Transcription
Chapter 10Resource MastersNew York, New YorkColumbus, OhioWoodland Hills, CaliforniaPeoria, Illinois
StudentWorksTM This CD-ROM includes the entire Student Edition along with theStudy Guide, Practice, and Enrichment masters.TeacherWorksTM All of the materials found in this booklet are included for viewingand printing in the Advanced Mathematical Concepts TeacherWorksCD-ROM.Copyright The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproducethe material contained herein on the condition that such material be reproduced onlyfor classroom use; be provided to students, teachers, and families without charge;and be used solely in conjunction with Glencoe Advanced Mathematical Concepts.Any other reproduction, for use or sale, is prohibited without prior writtenpermission of the publisher.Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027ISBN: 0-07-869137-01 2 3 4 5 6 7 8 9 10Advanced Mathematical ConceptsChapter 10 Resource MastersXXX11 10 09 08 07 06 05 04
ContentsVocabulary Builder . . . . . . . . . . . . . . . vii-viiiLesson 10-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 435Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 437Lesson 10-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 417Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 419Lesson 10-8Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 438Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 440Lesson 10-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 420Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 422Chapter 10 AssessmentChapter 10 Test, Form 1A . . . . . . . . . . . 441-442Chapter 10 Test, Form 1B . . . . . . . . . . . 443-444Chapter 10 Test, Form 1C . . . . . . . . . . . 445-446Chapter 10 Test, Form 2A . . . . . . . . . . . 447-448Chapter 10 Test, Form 2B . . . . . . . . . . . 449-450Chapter 10 Test, Form 2C . . . . . . . . . . . 451-452Chapter 10 Extended ResponseAssessment . . . . . . . . . . . . . . . . . . . . . . . 453Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 454Chapter 10 Quizzes A & B . . . . . . . . . . . . . . 455Chapter 10 Quizzes C & D. . . . . . . . . . . . . . 456Chapter 10 SAT and ACT Practice . . . . 457-458Chapter 10 Cumulative Review . . . . . . . . . . 459Lesson 10-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 423Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 425Lesson 10-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 426Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 428Lesson 10-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 429Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 431SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A17Lesson 10-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 432Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Glencoe/McGraw-HilliiiAdvanced Mathematical Concepts
A Teacher’s Guide to Using theChapter 10 Resource MastersThe Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 10 Resource Masters include the corematerials needed for Chapter 10. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.Vocabulary Builder Pages vii-viii include aPractice There is one master for each lesson.student study tool that presents the keyvocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.These problems more closely follow thestructure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.When to Use These provide additionalpractice options or may be used as homeworkfor second day teaching of the lesson.When to Use Give these pages to studentsbefore beginning Lesson 10-1. Remind them toadd definitions and examples as they completeeach lesson.Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they arelearning. These are not written exclusivelyfor honors students, but are accessible for usewith all levels of students.Study Guide There is one Study Guidemaster for each lesson.When to Use Use these masters asreteaching activities for students who needadditional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent. Glencoe/McGraw-HillWhen to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.ivAdvanced Mathematical Concepts
Assessment OptionsIntermediate AssessmentThe assessment section of the Chapter 10Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.Chapter Tests Forms 1A, 1B, and 1C Form 1 tests containmultiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with averagelevel students, and Form 1C is intended foruse with basic-level students. These testsare similar in format to offer comparabletesting situations.Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided forassessment. Glencoe/McGraw-Hill Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter. The SAT and ACT Practice offerscontinuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are providedon the master. The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.AnswersAll of the above tests include a challengingBonus question. A Mid-Chapter Test provides an option toassess the first half of the chapter. It iscomposed of free-response questions.Continuing AssessmentChapter Assessments v Page A1 is an answer sheet for the SAT andACT Practice questions that appear in theStudent Edition on page 693. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they mayencounter in test taking. The answers for the lesson-by-lessonmasters are provided as reduced pages withanswers appearing in red. Full-size answer keys are provided for theassessment options in this booklet.Advanced Mathematical Concepts
Chapter 10 Leveled WorksheetsGlencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow. Study Guide masters provide worked-out examples as well as practiceproblems. Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords. Practice masters provide average-level problems for students whoare moving at a regular pace. Enrichment masters offer students the opportunity to extend theirlearning.Five Different Options to Meet the Needs ofEvery Student in a Variety of Waysprimarily skillsprimarily conceptsprimarily applicationsBASICAVERAGE1Study Guide2Vocabulary Builder3Parent and Student Study Guide (online) dvanced Mathematical Concepts
NAME DATE PERIODChapter10Reading to Learn MathematicsVocabulary BuilderThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.Vocabulary TermFoundon PageDefinition/Description/Exampleanalytic geometryasymptotesaxis of symmetrycenterconic sectionconjugate axisdegenerate conicdirectrixeccentricityellipse(continued on the next page) Glencoe/McGraw-HillviiAdvanced Mathematical Concepts
NAME DATE PERIODChapter10Reading to Learn MathematicsVocabulary Builder (continued)Vocabulary TermFoundon PageDefinition/Description/Exampleequilateral hyperbolafocushyperbolalocusmajor axisminor axisrectangular hyperbolasemi-major axissemi-minor axistransverse axisvertex Glencoe/McGraw-HillviiiAdvanced Mathematical Concepts
NAME DATE PERIOD10-1Study GuideIntroduction to Analytic GeometryExample 1Find the distance between points at ( 2, 2) and(5, 4). Then find the midpoint of the segmentthat has endpoints at the given coordinates.d (x 2 x )2 (y 2 y )2 11Distance Formulad [5 ( 2 )] 2 ( [ 4 ) ]2 2Let (x1, y1 ) ( 2, 2) and(x2 , y2 ) (5, 4).d 7 2 ( 6 ) 2 or 8 5 x xy y11 2 , 2midpoint 22 Midpoint Formula 2 5 2 ( 4)The midpoint is at 2 , 2 or 32 , 1 .Example 2Determine whether quadrilateral ABCD withvertices A(1, 1), B(0, 1), C( 2, 0), and D( 1, 2) isa parallelogram.First, graph the figure. To determine if DA CB, find the slopes of DA and CB. slope of DAslope of CBy y21 m x x2Slope formula11 2 1 ( 1) 21 y y21 m x x2D( 1, 2) and A(1, 1)Slope formula1 1 0 0 ( 2) 12 C( 2, 0) and B(0, 1) Their slopes are equal. Therefore, DA CB. To determine if DA CB, use the distance formula to find DA and CB.DA [1 ( 1 )] 2 1( )2 2CB [0 ( 2 )] 2 ( 1 )0 2 5 5 The measures of DA and CB are equal. Therefore, DA CB. Since DA CB and DA CB, quadrilateral ABCD is aparallelogram. Glencoe/McGraw-Hill417Advanced Mathematical Concepts
NAME DATE PERIOD10-1PracticeIntroduction to Analytic GeometryFind the distance between each pair of points with the givencoordinates. Then find the midpoint of the segment that hasendpoints at the given coordinates.1. ( 2, 1), (3, 4)2. (1, 1), (9, 7)3. (3, 4),(5, 2)4. ( 1, 2), (5, 4)5. ( 7, 4), (2, 8)6. ( 4, 10), (4, 5)Determine whether the quadrilateral having vertices with thegiven coordinates is a parallelogram.7. (4, 4), (2, 2), ( 5, 1), ( 3, 5)8. (3, 5), ( 1, 1), ( 6, 2), ( 3, 7)9. (4, 1), (2, 5), ( 3, 3), ( 1, 1)10. (2, 6), (1, 2), ( 4, 4), ( 3, 9)11. Hiking Jenna and Maria are hiking to a campsite located at(2, 1) on a map grid, where each side of a square represents2.5 miles. If they start their hike at ( 3, 1), how far must theyhike to reach the campsite? Glencoe/McGraw-Hill418Advanced Mathematical Concepts
NAME DATE PERIOD10-1EnrichmentMathematics and History: HypatiaHypatia (A.D. 370–415) is the earliest woman mathematician whoselife is well documented. Born in Alexandria, Egypt, she was widelyknown for her keen intellect and extraordinary mathematical ability.Students from Europe, Asia, and Africa flocked to the university atAlexandria to attend her lectures on mathematics, astronomy,philosophy, and mechanics.Hypatia wrote several major treatises in mathematics. Perhaps themost significant of these was her commentary on the Arithmetica ofDiophantus, a mathematician who lived and worked in Alexandriain the third century. In her commentary, Hypatia offered several observations about the Arithmetica’s Diophantine problems—problems for which one was required to find only the rationalsolutions. It is believed that many of these observations weresubsequently incorporated into the original manuscript of theArithmetica.In modern mathematics, the solutions of a Diophantine equationare restricted to integers. In the exercises, you will explore somequestions involving simple Diophantine equations.For each equation, find three solutions that consist of an orderedpair of integers.1. 2x y 72. x 3y 53. 6x 5y 84. 11x 4y 65. Refer to your answers to Exercises 1–4. Suppose that the integerpair (x1, y1) is a solution of Ax By C. Describe how to findother integer pairs that are solutions of the equation.6. Explain why the equation 3x 6y 7 has no solutions that areinteger pairs.7. True or false: Any line on the coordinate plane must pass throughat least one point whose coordinates are integers. Explain. Glencoe/McGraw-Hill419Advanced Mathematical Concepts
NAME DATE PERIOD10-2Study GuideCirclesThe standard form of the equation of a circle with radius rand center at (h, k) is (x h)2 (y k)2 r2.Example 1Write the standard form of the equation of thecircle that is tangent to the x-axis and has itscenter at ( 4, 3). Then graph the equation.Since the circle is tangent to the x-axis, thedistance from the center to the x-axis is the radius.The center is 3 units above the x-axis. Therefore,the radius is 3.(x h)2 ( y k)2 r2[x ( 4)]2 ( y 3)2 32(x 4)2 ( y 3)2 9Example 2Standard form(h, k) ( 4, 3) and r 3Write the standard form of the equation of thecircle that passes through the points at (1, 1),(5, 3), and ( 3, 3). Then identify the center andradius of the circle.Substitute each ordered pair (x, y) in the generalform x2 y2 Dx Ey F 0 to create a systemof equations.(1)2 ( 1)2 D(1) E( 1) F 0(5)2 (3)2 D(5) E(3) F 0( 3)2 (3)2 D( 3) E(3) F 0(x, y) (1, 1)(x, y) (5, 3)(x, y) ( 3, 3)Simplify the system of equations.D E F 2 05D 3E F 34 0 3D 3E F 18 0The solution to the system is D 2, E 6, and F 6.The general form of the equation of the circle isx2 y2 2x 6y 6 0.x2 y2 2x 6y 6 0Group to form perfect square trinomials.(x2 2x ?) ( y2 6y ?) 6(x2 2x 1) ( y2 6y 9) 6 1 9 Complete the square.Factor the trinomials.(x 1)2 ( y 3)2 16After completing the square, the standard form of thecircle is (x 1)2 (y 3)2 16. Its center is at (1, 3), andits radius is 4. Glencoe/McGraw-Hill420Advanced Mathematical Concepts
NAME DATE PERIOD10-2PracticeCirclesWrite the standard form of the equation of each circle described.Then graph the equation.2. center at (2, 1), radius 41. center at (3, 3) tangent to the x-axisWrite the standard form of each equation. Then graph theequation.3. x 2 y 2 8x 6y 21 04. 4x 2 4y 2 16x 8y 5 0Write the standard form of the equation of the circle that passesthrough the points with the given coordinates. Then identify thecenter and radius.5. ( 3, 2), ( 2, 3), ( 4, 3)6. (0, 1), (2, 3), (4, 1)7. Geometry A square inscribed in a circle and centered at theorigin has points at (2, 2), ( 2, 2), (2, 2) and ( 2, 2). Whatis the equation of the circle that circumscribes the square? Glencoe/McGraw-Hill421Advanced Mathematical Concepts
NAME DATE PERIOD10-2EnrichmentSpheresThe set of all points in three-dimensional spacethat are a fixed distance r (the radius), from afixed point C (the center), is called a sphere. Theequation below is an algebraic representation ofthe sphere shown at the right.(x – h) 2 (y – k) 2 (z – l) 2 r 2A line segment containing the center of a sphereand having its endpoints on the sphere is called adiameter of the sphere. The endpoints of adiameter are called poles of the sphere. A greatcircle of a sphere is the intersection of the sphereand a plane containing the center of the sphere.1. If x 2 y 2 – 4y z 2 2z – 4 0 is an equation of asphere and (1, 4, –3) is one pole of the sphere, find thecoordinates of the opposite pole.2. a. On the coordinate system at the right, sketch thesphere described by the equation x 2 y 2 z 2 9.b. Is P(2, –2, –2) inside, outside, or on the sphere?c. Describe a way to tell if a point with coordinatesP(a, b, c) is inside, outside, or on the sphere withequation x 2 y 2 z 2 r 2.3. If x 2 y 2 z 2 – 4x 6y – 2z – 2 0 is an equation ofa sphere, find the circumference of a great circle, and thesurface area and volume of the sphere.4. The equation x2 y2 4 represents a set of points inthree-dimensional space. Describe that set of points inyour own words. Illustrate with a sketch on thecoordinate system at the right. Glencoe/McGraw-Hill422Advanced Mathematical Concepts
NAME DATE PERIOD10-3Study GuideEllipsesThe standard form of the equation of an ellipse is(x h)2 a2( y k)2b 1 when the major axis is horizontal.2In this case, a2 is in the denominator of the x term. The( y k)2a(x h)2standard form is b 1 when the major22axis is vertical. In this case, a2 is in the denominator ofthe y term. In both cases, c2 a2 b2.ExampleFind the coordinates of the center, the foci,and the vertices of the ellipse with the equation4x2 9y2 24x 36y 36 0. Then graph theequation.First write the equation in standard form.4x2 9y2 24x 36y 36 04(x2 6x ?) 9( y2 4y ?) 36 ? ?GCF of x terms is 4;GCF of y terms is 9.4(x2 6x 9) 9(y2 4y 4) 36 4(9) 9(4) Complete the square.4(x 3)2 9( y 2)2 36(x 3)(y 9422)2Factor. 1Divide each side by 36.Now determine the values of a, b, c, h, and k. In allellipses, a2 b2. Therefore, a2 9 and b2 4. Since a2is the denominator of the x term, the major axis isparallel to the x-axis.a 3b 2 2 b2 or 5 c acenter: ( 3, 2) , 2)foci: ( 3 5(h, k)(h c, k)major axis vertices:(0, 2) and ( 6, 2)(h a, k)minor axis vertices:( 3, 4) and ( 3, 0)(h, k b)h 3k 2Graph these ordered pairs. Then complete the ellipse. Glencoe/McGraw-Hill423Advanced Mathematical Concepts
NAME DATE PERIOD10-3PracticeEllipsesWrite the equation of each ellipse in standard form. Then find thecoordinates of its foci.1.2.For the equation of each ellipse, find the coordinates of the center,foci, and vertices. Then graph the equation.3. 4x 2 9y 2 8x 36y 4 04. 25x 2 9y 2 50x 90y 25 0Write the equation of the ellipse that meets each set of conditions.5. The center is at (1, 3), the major axis isparallel to the y-axis, and one vertex isat (1, 8), and b 3.6. The foci are at ( 2, 1) and( 2, 7), and a 5.7. Construction A semi elliptical arch is used to design aheadboard for a bed frame. The headboard will have a heightof 2 feet at the center and a width of 5 feet at the base. Whereshould the craftsman place the foci in order to sketch the arch? Glencoe/McGraw-Hill424Advanced Mathematical Concepts
NAME DATE PERIOD10-3EnrichmentSuperellipsesThe circle and the ellipse are members of an interesting family ofcurves that were first studied by the French physicist andmathematician Gabriel Lame′ (1795-1870). The general equation forthe family is x n a y n b 1, with a 0, b 0, and n 0.For even values of n greater than 2, the curves are calledsuperellipses.1. Consider two curves that are notsuperellipses. Graph each equation on thegrid at the right. State the type of curveproduced each time.a. 1b. 1x 2 2x 2 3 y 2 2y 2 22. In each of the following cases you are givenvalues of a, b, and n to use in the generalequation. Write the resulting equation.Then graph. Sketch each graph on the gridat the right.a. a 2, b 3, n 4b. a 2, b 3, n 6c. a 2, b 3, n 83. What shape will the graph of x n 2y n 3 1approximate for greater and greater even,whole-number values of n? Glencoe/McGraw-Hill425Advanced Mathematical Concepts
NAME DATE PERIOD10-4Study GuideHyperbolasThe standard form of the equation of a hyperbola is(x h)2( y k)2 1 when the transverse axis is horizontal, and22ab( y k)2(x h)2 b 2a2 1 when the transverse axis is vertical. In bothcases, b2 c2 a2.ExampleFind the coordinates of the center, foci, and vertices,and the equations of the asymptotes of the graph of25x2 9y2 100x 54y 206 0. Then graph the equation.Write the equation in standard form.25x2 9y2 100x 54y 206 025(x2 4x ?) 9(y2 6y ?) 206 ? ?GCF of x terms is 25;GCF of y terms is 9.25(x2 4x 4) 9(y2 6y 9) 206 25(4) ( 9)(9) Completethe square.25(x 2)2 9(y 3)2 225 Factor .(x 2)2(y 3)2 1925Divide each side by 225.From the equation, h 2, k 3,a 3, b 5, and c 3 4 . Thecenter is at ( 2, 3).Since the x terms are in the firstexpression, the hyperbola has ahorizontal transverse axis.The vertices are at (h a, k) or(1, 3) and ( 5, 3).The foci are at (h c, k) or 4 , 3).( 2 3The equations of the asymptotes arey k ab (x h) or y 3 53 (x 2).Graph the center, the vertices, and the rectangle guide,which is 2a units by 2b units. Next graph theasymptotes. Then sketch the hyperbola. Glencoe/McGraw-Hill426Advanced Mathematical Concepts
NAME DATE PERIOD10-4PracticeHyperbolasFor each equation, find the coordinates of the center, foci, and vertices, andthe equations of the asymptotes of its graph. Then graph the equation.1. x2 4y 2 4x 24y 36 02. y 2 4x2 8x 20Write the equation of each hyperbola.3.4.5. Write an equation of the hyperbola for which the length of thetransverse axis is 8 units, and the foci are at (6, 0) and ( 4, 0).6. Environmental Noise Two neighbors who live one mile aparthear an explosion while they are talking on the telephone. Oneneighbor hears the explosion two seconds before the other. Ifsound travels at 1100 feet per second, determine the equation ofthe hyperbola on which the explosion was located. Glencoe/McGraw-Hill427Advanced Mathematical Concepts
NAME DATE PERIOD10-4EnrichmentMoving FociRecall that the equation of a hyperbola with centerat the origin and horizontal transverse axis has theequationx2 a2–y2 b2 1. The foci are at (–c, 0) and(c, 0), where c2 a2 b2, the vertices are at (–a, 0)and (a, 0), and the asymptotes have equationsby x. Such a hyperbola is shown at the right.aWhat happens to the shape of the graph as c growsvery large or very small?Refer to the hyperbola described above.1. Write a convincing argument to show that as c approaches 0, thefoci, the vertices, and the center of the hyperbola become thesame point.2. Use a graphing calculator or computer to graph x 2 – y 2 1,x 2 – y 2 0.1, and x 2 – y 2 0.01. (Such hyperbolas correspondto smaller and smaller values of c.) Describe the changes in thegraphs. What shape do the graphs approach as c approaches 0?3. Suppose a is held fixed and c approaches infinity. How does thegraph of the hyperbola change?4. Suppose b is held fixed and c approaches infinity. How does thegraph of the hyperbola change? Glencoe/McGraw-Hill428Advanced Mathematical Concepts
NAME DATE PERIOD10-5Study GuideParabolasThe standard form of the equation of the parabola is (y k)2 4p(x h)when the parabola opens to the right. When p is negative, the parabolaopens to the left. The standard form is (x h)2 4p(y k) when theparabola opens upward. When p is negative, the parabola opens downward.Example 1Given the equation x2 12y 60, find the coordinatesof the focus and the vertex and the equations of thedirectrix and the axis of symmetry. Then graph theequation of the parabola.First write the equation in the form (x h)2 4p(y k).x2 12y 60Factor.x2 12( y 5)2(x 0) 4(3)( y 5) 4p 12, so p 3.In this form, we can see that h 0, k 5, and p 3.Vertex: (0, 5)(h, k)Focus: (0, 2)Directrix: y 8y k pAxis of Symmetry: x 0(h, k p)x hThe axis of symmetry is the y-axis.Since p is positive, the parabola opensupward. Graph the directrix, the vertex,and the focus. To determine the shape ofthe parabola, graph several otherordered pairs that satisfy the equationand connect them with a smooth curve.Example 2Write the equation y2 6y 8x 25 0 in standardform. Find the coordinates of the focus and thevertex, and the equations of the directrix and the axis ofsymmetry. Then graph the parabola.y2 6y 8x 25 0y2 6y 8x 252 6y ? 8x 25 ?yy2 6y 9 8x 25 9( y 3)2 8(x 2)Isolate the x terms and the y terms.Complete the square.Simplify and factor.From the standard form, we can see that h 2and k 3. Since 4p 8, p 2. Since y issquared, the directrix is parallel to the y-axis. Theaxis of symmetry is the x-axis. Since p is negative,the parabola opens to the left.Vertex: ( 2, 3)Focus: ( 4, 3)Directrix: x 0Axis of Symmetry: y 3 Glencoe/McGraw-Hill(h, k)(h p, k)x h py k429Advanced Mathematical Concepts
NAME DATE PERIOD10-5PracticeParabolasFor the equation of each parabola, find the coordinates of thevertex and focus, and the equations of the directrix and axis ofsymmetry. Then graph the equation.1. x 2 2x 8y 17 02. y 2 6y 9 12 12xWrite the equation of the parabola that meets each set ofconditions. Then graph the equation.3. The vertex is at ( 2, 4) andthe focus is at ( 2, 3).4. The focus is at (2, 1), and theequation of the directrix is x 2.5. Satellite Dish Suppose the receiver in a parabolic dish antennais 2 feet from the vertex and is located at the focus. Assume thatthe vertex is at the origin and that the dish is pointed upward.Find an equation that models a cross sectionof the dish. Glencoe/McGraw-Hill430Advanced Mathematical Concepts
NAME DATE PERIOD10-5EnrichmentTilted ParabolasThe diagram at the right shows a fixed point F(1, 1)and a line d whose equation is y – x 2. If P(x, y)satisfies the condition that PD PF, then P is on aparabola. Our objective is to find an equation for thetilted parabola; which is the locus of all points that arethe same distance from (1,1) and the line y – x 2.To do this, first find an equation for the line m throughP(x, y) and perpendicular to line d at D(a, b). Usingthis equation and the equation for line d, find thecoordinates (a, b) of point D in terms of x and y. Thenuse (PD) 2 (PF ) 2 to find an equation for the parabola.Refer to the discussion above.1. Find an equation for line m.2. Use the equations for lines m and d to show that the coordinatesof point D are D(a, b) D x y 2 y x 2 , 22 .3. Use the coordinates of F, P, and D, along with (PD) 2 (PF ) 2 tofind an equation of the parabola with focus F and directrix d.4. a. Every parabola has an axis of symmetry. Find an equation forthe axis of symmetry of the parabola described above. Justifyyour answer.b. Use your answer from part a to find the coordinates of thevertex of the parabola. Justify your answer. Glencoe/McGraw-Hill431Advanced Mathematical Concepts
NAME DATE PERIOD10-6Study GuideRectangular and Parametric Forms of Conic SectionsUse the table to identify a conic section given its equation ingeneral form.conicAx2 Bxy Cy2 Dx Ey F 0circleA CparabolaEither A or C is zero.ellipseA and C have the same sign and A C.hyperbolaA and C have opposite signs.Example 1Identify the conic section represented by theequation 5x2 4y2 10x 8y 18 0.A 5 and C 4. Since A and C have the samesigns and are not equal, the conic is an ellipse.Example 2Find the rectangular equation of the curve whoseparametric equations are x 2t and y 4t2 4t 1.Then identify the conic section represented bythe equation.Then substitute 2x for t in theFirst, solve the equation x 2tfor t.equation y 4t2 4t 1.y 4t2 4t 1x 2t2y 4 2x 4 2x 1 t x2 x t2y x2 2x 1Since C 0, the equation y x2 2x 1 is theequation of a parabola.Example 3Find the rectangular equation of the curve whoseparametric equations are x 3 cos t and y 5 sin t,where 0 t 2 . Then graph the equation usingarrows to indicate orientation.Solve the first equation for cos tand the second equation for sin t.ycos t 3x and sin t 5 Us
Feb 02, 2015 · Chapter 10 Resource Masters New York, New York C
