  Linear and Quadratic Functions  i  Example  Sketch the graph of  1  One way  Solution  Step  1  +  3.v  graph  to sketch the  To  =  2v  18.  to find the intercepts.  is  find the y-intercept.  x  let  +  3(0)  2? y  The  To  through  line passes  The Step 2  Plot (0. 9)  and  through them.  check  that  (6. 0). It is  2(0)  = =  x  =  y  let  +  through  line passes  0.  18  9  (0, 9), so the y-intercept is 9.  find the .v-intercept, 3.v  = =  18  6 so the .v-intercept  (6, 0).  Draw  0.  rectly. Select a different point its  to  t(0.9)  you have drawn the  and determine whether  y*\  a straight line  always a good idea  is 6.  line cor-  on the  \(4,3)  line  3-  \(6.0)  coordinates sat-  i  i  isfy the equation. In this case, (4, 3)  +  check since 3(4)  When  C  one of the constants A, B, or  in  line  is  A =  and the  line  C=  is 0,  and the  i  X  \  you can draw  line contains the  horizontal. In figure (c).  is  3  18.  Ax + By = C  certain conclusions about the graph. In figure (a), origin. In figure (b),  =  2(3)  O  does  B =  and the  vertical. v  (2, 3)  (7i.  •  3)  =  (4,V7)  3  -  1--  O  O..  (4, 1)  H 1  —(—  1  (4,0)  (0  (b)  (a)  .r  • v  1  i  Intersection of Lines  You can determine where two  lines intersect  their equations simultaneously.  2x 3.y  Chapter One  by drawing  their  graphs or by solving  Consider the following pair of linear equations:  + 5y = + 4v =  10  (1)  12  (2)  =4  You can make hand-drawn sketches or you can use a graphing calculator or computer to obtain the graphs shown  at the right. (If  be sure  to write  in  y  you use  a calculator or computer,  terms of x)  From  the figure  it  seems  x is a little less than 3 and (With a calculator's trace feature  that at the point of intersection,  y  is  a  little  less than  or a computer's  1.  zoom  mations to * and  y.)  II  you can get better approxiBe aware that solutions found by  II  feature,  graphing are not always exact.  An  1  X = 2.9  Y = .84 ssv  algebraic solution  yields the exact values.  To tion (1)  solve the equations simultaneously, you can multiply both sides of equa-  by  from the  3  and both sides of equation  3(2* 2(3*  + +  (2)  by  2.  I5y  = = =  24  =  6.  5y) Ay)  = =  3(10)  ->  6x  2(12)  ->  6x  + +  8y 7y  y  Now  Then  subtract the second equation  equation.  first  substitute  —  into equation (1)  30  6  7  and solve for  x:  2x  :  +  5(-y  )  v  =  10  =  20 7  Thus  (  —  ,  —J  is  the  common  solution of the  two equations  point of their graphs). Notice that the graphical estimate x the exact answer.  You can  ~  (or the intersection  3 and y  ~  1 is  close to  use a graphical estimate as a check on an algebraic  solution.  lel  When two linear equations have no common solution, their graphs are parallines. When two linear equations have infinitely many common solutions, the  equations have the same graph.  no common 6x 3x  + +  Ay 2y  solution:  = =  infinitely  many common  8  6x  1  3x  + +  Ay 2y  = =  solutions:  8  A  Linear and Quadratic Functions  We  A and B  denote the line segment with endpoints  AB. You can use the formulas below asked to derive these formulas  to find  AB  as  AB  and  its  length as  and the midpoint of AB. You are  Written Exercises 34 and 35.  in  The Distance and Midpoint Formulas Let  A=  B =  (xi, Ji),  V2),  (.y 2 ,  and  M be the  midpoint of AB.  Then:  AB = V(x 2 -x  M-  (  2 l  )  +  (y 2  —— —2—  -yi)  2  (midpoint formula)  ,  )  A = (-1,9) and B = a. the length of AB  Example 2  (distance formula)  If  (4,  -3), find:  AB  b. the coordinates of the midpoint of  Solution  Find the distance between  a.  AB = V(4= V25 + = 13  M=  b.  1  +  (-1))  2  A and  + (-3-  9)  A(-l,9)..  B. 2  144  4  9  +  fl(4,-3)  (-3) -,  3  CLASS EXERCISES Find the length and the coordinates of the midpoint of CD. D(8, 6)  1.  C(0,  3.  C(-3,4), D(3, -2)  5.  Let  0),  A =  (2, 3),  B =  of the midpoints of 6.  7.  Which of  AM  and  on (9,-1)  the following points are  (3,3) (2.5,3.5)  d. (-10.5, 12)  point (8, 4)  b. State  Name  on the horizontal on h.  +  3v  =15?  VJ  line h.  three other points  an equation of  The point (8,4) a.  is  C(7, -9), D(7, -1)  the line 2.r  c.  Name  4.  2),  and MB.  b.  The  C(4,  M be the midpoint of AB. Find the coordinates  a.  a.  8.  (6, 7),  D(6, 6)  2.  is  h  h.  on the  three other points  (8,4)  vertical line v.  on  V  v.  X  b. State an equation of v.  Exs.  Chapter One  7,  9.  Find the coordinates of the points where the  —  Ax 10.  =  3v  The diagram  at the right  +  equations x  =  v  shows  5 and 2x  the graphs of the  —y=  1  Estimate the coordinates of the point of  a.  line  18 intersects the axes.  inter-  section.  Find the exact coordinates of the point of  b.  inter-  section by solving the equations simultane-  ously.  Compare  c.  Ex.10  the solution with your estimate.  WRITTEN EXERCISES Find the length of D(l,  CD and  1.  C(l,  3.  C(-8, -3), D(7,5)  5 . c[  0),  i  8)  D( _ 2  |),  the coordinates of the midpoint of  ,  C(4.8, 2.2), D(4.8, -2.8)  9.  Which of  10.  C(-2, -1), D(4,  8.  c.  the following points are  (-1.2, 3.0)  a.  4.  3),  b.  (3,  9)  V|.-iM-ii  b. (8,4)  Which of  C(3,  C(1.7, 5.7), D(-2.3, 5.7)  —  the following points are on the graph of 3x  (9,6)  a.  D(15, 12)  2.  _i  7.  CD.  -|)  — Hf)  '  on the graph of  c.  15?  (3A " 3  d  (~™3'  =  2v  —5x + 4y =  (-18,24)  '  2)  (  e.  (3.6,9)  '  18?  (-6, -3)  d.  " 9 " 22)  e'  In Exercises 11 and 12, graph each equation. Label the origin and the x- and ^-intercepts as O, P, and Q, respectively. Find the area of AOPQ.  -  11. 3.v  13.  On  2y  =  12. 4.v  6  +  =  3y  24  a single set of axes, sketch the horizontal line through (4, 3)  vertical line through (5, -2).  What  is  the intersection of these lines?  and the  What  are  the equations of these lines? 14. Repeat Exercise 13 for the horizontal line through (-2, line  through (-2,  -1) and  the vertical  3).  In Exercises 15-18, solve the given pair of equations simultaneously. sketch the graphs of the equations and label the intersection point. 15.  3.v  x  17.  =  9  +y =  3  5y  x - 3y = 5x + v =  4  16.  2x 4.v  18.  + -  3y  9y  = =  15 3  -2.v- 6v= x  -  3v  Then  =  18  6  Linear and Quadratic Functions  19.  -l).andD(2,  Plot A(l, 7), 5(3,5), C(4,  figure  Use  1).  ABCD  of quadrilateral  that the opposite sides  show What kind of  the distance formula to  are equal in length.  ABCD?  is  A(-6, 3), B(-\. 6). C(2, 1), and D(-3, -2). Use the distance formula to show that quadrilateral ABCD is a square. (Hint: Show that the four sides are equal in length and that the two diagonals are equal in length.)  20. Plot  21. Plot ,4(5,  andD(-l,  5(7. -1), C(l. -3).  1).  ABCD  show that the diagonals of quadrilateral What kind of quadrilateral is ABCD? to  22. Plot A(2, 0), B(4, -6), C(9,  and D(7,  1),  each other. What kind of quadrilateral 23.  Given A( — 3, 3), 5(1, AB + BC = AC.  24. Repeat Exercise 23 for  PI  25. a.  Show  5(4, 2)  that  from A (9, b. If (2, A)  2)  is  Show  and 5(1,  that  b. If (3, k)  is  5  is  15),  Show  7).  that  AC  and  BD  bisect  ABCD?  show  5(-l,  have the same midpoint.  that  B  is  on  AC by  showing  that  and C(3, -2).  4),  equidistant 6).  A and  k.  5(1,4) is equidistant -3) and 5(-l, -5).  A and  equidistant from  5, find the value of 27.  7),  equidistant from  M-5,  from  A(-3, is  5. find the value of 26. a.  and C(3,  11),  is  -1). Use the midpoint formula  k.  a point on the .v-axis 13 units from  the point  — 3,  (  Find  5).  all  the possible  coordinates for P. 28.  Q  a point  is  on the v-axis  from the point  2VTo  Find  (6, 1).  units  and intersecting  Parallel  the  all  in  lines create visual interest  the Pyramid at the Louvre  Museum,  Paris.  possible coordinates of Q. 29.  Show x  —  that the three lines  2v  =  4 intersect  30. Determine  5x —  =  2v  31. a. Plot  {BC) b.  4.v  points  +  three  =  3v  A{-6,  + (AC) 2 =  x  +  3y  =  2x  19,  —  5v  =  and  5,  (AB)  lines  whose equations  3 intersect in  one  are  3a'  .  2v  =  4,  point.  5(6,3), and C(-2, -1). Then show What can you conclude about ZC?  7),  2  +  that  Give the coordinates of the midpoint, M, of AB. Verify these coordinates by  showing 32.  and  0,  are  one point.  in  whether the  the 2  whose equations  The  that  CM =  jAB.  area of a triangle with sides a, b, and c units long can be found using  Hero's (or Heron's) formula:  Area  =  \/s(s  —  a)(s  —  b)(s  —  c)  where  s  =  a  +  b  +  c  Find to the nearest tenth the area of the triangle with vertices 5(5, 17), and C(22, -4).  Chapter One  A(— 13,  2),  33. In  AABC,  F(5, 5)  D(7,  3)  the midpoint of  is  34. In this proof, you  The  the midpoint of  is  AC. Find  may assume  AB, £(10,  ence  is  is  the midpoint of  BC, and  the coordinates of A, B, and C.  the following:  distance between two points on the  vertical line  9)  VA  same B(x2 ,yz )  the absolute value of the differ-  v-coordinates; the distance between  in  two points on the same horizontal  line is the  absolute value of the difference in .v-coordinates.  Note  that the first quadrant is  used for  .  ACx,.*)  convenience.  Given:  A =  Prove:  AB = V(.v 2 -.v A = (a-,, y,), B =  35. Given:  1  M  is  2  +  )  (v 2  (A 2 ,y 2  is  the midpoint of  BC.  Q  is  the midpoint of  AC.  Prove: a.  P  b.  Q  c.  Points  +  i  ?i)  :  VA  )  M  A(A,,y,)  2  '>\ and  P  have the same  y-coordinate.  M and Q have the same  A-coordinate. e.  M=  A"l  +  A2  >'l  +  V2  36. Three vertices of a parallelogram have coordinates  Find the coordinates of the fourth vertex.  (  — 3,  How many  1),  (1,4),  and  (4, 3).  possible answers are  there?  1-2 Slopes of Lines {Objective]  To find the slope of a  line  and  to  determine whether two  lines are parallel, perpendicular, or neither.  The slope of  a nonvertical line  is  a  number measuring  steepness of the line relative to the a- axis. Let  be any two points on a  (jc 1s  y  x  )  the  and  The  difference in the y y values gives the rise, and the difference in the x values (jc 2 ,  2)  gives the run.  That  is,  The slope  the slope  m  is  v  C(.v :  v,)  :  )  v2  *2  2  d. Points  5U„  2  +  Ai  .V  AB.  P  v  .  (x2 ,y2 )  the midpoint of  A'2,  C(.v 2 y,)  O  B =  (ai,^),  _A  is  line.  the ratio of the rise to the run.  defined by:  m = yi-y\ X — Ai 2  Linear and Quadratic Functions  .  Some  important facts about the slope of a line  follow.  't 1  .  a.  Horizontal lines have a slope of 0, because  y2 b.  =0  V]  for all  — X\ =  for  The  figures  slope =  all x.  ho  2,  "  '  no slope  X  from having zero slope.  below show  lines with positive slope. Lines with positive slope rise  you look at points wit

Advanced Mathematics Mcdougal Littell Houghton Mifflin Answers

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