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Chapter 8Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 8 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828011-7 Algebra 2Chapter 8 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 8-1Study Guide and Intervention . . . . . . . . 455–456Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 457Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Reading to Learn Mathematics . . . . . . . . . . 459Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 460
Lesson 8-2Study Guide and Intervention . . . . . . . . 461–462Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 463Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 464Reading to Learn Mathematics . . . . . . . . . . 465Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 466
Lesson 8-3Study Guide and Intervention . . . . . . . . 467–468Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 469Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 470Reading to Learn Mathematics . . . . . . . . . . 471Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 472
Lesson 8-4Study Guide and Intervention . . . . . . . . 473–474Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 475Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Reading to Learn Mathematics . . . . . . . . . . 477Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 478
Lesson 8-5Study Guide and Intervention . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484
Lesson 8-6Study Guide and Intervention . . . . . . . . 485–486Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 487Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 488Reading to Learn Mathematics . . . . . . . . . . 489Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 490
Lesson 8-7Study Guide and Intervention . . . . . . . . 491–492Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 493Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 494Reading to Learn Mathematics . . . . . . . . . . 495Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 496
Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 497–498Chapter 8 Test, Form 2A . . . . . . . . . . . 499–500Chapter 8 Test, Form 2B . . . . . . . . . . . 501–502Chapter 8 Test, Form 2C . . . . . . . . . . . 503–504Chapter 8 Test, Form 2D . . . . . . . . . . . 505–506Chapter 8 Test, Form 3 . . . . . . . . . . . . 507–508Chapter 8 Open-Ended Assessment . . . . . . 509Chapter 8 Vocabulary Test/Review . . . . . . . 510Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 511Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 512Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 513Chapter 8 Cumulative Review . . . . . . . . . . . 514Chapter 8 Standardized Test Practice . . 515–516
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 8 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 8Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 468–469. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
88
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
asymptote
A·suhm(p)·TOHT
center of a circle
center of an ellipse
circle
conic section
conjugate axis
KAHN·jih·guht
directrix
duh·REHK·trihks
distance formula
ellipse
ih·LIHPS
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
foci of an ellipse
focus of a parabola
FOH·kuhs
hyperbola
hy·PUHR·buh·luh
latus rectum
LA·tuhs REHK·tuhm
major axis
midpoint formula
minor axis
parabola
puh·RA·buh·luh
tangent
TAN·juhnt
transverse axis
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
Study Guide and InterventionMidpoint and Distance Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 455 Glencoe Algebra 2
Less
on
8-1
The Midpoint Formula
Midpoint Formula The midpoint M of a segment with endpoints (x1, y1) and (x2, y2) is ! , ".y1 ! y2"2x1 ! x2"2
Find the midpoint of theline segment with endpoints at (4, !7) and (!2, 3).
! , " # ! , "# ! , " or (1, $2)
The midpoint of the segment is (1, $2).
$4"2
2"2
$7 ! 3"2
4 ! ($2)""2
y1 ! y2"2x1 ! x2"2
A diameter A!B! of a circlehas endpoints A(5, !11) and B(!7, 6).What are the coordinates of the centerof the circle?
The center of the circle is the midpoint of allof its diameters.
! , " # ! , "# ! , " or !$1, $2 "
The circle has center !$1, $2 ".1"2
1"2
$5"2
$2"2
$11 ! 6""2
5 ! ($7)""2
y1 ! y2"2
x1 ! x2"2
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the midpoint of each line segment with endpoints at the given coordinates.
1. (12, 7) and ($2, 11) 2. ($8, $3) and (10, 9) 3. (4, 15) and (10, 1)
(5, 9) (1, 3) (7, 8)
4. ($3, $3) and (3, 3) 5. (15, 6) and (12, 14) 6. (22, $8) and ($10, 6)
(0, 0) (13.5, 10) (6, !1)
7. (3, 5) and ($6, 11) 8. (8, $15) and ($7, 13) 9. (2.5, $6.1) and (7.9, 13.7)
"! , 8# " , !1# (5.2, 3.8)
10. ($7, $6) and ($1, 24) 11. (3, $10) and (30, $20) 12. ($9, 1.7) and ($11, 1.3)
(!4, 9) " , !15# (!10, 1.5)
13. Segment M#N# has midpoint P. If M has coordinates (14, $3) and P has coordinates ($8, 6), what are the coordinates of N? (!30, 15)
14. Circle R has a diameter S#T#. If R has coordinates ($4, $8) and S has coordinates (1, 4),what are the coordinates of T? (!9, !20)
15. Segment A#D# has midpoint B, and B#D# has midpoint C. If A has coordinates ($5, 4) and C has coordinates (10, 11), what are the coordinates of B and D?
B is "5, 8 #, D is "15, 13 #.1"
2"
33"
1"
3"
© Glencoe/McGraw-Hill 456 Glencoe Algebra 2
The Distance Formula
Distance FormulaThe distance between two points (x1, y1) and (x2, y2) is given by d # $(x2 $#x1)2 !# (y2 $# y1)2#.
What is the distance between (8, !2) and (!6, !8)?
d # $(x2 $#x1)2 !#( y2 $# y1)2# Distance Formula
# $($6 $# 8)2 !# [$8 $# ($2)]#2# Let (x1, y1) # (8, $2) and (x2, y2) # ($6, $8).
# $($14)#2 ! ($#6)2# Subtract.
# $196 !# 36# or $232# Simplify.
The distance between the points is $232# or about 15.2 units.
Find the perimeter and area of square PQRS with vertices P(!4, 1),Q(!2, 7), R(4, 5), and S(2, !1).
Find the length of one side to find the perimeter and the area. Choose P#Q#.
d # $(x2 $#x1)2 !#( y2 $# y1)2# Distance Formula
# $[$4 $# ($2)]#2 ! (1# $ 7)2# Let (x1, y1) # ($4, 1) and (x2, y2) # ($2, 7).
# $($2)2#! ($6#)2# Subtract.
# $40# or 2$10# Simplify.
Since one side of the square is 2$10#, the perimeter is 8$10# units. The area is (2$10#)2, or40 units2.
Find the distance between each pair of points with the given coordinates.
1. (3, 7) and ($1, 4) 2. ($2, $10) and (10, $5) 3. (6, $6) and ($2, 0)
5 units 13 units 10 units4. (7, 2) and (4, $1) 5. ($5, $2) and (3, 4) 6. (11, 5) and (16, 9)
3$2! units 10 units $41! units7. ($3, 4) and (6, $11) 8. (13, 9) and (11, 15) 9. ($15, $7) and (2, 12)
3$34! units 2$10! units 5$26! units
10. Rectangle ABCD has vertices A(1, 4), B(3, 1), C($3, $2), and D($5, 1). Find theperimeter and area of ABCD. 2$13! # 6$5! units; 3$65! units2
11. Circle R has diameter S#T# with endpoints S(4, 5) and T($2, $3). What are thecircumference and area of the circle? (Express your answer in terms of %.)10$ units; 25$ units2
Study Guide and Intervention (continued)
Midpoint and Distance Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeMidpoint and Distance Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 457 Glencoe Algebra 2
Less
on
8-1
Find the midpoint of each line segment with endpoints at the given coordinates.
1. (4, $1), ($4, 1) (0, 0) 2. ($1, 4), (5, 2) (2, 3)
3. (3, 4), (5, 4) (4, 4) 4. (6, 2), (2, $1) "4, #
5. (3, 9), ($2, $3) " , 3# 6. ($3, 5), ($3, $8) "!3, ! #
7. (3, 2), ($5, 0) (!1, 1) 8. (3, $4), (5, 2) (4, !1)
9. ($5, $9), (5, 4) "0, ! # 10. ($11, 14), (0, 4) "! , 9#
11. (3, $6), ($8, $3) "! , ! # 12. (0, 10), ($2, $5) "!1, #
Find the distance between each pair of points with the given coordinates.
13. (4, 12), ($1, 0) 13 units 14. (7, 7), ($5, $2) 15 units
15. ($1, 4), (1, 4) 2 units 16. (11, 11), (8, 15) 5 units
17. (1, $6), (7, 2) 10 units 18. (3, $5), (3, 4) 9 units
19. (2, 3), (3, 5) $5! units 20. ($4, 3), ($1, 7) 5 units
21. ($5, $5), (3, 10) 17 units 22. (3, 9), ($2, $3) 13 units
23. (6, $2), ($1, 3) $74! units 24. ($4, 1), (2, $4) $61! units
25. (0, $3), (4, 1) 4$2! units 26. ($5, $6), (2, 0) $85! units
5"
9"
5"
11"
5"
3"
1"
1"
© Glencoe/McGraw-Hill 458 Glencoe Algebra 2
Find the midpoint of each line segment with endpoints at the given coordinates.
1. (8, $3), ($6, $11) (1, !7) 2. ($14, 5), (10, 6) "!2, #3. ($7, $6), (1, $2) (!3, !4) 4. (8, $2), (8, $8) (8, !5)
5. (9, $4), (1, $1) "5, ! # 6. (3, 3), (4, 9) " , 6#7. (4, $2), (3, $7) " , ! # 8. (6, 7), (4, 4) "5, #9. ($4, $2), ($8, 2) (!6, 0) 10. (5, $2), (3, 7) "4, #
11. ($6, 3), ($5, $7) "! , !2# 12. ($9, $8), (8, 3) "! , ! #13. (2.6, $4.7), (8.4, 2.5) (5.5, !1.1) 14. !$ , 6", ! , 4" " , 5#15. ($2.5, $4.2), (8.1, 4.2) (2.8, 0) 16. ! , ", !$ , $ " "! , 0#
Find the distance between each pair of points with the given coordinates.
17. (5, 2), (2, $2) 5 units 18. ($2, $4), (4, 4) 10 units
19. ($3, 8), ($1, $5) $173! units 20. (0, 1), (9, $6) $130! units
21. ($5, 6), ($6, 6) 1 unit 22. ($3, 5), (12, $3) 17 units
23. ($2, $3), (9, 3) $157! units 24. ($9, $8), ($7, 8) 2$65! units
25. (9, 3), (9, $2) 5 units 26. ($1, $7), (0, 6) $170! units
27. (10, $3), ($2, $8) 13 units 28. ($0.5, $6), (1.5, 0) 2$10! units
29. ! , ", !1, " 1 unit 30. ($4$2#, $$5#), ($5$2#, 4$5#) $127! units
31. GEOMETRY Circle O has a diameter A#B#. If A is at ($6, $2) and B is at ($3, 4), find thecenter of the circle and the length of its diameter. "! , 1#; 3$5! units
32. GEOMETRY Find the perimeter of a triangle with vertices at (1, $3), ($4, 9), and ($2, 1).18 # 2$17! units
9"
7"5
3"5
2"5
1"
1"2
5"8
1"2
1"8
1"
2"3
1"3
5"
1"
11"
5"
11"
9"
7"
7"
5"
11"
Practice (Average)
Midpoint and Distance Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Reading to Learn MathematicsMidpoint and Distance Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 459 Glencoe Algebra 2
Less
on
8-1
Pre-Activity How are the Midpoint and Distance Formulas used in emergencymedicine?
Read the introduction to Lesson 8-1 at the top of page 412 in your textbook.
How do you find distances on a road map?
Sample answer: Use the scale of miles on the map. You mightalso use a ruler.
Reading the Lesson
1. a. Write the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2).
" , #b. Explain how to find the midpoint of a segment if you know the coordinates of the
endpoints. Do not use subscripts in your explanation.
Sample answer: To find the x-coordinate of the midpoint, add the x-coordinates of the endpoints and divide by two. To find the y-coordinate of the midpoint, do the same with the y-coordinates ofthe endpoints.
2. a. Write an expression for the distance between two points with coordinates (x1, y1) and(x2, y2). $(x2 !!x1)2 #! (y2 !! y1)2!
b. Explain how to find the distance between two points. Do not use subscripts in yourexplanation.
Sample answer: Find the difference between the x-coordinates and square it. Find the difference between the y-coordinates and square it. Add the squares. Then find the squareroot of the sum.
3. Consider the segment connecting the points ($3, 5) and (9, 11).
a. Find the midpoint of this segment. (3, 8)b. Find the length of the segment. Write your answer in simplified radical form. 6$5!
Helping You Remember
4. How can the “mid” in midpoint help you remember the midpoint formula?
Sample answer: The midpoint is the point in the middle of a segment. Itis halfway between the endpoints. The coordinates of the midpoint arefound by finding the average of the two x-coordinates (add them anddivide by 2) and the average of the two y-coordinates.
y1 # y2"2x1 # x2"2
© Glencoe/McGraw-Hill 460 Glencoe Algebra 2
Quadratic FormConsider two methods for solving the following equation.
(y $ 2)2 $ 5(y $ 2) ! 6 # 0
One way to solve the equation is to simplify first, then use factoring.
y2 $ 4y ! 4 $ 5y ! 10 ! 6 # 0y2 $ 9y ! 20 # 0
( y $ 4)( y $ 5) # 0
Thus, the solution set is {4, 5}.
Another way to solve the equation is first to replace y $ 2 by a single variable.This will produce an equation that is easier to solve than the original equation.Let t # y $ 2 and then solve the new equation.
( y $ 2)2 $ 5( y $ 2) ! 6 # 0t2 $ 5t ! 6 # 0
(t $ 2)(t $ 3) # 0
Thus, t is 2 or 3. Since t # y $ 2, the solution set of the original equation is {4, 5}.
Solve each equation using two different methods.
1. (z ! 2)2 ! 8(z ! 2) ! 7 # 0 2. (3x $ 1)2 $ (3x $ 1) $ 20 # 0
3. (2t ! 1)2 $ 4(2t ! 1) ! 3 # 0 4. ( y2 $ 1)2 $ ( y2 $ 1) $ 2 # 0
5. (a2 $ 2)2 $ 2(a2 $ 2) $ 3 # 0 6. (1 ! $c#)2 ! (1 ! $c#) $ 6 # 0
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Study Guide and InterventionParabolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 461 Glencoe Algebra 2
Less
on
8-2
Equations of Parabolas A parabola is a curve consisting of all points in thecoordinate plane that are the same distance from a given point (the focus) and a given line(the directrix). The following chart summarizes important information about parabolas.
Standard Form of Equation y # a(x $ h)2 ! k x # a(y $ k)2 ! h
Axis of Symmetry x # h y # k
Vertex (h, k ) (h, k )
Focus !h, k ! " !h ! , k"Directrix y # k $ x # h $
Direction of Opening upward if a & 0, downward if a ' 0 right if a & 0, left if a ' 0
Length of Latus Rectum units units
Identify the coordinates of the vertex and focus, the equations ofthe axis of symmetry and directrix, and the direction of opening of the parabolawith equation y % 2x2 ! 12x ! 25.
y # 2x2 $ 12x $ 25 Original equationy # 2(x2 $ 6x) $ 25 Factor 2 from the x-terms.y # 2(x2 $ 6x ! ■) $ 25 $ 2(■) Complete the square on the right side.y # 2(x2 $ 6x ! 9) $ 25 $ 2(9) The 9 added to complete the square is multiplied by 2.y # 2(x $ 3)2 $ 43 Write in standard form.
The vertex of this parabola is located at (3, $43), the focus is located at !3, $42 ", the
equation of the axis of symmetry is x # 3, and the equation of the directrix is y # $43 .The parabola opens upward.
Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation.
1. y # x2 ! 6x $ 4 2. y # 8x $ 2x2 ! 10 3. x # y2 $ 8y ! 6
(!3, !13), (2, 18), "2, 17 #, (!10, 4), "!9 , 4#,"!3, !12 #, x % !3, x % 2, y % 18 , y % 4, x % !10 ,
y % !13 , up down right
Write an equation of each parabola described below.
4. focus ($2, 3), directrix x # $2 5. vertex (5, 1), focus !4 , 1"x % 6(y ! 3)2 ! 2 x % !3(y ! 1)2 # 51
"
11"12
1"12
1"
1"
1"
3"
3"
1"
1"8
7"8
1"a
1"a
1"4a
1"4a
1"4a
1"4a
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 462 Glencoe Algebra 2
Graph Parabolas To graph an equation for a parabola, first put the given equation instandard form.
y # a(x $ h)2 ! k for a parabola opening up or down, orx # a(y $ k)2 ! h for a parabola opening to the left or right
Use the values of a, h, and k to determine the vertex, focus, axis of symmetry, and length ofthe latus rectum. The vertex and the endpoints of the latus rectum give three points on theparabola. If you need more points to plot an accurate graph, substitute values for pointsnear the vertex.
Graph y % (x ! 1)2 # 2.
In the equation, a # , h # 1, k # 2.
The parabola opens up, since a & 0.vertex: (1, 2)axis of symmetry: x # 1
focus: !1, 2 ! " or !1, 2 "
length of latus rectum: or 3 units
endpoints of latus rectum: !2 , 2 ", !$ , 2 "
The coordinates of the focus and the equation of the directrix of a parabola aregiven. Write an equation for each parabola and draw its graph.
1. (3, 5), y # 1 2. (4, $4), y# $6 3. (5, $1), x # 3
y % (x ! 3)2 # 3 y % (x ! 4)2 ! 5 x % (y # 1)2 # 41"
1"
1"
x
y
Ox
y
O
x
y
O
3"4
1"2
3"4
1"2
1"
"13"
3"4
1"4!"
13""
x
y
O
1"3
1"3
Study Guide and Intervention (continued)
Parabolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
ExampleExample
ExercisesExercises
Skills PracticeParabolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 463 Glencoe Algebra 2
Less
on
8-2
Write each equation in standard form.
1. y # x2 ! 2x ! 2 2. y # x2 $ 2x ! 4 3. y # x2 ! 4x ! 1
y % [x ! (!1)]2 # 1 y % (x ! 1)2 # 3 y % [x ! (!2)]2 # (!3)
Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation. Then find the length of the latus rectum and graph the parabola.
4. y # (x $ 2)2 5. x # (y $ 2)2 ! 3 6. y # $(x ! 3)2 ! 4
vertex: (2, 0); vertex: (3, 2); vertex: (!3, 4); focus: "2, #; focus: "3 , 2#; focus: "!3, 3 #;axis of symmetry: axis of symmetry: axis of symmetry: x % 2; y % 2; x % !3;directrix: y % ! ; directrix: x % 2 ; directrix: y % 4 ; opens up; opens right; opens down;latus rectum: 1 unit latus rectum: 1 unit latus rectum: 1 unit
Write an equation for each parabola described below. Then draw the graph.
7. vertex (0, 0), 8. vertex (5, 1), 9. vertex (1, 3),
focus !0, $ " focus !5, " directrix x #
y % !3x2 y % (x ! 5)2 # 1 x % 2(y ! 3)2 # 1
x
y
Ox
y
O
x
y
O
7"8
5"4
1"12
1"
3"
1"
3"
1"
1"
x
y
Ox
y
O
x
y
O
© Glencoe/McGraw-Hill 464 Glencoe Algebra 2
Write each equation in standard form.
1. y # 2x2 $ 12x ! 19 2. y # x2 ! 3x ! 3. y # $3x2 $ 12x $ 7
y % 2(x ! 3)2 # 1 y % [x ! (!3)]2 # (!4) y % !3[x ! (!2)]2 # 5Identify the coordinates of the vertex and focus, the equations of the axis ofsymmetry and directrix, and the direction of opening of the parabola with thegiven equation. Then find the length of the latus rectum and graph the parabola.
4. y # (x $ 4)2 ! 3 5. x # $ y2 ! 1 6. x # 3(y ! 1)2 $ 3
vertex: (4, 3); vertex: (1, 0); vertex: (!3, !1); focus: "4, 3 #; focus: " , 0#; focus: "!2 , !1#;axis: x % 4; axis: y % 0; axis: y % !1;directrix: y % 2 ; directrix: x % 1 ; directrix: x % !3 ; opens up; opens left; opens right;latus rectum: 1 unit latus rectum: 3 units latus rectum: unit
Write an equation for each parabola described below. Then draw the graph.
7. vertex (0, $4), 8. vertex ($2, 1), 9. vertex (1, 3),
focus !0, $3 " directrix x # $3 axis of symmetry x # 1,latus rectum: 2 units,a ' 0
y % 2x2 ! 4 x % (y ! 1)2 ! 2 y % ! (x ! 1)2 # 3
10. TELEVISION Write the equation in the form y # ax2 for a satellite dish. Assume that thebottom of the upward-facing dish passes through (0, 0) and that the distance from thebottom to the focus point is 8 inches. y % x21
"
x
y
Ox
y
O
1"
1"
7"8
1"
1"
3"
3"
11"
1"
1"
x
y
O
x
y
O
1"3
1"
1"2
1"2
Practice (Average)
Parabolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Reading to Learn MathematicsParabolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 465 Glencoe Algebra 2
Less
on
8-2
Pre-Activity How are parabolas used in manufacturing?
Read the introduction to Lesson 8-2 at the top of page 419 in your textbook.
Name at least two reflective objects that might have the shape of aparabola.
Sample answer: telescope mirror, satellite dish
Reading the Lesson
1. In the parabola shown in the graph, the point (2, $2) is called
the and the point (2, 0) is called the
. The line y # $4 is called the
, and the line x # 2 is called the
.
2. a. Write the standard form of the equation of a parabola that opens upward ordownward. y % a(x ! h)2 # k
b. The parabola opens downward if and opens upward if . The
equation of the axis of symmetry is , and the coordinates of the vertex are
.
3. A parabola has equation x # $ ( y $ 2)2 ! 4. This parabola opens to the .
It has vertex and focus . The directrix is . The length
of the latus rectum is units.
Helping You Remember
4. How can the way in which you plot points in a rectangular coordinate system help you toremember what the sign of a tells you about the direction in which a parabola opens?Sample answer: In plotting points, a positive x-coordinate tells you tomove to the right and a negative x-coordinate tells you to move to theleft. This is like a parabola whose equation is of the form “x % …”; itopens to the right if a & 0 and to the left if a ' 0. Likewise, a positive y-coordinate tells you to move up and a negative y-coordinate tells youto move down. This is like a parabola whose equation is of the form “y % …”; it opens upward if a & 0 and downward if a ' 0.
8x % 6(2, 2)(4, 2)
left1"8
(h, k)x % h
a & 0a ' 0
axis of symmetrydirectrix
focusvertex
x
y
O
(2, –2)
(2, 0)
y % –4
© Glencoe/McGraw-Hill 466 Glencoe Algebra 2
Tangents to ParabolasA line that intersects a parabola in exactly one point without crossing the curve is a tangent to the parabola. The point where a tangent line touches a parabola is the point of tangency. The line perpendicular to a tangent to a parabola at the point of tangency is called the normal to the parabola at that point. In the diagram, line ! is tangent to the
parabola that is the graph of y # x2 at !"32", "
94"". The
x-axis is tangent to the parabola at O, and the y-axis is the normal to the parabola at O.
Solve each problem.
1. Find an equation for line ! in the diagram. Hint: A nonvertical line with anequation of the form y # mx ! b will be tangent to the graph of y # x2 at
!"32", "
94"" if and only if !"
32", "
94"" is the only pair of numbers that satisfies both
y # x2 and y # mx ! b.
2. If a is any real number, then (a, a2) belongs to the graph of y # x2. Express m and b in terms of a to find an equation of the form y # mx ! b for the linethat is tangent to the graph of y # x2 at (a, a2).
3. Find an equation for the normal to the graph of y # x2 at !"32", "
94"".
4. If a is a nonzero real number, find an equation for the normal to the graph ofy # x2 at (a, a2).
x
y
O
!
y % x2
1–1–2–3 2
6
5
4
3
2
1
3
!3–2, 9–4"
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Study Guide and InterventionCircles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 467 Glencoe Algebra 2
Less
on
8-3
Equations of Circles The equation of a circle with center (h, k) and radius r units is (x $ h)2 ! (y $ k)2 # r2.
Write an equation for a circle if the endpoints of a diameter are at(!4, 5) and (6, !3).
Use the midpoint formula to find the center of the circle.
(h, k) # ! , " Midpoint formula
# ! , " (x1, y1) # ($4, 5), (x2, y2) # (6, $3)
# ! , " or (1, 1) Simplify.
Use the coordinates of the center and one endpoint of the diameter to find the radius.
r # $(x2 $x#1)2 !#( y2 $# y1)2# Distance formula
r # $($4 $# 1)2 !# (5 $#1)2# (x1, y1) # (1, 1), (x2, y2) # ($4, 5)
# $($5)2# ! 42# # $41# Simplify.
The radius of the circle is $41#, so r2 # 41.
An equation of the circle is (x $ 1)2 ! (y $ 1)2 # 41.
Write an equation for the circle that satisfies each set of conditions.
1. center (8, $3), radius 6 (x ! 8)2 # (y # 3)2 % 36
2. center (5, $6), radius 4 (x ! 5)2 # (y # 6)2 % 16
3. center ($5, 2), passes through ($9, 6) (x # 5)2 # (y ! 2)2 % 32
4. endpoints of a diameter at (6, 6) and (10, 12) (x ! 8)2 # (y ! 9)2 % 13
5. center (3, 6), tangent to the x-axis (x ! 3)2 # (y ! 6)2 % 36
6. center ($4, $7), tangent to x # 2 (x # 4)2 # (y # 7)2 % 36
7. center at ($2, 8), tangent to y # $4 (x # 2)2 # (y ! 8)2 % 144
8. center (7, 7), passes through (12, 9) (x ! 7)2 # (y ! 7)2 % 29
9. endpoints of a diameter are ($4, $2) and (8, 4) (x ! 2)2 # (y ! 1)2 % 45
10. endpoints of a diameter are ($4, 3) and (6, $8) (x ! 1)2 # (y # 2.5)2 % 55.25
2"2
2"2
5 ! ($3)""2
$4 ! 6"2
y1 ! y2"2
x1 ! x2"2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 468 Glencoe Algebra 2
Graph Circles To graph a circle, write the given equation in the standard form of theequation of a circle, (x $ h)2 ! (y $ k)2 # r2.
Plot the center (h, k) of the circle. Then use r to calculate and plot the four points (h ! r, k),(h $ r, k), (h, k ! r), and (h, k $ r), which are all points on the circle. Sketch the circle thatgoes through those four points.
Find the center and radius of the circle whose equation is x2 # 2x # y2 # 4y % 11. Then graph the circle.
x2 ! 2x ! y2 ! 4y # 11x2 ! 2x ! ■ ! y2 ! 4y ! ■ # 11 !■
x2 ! 2x ! 1 ! y2 ! 4y ! 4 # 11 ! 1 ! 4(x ! 1)2 ! ( y ! 2)2 # 16
Therefore, the circle has its center at ($1, $2) and a radius of $16# # 4. Four points on the circle are (3, $2), ($5, $2), ($1, 2),and ($1, $6).
Find the center and radius of the circle with the given equation. Then graph thecircle.
1. (x $ 3)2 ! y2 # 9 2. x2 ! (y ! 5)2 # 4 3. (x $ 1)2 ! (y ! 3)2 # 9
(3, 0), r % 3 (0, !5), r % 2 (1, !3), r % 3
4. (x $ 2)2 ! (y ! 4)2 # 16 5. x2 ! y2 $ 10x ! 8y ! 16 # 0 6. x2 ! y2 $ 4x ! 6y # 12
(2, !4), r % 4 (5, !4), r % 5 (2, !3), r % 5
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x
y
O
x2 # 2x # y2 # 4y % 11
Study Guide and Intervention (continued)
Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
ExampleExample
ExercisesExercises
Skills PracticeCircles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 469 Glencoe Algebra 2
Less
on
8-3
Write an equation for the circle that satisfies each set of conditions.
1. center (0, 5), radius 1 unit 2. center (5, 12), radius 8 unitsx2 # (y ! 5)2 % 1 (x ! 5)2 # (y ! 12)2 % 64
3. center (4, 0), radius 2 units 4. center (2, 2), radius 3 units(x ! 4)2 # y2 % 4 (x ! 2)2 # (y ! 2)2 % 9
5. center (4, $4), radius 4 units 6. center ($6, 4), radius 5 units(x ! 4)2 # (y # 4)2 % 16 (x # 6)2 # (y ! 4)2 % 25
7. endpoints of a diameter at ($12, 0) and (12, 0) x2 # y2 % 1448. endpoints of a diameter at ($4, 0) and ($4, $6) (x # 4)2 # (y # 3)2 % 99. center at (7, $3), passes through the origin (x ! 7)2 # (y # 3)2 % 58
10. center at ($4, 4), passes through ($4, 1) (x # 4)2 # (y ! 4)2 % 911. center at ($6, $5), tangent to y-axis (x # 6)2 # (y # 5)2 % 3612. center at (5, 1), tangent to x-axis (x ! 5)2 # (y ! 1)2 % 1
Find the center and radius of the circle with the given equation. Then graph thecircle.
13. x2 ! y2 # 9 14. (x $ 1)2 ! (y $ 2)2 # 4 15. (x ! 1)2 ! y2 # 16
(0, 0), 3 units (1, 2), 2 units (!1, 0), 4 units
16. x2 ! (y ! 3)2 # 81 17. (x $ 5)2 ! (y ! 8)2 # 49 18. x2 ! y2 $ 4y $ 32 # 0
(0, !3), 9 units (5, !8), 7 units (0, 2), 6 units
x
y
O 4 8
8
4
–4
–8
–4–8
x
y
O 4 8 12
–4
–8
–12
x
y
O 6 12
12
6
–6
–12
–6–12
x
y
Ox
y
Ox
y
O
© Glencoe/McGraw-Hill 470 Glencoe Algebra 2
Write an equation for the circle that satisfies each set of conditions.
1. center ($4, 2), radius 8 units 2. center (0, 0), radius 4 units(x # 4)2 # (y ! 2)2 % 64 x2 # y2 % 16
3. center !$ , $$3#", radius 5$2# units 4. center (2.5, 4.2), radius 0.9 unit
"x # #2 # (y # $3!)2 % 50 (x ! 2.5)2 # (y ! 4.2)2 % 0.815. endpoints of a diameter at ($2, $9) and (0, $5) (x # 1)2 # (y # 7)2 % 56. center at ($9, $12), passes through ($4, $5) (x # 9)2 # (y # 12)2 % 747. center at ($6, 5), tangent to x-axis (x # 6)2 # (y ! 5)2 % 25
Find the center and radius of the circle with the given equation. Then graph thecircle.
8. (x ! 3)2 ! y2 # 16 9. 3x2 ! 3y2 # 12 10. x2 ! y2 ! 2x ! 6y # 26(!3, 0), 4 units (0, 0), 2 units (!1, !3), 6 units
11. (x $ 1)2 ! y2 ! 4y # 12 12. x2 $ 6x ! y2 # 0 13. x2 ! y2 ! 2x ! 6y # $1(1, !2), 4 units (3, 0), 3 units (!1, !3), 3 units
WEATHER For Exercises 14 and 15, use the following information.On average, the circular eye of a hurricane is about 15 miles in diameter. Gale winds canaffect an area up to 300 miles from the storm’s center. In 1992, Hurricane Andrew devastatedsouthern Florida. A satellite photo of Andrew’s landfall showed the center of its eye on onecoordinate system could be approximated by the point (80, 26).
14. Write an equation to represent a possible boundary of Andrew’s eye.(x ! 80)2 # (y ! 26)2 % 56.25
15. Write an equation to represent a possible boundary of the area affected by gale winds.(x ! 80)2 # (y ! 26)2 % 90,000
x
y
O 4 8
4
–4
–8
–4–8x
y
Ox
y
O
1"
1"4
Practice (Average)
Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Reading to Learn MathematicsCircles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 471 Glencoe Algebra 2
Less
on
8-3
Pre-Activity Why are circles important in air traffic control?
Read the introduction to Lesson 8-3 at the top of page 426 in your textbook.
A large home improvement chain is planning to enter a new metropolitanarea and needs to select locations for its stores. Market research has shownthat potential customers are willing to travel up to 12 miles to shop at oneof their stores. How can circles help the managers decide where to placetheir store?
Sample answer: A store will draw customers who live inside acircle with center at the store and a radius of 12 miles. The management should select locations for whichas many people as possible live within a circle of radius 12 miles around one of the stores.
Reading the Lesson
1. a. Write the equation of the circle with center (h, k) and radius r.(x ! h)2 # (y ! k)2 % r 2
b. Write the equation of the circle with center (4, $3) and radius 5.(x ! 4)2 # (y # 3)2 % 25
c. The circle with equation (x ! 8)2 ! y2 # 121 has center and radius
.
d. The circle with equation (x $ 10)2 ! ( y ! 10)2 # 1 has center and
radius .
2. a. In order to find center and radius of the circle with equation x2 ! y2 ! 4x $ 6y $3 # 0,
it is necessary to . Fill in the missing parts of thisprocess.
x2 ! y2 ! 4x $ 6y $ 3 # 0
x2 ! y2 ! 4x $ 6y #
x2 ! 4x ! ! y2 $ 6y ! # ! !
(x ! )2 ! ( y $ )2 #
b. This circle has radius 4 and center at .
Helping You Remember
3. How can the distance formula help you to remember the equation of a circle?Sample answer: Write the distance formula. Replace (x1, y1) with (h, k)and (x2, y2) with (x, y). Replace d with r. Square both sides. Now youhave the equation of a circle.
(!2, 3)1632
943943
complete the square
1(10, !10)
11(!8, 0)
© Glencoe/McGraw-Hill 472 Glencoe Algebra 2
Tangents to CirclesA line that intersects a circle in exactly one point is a tangent to the circle. In the diagram, line ! is tangent to the circle with equation x2 ! y2 # 25 at the point whose coordinates are (3, 4).
A line is tangent to a circle at a point P on the circle if and only if the line is perpendicular to the radius from the center of the circle to point P. This fact enables you to find an equation of the tangent to a circle at a point P if you know an equation for the circle and the coordinates of P.
Use the diagram above to solve each problem.
1. What is the slope of the radius to the point with coordinates (3, 4)? What isthe slope of the tangent to that point?
2. Find an equation of the line ! that is tangent to the circle at (3, 4).
3. If k is a real number between $5 and 5, how many points on the circle have x-coordinate k? State the coordinates of these points in terms of k.
4. Describe how you can find equations for the tangents to the points you namedfor Exercise 3.
5. Find an equation for the tangent at ($3, 4).
5
–5
–5
5
(3, 4)
y
xO
!x2 # y2 % 25
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Study Guide and InterventionEllipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 473 Glencoe Algebra 2
Less
on
8-4
Equations of Ellipses An ellipse is the set of all points in a plane such that the sumof the distances from two given points in the plane, called the foci, is constant. An ellipsehas two axes of symmetry which contain the major and minor axes. In the table, thelengths a, b, and c are related by the formula c2 # a2 $ b2.
Standard Form of Equation ! # 1 ! # 1
Center (h, k) (h, k)
Direction of Major Axis Horizontal Vertical
Foci (h ! c, k ), (h $ c, k ) (h, k $ c), (h, k ! c)
Length of Major Axis 2a units 2a units
Length of Minor Axis 2b units 2b units
Write an equation for the ellipse shown.
The length of the major axis is the distance between ($2, $2) and ($2, 8). This distance is 10 units.
2a # 10, so a # 5The foci are located at ($2, 6) and ($2, 0), so c # 3.
b2 # a2 $ c2
# 25 $ 9# 16
The center of the ellipse is at ($2, 3), so h # $2, k # 3,a2 # 25, and b2 # 16. The major axis is vertical.
An equation of the ellipse is ! # 1.
Write an equation for the ellipse that satisfies each set of conditions.
1. endpoints of major axis at ($7, 2) and (5, 2), endpoints of minor axis at ($1, 0) and ($1, 4)
# % 1
2. major axis 8 units long and parallel to the x-axis, minor axis 2 units long, center at ($2, $5)
# (y # 5)2 % 1
3. endpoints of major axis at ($8, 4) and (4, 4), foci at ($3, 4) and ($1, 4)
# % 1
4. endpoints of major axis at (3, 2) and (3, $14), endpoints of minor axis at ($1, $6) and (7, $6)
# % 1
5. minor axis 6 units long and parallel to the x-axis, major axis 12 units long, center at (6, 1)
# % 1(x ! 6)2"9
(y ! 1)2"36
(x ! 3)2"16
(y # 6)2"64
(y ! 4)2"35
(x # 2)2"36
(x # 2)2"16
(y ! 2)2"4
(x # 1)2"36
(x ! 2)2"16
( y $ 3)2"25
x
F1
F2O
y
(x $ h)2"
b2(y $ k)2"
a2(y $ k)2"
b2(x $ h)2"
a2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 474 Glencoe Algebra 2
Graph Ellipses To graph an ellipse, if necessary, write the given equation in thestandard form of an equation for an ellipse.
! # 1 (for ellipse with major axis horizontal) or
! # 1 (for ellipse with major axis vertical)
Use the center (h, k) and the endpoints of the axes to plot four points of the ellipse. To makea more accurate graph, use a calculator to find some approximate values for x and y thatsatisfy the equation.
Graph the ellipse 4x2 # 6y2 # 8x ! 36y % !34.
4x2 ! 6y2 ! 8x $ 36y # $344x2 ! 8x ! 6y2 $ 36y # $ 34
4(x2 ! 2x ! ■) ! 6( y2 $ 6y ! ■) # $34 ! ■4(x2 ! 2x ! 1) ! 6( y2 $ 6y ! 9) # $34 ! 58
4(x ! 1)2 ! 6( y $ 3)2 # 24
! # 1
The center of the ellipse is ($1, 3). Since a2 # 6, a # $6#.Since b2 # 4, b # 2.The length of the major axis is 2$6#, and the length of the minor axis is 4. Since the x-termhas the greater denominator, the major axis is horizontal. Plot the endpoints of the axes.Then graph the ellipse.
Find the coordinates of the center and the lengths of the major and minor axesfor the ellipse with the given equation. Then graph the ellipse.
1. ! # 1 (0, 0), 4$3!, 6 2. ! # 1 (0, 0), 10, 4
3. x2 ! 4y2 ! 24y # $32 (0, !3), 4, 2 4. 9x2 ! 6y2 $ 36x ! 12y # 12 (2, !1), 6, 2$6!
x
y
Ox
y
O
x
y
Ox
y
O
y2"4
x2"25
x2"9
y2"12
( y $ 3)2"4
(x ! 1)2"6 xO
y
4x2 # 6y2 # 8x ! 36y % !34
(x $ h)2"
b2( y $ k)2"
a2
( y $ k)2"
b2(x $ h)2"
a2
Study Guide and Intervention (continued)
Ellipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
ExampleExample
ExercisesExercises
Skills PracticeEllipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 475 Glencoe Algebra 2
Less
on
8-4
Write an equation for each ellipse.
1. 2. 3.
# % 1 # % 1 # % 1
Write an equation for the ellipse that satisfies each set of conditions.
4. endpoints of major axis 5. endpoints of major axis 6. endpoints of major axis at (0, 6) and (0, $6), at (2, 6) and (8, 6), at (7, 3) and (7, 9),endpoints of minor axis endpoints of minor axis endpoints of minor axis at ($3, 0) and (3, 0) at (5, 4) and (5, 8) at (5, 6) and (9, 6)
# % 1 # % 1 # % 1
7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis atand parallel to x-axis, at ($6, 0) and (6, 0), foci (0, 12) and (0, $12), foci atminor axis 4 units long, at ($$32#, 0) and ($32#, 0) (0, $23# ) and (0, $$23# )center at (0, 0)
# % 1 # % 1 # % 1
Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.
10. ! # 1 11. ! # 1 12. ! # 1
(0, 0); (0, ($19!); (0, 0); ((6$2!, 0); (0, 0), (0, (2$6!); 20; 18 18; 6 14; 10
x
y
O 4 8
8
4
–4
–8
–4–8x
y
O 4 8
8
4
–4
–8
–4–8x
y
O 4 8
8
4
–4
–8
–4–8
x2"25
y2"49
y2"9
x2"81
x2"81
y2"100
x2"
y2"
y2"
x2"
y2"
x2"
(x ! 7)2"4
(y ! 6)2"9
(y ! 6)2"4
(x ! 5)2"9
x2"
y2"
(y ! 2)2"9
x2"
x2"
y2"
y2"
x2"
xO
y(0, 5)
(0, –1)
(–4, 2) (4, 2)
xO
y
(0, 3)
(0, –3)
(0, –5)
(0, 5)
xO
y
(0, 2)
(0, –2)
(–3, 0)(3, 0)
© Glencoe/McGraw-Hill 476 Glencoe Algebra 2
Write an equation for each ellipse.
1. 2. 3.
# % 1 # % 1 # % 1
Write an equation for the ellipse that satisfies each set of conditions.
4. endpoints of major axis 5. endpoints of major axis 6. major axis 20 units long at ($9, 0) and (9, 0), at (4, 2) and (4, $8), and parallel to x-axis,endpoints of minor axis endpoints of minor axis minor axis 10 units long,at (0, 3) and (0, $3) at (1, $3) and (7, $3) center at (2, 1)
# % 1 # % 1 # % 1
7. major axis 10 units long, 8. major axis 16 units long, 9. endpoints of minor axis minor axis 6 units long center at (0, 0), foci at at (0, 2) and (0, $2), foci and parallel to x-axis, (0, 2$15# ) and (0, $2$15# ) at ($4, 0) and (4, 0)center at (2, $4)
# % 1 # % 1 # % 1
Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.
10. ! # 1 11. ! # 1 12. ! # 1
(0, 0); (0, ($7!); 8; 6 (3, 1); (3, 1 ( $35! ); (!4, !3); 12; 2 (!4 ( 2$6!, !3); 14;
10
13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that thecenter of the first loop is at the origin, with the second loop to its right. Write an equationto model the first loop if its major axis (along the x-axis) is 12 feet long and its minoraxis is 6 feet long. Write another equation to model the second loop.
x
y
O 4 8
8
4
–4
–8
–4–8x
y
O
( y ! 3)2"25
(x ! 4)2"49
(x $ 3)2"1
( y $ 1)2"36
x2"9
y2"16
y2"
x2"
x2"
y2"
(x ! 2)2"9
(y # 4)2"25
(y ! 1)2"25
(x ! 2)2"100
(x ! 4)2"9
(y # 3)2"25
y2"
x2"
(y ! 3)2"9
(x # 1)2"25
x2"
(y ! 2)2"9
y2"
x2"
xO
y
(–5, 3)
(–6, 3)
(3, 3)
(4, 3)
xO
y
(0, 2 ! $%5)
(0, 2 # $%5)
(0, –1)
(0, 5)
xO
y(0, 3)
(0, –3)
(–11, 0) (11, 0)6 12
2
–2
–6–12
Practice (Average)
Ellipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Reading to Learn MathematicsEllipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 477 Glencoe Algebra 2
Less
on
8-4
Pre-Activity Why are ellipses important in the study of the solar system?
Read the introduction to Lesson 8-4 at the top of page 433 in your textbook.
Is the Earth always the same distance from the Sun? Explain your answerusing the words circle and ellipse. No; if the Earth’s orbit were acircle, it would always be the same distance from the Sunbecause every point on a circle is the same distance from thecenter. However, the Earth’s orbit is an ellipse, and the pointson an ellipse are not all the same distance from the center.
Reading the Lesson1. An ellipse is the set of all points in a plane such that the of the
distances from two fixed points is . The two fixed points are called the
of the ellipse.
2. Consider the ellipse with equation ! # 1.
a. For this equation, a # and b # .
b. Write an equation that relates the values of a, b, and c. c2 % a2 ! b2
c. Find the value of c for this ellipse. $5!
3. Consider the ellipses with equations ! # 1 and ! # 1. Complete the
following table to describe characteristics of their graphs.
Standard Form of Equation ! # 1 ! # 1
Direction of Major Axis vertical horizontal
Direction of Minor Axis horizontal vertical
Foci (0, 3), (0, !3) ($5!, 0), (!$5!, 0)Length of Major Axis 10 units 6 units
Length of Minor Axis 8 units 4 units
Helping You Remember4. Some students have trouble remembering the two standard forms for the equation of an
ellipse. How can you remember which term comes first and where to place a and b inthese equations? The x-axis is horizontal. If the major axis is horizontal, the first term is . The y-axis is vertical. If the major axis is vertical, the
first term is . a is always the larger of the numbers a and b.y2"
x2"
y2"4
x2"9
x2"16
y2"25
y2"4
x2"9
x2"16
y2"25
23
y2"4
x2"9
fociconstant
sum
© Glencoe/McGraw-Hill 478 Glencoe Algebra 2
Eccentricity In an ellipse, the ratio "d
c" is called the eccentricity and is denoted by the
letter e. Eccentricity measures the elongation of an ellipse. The closer e is to 0,the more an ellipse looks like a circle. The closer e is to 1, the more elongated
it is. Recall that the equation of an ellipse is "ax2
2" ! "by2
2" # 1 or "bx2
2" ! "ay2
2" # 1
where a is the length of the major axis, and that c # $a2 $ b#2#.
Find the eccentricity of each ellipse rounded to the nearesthundredth.
1. "x92" ! "3
y6
2" # 1 2. "8
x12" ! "
y9
2" # 1 3. "
x42" ! "
y9
2" # 1
0.87 0.94 0.75
4. "1x62" ! "
y9
2" # 1 5. "3
x62" ! "1
y6
2" # 1 6. "
x42" ! "3
y6
2" # 1
0.66 0.75 0.94
7. Is a circle an ellipse? Explain your reasoning.
Yes; it is an ellipse with eccentricity 0.
8. The center of the sun is one focus of Earth's orbit around the sun. Thelength of the major axis is 186,000,000 miles, and the foci are 3,200,000miles apart. Find the eccentricity of Earth's orbit.
approximately 0.17
9. An artificial satellite orbiting the earth travels at an altitude that variesbetween 132 miles and 583 miles above the surface of the earth. If thecenter of the earth is one focus of its elliptical orbit and the radius of theearth is 3950 miles, what is the eccentricity of the orbit?
approximately 0.052
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Study Guide and InterventionHyperbolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 479 Glencoe Algebra 2
Less
on
8-5
Equations of Hyperbolas A hyperbola is the set of all points in a plane such thatthe absolute value of the difference of the distances from any point on the hyperbola to anytwo given points in the plane, called the foci, is constant.
In the table, the lengths a, b, and c are related by the formula c2 # a2 ! b2.
Standard Form of Equation $ # 1 $ # 1
Equations of the Asymptotes y $ k # ( (x $ h) y $ k # ( (x $ h)
Transverse Axis Horizontal Vertical
Foci (h $ c, k), (h ! c, k) (h, k $ c), (h, k ! c)
Vertices (h $ a, k), (h ! a, k) (h, k $ a), (h, k ! a)
Write an equation for the hyperbola with vertices (!2, 1) and (6, 1)and foci (!4, 1) and (8, 1).
Use a sketch to orient the hyperbola correctly. The center of the hyperbola is the midpoint of the segment joining the two
vertices. The center is ( , 1), or (2, 1). The value of a is the
distance from the center to a vertex, so a # 4. The value of c is the distance from the center to a focus, so c # 6.
c2 # a2 ! b2
62 # 42 ! b2
b2 # 36 $ 16 # 20
Use h, k, a2, and b2 to write an equation of the hyperbola.
$ # 1
Write an equation for the hyperbola that satisfies each set of conditions.
1. vertices ($7, 0) and (7, 0), conjugate axis of length 10 ! % 1
2. vertices ($2, $3) and (4, $3), foci ($5, $3) and (7, $3) ! % 1
3. vertices (4, 3) and (4, $5), conjugate axis of length 4 ! % 1
4. vertices ($8, 0) and (8, 0), equation of asymptotes y # ( x ! % 1
5. vertices ($4, 6) and ($4, $2), foci ($4, 10) and ($4, $6) ! % 1(x # 4)2"48
(y ! 2)2"16
9y2"
x2"
1"6
(x ! 4)2"4
(y # 1)2"16
(y # 3)2"27
(x ! 1)2"9
y2"
x2"
( y $ 1)2"20
(x $ 2)2"16
$2 ! 6"2
x
y
O
a"b
b"a
(x $ h)2"
b2(y $ k)2"
a2(y $ k)2"
b2(x $ h)2"
a2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 480 Glencoe Algebra 2
Graph Hyperbolas To graph a hyperbola, write the given equation in the standardform of an equation for a hyperbola
$ # 1 if the branches of the hyperbola open left and right, or
$ # 1 if the branches of the hyperbola open up and down
Graph the point (h, k), which is the center of the hyperbola. Draw a rectangle withdimensions 2a and 2b and center (h, k). If the hyperbola opens left and right, the verticesare (h $ a, k) and (h ! a, k). If the hyperbola opens up and down, the vertices are (h, k $ a)and (h, k ! a).
Draw the graph of 6y2 ! 4x2 ! 36y ! 8x % !26.
Complete the squares to get the equation in standard form.6y2 $ 4x2 $ 36y $ 8x # $266( y2 $ 6y ! ■) $ 4(x2 ! 2x ! ■) # $26 ! ■6( y2 $ 6y ! 9) $ 4(x2 ! 2x ! 1) # $26 ! 506( y $ 3)2 $ 4(x ! 1)2 # 24
$ # 1
The center of the hyperbola is ($1, 3).According to the equation, a2 # 4 and b2 # 6, so a # 2 and b # $6#.The transverse axis is vertical, so the vertices are ($1, 5) and ($1, 1). Draw a rectangle withvertical dimension 4 and horizontal dimension 2$6# % 4.9. The diagonals of this rectangleare the asymptotes. The branches of the hyperbola open up and down. Use the vertices andthe asymptotes to sketch the hyperbola.
Find the coordinates of the vertices and foci and the equations of the asymptotesfor the hyperbola with the given equation. Then graph the hyperbola.
1. $ # 1 2. ( y $ 3)2 $ # 1 3. $ # 1
(2, 0), (!2, 0); (!2, 4), (!2, 2); (0, 4), (0, !4); (2$5!, 0), (!2$5!, 0); (!2, 3 # $10! ), (0, 5), (0, !5); y % (2x (!2, 3 ! $10! ); y % ( x
y % x # 3 ,
y % ! x # 2 1"
1"
2"
1"
xO
y
4"
x2"9
y2"16
(x ! 2)2"9
y2"16
x2"4
(x ! 1)2"6
( y $ 3)2"4 xO
y
(x $ h)2"
b2( y $ k)2"
a2
( y $ k)2""
b2(x $ h)2"
a2
Study Guide and Intervention (continued)
Hyperbolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
ExampleExample
ExercisesExercises
xO
y
xO
y
Skills PracticeHyperbolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 481 Glencoe Algebra 2
Less
on
8-5
Write an equation for each hyperbola.
1. 2. 3.
! % 1 ! % 1 ! % 1
Write an equation for the hyperbola that satisfies each set of conditions.
4. vertices ($4, 0) and (4, 0), conjugate axis of length 8 ! % 1
5. vertices (0, 6) and (0, $6), conjugate axis of length 14 ! % 1
6. vertices (0, 3) and (0, $3), conjugate axis of length 10 ! % 1
7. vertices ($2, 0) and (2, 0), conjugate axis of length 4 ! % 1
8. vertices ($3, 0) and (3, 0), foci ((5, 0) ! % 1
9. vertices (0, 2) and (0, $2), foci (0, (3) ! % 1
10. vertices (0, $2) and (6, $2), foci (3 ( $13#, $2) ! % 1
Find the coordinates of the vertices and foci and the equations of the asymptotesfor the hyperbola with the given equation. Then graph the hyperbola.
11. $ # 1 12. $ # 1 13. $ # 1
((3, 0); ((3$5!, 0); (0, (7); (0, ($58! ); ((4, 0); (($17!, 0);y % (2x y % ( x y % ( x
xO
y
4 8
8
4
–4
–8
–4–8xO
y
4 8
8
4
–4
–8
–4–8xO
y
1"
7"
y2"1
x2"16
x2"9
y2"49
y2"36
x2"9
(y # 2)2"4
(x ! 3)2"9
x2"
y2"
y2"
x2"
y2"
x2"
x2"
y2"
x2"
y2"
y2"
x2"
y2"
x2"
x2"
y2"
y2"
x2"
x
y
O
($!29, 0)(–$!29, 0)
(2, 0)(–2, 0)
4 8
8
4
–4
–8
–4–8x
y
O
(0, $!61)
(0, –$!61)
(0, 6)
(0, –6)
4 8
8
4
–4
–8
–4–8x
y
O
($!41, 0)(–$!41, 0)
(5, 0)
(–5, 0)
4 8
8
4
–4
–8
–4–8
© Glencoe/McGraw-Hill 482 Glencoe Algebra 2
Write an equation for each hyperbola.
1. 2. 3.
! % 1 ! % 1 ! % 1
Write an equation for the hyperbola that satisfies each set of conditions.
4. vertices (0, 7) and (0, $7), conjugate axis of length 18 units ! % 1
5. vertices (1, $1) and (1, $9), conjugate axis of length 6 units ! % 1
6. vertices ($5, 0) and (5, 0), foci (($26#, 0) ! % 1
7. vertices (1, 1) and (1, $3), foci (1, $1 ( $5#) ! % 1
Find the coordinates of the vertices and foci and the equations of the asymptotesfor the hyperbola with the given equation. Then graph the hyperbola.
8. $ # 1 9. $ # 1 10. $ # 1
(0, (4); (0, (2$5!); (1, 3), (1, 1); (3, 0), (3, !4); y % (2x (1, 2 ( $5!); (3, !2 ( 2$2!);
y ! 2 % ( (x ! 1) y # 2 % ((x ! 3)
11. ASTRONOMY Astronomers use special X-ray telescopes to observe the sources ofcelestial X rays. Some X-ray telescopes are fitted with a metal mirror in the shape of ahyperbola, which reflects the X rays to a focus. Suppose the vertices of such a mirror arelocated at ($3, 0) and (3, 0), and one focus is located at (5, 0). Write an equation thatmodels the hyperbola formed by the mirror. ! % 1y2
"x2"
xO
y
xO
y
4 8
8
4
–4
–8
–4–8
1"
(x $ 3)2"4
( y ! 2)2"4
(x $ 1)2"4
( y $ 2)2"1
x2"4
y2"16
(x ! 1)2"1
(y # 1)2"4
y2"
x2"
(x ! 1)2"9
(y # 5)2"16
x2"
y2"
(y # 2)2"16
(x ! 1)2"4
(x # 3)2"25
(y ! 2)2"9
x2"
y2"
x
y
O(–1, –2)
(1, –2)
(3, –2)x
y
O
(–3, 2 # $!34)
(–3, 2 ! $!34)
(–3, –1)(–3, 5)
4
8
4
–4
–4–8x
y
O
(0, 3$%5)
(0, –3$%5)
(0, 3)
(0, –3)
4 8
8
4
–4
–8
–4–8
Practice (Average)
Hyperbolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
Reading to Learn MathematicsHyperbolas
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 483 Glencoe Algebra 2
Less
on
8-5
Pre-Activity How are hyperbolas different from parabolas?
Read the introduction to Lesson 8-5 at the top of page 441 in your textbook.
Look at the sketch of a hyperbola in the introduction to this lesson. Listthree ways in which hyperbolas are different from parabolas.Sample answer: A hyperbola has two branches, while aparabola is one continuous curve. A hyperbola has two foci,while a parabola has one focus. A hyperbola has two vertices,while a parabola has one vertex.
Reading the Lesson
1. The graph at the right shows the hyperbola whose
equation in standard form is $ # 1.
The point (0, 0) is the of the hyperbola.
The points (4, 0) and ($4, 0) are the of the hyperbola.
The points (5, 0) and ($5, 0) are the of the hyperbola.
The segment connecting (4, 0) and ($4, 0) is called the axis.
The segment connecting (0, 3) and (0, $3) is called the axis.
The lines y # x and y # $ x are called the .
2. Study the hyperbola graphed at the right.
The center is .
The value of a is .
The value of c is .
To find b2, solve the equation # ! .
The equation in standard form for this hyperbola is .
Helping You Remember
3. What is an easy way to remember the equation relating the values of a, b, and c for ahyperbola? This equation looks just like the Pythagorean Theorem,although the variables represent different lengths in a hyperbola than ina right triangle.
"x42" ! "1
y22" % 1
b2a2c242
(0, 0)
x
y
O
asymptotes3"4
3"4
conjugatetransverse
foci
vertices
center
y2"9
x2"16
x
y
O(–4, 0) (4, 0)(–5, 0) (5, 0)
y % 34xy % – 34x
© Glencoe/McGraw-Hill 484 Glencoe Algebra 2
Rectangular Hyperbolas A rectangular hyperbola is a hyperbola with perpendicular asymptotes.For example, the graph of x2 $ y2 # 1 is a rectangular hyperbola. A hyperbolawith asymptotes that are not perpendicular is called a nonrectangularhyperbola. The graphs of equations of the form xy # c, where c is a constant,are rectangular hyperbolas.
Make a table of values and plot points to graph each rectangularhyperbola below. Be sure to consider negative values for thevariables. See students’ tables.1. xy # $4 2. xy # 3
3. xy # $1 4. xy # 8
5. Make a conjecture about the asymptotes of rectangular hyperbolas.
The coordinate axes are the asymptotes.
x
y
Ox
y
O
x
y
Ox
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
Study Guide and InterventionConic Sections
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 485 Glencoe Algebra 2
Less
on
8-6
Standard Form Any conic section in the coordinate plane can be described by anequation of the form
Ax2 ! Bxy ! Cy2 ! Dx ! Ey ! F # 0, where A, B, and C are not all zero.One way to tell what kind of conic section an equation represents is to rearrange terms andcomplete the square, if necessary, to get one of the standard forms from an earlier lesson.This method is especially useful if you are going to graph the equation.
Write the equation 3x2 ! 4y2 ! 30x ! 8y # 59 % 0 in standard form.State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
3x2 $ 4y2 $ 30x $ 8y ! 59 # 0 Original equation3x2 $ 30x $ 4y2 $ 8y # $59 Isolate terms.
3(x2 $ 10x ! ■) $ 4( y2 ! 2y ! ■) # $59 ! ■ ! ■ Factor out common multiples.3(x2 $ 10x ! 25) $ 4( y2 ! 2y ! 1) # $59 ! 3(25) ! ($4)(1) Complete the squares.
3(x $ 5)2 $ 4( y ! 1)2 # 12 Simplify.
$ # 1 Divide each side by 12.
The graph of the equation is a hyperbola with its center at (5, $1). The length of the transverse axis is 4 units and the length of the conjugate axis is 2$3# units.
Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola.
1. x2 ! y2 $ 6x ! 4y ! 3 # 0 2. x2 ! 2y2 ! 6x $ 20y ! 53 # 0
(x ! 3)2 # (y # 2)2 % 10; circle # % 1; ellipse
3. 6x2 $ 60x $ y ! 161 # 0 4. x2 ! y2 $ 4x $14y ! 29 # 0
y % 6(x ! 5)2 # 11; parabola (x ! 2)2 # (y ! 7)2 % 24; circle
5. 6x2 $ 5y2 ! 24x ! 20y $ 56 # 0 6. 3y2 ! x $ 24y ! 46 # 0
! % 1; hyperbola x % !3(y ! 4)2 # 2; parabola
7. x2 $ 4y2 $ 16x ! 24y $ 36 # 0 8. x2 ! 2y2 ! 8x ! 4y ! 2 # 0
! % 1; hyperbola # % 1; ellipse
9. 4x2 ! 48x ! y ! 158 # 0 10. 3x2 ! y2 $ 48x $ 4y ! 184 # 0
y % !4(x # 6)2 ! 14; parabola # % 1; ellipse
11. $3x2 ! 2y2 $ 18x ! 20y ! 5 # 0 12. x2 ! y2 ! 8x ! 2y ! 8 # 0
! % 1; hyperbola (x # 4)2 # (y # 1)2 % 9; circle(x # 3)2"6
(y # 5)2"9
(y ! 2)2"12
(x ! 8)2"4
(y # 1)2"8
(x # 4)2"16
(y ! 3)2"16
(x ! 8)2"64
(y ! 2)2"12
(x # 2)2"10
(y ! 5)2"3
(x # 3)2"6
( y ! 1)2"3
(x $ 5)2"4
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 486 Glencoe Algebra 2
Identify Conic Sections If you are given an equation of the formAx2 ! Bxy ! Cy2 ! Dx ! Ey ! F # 0, with B # 0,
you can determine the type of conic section just by considering the values of A and C. Referto the following chart.
Relationship of A and C Type of Conic Section
A # 0 or C # 0, but not both. parabola
A # C circle
A and C have the same sign, but A ) C. ellipse
A and C have opposite signs. hyperbola
Without writing the equation in standard form, state whether thegraph of each equation is a parabola, circle, ellipse, or hyperbola.
Study Guide and Intervention (continued)
Conic Sections
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
ExampleExample
a. 3x2 ! 3y2 # 5x # 12 % 0A # 3 and C # $3 have opposite signs, sothe graph of the equation is a hyperbola.
b. y2 % 7y ! 2x # 13A # 0, so the graph of the equation isa parabola.
ExercisesExercises
Without writing the equation in standard form, state whether the graph of eachequation is a parabola, circle, ellipse, or hyperbola.
1. x2 # 17x $ 5y ! 8 2. 2x2 ! 2y2 $ 3x ! 4y # 5parabola circle
3. 4x2 $ 8x # 4y2 $ 6y ! 10 4. 8(x $ x2) # 4(2y2 $ y) $ 100hyperbola circle
5. 6y2 $ 18 # 24 $ 4x2 6. y # 27x $ y2
ellipse parabola7. x2 # 4( y $ y2) ! 2x $ 1 8. 10x $ x2 $ 2y2 # 5y
ellipse ellipse9. x # y2 $ 5y ! x2 $ 5 10. 11x2 $ 7y2 # 77
circle hyperbola11. 3x2 ! 4y2 # 50 ! y2 12. y2 # 8x $ 11
circle parabola13. 9y2 $ 99y # 3(3x $ 3x2) 14. 6x2 $ 4 # 5y2 $ 3
circle hyperbola15. 111 # 11x2 ! 10y2 16. 120x2 $ 119y2 ! 118x $ 117y # 0
ellipse hyperbola17. 3x2 # 4y2 ! 12 18. 150 $ x2 # 120 $ y
hyperbola parabola
Skills PracticeConic Sections
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 487 Glencoe Algebra 2
Less
on
8-6
Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola. Then graph the equation.
1. x2 $ 25y2 # 25 hyperbola 2. 9x2 ! 4y2 # 36 ellipse 3. x2 ! y2 $ 16 # 0 circle! % 1 # % 1 x2 # y2 % 16
4. x2 ! 8x ! y2 # 9 circle 5. x2 ! 2x $ 15 # y parabola 6. 100x2 ! 25y2 # 400ellipse(x # 4)2 # y2 % 25 y % (x # 1)2 ! 16 # % 1
Without writing the equation in standard form, state whether the graph of eachequation is a parabola, circle, ellipse, or hyperbola.
7. 9x2 ! 4y2 # 36 ellipse 8. x2 ! y2 # 25 circle
9. y # x2 ! 2x parabola 10. y # 2x2 $ 4x $ 4 parabola
11. 4y2 $ 25x2 # 100 hyperbola 12. 16x2 ! y2 # 16 ellipse
13. 16x2 $ 4y2 # 64 hyperbola 14. 5x2 ! 5y2 # 25 circle
15. 25y2 ! 9x2 # 225 ellipse 16. 36y2 $ 4x2 # 144 hyperbola
17. y # 4x2 $ 36x $ 144 parabola 18. x2 ! y2 $ 144 # 0 circle
19. (x ! 3)2 ! ( y $ 1)2 # 4 circle 20. 25y2 $ 50y ! 4x2 # 75 ellipse
21. x2 $ 6y2 ! 9 # 0 hyperbola 22. x # y2 ! 5y $ 6 parabola
23. (x ! 5)2 ! y2 # 10 circle 24. 25x2 ! 10y2 $ 250 # 0 ellipse
x
y
O
xy
O 4 8
–4
–8
–12
–16
–4–8
x
y
O 4 8
8
4
–4
–8
–4–8
y2"
x2"
x
y
Ox
y
OxO
y
4 8
4
2
–2
–4
–4–8
y2"
x2"
y2"
x2"
© Glencoe/McGraw-Hill 488 Glencoe Algebra 2
Write each equation in standard form. State whether the graph of the equation isa parabola, circle, ellipse, or hyperbola. Then graph the equation.
1. y2 # $3x 2. x2 ! y2 ! 6x # 7 3. 5x2 $ 6y2 $ 30x $ 12y # $9parabola circle hyperbolax % ! y 2 (x # 3)2 # y2 % 16 ! % 1
4. 196y2 # 1225 $ 100x2 5. 3x2 # 9 $ 3y2 $ 6y 6. 9x2 ! y2 ! 54x $ 6y # $81ellipse circle ellipse
# % 1 x2 # (y # 1)2 % 4 # % 1
Without writing the equation in standard form, state whether the graph of eachequation is a parabola, circle, ellipse, or hyperbola.
7. 6x2 ! 6y2 # 36 8. 4x2 $ y2 # 16 9. 9x2 ! 16y2 $ 64y $ 80 # 0 circle hyperbola ellipse
10. 5x2 ! 5y2 $ 45 # 0 11. x2 ! 2x # y 12. 4y2 $ 36x2 ! 4x $ 144 # 0circle parabola hyperbola
13. ASTRONOMY A satellite travels in an hyperbolic orbit. It reaches the vertex of its orbit
at (5, 0) and then travels along a path that gets closer and closer to the line y # x.
Write an equation that describes the path of the satellite if the center of its hyperbolicorbit is at (0, 0).
! % 1y2"
x2"
2"5
(y ! 3)2"9
(x # 3)2"1
y2"
x2"
xO
y
x
y
Ox
y
O
(y # 1)2"5
(x ! 3)2"6
1"
Practice (Average)
Conic Sections
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
Reading to Learn MathematicsConic Sections
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 489 Glencoe Algebra 2
Less
on
8-6
Pre-Activity How can you use a flashlight to make conic sections?
Read the introduction to Lesson 8-6 at the top of page 449 in your textbook.
The figures in the introduction show how a plane can slice a double cone toform the conic sections. Name the conic section that is formed if the planeslices the double cone in each of the following ways:
• The plane is parallel to the base of the double cone and slices throughone of the cones that form the double cone. circle
• The plane is perpendicular to the base of the double cone and slicesthrough both of the cones that form the double cone. hyperbola
Reading the Lesson
1. Name the conic section that is the graph of each of the following equations. Give thecoordinates of the vertex if the conic section is a parabola and of the center if it is acircle, an ellipse, or a hyperbola.
a. ! # 1 ellipse; (3, !5)
b. x # $2( y ! 1)2 ! 7 parabola; (7, !1)c. (x $ 5)2 $ ( y ! 5)2 # 1 hyperbola; (5, !5)d. (x ! 6)2 ! ( y $ 2)2 # 1 circle; (!6, 2)
2. Each of the following is the equation of a conic section. For each equation, identify thevalues of A and C. Then, without writing the equation in standard form, state whetherthe graph of each equation is a parabola, circle, ellipse, or hyperbola.
a. 2x2 ! y2 $ 6x ! 8y ! 12 # 0 A # ; C # ; type of graph:
b. 2x2 ! 3x $ 2y $ 5 # 0 A # ; C # ; type of graph:
c. 5x2 ! 10x ! 5y2 $ 20y ! 1 # 0 A # ; C # ; type of graph:
d. x2 $ y2 ! 4x ! 2y $ 5 # 0 A # ; C # ; type of graph:
Helping You Remember
3. What is an easy way to recognize that an equation represents a parabola rather thanone of the other conic sections?
If the equation has an x2 term and y term but no y2 term, then the graphis a parabola. Likewise, if the equation has a y2 term and x term but nox2 term, then the graph is a parabola.
hyperbola!11circle55
parabola02ellipse12
( y ! 5)2"15
(x $ 3)2"36
© Glencoe/McGraw-Hill 490 Glencoe Algebra 2
LociA locus (plural, loci) is the set of all points, and only those points, that satisfya given set of conditions. In geometry, figures often are defined as loci. Forexample, a circle is the locus of points of a plane that are a given distancefrom a given point. The definition leads naturally to an equation whose graphis the curve described.
Write an equation of the locus of points that are thesame distance from (3, 4) and y % !4.
Recognizing that the locus is a parabola with focus (3, 4) and directrix y # $4,you can find that h # 3, k # 0, and a # 4 where (h, k) is the vertex and 4 unitsis the distance from the vertex to both the focus and directrix.
Thus, an equation for the parabola is y # "116"(x $ 3)2.
The problem also may be approached analytically as follows:
Let (x, y) be a point of the locus.
The distance from (3, 4) to (x, y) # the distance from y # $4 to (x, y).
$(x $ 3#)2 ! (#y $ 4)#2# # $(x $ x#)2 ! (#y $ ($#4))2#(x $ 3)2 ! y2 $ 8y ! 16 # y2 ! 8y ! 16
(x $ 3)2 # 16y
"116"(x $ 3)2 # y
Describe each locus as a geometric figure. Then write an equationfor the locus.
1. All points that are the same distance from (0, 5) and (4, 5).
2. All points that are 4 units from the origin.
3. All points that are the same distance from ($2, $1) and x # 2.
4. The locus of points such that the sum of the distances from ($2, 0) and (2, 0) is 6.
5. The locus of points such that the absolute value of the difference of the distances from ($3, 0) and (3, 0) is 2.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
ExampleExample
Study Guide and InterventionSolving Quadratic Systems
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 491 Glencoe Algebra 2
Less
on
8-7
Systems of Quadratic Equations Like systems of linear equations, systems ofquadratic equations can be solved by substitution and elimination. If the graphs are a conicsection and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conicsections, the system will have 0, 1, 2, 3, or 4 solutions.
Solve the system of equations. y % x2 ! 2x ! 15x # y % !3
Rewrite the second equation as y # $x $ 3 and substitute into the first equation.
$x $ 3 # x2 $ 2x $ 150 # x2 $ x $ 12 Add x ! 3 to each side.
0 # (x $ 4)(x ! 3) Factor.
Use the Zero Product property to getx # 4 or x # $3.
Substitute these values for x in x ! y # $3:
4 ! y # $3 or $3 ! y # $3y # $7 y # 0
The solutions are (4, $7) and ($3, 0).
Find the exact solution(s) of each system of equations.
1. y# x2 $ 5 2. x2 ! ( y $ 5)2 # 25y# x $ 3 y # $x2
(2, !1), (!1, !4) (0, 0)
3. x2 ! ( y $ 5)2 # 25 4. x2 ! y2 # 9y # x2 x2 ! y # 3
(0, 0), (3, 9), (!3, 9) (0, 3), ($5!, !2), (!$5!, !2)
5. x2 $ y2 # 1 6. y # x $ 3x2 ! y2 # 16 x # y2 $ 4
" , #, " , ! #, " , #, "! , #, "! , ! # " , #1 ! $29!""2
7 ! $29!""2$30!"$34!"$30!"$34!"
1 # $29!""27 # $29!""2
$30!"$34!"$30!"$34!"
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 492 Glencoe Algebra 2
Systems of Quadratic Inequalities Systems of quadratic inequalities can be solvedby graphing.
Solve the system of inequalities by graphing.x2 # y2 ) 25
"x ! #2# y2 *
The graph of x2 ! y2 * 25 consists of all points on or inside the circle with center (0, 0) and radius 5.The graph of
!x $ "2! y2 + consists of all points on or outside the
circle with center ! , 0" and radius . The solution of the
system is the set of points in both regions.
Solve the system of inequalities by graphing.x2 # y2 ) 25
! & 1
The graph of x2 ! y2 * 25 consists of all points on or inside the circle with center (0, 0) and radius 5.The graph of
$ & 1 are the points “inside” but not on the branches of
the hyperbola shown. The solution of the system is the set ofpoints in both regions.
Solve each system of inequalities below by graphing.
1. ! * 1 2. x2 ! y2 * 169 3. y + (x $ 2)2
y & x $ 2x2 ! 9y2 + 225 (x ! 1)2 ! ( y ! 1)2 * 16
x
y
Ox
y
O 6 12
12
6
–6
–12
–6–12x
y
O
1"2
y2"4
x2"16
x2"9
y2"4
x2"9
y2"4
x
y
O
5"2
5"2
25"4
5"2
25"4
5"2
x
y
O
Study Guide and Intervention (continued)
Solving Quadratic Systems
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeSolving Quadratic Systems
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 493 Glencoe Algebra 2
Less
on
8-7
Find the exact solution(s) of each system of equations.
1. y # x $ 2 (0, !2), (1, !1) 2. y # x ! 3 (!1, 2), 3. y # 3x (0, 0)y # x2 $ 2 y # 2x2 (1.5, 4.5) x # y2
4. y # x ($2!, $2!), 5. x # $5 (!5, 0) 6. y # 7 no solutionx2 ! y2 # 4 (!$2!, !$2!) x2 ! y2 # 25 x2 ! y2 # 9
7. y # $2x ! 2 (2, !2), 8. x $ y ! 1 # 0 (1, 2) 9. y # 2 $ x (0, 2), (3,!1)y2 # 2x " , 1# y2 # 4x y # x2 $ 4x ! 2
10. y # x $ 1 no solution 11. y # 3x2 (0, 0) 12. y # x2 ! 1 (!1, 2), y # x2 y # $3x2 y # $x2 ! 3 (1, 2)
13. y # 4x (!1, !4), (1, 4) 14. y # $1 (0, !1) 15. 4x2 ! 9y2 # 36 (!3, 0), 4x2 ! y2 # 20 4x2 ! y2 # 1 x2 $ 9y2 # 9 (3, 0)
16. 3( y ! 2)2 $ 4(x $ 3)2 # 12 17. x2 $ 4y2 # 4 (!2, 0), 18. y2 $ 4x2 # 4 no y # $2x ! 2 (0, 2), (3, !4) x2 ! y2 # 4 (2, 0) y # 2x solution
Solve each system of inequalities by graphing.
19. y * 3x $ 2 20. y * x 21. 4y2 ! 9x2 ' 144x2 ! y2 ' 16 y + $2x2 ! 4 x2 ! 8y2 ' 16
x
y
O 4 8
8
4
–4
–8
–4–8x
y
Ox
y
O
1"
© Glencoe/McGraw-Hill 494 Glencoe Algebra 2
Find the exact solution(s) of each system of equations.
1. (x $ 2)2 ! y2 # 5 2. x # 2( y ! 1)2 $ 6 3. y2 $ 3x2 # 6 4. x2 ! 2y2 # 1x $ y # 1 x ! y # 3 y # 2x $ 1 y # $x ! 1
(0, !1), (3, 2) (2, 1), (6.5, !3.5) (!1, !3), (5, 9) (1, 0), " , #5. 4y2 $ 9x2 # 36 6. y # x2 $ 3 7. x2 ! y2 # 25 8. y2 # 10 $ 6x2
4x2 $ 9y2 # 36 x2 ! y2 # 9 4y # 3x 4y2 # 40 $ 2x2
no solution (0, !3), (($5!, 2) (4, 3), (!4, !3) (0, ($10! )9. x2 ! y2 # 25 10. 4x2 ! 9y2 # 36 11. x # $( y $ 3)2 ! 2 12. $ # 1
x # 3y $ 5 2x2 $ 9y2 # 18 x # ( y $ 3)2 ! 3x2 ! y2 # 9
(!5, 0), (4, 3) ((3, 0) no solution ((3, 0)
13. 25x2 ! 4y2 # 100 14. x2 ! y2 # 4 15. x2 $ y2 # 3
x # $ ! # 1 y2 $ x2 # 3
no solution ((2, 0) no solution
16. ! # 1 17. x ! 2y # 3 18. x2 ! y2 # 64
3x2 $ y2 # 9x2 ! y2 # 9 x2 $ y2 # 8
((2, ($3!) (3, 0), "! , # ((6, (2$7!)Solve each system of inequalities by graphing.
19. y + x2 20. x2 ! y2 ' 36 21. ! * 1y & $x ! 2 x2 ! y2 + 16
(x ! 1)2 ! ( y $ 2)2 * 4
22. GEOMETRY The top of an iron gate is shaped like half an ellipse with two congruent segments from the center of theellipse to the ellipse as shown. Assume that the center ofthe ellipse is at (0, 0). If the ellipse can be modeled by theequation x2 ! 4y2 # 4 for y + 0 and the two congruent
segments can be modeled by y # x and y # $ x,
what are the coordinates of points A and B?
$3#"2
$3#"2
BA
(0, 0)
x
y
O
x
y
O 4 8
8
4
–4
–8
–4–8
x
y
O
(x ! 2)2"4
( y $ 3)2"16
12"
9"
y2"7
x2"7
y2"8
x2"4
5"2
y2"16
x2"9
2"
1"
Practice (Average)
Solving Quadratic Systems
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
"!1, # and "1, #$3!"$3!"
Reading to Learn MathematicsSolving Quadratic Systems
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 495 Glencoe Algebra 2
Less
on
8-7
Pre-Activity How do systems of equations apply to video games?
Read the introduction to Lesson 8-7 at the top of page 455 in your textbook.
The figure in your textbook shows that the spaceship hits the circular forcefield in two points. Is it possible for the spaceship to hit the force field ineither fewer or more than two points? State all possibilities and explainhow these could happen. Sample answer: The spaceship could hitthe force field in zero points if the spaceship missed the forcefield all together. The spaceship could also hit the force fieldin one point if the spaceship just touched the edge of theforce field.
Reading the Lesson
1. Draw a sketch to illustrate each of the following possibilities.
a. a parabola and a line b. an ellipse and a circle c. a hyperbola and athat intersect in that intersect in line that intersect in2 points 4 points 1 point
2. Consider the following system of equations.
x2 # 3 ! y2
2x2 ! 3y2 # 11
a. What kind of conic section is the graph of the first equation? hyperbolab. What kind of conic section is the graph of the second equation? ellipsec. Based on your answers to parts a and b, what are the possible numbers of solutions
that this system could have? 0, 1, 2, 3, or 4
Helping You Remember
3. Suppose that the graph of a quadratic inequality is a region whose boundary is a circle.How can you remember whether to shade the interior or the exterior of the circle?Sample answer: The solutions of an inequality of the form x2 # y2 ' r2
are all points that are less than r units from the origin, so the graph isthe interior of the circle. The solutions of an inequality of the form x2 # y2 & r2 are the points that are more than r units from the origin, sothe graph is the exterior of the circle.
© Glencoe/McGraw-Hill 496 Glencoe Algebra 2
Graphing Quadratic Equations with xy-TermsYou can use a graphing calculator to examine graphs of quadratic equations that contain xy-terms.
Use a graphing calculator to display the graph of x2 # xy # y2 % 4.
Solve the equation for y in terms of x by using the quadratic formula.
y2 ! xy ! (x2 $ 4) # 0
To use the formula, let a # 1, b # x, and c # (x2 $ 4).
y #
y #
To graph the equation on the graphing calculator, enter the two equations:
y # and y #
Use a graphing calculator to graph each equation. State the type of curve each graph represents.
1. y2 ! xy # 8 2. x2 ! y2 $ 2xy $ x # 0
3. x2 $ xy ! y2 # 15 4. x2 ! xy ! y2 # $9
5. 2x2 $ 2xy $ y2 ! 4x # 20 6. x2 $ xy $ 2y2 ! 2x ! 5y $ 3 # 0
$x $ $16 $#3x2#"""2
$x ! $16 $#3x2#"""2
$x ( $16 $#3x2#"""2
$x ( $x2 $ 4#(1)(x2#$ 4)#"""2
x
y
O 1–1–2 2
2
1
–1
–2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
ExampleExample
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 8-1)
Stu
dy
Gu
ide
and I
nte
rven
tion
Mid
poin
t and
Dis
tanc
e Fo
rmul
as
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-Hi
ll45
5G
lenc
oe A
lgeb
ra 2
Lesson 8-1
The
Mid
po
int
Form
ula
Mid
poin
t For
mul
aTh
e m
idpo
int M
of a
seg
men
t with
end
poin
ts (x
1, y 1
) and
(x2,
y 2) i
s !
, ".
y 1!
y 2"
2x 1
!x 2
"2
Fin
d t
he
mid
poi
nt
of t
he
lin
e se
gmen
t w
ith
en
dp
oin
ts a
t (4
,!7)
an
d (
!2,
3).
!,
"#!
,"
#!
,"o
r (1
,$2)
The
mid
poin
t of
the
seg
men
t is
(1,
$2)
.
$4
"2
2 " 2
$7
!3
"2
4 !
($2)
"" 2
y 1!
y 2"
2x 1
!x 2
"2
A d
iam
eter
A!B!
of a
cir
cle
has
en
dp
oin
ts A
(5,!
11)
and
B(!
7,6)
.W
hat
are
th
e co
ord
inat
es o
f th
e ce
nte
rof
th
e ci
rcle
?
The
cen
ter
of t
he c
ircl
e is
the
mid
poin
t of
all
of it
s di
amet
ers.
!,
"#!
,"
#!
,"o
r !$
1,$
2"
The
cir
cle
has
cent
er !$
1,$
2".
1 " 2
1 " 2$
5"2
$2
"2
$11
!6
"" 2
5 !
($7)
"" 2
y 1!
y 2"
2x 1
!x 2
"2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
mid
poi
nt
of e
ach
lin
e se
gmen
t w
ith
en
dp
oin
ts a
t th
e gi
ven
coo
rdin
ates
.
1.(1
2,7)
and
($
2,11
)2.
($8,
$3)
and
(10
,9)
3.(4
,15)
and
(10
,1)
(5,9
)(1
,3)
(7,8
)
4.($
3,$
3) a
nd (
3,3)
5.(1
5,6)
and
(12
,14)
6.(2
2,$
8) a
nd (
$10
,6)
(0,0
)(1
3.5,
10)
(6,!
1)
7.(3
,5)
and
($6,
11)
8.(8
,$15
) an
d ($
7,13
)9.
(2.5
,$6.
1) a
nd (
7.9,
13.7
)
"!,8
#"
,!1 #
(5.2
,3.8
)
10.(
$7,
$6)
and
($
1,24
) 11
.(3,
$10
) an
d (3
0,$
20)
12.(
$9,
1.7)
and
($
11,1
.3)
(!4,
9)"
,!15
#(!
10,1
.5)
13.S
egm
ent
M#N#
has
mid
poin
t P
.If
Mha
s co
ordi
nate
s (1
4,$
3) a
nd P
has
coor
dina
tes
($8,
6),w
hat
are
the
coor
dina
tes
of N
?(!
30,1
5)
14.C
ircl
e R
has
a di
amet
er S#
T#.I
f R
has
coor
dina
tes
($4,
$8)
and
Sha
s co
ordi
nate
s (1
,4),
wha
t ar
e th
e co
ordi
nate
s of
T?
(!9,
!20
)
15.S
egm
ent
A#D#
has
mid
poin
t B
,and
B#D#
has
mid
poin
t C
.If A
has
coor
dina
tes
($5,
4) a
nd
Cha
s co
ordi
nate
s (1
0,11
),w
hat
are
the
coor
dina
tes
of B
and
D?
Bis
"5,8
#,Dis
"15,
13#.1 " 3
2 " 3
33 " 21 " 23 " 2
©G
lenc
oe/M
cGra
w-Hi
ll45
6G
lenc
oe A
lgeb
ra 2
The
Dis
tan
ce F
orm
ula
Dist
ance
For
mul
aTh
e di
stan
ce b
etwe
en tw
o po
ints
(x1,
y 1) a
nd (x
2, y 2
) is
give
n by
d
#$
(x2
$#
x 1)2
!#
(y2
$#
y 1)2
#.
Wh
at i
s th
e d
ista
nce
bet
wee
n (
8,!
2) a
nd
(!
6,!
8)?
d#
$(x
2$
#x 1
)2!
#(y
2$
#y 1
)2#
Dist
ance
For
mul
a
#$
($6
$#
8)2
!#
[$8
$#
($2)
]#
2 #Le
t (x 1
, y1)
#(8
, $2)
and
(x2,
y 2) #
($6,
$8)
.
#$
($14
)#
2!
($#
6)2
#Su
btra
ct.
#$
196
!#
36#or
$23
2#
Sim
plify
.
The
dis
tanc
e be
twee
n th
e po
ints
is $
232
#or
abo
ut 1
5.2
unit
s.
Fin
d t
he
per
imet
er a
nd
are
a of
squ
are
PQ
RS
wit
h v
erti
ces
P(!
4,1)
,Q
(!2,
7),R
(4,5
),an
d S
(2,!
1).
Fin
d th
e le
ngth
of
one
side
to
find
the
per
imet
er a
nd t
he a
rea.
Cho
ose
P#Q#.
d#
$(x
2$
#x 1
)2!
#(y
2$
#y 1
)2#
Dist
ance
For
mul
a
#$
[$4
$#
($2)
]#
2!
(1#
$ 7
)2#
Let (
x 1, y
1) #
($4,
1) a
nd (x
2, y 2
) #($
2, 7
).
#$
($2)
2#
!($
6#
)2 #Su
btra
ct.
#$
40#or
2$
10#Si
mpl
ify.
Sinc
e on
e si
de o
f th
e sq
uare
is 2
$10#
,the
per
imet
er is
8$
10#un
its.
The
are
a is
(2$
10#)2 ,o
r40
uni
ts2 .
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
.
1.(3
,7)
and
($1,
4)
2.($
2,$
10)
and
(10,
$5)
3.
(6,$
6) a
nd (
$2,
0)
5 un
its13
uni
ts10
uni
ts4.
(7,2
) an
d (4
,$1)
5.
($5,
$2)
and
(3,
4)
6.(1
1,5)
and
(16
,9)
3$2!
units
10 u
nits
$41!
units
7.($
3,4)
and
(6,
$11
) 8.
(13,
9) a
nd (
11,1
5)
9.($
15,$
7) a
nd (
2,12
)
3$34!
units
2$10!
units
5$26!
units
10.R
ecta
ngle
AB
CD
has
vert
ices
A(1
,4),
B(3
,1),
C($
3,$
2),a
nd D
($5,
1).F
ind
the
peri
met
er a
nd a
rea
of A
BC
D.
2 $13 !
#6 $
5 !un
its;3
$65 !
units
2
11.C
ircl
e R
has
diam
eter
S#T#
wit
h en
dpoi
nts
S(4
,5)
and
T($
2,$
3).W
hat
are
the
circ
umfe
renc
e an
d ar
ea o
f th
e ci
rcle
? (E
xpre
ss y
our
answ
er in
ter
ms
of %
.)10
$un
its;2
5$un
its2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Mid
poin
t and
Dis
tanc
e Fo
rmul
as
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-1)
Skil
ls P
ract
ice
Mid
poin
t and
Dis
tanc
e Fo
rmul
as
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-Hi
ll45
7G
lenc
oe A
lgeb
ra 2
Lesson 8-1
Fin
d t
he
mid
poi
nt
of e
ach
lin
e se
gmen
t w
ith
en
dp
oin
ts a
t th
e gi
ven
coo
rdin
ates
.
1.(4
,$1)
,($
4,1)
(0,0
)2.
($1,
4),(
5,2)
(2,3
)
3.(3
,4),
(5,4
)(4
,4)
4.(6
,2),
(2,$
1)"4,
#
5.(3
,9),
($2,
$3)
",3
#6.
($3,
5),(
$3,
$8)
"!3,
!#
7.(3
,2),
($5,
0)(!
1,1)
8.(3
,$4)
,(5,
2)(4
,!1)
9.($
5,$
9),(
5,4)
"0,!
#10
.($
11,1
4),(
0,4)
"!,9
#
11.(
3,$
6),(
$8,
$3)
"!,!
#12
.(0,
10),
($2,
$5)
"!1,
#
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
.
13.(
4,12
),($
1,0)
13 u
nits
14.(
7,7)
,($
5,$
2)15
uni
ts
15.(
$1,
4),(
1,4)
2 un
its16
.(11
,11)
,(8,
15)
5 un
its
17.(
1,$
6),(
7,2)
10 u
nits
18.(
3,$
5),(
3,4)
9 un
its
19.(
2,3)
,(3,
5)$
5!un
its20
.($
4,3)
,($
1,7)
5 un
its
21.(
$5,
$5)
,(3,
10)
17 u
nits
22.(
3,9)
,($
2,$
3)13
uni
ts
23.(
6,$
2),(
$1,
3)$
74!un
its24
.($
4,1)
,(2,
$4)
$61!
units
25.(
0,$
3),(
4,1)
4$2!
units
26.(
$5,
$6)
,(2,
0)$
85!un
its
5 " 29 " 2
5 " 2
11 " 25 " 2
3 " 21 " 2
1 " 2
©G
lenc
oe/M
cGra
w-Hi
ll45
8G
lenc
oe A
lgeb
ra 2
Fin
d t
he
mid
poi
nt
of e
ach
lin
e se
gmen
t w
ith
en
dp
oin
ts a
t th
e gi
ven
coo
rdin
ates
.
1.(8
,$3)
,($
6,$
11)
(1,!
7)2.
($14
,5),
(10,
6)"!
2,#
3.($
7,$
6),(
1,$
2)(!
3,!
4)4.
(8,$
2),(
8,$
8)(8
,!5)
5.(9
,$4)
,(1,
$1)
"5,!
#6.
(3,3
),(4
,9)"
,6#
7.(4
,$2)
,(3,
$7)
",!
#8.
(6,7
),(4
,4)"5,
#9.
($4,
$2)
,($
8,2)
(!6,
0)10
.(5,
$2)
,(3,
7)"4,
#11
.($
6,3)
,($
5,$
7)"!
,!2 #
12.(
$9,
$8)
,(8,
3)"!
,!#
13.(
2.6,
$4.
7),(
8.4,
2.5)
(5.5
,!1.
1)14
. !$,6
", !,4
" ",5
#15
.($
2.5,
$4.
2),(
8.1,
4.2)
(2.8
,0)
16. !
,", !
$,$
" "!
,0#
Fin
d t
he
dis
tan
ce b
etw
een
eac
h p
air
of p
oin
ts w
ith
th
e gi
ven
coo
rdin
ates
.
17.(
5,2)
,(2,
$2)
5 un
its18
.($
2,$
4),(
4,4)
10 u
nits
19.(
$3,
8),(
$1,
$5)
$17
3!
units
20.(
0,1)
,(9,
$6)
$13
0!
units
21.(
$5,
6),(
$6,
6)1
unit
22.(
$3,
5),(
12,$
3)17
uni
ts
23.(
$2,
$3)
,(9,
3)$
157
!un
its24
.($
9,$
8),(
$7,
8)2$
65!un
its
25.(
9,3)
,(9,
$2)
5 un
its26
.($
1,$
7),(
0,6)
$17
0!
units
27.(
10,$
3),(
$2,
$8)
13 u
nits
28.(
$0.
5,$
6),(
1.5,
0)2$
10!un
its
29. !
,", !
1,"1
uni
t30
.($
4$2#,
$$
5#),(
$5$
2#,4$
5#)$
127
!un
its
31.G
EOM
ETRY
Cir
cle
Oha
s a
diam
eter
A#B#
.If
Ais
at
($6,
$2)
and
Bis
at
($3,
4),f
ind
the
cent
er o
f th
e ci
rcle
and
the
leng
th o
f it
s di
amet
er."!
,1#;3
$5!
units
32.G
EOM
ETRY
Fin
d th
e pe
rim
eter
of a
tri
angl
e w
ith
vert
ices
at
(1,$
3),(
$4,
9),a
nd ($
2,1)
.18
#2$
17!un
its
9 " 2
7 " 53 " 5
2 " 5
1 " 41 " 2
5 " 81 " 2
1 " 8
1 " 62 " 3
1 " 3
5 " 21 " 2
11 " 2
5 " 2
11 " 29 " 2
7 " 2
7 " 25 " 2
11 " 2
Pra
ctic
e (A
vera
ge)
Mid
poin
t and
Dis
tanc
e Fo
rmul
as
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 8-1)
Rea
din
g t
o L
earn
Math
emati
csM
idpo
int a
nd D
ista
nce
Form
ulas
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-Hi
ll45
9G
lenc
oe A
lgeb
ra 2
Lesson 8-1
Pre-
Act
ivit
yH
ow a
re t
he
Mid
poi
nt
and
Dis
tan
ce F
orm
ula
s u
sed
in
em
erge
ncy
med
icin
e?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
1 at
the
top
of
page
412
in y
our
text
book
.
How
do
you
find
dis
tanc
es o
n a
road
map
?
Sam
ple
answ
er:U
se th
e sc
ale
of m
iles
on th
e m
ap.Y
ou m
ight
also
use
a ru
ler.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he c
oord
inat
es o
f th
e m
idpo
int
of a
seg
men
t w
ith
endp
oint
s (x
1,y 1
) an
d (x
2,y 2
).
",
#b.
Exp
lain
how
to
find
the
mid
poin
t of
a s
egm
ent
if y
ou k
now
the
coo
rdin
ates
of
the
endp
oint
s.D
o no
t us
e su
bscr
ipts
in y
our
expl
anat
ion.
Sam
ple
answ
er:T
o fin
d th
e x-
coor
dina
te o
f the
mid
poin
t,ad
d th
e x-
coor
dina
tes
of th
e en
dpoi
nts
and
divi
de b
y tw
o.To
find
the
y-co
ordi
nate
of t
he m
idpo
int,
do th
e sa
me
with
the
y-co
ordi
nate
s of
the
endp
oint
s.
2.a.
Wri
te a
n ex
pres
sion
for
the
dis
tanc
e be
twee
n tw
o po
ints
wit
h co
ordi
nate
s (x
1,y 1
) an
d(x
2,y 2
).$
(x2
!!
x 1)2
#!
(y2
!!
y 1)2
!b.
Exp
lain
how
to
find
the
dis
tanc
e be
twee
n tw
o po
ints
.Do
not
use
subs
crip
ts in
you
rex
plan
atio
n.
Sam
ple
answ
er:F
ind
the
diffe
renc
e be
twee
n th
e x-
coor
dina
tes
and
squa
re it
.Fin
d th
e di
ffere
nce
betw
een
the
y-co
ordi
nate
s an
d sq
uare
it.A
dd th
e sq
uare
s.Th
en fi
nd th
e sq
uare
root
of t
he s
um.
3.C
onsi
der
the
segm
ent
conn
ecti
ng t
he p
oint
s ($
3,5)
and
(9,
11).
a.F
ind
the
mid
poin
t of
thi
s se
gmen
t.(3
,8)
b.F
ind
the
leng
th o
f th
e se
gmen
t.W
rite
you
r an
swer
in s
impl
ifie
d ra
dica
l for
m.
6$5!
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an t
he “
mid
”in
mid
poin
t he
lp y
ou r
emem
ber
the
mid
poin
t fo
rmul
a?
Sam
ple
answ
er:T
he m
idpo
inti
s th
e po
int i
n th
e m
iddl
eof
a s
egm
ent.
Itis
hal
fway
bet
wee
n th
e en
dpoi
nts.
The
coor
dina
tes
of th
e m
idpo
int a
refo
und
by fi
ndin
g th
e av
erag
e of
the
two
x-co
ordi
nate
s (a
dd th
em a
nddi
vide
by
2) a
nd th
e av
erag
e of
the
two
y-co
ordi
nate
s.
y 1#
y 2"
2x 1
#x 2
"2
©G
lenc
oe/M
cGra
w-Hi
ll46
0G
lenc
oe A
lgeb
ra 2
Qua
drat
ic F
orm
Con
side
r tw
o m
etho
ds f
or s
olvi
ng t
he f
ollo
win
g eq
uati
on.
(y$
2)2
$5(
y$
2) !
6#
0
One
way
to
solv
e th
e eq
uati
on is
to
sim
plif
y fi
rst,
then
use
fac
tori
ng.
y2$
4y!
4 $
5y!
10 !
6#
0y2
$9y
!20
#0
(y$
4)(y
$5)
#0
Thu
s,th
e so
luti
on s
et is
{4,
5}.
Ano
ther
way
to
solv
e th
e eq
uati
on is
fir
st t
o re
plac
e y
$2
by a
sin
gle
vari
able
.T
his
will
pro
duce
an
equa
tion
tha
t is
eas
ier
to s
olve
tha
n th
e or
igin
al e
quat
ion.
Let
t#
y$
2 an
d th
en s
olve
the
new
equ
atio
n.
(y$
2)2
$5(
y$
2) !
6#
0t2
$5t
!6
#0
(t$
2)(t
$3)
#0
Thu
s,t
is 2
or
3.Si
nce
t#
y$
2,th
e so
luti
on s
et o
f th
e or
igin
al e
quat
ion
is {
4,5}
.
Sol
ve e
ach
equ
atio
n u
sin
g tw
o d
iffe
ren
t m
eth
ods.
1.(z
!2)
2!
8(z
!2)
!7
#0
2.(3
x$
1)2
$(3
x$
1) $
20 #
0
{!3,
!9}
{2,!
1}
3.(2
t!
1)2
$4(
2t!
1) !
3 #
04.
(y2
$1)
2$
(y2
$1)
$2
#0
{0,1
}&0,
($
3!'
5.(a
2$
2)2
$2(
a2$
2) $
3 #
06.
(1 !
$c#)
2!
(1 !
$c#)
$6
#0
&(1,
($
5!'{1
}
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-1
8-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-2)
Stu
dy
Gu
ide
and I
nte
rven
tion
Para
bola
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-Hi
ll46
1G
lenc
oe A
lgeb
ra 2
Lesson 8-2
Equ
atio
ns
of
Para
bo
las
A p
arab
ola
is a
cur
ve c
onsi
stin
g of
all
poin
ts in
the
coor
dina
te p
lane
tha
t ar
e th
e sa
me
dist
ance
fro
m a
giv
en p
oint
(th
e fo
cus)
and
a g
iven
line
(the
dir
ectr
ix).
The
fol
low
ing
char
t su
mm
ariz
es im
port
ant
info
rmat
ion
abou
t pa
rabo
las.
Stan
dard
For
m o
f Equ
atio
ny
#a(
x$
h)2
!k
x#
a(y
$k)
2!
h
Axis
of S
ymm
etry
x#
hy
#k
Vert
ex(h
, k)
(h, k
)
Focu
s!h,
k!
"!h
!, k
"Di
rect
rixy
#k
$x
#h
$
Dire
ctio
n of
Ope
ning
upwa
rd if
a&
0, d
ownw
ard
if a
'0
right
if a
&0,
left
if a
'0
Leng
th o
f Lat
us R
ectu
m
uni
ts
uni
ts
Iden
tify
th
e co
ord
inat
es o
f th
e ve
rtex
an
d f
ocu
s,th
e eq
uat
ion
s of
the
axis
of
sym
met
ry a
nd
dir
ectr
ix,a
nd
th
e d
irec
tion
of
open
ing
of t
he
par
abol
aw
ith
equ
atio
n y
%2x
2!
12x
!25
.
y#
2x2
$12
x$
25O
rigin
al e
quat
ion
y#
2(x2
$6x
) $25
Fact
or 2
from
the
x-te
rms.
y#
2(x2
$6x
!■
) $25
$2(■
)Co
mpl
ete
the
squa
re o
n th
e rig
ht s
ide.
y#
2(x2
$6x
!9)
$25
$2(
9)Th
e 9
adde
d to
com
plet
e th
e sq
uare
is m
ultip
lied
by 2
.y
#2(
x$
3)2
$43
Writ
e in
sta
ndar
d fo
rm.
The
ver
tex
of t
his
para
bola
is lo
cate
d at
(3,
$43
),th
e fo
cus
is lo
cate
d at
!3,$
42",t
he
equa
tion
of
the
axis
of
sym
met
ry is
x#
3,an
d th
e eq
uati
on o
f th
e di
rect
rix
is y
#$
43.
The
par
abol
a op
ens
upw
ard.
Iden
tify
th
e co
ord
inat
es o
f th
e ve
rtex
an
d f
ocu
s,th
e eq
uat
ion
s of
th
e ax
is o
fsy
mm
etry
an
d d
irec
trix
,an
d t
he
dir
ecti
on o
f op
enin
g of
th
e p
arab
ola
wit
h t
he
give
n e
quat
ion
.
1.y
#x2
!6x
$4
2.y
#8x
$2x
2!
103.
x#
y2$
8y!
6
(!3,
!13
),(2
,18)
, "2,1
7#,
(!10
,4),
"!9
,4#,
"!3,
!12
#,x%
!3,
x%
2,y
%18
,y
%4,
x%
!10
,
y%
!13
,up
dow
nrig
ht
Wri
te a
n e
quat
ion
of
each
par
abol
a d
escr
ibed
bel
ow.
4.fo
cus
($2,
3),d
irec
trix
x#
$2
5.ve
rtex
(5,
1),f
ocus
!4,1
"x
%6(
y!
3)2
!2
x%
!3(
y!
1)2
#5
1 " 24
11 " 121 " 12
1 " 4
1 " 41 " 8
3 " 4
3 " 41 " 8
1 " 8
7 " 8
1 " a1 " a
1 " 4a1 " 4a
1 " 4a1 " 4a
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll46
2G
lenc
oe A
lgeb
ra 2
Gra
ph
Par
abo
las
To g
raph
an
equa
tion
for
a p
arab
ola,
firs
t pu
t th
e gi
ven
equa
tion
inst
anda
rd f
orm
.
y#
a(x
$h)
2!
kfo
r a
para
bola
ope
ning
up
or d
own,
orx
#a(
y$
k)2
!h
for
a pa
rabo
la o
peni
ng t
o th
e le
ft o
r ri
ght
Use
the
val
ues
of a
,h,a
nd k
to d
eter
min
e th
e ve
rtex
,foc
us,a
xis
of s
ymm
etry
,and
leng
th o
fth
e la
tus
rect
um.T
he v
erte
x an
d th
e en
dpoi
nts
of t
he la
tus
rect
um g
ive
thre
e po
ints
on
the
para
bola
.If
you
need
mor
e po
ints
to
plot
an
accu
rate
gra
ph,s
ubst
itut
e va
lues
for
poi
nts
near
the
ver
tex.
Gra
ph
y%
(x!
1)2
#2.
In t
he e
quat
ion,
a#
,h#
1,k
#2.
The
par
abol
a op
ens
up,s
ince
a&
0.ve
rtex
:(1,
2)ax
is o
f sy
mm
etry
:x#
1
focu
s:!1,
2 !
"or !1
,2"
leng
th o
f la
tus
rect
um:
or 3
uni
ts
endp
oint
s of
latu
s re
ctum
: !2,2
", !$
,2"
Th
e co
ord
inat
es o
f th
e fo
cus
and
th
e eq
uat
ion
of
the
dir
ectr
ix o
f a
par
abol
a ar
egi
ven
.Wri
te a
n e
quat
ion
for
eac
h p
arab
ola
and
dra
w i
ts g
rap
h.
1.(3
,5),
y#
12.
(4,$
4),y
#$
63.
(5,$
1),x
#3
y%
(x!
3)2
#3
y%
(x!
4)2
!5
x%
(y#
1)2
#4
1 " 41 " 4
1 " 8
x
y
Ox
y
O
x
y
O
3 " 41 " 2
3 " 41 " 2
1 " "1 3"
3 " 41
" 4 !"1 3" "x
y
O
1 " 3
1 " 3
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Para
bola
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 8-2)
Skil
ls P
ract
ice
Para
bola
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-Hi
ll46
3G
lenc
oe A
lgeb
ra 2
Lesson 8-2
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.
1.y
#x2
!2x
!2
2.y
#x2
$2x
!4
3.y
#x2
!4x
!1
y%
[x!
(!1)
]2#
1y
%(x
!1)
2#
3y
%[x
!(!
2)]2
#(!
3)
Iden
tify
th
e co
ord
inat
es o
f th
e ve
rtex
an
d f
ocu
s,th
e eq
uat
ion
s of
th
e ax
is o
fsy
mm
etry
an
d d
irec
trix
,an
d t
he
dir
ecti
on o
f op
enin
g of
th
e p
arab
ola
wit
h t
he
give
n e
quat
ion
.Th
en f
ind
th
e le
ngt
h o
f th
e la
tus
rect
um
an
d g
rap
h t
he
par
abol
a.
4.y
#(x
$2)
25.
x#
(y$
2)2
!3
6.y
#$
(x!
3)2
!4
vert
ex:(
2,0)
;ve
rtex
:(3,
2);
vert
ex:(
!3,
4);
focu
s:"2,
#;fo
cus:
"3,2
#;fo
cus:
"!3,3
#;ax
is o
f sym
met
ry:
axis
of s
ymm
etry
:ax
is o
f sym
met
ry:
x%
2;y
%2;
x%
!3;
dire
ctrix
:y%
!;
dire
ctrix
:x%
2;
dire
ctrix
:y%
4;
open
s up
;op
ens
right
;op
ens
dow
n;la
tus
rect
um:1
uni
tla
tus
rect
um:1
uni
tla
tus
rect
um:1
uni
t
Wri
te a
n e
quat
ion
for
eac
h p
arab
ola
des
crib
ed b
elow
.Th
en d
raw
th
e gr
aph
.
7.ve
rtex
(0,
0),
8.ve
rtex
(5,
1),
9.ve
rtex
(1,
3),
focu
s !0,
$"
focu
s !5,
"di
rect
rix
x#
y%
!3x
2y
%(x
!5)
2#
1x
%2(
y!
3)2
#1 x
y
Ox
y
O
x
y
O
7 " 85 " 4
1 " 12
1 " 43 " 4
1 " 4
3 " 41 " 4
1 " 4
x
y
Ox
y
O
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll46
4G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.
1.y
#2x
2$
12x
!19
2.y
#x2
!3x
!3.
y#
$3x
2$
12x
$7
y%
2(x
!3)
2#
1y
%[x
!(!
3)]2
#(!
4)y
%!
3[x
!(!
2)]2
#5
Iden
tify
th
e co
ord
inat
es o
f th
e ve
rtex
an
d f
ocu
s,th
e eq
uat
ion
s of
th
e ax
is o
fsy
mm
etry
an
d d
irec
trix
,an
d t
he
dir
ecti
on o
f op
enin
g of
th
e p
arab
ola
wit
h t
he
give
n e
quat
ion
.Th
en f
ind
th
e le
ngt
h o
f th
e la
tus
rect
um
an
d g
rap
h t
he
par
abol
a.
4.y
#(x
$4)
2!
35.
x#
$y2
!1
6.x
#3(
y!
1)2
$3
vert
ex:(
4,3)
;ve
rtex
:(1,
0);
vert
ex:(
!3,
!1)
;fo
cus:
"4,3
#;fo
cus:
",0
#;fo
cus:
"!2
,!1 #;
axis
:x%
4;ax
is:y
%0;
axis
:y%
!1;
dire
ctrix
:y%
2;
dire
ctrix
:x%
1;
dire
ctrix
:x%
!3
;op
ens
up;
open
s le
ft;op
ens
right
;la
tus
rect
um:1
uni
tla
tus
rect
um:3
uni
tsla
tus
rect
um:
unit
Wri
te a
n e
quat
ion
for
eac
h p
arab
ola
des
crib
ed b
elow
.Th
en d
raw
th
e gr
aph
.
7.ve
rtex
(0,
$4)
,8.
vert
ex ($
2,1)
,9.
vert
ex (
1,3)
,
focu
s !0,
$3
"di
rect
rix
x#
$3
axis
of
sym
met
ry x
#1,
latu
s re
ctum
:2 u
nits
,a
'0
y%
2x2
!4
x %
(y !
1)2
!2
y %
!(x
!1)
2#
3
10.T
ELEV
ISIO
NW
rite
the
equ
atio
n in
the
form
y#
ax2
for
a sa
telli
te d
ish.
Ass
ume
that
the
bott
om o
f th
e up
war
d-fa
cing
dis
h pa
sses
thr
ough
(0,
0) a
nd t
hat
the
dist
ance
fro
m t
hebo
ttom
to
the
focu
s po
int
is 8
inch
es.
y%
x21 " 32
x
y
Ox
y
Ox
y
O
1 " 21 " 4
7 " 8
1 " 3
1 " 123 " 4
3 " 4
11 " 121 " 4
1 " 4
x
y
Ox
y
O
x
y
O
1 " 3
1 " 2
1 " 21 " 2
Pra
ctic
e (A
vera
ge)
Para
bola
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-2)
Rea
din
g t
o L
earn
Math
emati
csPa
rabo
las
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-Hi
ll46
5G
lenc
oe A
lgeb
ra 2
Lesson 8-2
Pre-
Act
ivit
yH
ow a
re p
arab
olas
use
d i
n m
anu
fact
uri
ng?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
2 at
the
top
of
page
419
in y
our
text
book
.
Nam
e at
leas
t tw
o re
flec
tive
obj
ects
tha
t m
ight
hav
e th
e sh
ape
of a
para
bola
.
Sam
ple
answ
er:t
eles
cope
mirr
or,s
atel
lite
dish
Rea
din
g t
he
Less
on
1.In
the
par
abol
a sh
own
in t
he g
raph
,the
poi
nt (
2,$
2) is
cal
led
the
and
the
poin
t (2
,0)
is c
alle
d th
e
.The
line
y#
$4
is c
alle
d th
e
,and
the
line
x#
2 is
cal
led
the
.
2.a.
Wri
te t
he s
tand
ard
form
of
the
equa
tion
of
a pa
rabo
la t
hat
open
s up
war
d or
dow
nwar
d.y
%a(
x!
h)2
#k
b.T
he p
arab
ola
open
s do
wnw
ard
if
and
open
s up
war
d if
.T
he
equa
tion
of
the
axis
of
sym
met
ry is
,a
nd t
he c
oord
inat
es o
f th
e ve
rtex
are
.
3.A
par
abol
a ha
s eq
uati
on x
#$
(y$
2)2
!4.
Thi
s pa
rabo
la o
pens
to
the
.
It h
as v
erte
x an
d fo
cus
.The
dir
ectr
ix is
.T
he le
ngth
of t
he la
tus
rect
um is
un
its.
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an t
he w
ay in
whi
ch y
ou p
lot
poin
ts in
a r
ecta
ngul
ar c
oord
inat
e sy
stem
hel
p yo
u to
rem
embe
r w
hat
the
sign
of a
tells
you
abo
ut t
he d
irec
tion
in w
hich
a p
arab
ola
open
s?Sa
mpl
e an
swer
:In
plot
ting
poin
ts,a
pos
itive
x-c
oord
inat
e te
lls y
ou to
mov
e to
the
right
and
a ne
gativ
e x-
coor
dina
te te
lls y
ou to
mov
e to
the
left.
This
is li
ke a
par
abol
a w
hose
equ
atio
n is
of t
he fo
rm “x
%…
”;it
open
s to
the
right
if a
&0
and
to th
e le
ftif
a'
0.Li
kew
ise,
a po
sitiv
e y-
coor
dina
te te
lls y
ou to
mov
e up
and
a ne
gativ
e y-
coor
dina
te te
lls y
outo
mov
e do
wn.
This
is li
ke a
par
abol
a w
hose
equ
atio
n is
of t
he fo
rm
“y%
…”;
it op
ens
upw
ard
if a
&0
and
dow
nwar
dif
a'
0.
8x
%6
(2,2
)(4
,2)
left
1 " 8
(h,k
)x
%h
a&
0a
'0
axis
of s
ymm
etry
dire
ctrix
focu
svert
exx
y O
( 2, –
2)
( 2, 0
)
y % –
4
©G
lenc
oe/M
cGra
w-Hi
ll46
6G
lenc
oe A
lgeb
ra 2
Tang
ents
to P
arab
olas
A li
ne t
hat
inte
rsec
ts a
par
abol
a in
exa
ctly
one
poi
nt
wit
hout
cro
ssin
g th
e cu
rve
is a
tan
gen
tto
the
pa
rabo
la.T
he p
oint
whe
re a
tan
gent
line
tou
ches
a
para
bola
is t
he p
oin
t of
tan
gen
cy.T
he li
ne
perp
endi
cula
r to
a t
ange
nt t
o a
para
bola
at
the
poin
t of
tan
genc
yis
cal
led
the
nor
mal
to t
he p
arab
ola
at
that
poi
nt.I
n th
e di
agra
m,l
ine
!is
tan
gent
to
the
para
bola
tha
t is
the
gra
ph o
f y#
x2at
!"3 2" ,"9 4" ".
The
x-ax
is is
tan
gent
to
the
para
bola
at
O,a
nd t
he y
-axi
s is
the
nor
mal
to
the
para
bola
at
O.
Sol
ve e
ach
pro
blem
.
1.F
ind
an e
quat
ion
for
line
!in
the
dia
gram
.Hin
t:A
non
vert
ical
line
wit
h an
equa
tion
of
the
form
y#
mx
!b
will
be
tang
ent
to t
he g
raph
of y
#x2
at
!"3 2" ,"9 4" "i
f an
d on
ly if
!"3 2" ,"9 4" "i
s th
e on
ly p
air
of n
umbe
rs t
hat
sati
sfie
s bo
th
y#
x2an
d y
#m
x!
b.
m%
3,b
%!
"9 4" ,y
%3x
!"9 4"
2.If
ais
any
rea
l num
ber,
then
(a,a
2 ) b
elon
gs t
o th
e gr
aph
of y
#x2
.Exp
ress
m
and
bin
ter
ms
of a
to f
ind
an e
quat
ion
of t
he f
orm
y#
mx
!b
for
the
line
that
is t
ange
nt t
o th
e gr
aph
of y
#x2
at (
a,a2
).
m%
2a,b
%a2
,y%
(2a)
x#
(!a2
)or y
%2a
x!
a2
3.F
ind
an e
quat
ion
for
the
norm
al t
o th
e gr
aph
of y
#x2
at !"3 2" ,
"9 4" ".y
%!
"1 3" x#
"1 41 "
4.If
ais
a n
onze
ro r
eal n
umbe
r,fi
nd a
n eq
uati
on f
or t
he n
orm
al t
o th
e gr
aph
ofy
#x2
at (
a,a2
).
y%
"!" 21 a"
#x#
"a2#
"1 2" #
x
y
O
!
y %
x2
1–1
–2–3
2
6 5 4 3 2 1
3
!3 – 2, 9 – 4"
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-2
8-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 8-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Circ
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-Hi
ll46
7G
lenc
oe A
lgeb
ra 2
Lesson 8-3
Equ
atio
ns
of
Cir
cles
The
equ
atio
n of
a c
ircl
e w
ith
cent
er (h
,k)
and
radi
us r
unit
s is
(x
$h)
2!
(y$
k)2
#r2
.
Wri
te a
n e
quat
ion
for
a c
ircl
e if
th
e en
dp
oin
ts o
f a
dia
met
er a
re a
t(!
4,5)
an
d (
6,!
3).
Use
the
mid
poin
t fo
rmul
a to
fin
d th
e ce
nter
of
the
circ
le.
(h,k
) #!
,"
Mid
poin
t for
mul
a
#!
,"
(x1,
y 1) #
($4,
5),
(x2,
y 2) #
(6, $
3)
#!
,"o
r (1
,1)
Sim
plify
.
Use
the
coo
rdin
ates
of
the
cent
er a
nd o
ne e
ndpo
int
of t
he d
iam
eter
to
find
the
rad
ius.
r#
$(x
2$
x#
1)2
!#
(y2
$#
y 1)2
#Di
stan
ce fo
rmul
a
r#
$($
4 $
#1)
2!
#(5
$#
1)2
#(x
1, y 1
) #(1
, 1),
(x2,
y 2) #
($4,
5)
#$
($5)
2#
!42
##
$41#
Sim
plify
.
The
rad
ius
of t
he c
ircl
e is
$41#
,so
r2#
41.
An
equa
tion
of
the
circ
le is
(x
$1)
2!
(y$
1)2
#41
.
Wri
te a
n e
quat
ion
for
th
e ci
rcle
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
1.ce
nter
(8,
$3)
,rad
ius
6(x
!8)
2#
(y#
3)2
%36
2.ce
nter
(5,
$6)
,rad
ius
4(x
!5)
2#
(y#
6)2
%16
3.ce
nter
($5,
2),p
asse
s th
roug
h ($
9,6)
(x#
5)2
#(y
!2)
2%
32
4.en
dpoi
nts
of a
dia
met
er a
t (6
,6)
and
(10,
12)
(x!
8)2
#(y
!9)
2%
13
5.ce
nter
(3,
6),t
ange
nt t
o th
e x-
axis
(x!
3)2
#(y
!6)
2%
36
6.ce
nter
($4,
$7)
,tan
gent
to
x#
2(x
#4)
2#
(y#
7)2
%36
7.ce
nter
at
($2,
8),t
ange
nt t
o y
#$
4(x
#2)
2#
(y !
8)2
%14
4
8.ce
nter
(7,
7),p
asse
s th
roug
h (1
2,9)
(x!
7)2
#(y
!7)
2%
29
9.en
dpoi
nts
of a
dia
met
er a
re ($
4,$
2) a
nd (
8,4)
(x!
2)2
#(y
!1)
2%
45
10.e
ndpo
ints
of
a di
amet
er a
re ($
4,3)
and
(6,
$8)
(x!
1)2
#(y
#2.
5)2
%55
.25
2 " 22 " 2
5 !
($3)
"" 2
$4
!6
"2
y 1!
y 2"
2x 1
!x 2
"2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll46
8G
lenc
oe A
lgeb
ra 2
Gra
ph
Cir
cles
To g
raph
a c
ircl
e,w
rite
the
giv
en e
quat
ion
in t
he s
tand
ard
form
of
the
equa
tion
of
a ci
rcle
,(x
$h)
2!
(y$
k)2
#r2
.
Plo
t th
e ce
nter
(h,k
) of
the
cir
cle.
The
n us
e r
to c
alcu
late
and
plo
t th
e fo
ur p
oint
s (h
!r,
k),
(h$
r,k)
,(h,
k!
r),a
nd (
h,k
$r)
,whi
ch a
re a
ll po
ints
on
the
circ
le.S
ketc
h th
e ci
rcle
tha
tgo
es t
hrou
gh t
hose
fou
r po
ints
.
Fin
d t
he
cen
ter
and
rad
ius
of t
he
circ
le
wh
ose
equ
atio
n i
s x2
#2x
#y2
#4y
%11
.Th
en g
rap
h
the
circ
le. x2
!2x
!y2
!4y
#11
x2!
2x!
■!
y2!
4y!
■#
11 !
■
x2!
2x!
1 !
y2!
4y!
4 #
11 !
1 !
4(x
!1)
2!
(y!
2)2
#16
The
refo
re,t
he c
ircl
e ha
s it
s ce
nter
at
($1,
$2)
and
a r
adiu
s of
$
16##
4.Fo
ur p
oint
s on
the
cir
cle
are
(3,$
2),(
$5,
$2)
,($
1,2)
,an
d ($
1,$
6).
Fin
d t
he
cen
ter
and
rad
ius
of t
he
circ
le w
ith
th
e gi
ven
equ
atio
n.T
hen
gra
ph
th
eci
rcle
.
1.(x
$3)
2!
y2#
92.
x2!
(y!
5)2
#4
3.(x
$1)
2!
(y!
3)2
#9
(3,0
),r%
3(0
,!5)
,r%
2(1
,!3)
,r%
3
4.(x
$2)
2!
(y!
4)2
#16
5.x2
!y2
$10
x!
8y!
16 #
06.
x2!
y2$
4x!
6y#
12
(2,!
4),r
%4
(5,!
4),r
%5
(2,!
3),r
%5
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x
y
O
x2 #
2x #
y2 #
4y %
11
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Circ
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-3
8-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-3)
Skil
ls P
ract
ice
Circ
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-Hi
ll46
9G
lenc
oe A
lgeb
ra 2
Lesson 8-3
Wri
te a
n e
quat
ion
for
th
e ci
rcle
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
1.ce
nter
(0,
5),r
adiu
s 1
unit
2.ce
nter
(5,
12),
radi
us 8
uni
tsx2
#(y
!5)
2%
1(x
!5)
2#
(y!
12)2
%64
3.ce
nter
(4,
0),r
adiu
s 2
unit
s4.
cent
er (
2,2)
,rad
ius
3 un
its
(x!
4)2
#y2
%4
(x!
2)2
#(y
!2)
2%
9
5.ce
nter
(4,
$4)
,rad
ius
4 un
its
6.ce
nter
($6,
4),r
adiu
s 5
unit
s(x
!4)
2#
(y#
4)2
%16
(x#
6)2
#(y
!4)
2%
257.
endp
oint
s of
a d
iam
eter
at
($12
,0)
and
(12,
0)x2
#y2
%14
48.
endp
oint
s of
a d
iam
eter
at
($4,
0) a
nd ($
4,$
6)(x
#4)
2#
(y#
3)2
%9
9.ce
nter
at
(7,$
3),p
asse
s th
roug
h th
e or
igin
(x!
7)2
#(y
#3)
2%
5810
.cen
ter
at ($
4,4)
,pas
ses
thro
ugh
($4,
1)(x
#4)
2#
(y!
4)2
%9
11.c
ente
r at
($6,
$5)
,tan
gent
to
y-ax
is(x
#6)
2#
(y#
5)2
%36
12.c
ente
r at
(5,
1),t
ange
nt t
o x-
axis
(x!
5)2
#(y
!1)
2%
1
Fin
d t
he
cen
ter
and
rad
ius
of t
he
circ
le w
ith
th
e gi
ven
equ
atio
n.T
hen
gra
ph
th
eci
rcle
.
13.x
2!
y2#
914
.(x
$1)
2!
(y$
2)2
#4
15.(
x!
1)2
!y2
#16
(0,0
),3
units
(1,2
),2
units
(!1,
0),4
uni
ts
16.x
2!
(y!
3)2
#81
17.(
x$
5)2
!(y
!8)
2#
4918
.x2
!y2
$4y
$32
#0
(0,!
3),9
uni
ts(5
,!8)
,7 u
nits
(0,2
),6
units
x
y
O4
8
8 4 –4 –8
–4–8
x
y
O4
812
–4 –8 –12
x
y
O6
12
12 6 –6 –12
–6–1
2
x
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-Hi
ll47
0G
lenc
oe A
lgeb
ra 2
Wri
te a
n e
quat
ion
for
th
e ci
rcle
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
1.ce
nter
($4,
2),r
adiu
s 8
unit
s2.
cent
er (
0,0)
,rad
ius
4 un
its
(x#
4)2
#(y
!2)
2%
64x2
#y2
%16
3.ce
nter
!$,$
$3# ",
radi
us 5
$2#
unit
s4.
cent
er (
2.5,
4.2)
,rad
ius
0.9
unit
"x#
#2#
( y#
$3!)
2%
50(x
!2.
5)2
#(y
!4.
2)2
%0.
815.
endp
oint
s of
a d
iam
eter
at
($2,
$9)
and
(0,
$5)
(x#
1)2
#(y
#7)
2%
56.
cent
er a
t ($
9,$
12),
pass
es t
hrou
gh ($
4,$
5)(x
#9)
2#
(y#
12)2
%74
7.ce
nter
at
($6,
5),t
ange
nt t
o x-
axis
(x#
6)2
#(y
!5)
2%
25
Fin
d t
he
cen
ter
and
rad
ius
of t
he
circ
le w
ith
th
e gi
ven
equ
atio
n.T
hen
gra
ph
th
eci
rcle
.
8.(x
!3)
2!
y2#
169.
3x2
!3y
2#
1210
.x2
!y2
!2x
!6y
#26
(!3,
0),4
uni
ts(0
,0),
2 un
its(!
1,!
3),6
uni
ts
11.(
x $
1)2
!y2
!4y
#12
12.x
2$
6x!
y2#
013
.x2
!y2
!2x
!6y
#$
1(1
,!2)
,4 u
nits
(3,0
),3
units
(!1,
!3)
,3 u
nits
WEA
THER
For
Exe
rcis
es 1
4 an
d 1
5,u
se t
he
foll
owin
g in
form
atio
n.
On
aver
age,
the
circ
ular
eye
of a
hur
rica
ne is
abo
ut 1
5 m
iles
in d
iam
eter
.Gal
e w
inds
can
affe
ct a
n ar
ea u
p to
300
mile
s fr
om t
he s
torm
’s ce
nter
.In
1992
,Hur
rica
ne A
ndre
w d
evas
tate
dso
uthe
rn F
lori
da.A
sat
ellit
e ph
oto
of A
ndre
w’s
land
fall
show
ed t
he c
ente
r of
its
eye
on o
neco
ordi
nate
sys
tem
cou
ld b
e ap
prox
imat
ed b
y th
e po
int
(80,
26).
14.W
rite
an
equa
tion
to
repr
esen
t a
poss
ible
bou
ndar
y of
And
rew
’s e
ye.
(x!
80)2
#(y
!26
)2%
56.2
515
.Wri
te a
n eq
uati
on t
o re
pres
ent
a po
ssib
le b
ound
ary
of t
he a
rea
affe
cted
by
gale
win
ds.
(x!
80)2
#(y
!26
)2%
90,0
00
x
y
O
x
y
O
x
y
O
x
y
O4
8
4 –4 –8
–4–8
x
y
Ox
y
O
1 " 4
1 " 4
Pra
ctic
e (A
vera
ge)
Circ
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-3
8-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 8-3)
Rea
din
g t
o L
earn
Math
emati
csC
ircle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-Hi
ll47
1G
lenc
oe A
lgeb
ra 2
Lesson 8-3
Pre-
Act
ivit
yW
hy
are
circ
les
imp
orta
nt
in a
ir t
raff
ic c
ontr
ol?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
3 at
the
top
of
page
426
in y
our
text
book
.
A la
rge
hom
e im
prov
emen
t ch
ain
is p
lann
ing
to e
nter
a n
ew m
etro
polit
anar
ea a
nd n
eeds
to
sele
ct lo
cati
ons
for
its
stor
es.M
arke
t re
sear
ch h
as s
how
nth
at p
oten
tial
cus
tom
ers
are
will
ing
to t
rave
l up
to 1
2 m
iles
to s
hop
at o
neof
the
ir s
tore
s.H
ow c
an c
ircl
es h
elp
the
man
ager
s de
cide
whe
re t
o pl
ace
thei
r st
ore?
Sam
ple
answ
er:A
sto
re w
ill d
raw
cus
tom
ers
who
live
insi
de a
circ
le w
ith c
ente
r at t
he s
tore
and
a ra
dius
of
12 m
iles.
The
man
agem
ent s
houl
d se
lect
loca
tions
for w
hich
as m
any
peop
le a
s po
ssib
le li
ve w
ithin
a c
ircle
of r
adiu
s 12
mile
s ar
ound
one
of t
he s
tore
s.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he e
quat
ion
of t
he c
ircl
e w
ith
cent
er (h
,k)
and
radi
us r
.(x
!h)
2#
(y!
k)2
%r2
b.W
rite
the
equ
atio
n of
the
cir
cle
wit
h ce
nter
(4,
$3)
and
rad
ius
5.(x
!4)
2#
(y#
3)2
%25
c.T
he c
ircl
e w
ith
equa
tion
(x !
8)2
!y2
#12
1 ha
s ce
nter
an
d ra
dius
.
d.T
he c
ircl
e w
ith
equa
tion
(x$
10)2
!(y
!10
)2#
1 ha
s ce
nter
an
d
radi
us
.
2.a.
In o
rder
to
find
cent
er a
nd r
adiu
s of
the
cir
cle
wit
h eq
uati
on x
2!
y2!
4x$
6y$
3 #
0,
it is
nec
essa
ry t
o .F
ill in
the
mis
sing
par
ts o
f th
ispr
oces
s.
x2!
y2!
4x$
6y$
3 #
0
x2!
y2!
4x$
6y#
x2!
4x!
!y2
$6y
!#
!!
(x!
)2!
(y$
)2#
b.T
his
circ
le h
as r
adiu
s 4
and
cent
er a
t .
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an t
he d
ista
nce
form
ula
help
you
to
rem
embe
r th
e eq
uati
on o
f a
circ
le?
Sam
ple
answ
er:W
rite
the
dist
ance
form
ula.
Repl
ace
(x1,
y 1) w
ith (h
,k)
and
(x2,
y 2) w
ith (x
,y).
Repl
ace
dw
ith r.
Squa
re b
oth
side
s.No
w y
ouha
ve th
e eq
uatio
n of
a c
ircle
.
(!2,
3)16
32
94
39
43
com
plet
e th
e sq
uare
1(1
0,!
10)
11(!
8,0)
©G
lenc
oe/M
cGra
w-Hi
ll47
2G
lenc
oe A
lgeb
ra 2
Tang
ents
to C
ircle
sA
line
tha
t in
ters
ects
a c
ircl
e in
exa
ctly
one
poi
nt is
a
tan
gen
tto
the
cir
cle.
In t
he d
iagr
am,l
ine
!is
ta
ngen
t to
the
cir
cle
wit
h eq
uati
on x
2!
y2#
25 a
t th
e po
int
who
se c
oord
inat
es a
re (
3,4)
.
A li
ne is
tan
gent
to
a ci
rcle
at
a po
int
Pon
the
cir
cle
if a
nd o
nly
if t
he li
ne is
per
pend
icul
ar t
o th
e ra
dius
fr
om t
he c
ente
r of
the
cir
cle
to p
oint
P.T
his
fact
en
able
s yo
u to
fin
d an
equ
atio
n of
the
tan
gent
to
a ci
rcle
at
a po
int
Pif
you
kno
w a
n eq
uati
on f
or t
he
circ
le a
nd t
he c
oord
inat
es o
f P.
Use
th
e d
iagr
am a
bove
to
solv
e ea
ch p
robl
em.
1.W
hat
is t
he s
lope
of
the
radi
us t
o th
e po
int
wit
h co
ordi
nate
s (3
,4)?
Wha
t is
the
slop
e of
the
tan
gent
to
that
poi
nt?
"4 3" ,!
"3 4"
2.F
ind
an e
quat
ion
of t
he li
ne !
that
is t
ange
nt t
o th
e ci
rcle
at
(3,4
).
y%
!"3 4" x
#"2 45 "
3.If
kis
a r
eal n
umbe
r be
twee
n $
5 an
d 5,
how
man
y po
ints
on
the
circ
le h
ave
x-co
ordi
nate
k?
Stat
e th
e co
ordi
nate
s of
the
se p
oint
s in
ter
ms
of k
.
two,
( k,(
$25
!!
k2 !)
4.D
escr
ibe
how
you
can
fin
d eq
uati
ons
for
the
tang
ents
to
the
poin
ts y
ou n
amed
for
Exe
rcis
e 3.
Use
the
coor
dina
tes
of (0
,0) a
nd o
f one
of t
he g
iven
poi
nts.
Find
the
slop
e of
the
radi
us to
that
poi
nt.U
se th
e sl
ope
of th
e ra
dius
to fi
nd w
hat
the
slop
e of
the
tang
ent m
ust b
e.Us
e th
e sl
ope
of th
e ta
ngen
t and
the
coor
dina
tes
of th
e po
int o
n th
e ci
rcle
to fi
nd a
n eq
uatio
n fo
r the
tang
ent.
5.F
ind
an e
quat
ion
for
the
tang
ent
at ($
3,4)
.
y%
"3 4" x#
"2 x5 "
5
–5
–5
5
(3, 4
)
y
xO
!x2
# y
2 % 2
5
En
rich
men
t
NAM
E__
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____
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____
DATE
____
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PERI
OD
____
_
8-3
8-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-4)
Stu
dy
Gu
ide
and I
nte
rven
tion
Ellip
ses
NAM
E__
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OD
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_
8-4
8-4
©G
lenc
oe/M
cGra
w-Hi
ll47
3G
lenc
oe A
lgeb
ra 2
Lesson 8-4
Equ
atio
ns
of
Ellip
ses
An
elli
pse
is t
he s
et o
f al
l poi
nts
in a
pla
ne s
uch
that
the
sum
of t
he d
ista
nces
fro
m t
wo
give
n po
ints
in t
he p
lane
,cal
led
the
foci
,is
cons
tant
.An
ellip
seha
s tw
o ax
es o
f sy
mm
etry
whi
ch c
onta
in t
he m
ajor
and
min
or a
xes.
In t
he t
able
,the
leng
ths
a,b,
and
car
e re
late
d by
the
for
mul
a c2
#a2
$ b
2 .
Stan
dard
For
m o
f Equ
atio
n!
#1
!#
1
Cent
er(h
, k)
(h, k
)
Dire
ctio
n of
Maj
or A
xis
Horiz
onta
lVe
rtica
l
Foci
(h!
c, k
), (h
$c,
k)
(h, k
$c)
, (h,
k!
c)
Leng
th o
f Maj
or A
xis
2aun
its2a
units
Leng
th o
f Min
or A
xis
2bun
its2b
units
Wri
te a
n e
quat
ion
for
th
e el
lip
se s
how
n.
The
leng
th o
f th
e m
ajor
axi
s is
the
dis
tanc
e be
twee
n ($
2,$
2)
and
($2,
8).T
his
dist
ance
is 1
0 un
its.
2a#
10,s
o a
#5
The
foc
i are
loca
ted
at ($
2,6)
and
($2,
0),s
o c
#3.
b2#
a2$
c2#
25 $
9#
16T
he c
ente
r of
the
elli
pse
is a
t ($
2,3)
,so
h#
$2,
k#
3,a2
#25
,and
b2
#16
.The
maj
or a
xis
is v
erti
cal.
An
equa
tion
of
the
ellip
se is
!
#1.
Wri
te a
n e
quat
ion
for
th
e el
lip
se t
hat
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.en
dpoi
nts
of m
ajor
axi
s at
($7,
2) a
nd (5
,2),
endp
oint
s of
min
or a
xis
at ($
1,0)
and
($1,
4)
#%
1
2.m
ajor
axi
s 8
unit
s lo
ng a
nd p
aral
lel t
o th
e x-
axis
,min
or a
xis
2 un
its
long
,cen
ter
at ($
2,$
5)
#(y
#5)
2%
1
3.en
dpoi
nts
of m
ajor
axi
s at
($8,
4) a
nd (
4,4)
,foc
i at
($3,
4) a
nd ($
1,4)
#%
1
4.en
dpoi
nts
of m
ajor
axi
s at
(3,2
) and
(3,$
14),
endp
oint
s of
min
or a
xis
at ($
1,$
6) a
nd (7
,$6)
#%
1
5.m
inor
axi
s 6
unit
s lo
ng a
nd p
aral
lel t
o th
e x-
axis
,maj
or a
xis
12 u
nits
long
,cen
ter
at (6
,1)
#%
1(x
!6)
2"
9(y
!1)
2"
36
(x!
3)2
"16
(y#
6)2
"64
(y!
4)2
"35
(x#
2)2
"36
(x#
2)2
"16
(y!
2)2
"4
(x#
1)2
"36
(x!
2)2
"16
(y$
3)2
"25
x
F 1 F 2O
y
(x$
h)2
"b2
(y$
k)2
"a2
(y$
k)2
"b2
(x$
h)2
"a2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll47
4G
lenc
oe A
lgeb
ra 2
Gra
ph
Elli
pse
sTo
gra
ph a
n el
lipse
,if
nece
ssar
y,w
rite
the
giv
en e
quat
ion
in t
hest
anda
rd f
orm
of
an e
quat
ion
for
an e
llips
e.
!#
1 (f
or e
llips
e w
ith
maj
or a
xis
hori
zont
al)
or
!#
1 (f
or e
llips
e w
ith
maj
or a
xis
vert
ical
)
Use
the
cen
ter
(h,k
) an
d th
e en
dpoi
nts
of t
he a
xes
to p
lot
four
poi
nts
of t
he e
llips
e.To
mak
ea
mor
e ac
cura
te g
raph
,use
a c
alcu
lato
r to
fin
d so
me
appr
oxim
ate
valu
es f
or x
and
yth
atsa
tisf
y th
e eq
uati
on.
Gra
ph
th
e el
lip
se 4
x2#
6y2
#8x
!36
y%
!34
.
4x2
!6y
2!
8x$
36y
#$
344x
2!
8x!
6y2
$36
y#
$34
4(x2
!2x
!■
) !6(
y2$
6y!
■) #
$34
!■
4(x2
!2x
!1)
!6(
y2$
6y!
9) #
$34
!58
4(x
!1)
2!
6(y
$3)
2#
24
!#
1
The
cen
ter
of t
he e
llips
e is
($1,
3).S
ince
a2
#6,
a#
$6#.
Sinc
e b2
#4,
b#
2.T
he le
ngth
of
the
maj
or a
xis
is 2
$6#,
and
the
leng
th o
f th
e m
inor
axi
s is
4.S
ince
the
x-t
erm
has
the
grea
ter
deno
min
ator
,the
maj
or a
xis
is h
oriz
onta
l.P
lot
the
endp
oint
s of
the
axe
s.T
hen
grap
h th
e el
lipse
.
Fin
d t
he
coor
din
ates
of
the
cen
ter
and
th
e le
ngt
hs
of t
he
maj
or a
nd
min
or a
xes
for
the
elli
pse
wit
h t
he
give
n e
quat
ion
.Th
en g
rap
h t
he
elli
pse
.
1.!
#1
(0,0
),4$
3!,6
2.!
#1
(0,0
),10
,4
3.x2
!4y
2!
24y
#$
32(0
,!3)
,4,2
4.9x
2!
6y2
$36
x!
12y
#12
(2,!
1),6
,2$
6!
x
y
Ox
y
O
x
y
Ox
y
O
y2" 4
x2" 25
x2" 9
y2" 12
(y$
3)2
"4
(x!
1)2
"6
xO
y
4x2 #
6y2
# 8
x ! 3
6y %
!34
(x$
h)2
"b2
(y$
k)2
"a2
(y$
k)2
"b2
(x$
h)2
"a2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Ellip
ses
NAM
E__
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____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-4
8-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lencoe/McG
raw-HillA12
Glencoe Algebra 2
Answers
(Lesson 8-4)
Skills PracticeEllipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 475 Glencoe Algebra 2
Less
on
8-4
Write an equation for each ellipse.
1. 2. 3.
# % 1 # % 1 # % 1
Write an equation for the ellipse that satisfies each set of conditions.
4. endpoints of major axis 5. endpoints of major axis 6. endpoints of major axis at (0, 6) and (0, $6), at (2, 6) and (8, 6), at (7, 3) and (7, 9),endpoints of minor axis endpoints of minor axis endpoints of minor axis at ($3, 0) and (3, 0) at (5, 4) and (5, 8) at (5, 6) and (9, 6)
# % 1 # % 1 # % 1
7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis atand parallel to x-axis, at ($6, 0) and (6, 0), foci (0, 12) and (0, $12), foci atminor axis 4 units long, at ($$32#, 0) and ($32#, 0) (0, $23# ) and (0, $$23# )center at (0, 0)
# % 1 # % 1 # % 1
Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.
10. ! # 1 11. ! # 1 12. ! # 1
(0, 0); (0, ($19!); (0, 0); ((6$2!, 0); (0, 0), (0, (2$6!);20; 18 18; 6 14; 10
x
y
O 4 8
8
4
–4
–8
–4–8x
y
O 4 8
8
4
–4
–8
–4–8x
y
O 4 8
8
4
–4
–8
–4–8
x2"25
y2"49
y2"9
x2"81
x2"81
y2"100
x2"121
y2"144
y2"4
x2"36
y2"4
x2"36
(x ! 7)2"4
(y ! 6)2"9
(y ! 6)2"4
(x ! 5)2"9
x2"9
y2"36
(y ! 2)2"9
x2"16
x2"16
y2"25
y2"4
x2"9
xO
y(0, 5)
(0, –1)
(–4, 2) (4, 2)
xO
y
(0, 3)
(0, –3)
(0, –5)
(0, 5)
xO
y
(0, 2)
(0, –2)
(–3, 0)(3, 0)
© Glencoe/McGraw-Hill 476 Glencoe Algebra 2
Write an equation for each ellipse.
1. 2. 3.
# % 1 # % 1 # % 1
Write an equation for the ellipse that satisfies each set of conditions.
4. endpoints of major axis 5. endpoints of major axis 6. major axis 20 units long at ($9, 0) and (9, 0), at (4, 2) and (4, $8), and parallel to x-axis,endpoints of minor axis endpoints of minor axis minor axis 10 units long,at (0, 3) and (0, $3) at (1, $3) and (7, $3) center at (2, 1)
# % 1 # % 1 # % 1
7. major axis 10 units long, 8. major axis 16 units long, 9. endpoints of minor axis minor axis 6 units long center at (0, 0), foci at at (0, 2) and (0, $2), foci and parallel to x-axis, (0, 2$15# ) and (0, $2$15# ) at ($4, 0) and (4, 0)center at (2, $4)
# % 1 # % 1 # % 1
Find the coordinates of the center and foci and the lengths of the major andminor axes for the ellipse with the given equation. Then graph the ellipse.
10. ! # 1 11. ! # 1 12. ! # 1
(0, 0); (0, ($7!); 8; 6 (3, 1); (3, 1 ( $35! ); (!4, !3);12; 2 (!4 ( 2$6!, !3); 14; 10
13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that thecenter of the first loop is at the origin, with the second loop to its right. Write an equationto model the first loop if its major axis (along the x-axis) is 12 feet long and its minoraxis is 6 feet long. Write another equation to model the second loop.
# % 1; # % 1y2"9
(x ! 12)2""36
y2"9
x2"36
4
4
–4
–8
–12
–4–8 x
y
O
x
y
O 4 8
8
4
–4
–8
–4–8x
y
O
( y ! 3)2"25
(x ! 4)2"49
(x $ 3)2"1
( y $ 1)2"36
x2"9
y2"16
y2"4
x2"20
x2"4
y2"64
(x ! 2)2"9
(y # 4)2"25
(y ! 1)2"25
(x ! 2)2"100
(x ! 4)2"9
(y # 3)2"25
y2"9
x2"81
(y ! 3)2"9
(x # 1)2"25
x2"4
(y ! 2)2"9
y2"9
x2"121
xO
y
(–5, 3)
(–6, 3)
(3, 3)
(4, 3)
xO
y
(0, 2 ! $%5)
(0, 2 # $%5)
(0, –1)
(0, 5)
xO
y(0, 3)
(0, –3)
(–11, 0) (11, 0)6 12
2
–2
–6–12
Practice (Average)
Ellipses
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-4)
Rea
din
g t
o L
earn
Math
emati
csEl
lipse
s
NAM
E__
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____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-Hi
ll47
7G
lenc
oe A
lgeb
ra 2
Lesson 8-4
Pre-
Act
ivit
yW
hy
are
elli
pse
s im
por
tan
t in
th
e st
ud
y of
th
e so
lar
syst
em?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
4 at
the
top
of
page
433
in y
our
text
book
.
Is t
he E
arth
alw
ays
the
sam
e di
stan
ce f
rom
the
Sun
? E
xpla
in y
our
answ
erus
ing
the
wor
ds c
ircl
ean
d el
lips
e.No
;if t
he E
arth
’s o
rbit
wer
e a
circ
le,i
t wou
ld a
lway
s be
the
sam
e di
stan
ce fr
om th
e Su
nbe
caus
e ev
ery
poin
t on
a ci
rcle
is th
e sa
me
dist
ance
from
the
cent
er.H
owev
er,t
he E
arth
’s o
rbit
is a
n el
lipse
,and
the
poin
tson
an
ellip
se a
re n
ot a
ll th
e sa
me
dist
ance
from
the
cent
er.
Rea
din
g t
he
Less
on
1.A
n el
lipse
is t
he s
et o
f al
l poi
nts
in a
pla
ne s
uch
that
the
of
the
dist
ance
s fr
om t
wo
fixe
d po
ints
is
.The
tw
o fi
xed
poin
ts a
re c
alle
d th
e
of t
he e
llips
e.
2.C
onsi
der
the
ellip
se w
ith
equa
tion
!
#1.
a.Fo
r th
is e
quat
ion,
a#
and
b#
.
b.W
rite
an
equa
tion
tha
t re
late
s th
e va
lues
of a
,b,a
nd c
.c2
%a2
!b2
c.F
ind
the
valu
e of
cfo
r th
is e
llips
e.$
5!
3.C
onsi
der
the
ellip
ses
wit
h eq
uati
ons
!#
1 an
d !
#1.
Com
plet
e th
e
follo
win
g ta
ble
to d
escr
ibe
char
acte
rist
ics
of t
heir
gra
phs.
Stan
dard
For
m o
f Equ
atio
n!
#1
!#
1
Dire
ctio
n of
Maj
or A
xis
vert
ical
horiz
onta
l
Dire
ctio
n of
Min
or A
xis
horiz
onta
lve
rtic
al
Foci
(0,3
),(0
,!3)
( $5!,
0) ,( !
$5!,
0)Le
ngth
of M
ajor
Axi
s10
uni
ts6
units
Leng
th o
f Min
or A
xis
8 un
its4
units
Hel
pin
g Y
ou
Rem
emb
er4.
Som
e st
uden
ts h
ave
trou
ble
rem
embe
ring
the
tw
o st
anda
rd f
orm
s fo
r th
e eq
uati
on o
f an
ellip
se.H
ow c
an y
ou r
emem
ber
whi
ch t
erm
com
es f
irst
and
whe
re t
o pl
ace
a an
d b
inth
ese
equa
tion
s?Th
e x-
axis
is h
oriz
onta
l.If
the
maj
or a
xis
is h
oriz
onta
l,th
e fir
st te
rm is
.T
he y
-axi
s is
ver
tical
.If t
he m
ajor
axi
s is
ver
tical
,the
first
term
is
.ais
alw
ays
the
larg
er o
f the
num
bers
aan
d b.
y2" a2x2" a2
y2" 4
x2" 9
x2" 16
y2" 25
y2" 4
x2" 9
x2" 16
y2" 25
23
y2" 4
x2" 9
foci
cons
tant
sum
©G
lenc
oe/M
cGra
w-Hi
ll47
8G
lenc
oe A
lgeb
ra 2
Ecce
ntric
ity
In a
n el
lipse
,the
rat
io " dc "
is c
alle
d th
e ec
cen
tric
ity
and
is d
enot
ed b
y th
e
lett
er e
.Ecc
entr
icit
y m
easu
res
the
elon
gati
on o
f an
ellip
se.T
he c
lose
r e
is t
o 0,
the
mor
e an
elli
pse
look
s lik
e a
circ
le.T
he c
lose
r e
is t
o 1,
the
mor
e el
onga
ted
it is
.Rec
all t
hat
the
equa
tion
of
an e
llips
e is
" ax2 2"!
" by2 2"#
1 or
" bx2 2"!
" ay2 2"#
1
whe
re a
is t
he le
ngth
of
the
maj
or a
xis,
and
that
c#
$a2
$b
#2 #.
Fin
d t
he
ecce
ntr
icit
y of
eac
h e
llip
se r
oun
ded
to
the
nea
rest
hu
nd
red
th.
1."x 92 "
!" 3y 62 "
#1
2." 8x 12 "
!"y 92 "
#1
3."x 42 "
!"y 92 "
#1
0.87
0.94
0.75
4." 1x 62 "
!"y 92 "
#1
5." 3x 62 "
!" 1y 62 "
#1
6."x 42 "
!" 3y 62 "
#1
0.66
0.75
0.94
7.Is
a c
ircl
e an
elli
pse?
Exp
lain
you
r re
ason
ing.
Yes;
it is
an
ellip
se w
ith e
ccen
trici
ty 0
.
8.T
he c
ente
r of
the
sun
is o
ne f
ocus
of
Ear
th's
orb
it a
roun
d th
e su
n.T
hele
ngth
of
the
maj
or a
xis
is 1
86,0
00,0
00 m
iles,
and
the
foci
are
3,2
00,0
00m
iles
apar
t.F
ind
the
ecce
ntri
city
of
Ear
th's
orb
it.
appr
oxim
atel
y 0.
17
9.A
n ar
tifi
cial
sat
ellit
e or
biti
ng t
he e
arth
tra
vels
at
an a
ltit
ude
that
var
ies
betw
een
132
mile
s an
d 58
3 m
iles
abov
e th
e su
rfac
e of
the
ear
th.I
f th
ece
nter
of
the
eart
h is
one
foc
us o
f it
s el
lipti
cal o
rbit
and
the
rad
ius
of t
heea
rth
is 3
950
mile
s,w
hat
is t
he e
ccen
tric
ity
of t
he o
rbit
?
appr
oxim
atel
y 0.
052
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-4
8-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 8-5)
Stu
dy
Gu
ide
and I
nte
rven
tion
Hyp
erbo
las
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-Hi
ll47
9G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Equ
atio
ns
of
Hyp
erb
ola
sA
hyp
erbo
lais
the
set
of
all p
oint
s in
a p
lane
suc
h th
atth
e ab
solu
te v
alue
of
the
diff
eren
ceof
the
dis
tanc
es f
rom
any
poi
nt o
n th
e hy
perb
ola
to a
nytw
o gi
ven
poin
ts in
the
pla
ne,c
alle
d th
e fo
ci,i
s co
nsta
nt.
In t
he t
able
,the
leng
ths
a,b,
and
car
e re
late
d by
the
for
mul
a c2
#a2
!b2
.
Stan
dard
For
m o
f Equ
atio
n$
#1
$#
1
Equa
tions
of t
he A
sym
ptot
esy
$k
#(
(x$
h)y
$k
#(
(x$
h)
Tran
sver
se A
xis
Horiz
onta
lVe
rtica
l
Foci
(h$
c, k
), (h
!c,
k)
(h, k
$c)
, (h,
k!
c)
Vert
ices
(h$
a, k
), (h
!a,
k)
(h, k
$a)
, (h,
k!
a)
Wri
te a
n e
quat
ion
for
th
e h
yper
bola
wit
h v
erti
ces
(!2,
1) a
nd
(6,
1)an
d f
oci
(!4,
1) a
nd
(8,
1).
Use
a s
ketc
h to
ori
ent
the
hype
rbol
a co
rrec
tly.
The
cen
ter
of
the
hype
rbol
a is
the
mid
poin
t of
the
seg
men
t jo
inin
g th
e tw
o
vert
ices
.The
cen
ter
is (
,1),
or (
2,1)
.The
val
ue o
f ais
the
dist
ance
fro
m t
he c
ente
r to
a v
erte
x,so
a#
4.T
he v
alue
of c
is
the
dist
ance
fro
m t
he c
ente
r to
a f
ocus
,so
c#
6.
c2#
a2!
b2
62#
42!
b2
b2#
36 $
16 #
20
Use
h,k
,a2 ,
and
b2to
wri
te a
n eq
uati
on o
f th
e hy
perb
ola.
$#
1
Wri
te a
n e
quat
ion
for
th
e h
yper
bola
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
1.ve
rtic
es ($
7,0)
and
(7,
0),c
onju
gate
axi
s of
leng
th 1
0!
%1
2.ve
rtic
es (
$2,
$3)
and
(4,
$3)
,foc
i ($
5,$
3) a
nd (
7,$
3)!
%1
3.ve
rtic
es (
4,3)
and
(4,
$5)
,con
juga
te a
xis
of le
ngth
4!
%1
4.ve
rtic
es (
$8,
0) a
nd (
8,0)
,equ
atio
n of
asy
mpt
otes
y#
(x
!%
1
5.ve
rtic
es (
$4,
6) a
nd ($
4,$
2),f
oci (
$4,
10)
and
($4,
$6)
!%
1(x
#4)
2"
48(y
!2)
2"
16
9y2
" 16x2 " 64
1 " 6
(x!
4)2
"4
(y#
1)2
"16
(y#
3)2
"27
(x!
1)2
"9
y2 " 25x2 " 49
(y$
1)2
"20
(x$
2)2
"16
$2
!6
"2
x
y
O
a " bb " a
(x$
h)2
"b2
(y$
k)2
"a2
(y$
k)2
"b2
(x$
h)2
"a2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll48
0G
lenc
oe A
lgeb
ra 2
Gra
ph
Hyp
erb
ola
sTo
gra
ph a
hyp
erbo
la,w
rite
the
giv
en e
quat
ion
in t
he s
tand
ard
form
of
an e
quat
ion
for
a hy
perb
ola
$#
1 if
the
bra
nche
s of
the
hyp
erbo
la o
pen
left
and
rig
ht,o
r
$#
1 if
the
bra
nche
s of
the
hyp
erbo
la o
pen
up a
nd d
own
Gra
ph t
he p
oint
(h,k
),w
hich
is t
he c
ente
r of
the
hyp
erbo
la.D
raw
a r
ecta
ngle
wit
hdi
men
sion
s 2a
and
2ban
d ce
nter
(h,k
).If
the
hyp
erbo
la o
pens
left
and
rig
ht,t
he v
erti
ces
are
(h$
a,k)
and
(h
!a,
k).I
f th
e hy
perb
ola
open
s up
and
dow
n,th
e ve
rtic
es a
re (h
,k$
a)an
d (h
,k!
a).
Dra
w t
he
grap
h o
f 6y
2!
4x2
!36
y!
8x%
!26
.
Com
plet
e th
e sq
uare
s to
get
the
equ
atio
n in
sta
ndar
d fo
rm.
6y2
$4x
2$
36y
$8x
#$
266(
y2$
6y!
■) $
4(x2
!2x
!■
) #$
26 !
■6(
y2$
6y!
9) $
4(x2
!2x
!1)
#$
26 !
506(
y$
3)2
$4(
x!
1)2
#24
$#
1
The
cen
ter
of t
he h
yper
bola
is ($
1,3)
.A
ccor
ding
to
the
equa
tion
,a2
#4
and
b2#
6,so
a#
2 an
d b
#$
6#.T
he t
rans
vers
e ax
is is
ver
tica
l,so
the
ver
tice
s ar
e ($
1,5)
and
($1,
1).D
raw
a r
ecta
ngle
wit
hve
rtic
al d
imen
sion
4 a
nd h
oriz
onta
l dim
ensi
on 2
$6#
%4.
9.T
he d
iago
nals
of
this
rec
tang
lear
e th
e as
ympt
otes
.The
bra
nche
s of
the
hyp
erbo
la o
pen
up a
nd d
own.
Use
the
ver
tice
s an
dth
e as
ympt
otes
to
sket
ch t
he h
yper
bola
.
Fin
d t
he
coor
din
ates
of
the
vert
ices
an
d f
oci
and
th
e eq
uat
ion
s of
th
e as
ymp
tote
sfo
r th
e h
yper
bola
wit
h t
he
give
n e
quat
ion
.Th
en g
rap
h t
he
hyp
erbo
la.
1.$
#1
2.(y
$3)
2$
#1
3.$
#1
(2,0
),(!
2,0)
;(!
2,4)
,(!
2,2)
;(0
,4),
(0,!
4);
( 2$5!,
0) ,( !
2$5!,
0) ;( !
2,3
#$
10!) ,
(0,5
),(0
,!5)
;y
%(
2x( !
2,3
!$
10!) ;
y%
(x
y %
x #
3,
y %
!x
#21 " 3
1 " 3
2 " 31 " 3
xO
y
4 " 3
x2" 9
y2" 16
(x!
2)2
"9
y2" 16
x2" 4
(x!
1)2
"6
(y$
3)2
"4
xO
y
(x$
h)2
"b2
(y$
k)2
"a2
(y$
k)2
"" b2
(x$
h)2
"a2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Hyp
erbo
las
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
xO
y
xO
y
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-5)
Skil
ls P
ract
ice
Hyp
erbo
las
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-Hi
ll48
1G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Wri
te a
n e
quat
ion
for
eac
h h
yper
bola
.
1.2.
3.
!%
1!
%1
!%
1
Wri
te a
n e
quat
ion
for
th
e h
yper
bola
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
4.ve
rtic
es ($
4,0)
and
(4,
0),c
onju
gate
axi
s of
leng
th 8
!%
1
5.ve
rtic
es (
0,6)
and
(0,
$6)
,con
juga
te a
xis
of le
ngth
14
!%
1
6.ve
rtic
es (
0,3)
and
(0,
$3)
,con
juga
te a
xis
of le
ngth
10
!%
1
7.ve
rtic
es (
$2,
0) a
nd (
2,0)
,con
juga
te a
xis
of le
ngth
4!
%1
8.ve
rtic
es (
$3,
0) a
nd (
3,0)
,foc
i ((
5,0)
!%
1
9.ve
rtic
es (
0,2)
and
(0,
$2)
,foc
i (0,
(3)
!%
1
10.v
erti
ces
(0,$
2) a
nd (
6,$
2),f
oci (
3 (
$13#
,$2)
!%
1
Fin
d t
he
coor
din
ates
of
the
vert
ices
an
d f
oci
and
th
e eq
uat
ion
s of
th
e as
ymp
tote
sfo
r th
e h
yper
bola
wit
h t
he
give
n e
quat
ion
.Th
en g
rap
h t
he
hyp
erbo
la.
11.
$#
112
.$
#1
13.
$#
1
((3,
0);( (
3$5!,
0) ;(0
,(7)
;( 0,(
$58!
) ;((
4,0)
;( ($
17!,0
) ;y
%(
2xy
%(
xy
%(
x
xO
y
48
8 4 –4 –8
–4–8
xO
y
48
8 4 –4 –8
–4–8
xO
y
1 " 47 " 3
y2" 1
x2" 16
x2" 9
y2" 49
y2" 36
x2" 9
(y#
2)2
"4
(x!
3)2
"9
x2" 5
y2 " 4
y2 " 16x2" 9
y2 " 4x2" 4
x2" 25
y2 " 9
x2" 49
y2 " 36
y2 " 16x2" 16
y2" 25
x2" 4
x2" 25
y2 " 36y2 " 16
x2" 25
x
y
O
( $!29
, 0)
( –$
!29, 0
)
( 2, 0
)(–
2, 0
)
48
8 4 –4 –8
–4–8
x
y
O
( 0, $
!61)
( 0, –
$!61
)
( 0, 6
)
(0, –
6)48
8 4 –4 –8
–4–8
x
y
O
( $!41
, 0)
( –$
!41, 0
)
( 5, 0
)
(–5,
0)
48
8 4 –4 –8
–4–8
©G
lenc
oe/M
cGra
w-Hi
ll48
2G
lenc
oe A
lgeb
ra 2
Wri
te a
n e
quat
ion
for
eac
h h
yper
bola
.
1.2.
3.
!%
1!
%1
!%
1
Wri
te a
n e
quat
ion
for
th
e h
yper
bola
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
4.ve
rtic
es (
0,7)
and
(0,
$7)
,con
juga
te a
xis
of le
ngth
18
unit
s!
%1
5.ve
rtic
es (
1,$
1) a
nd (
1,$
9),c
onju
gate
axi
s of
leng
th 6
uni
ts!
%1
6.ve
rtic
es ($
5,0)
and
(5,
0),f
oci (
($
26#,0
)!
%1
7.ve
rtic
es (
1,1)
and
(1,
$3)
,foc
i (1,
$1
($
5#)!
%1
Fin
d t
he
coor
din
ates
of
the
vert
ices
an
d f
oci
and
th
e eq
uat
ion
s of
th
e as
ymp
tote
sfo
r th
e h
yper
bola
wit
h t
he
give
n e
quat
ion
.Th
en g
rap
h t
he
hyp
erbo
la.
8.$
#1
9.$
#1
10.
$#
1
(0,(
4);( 0
,(2$
5!) ;(1
,3),
(1,1
);(3
,0),
(3,!
4);
y%
(2x
( 1,2
($
5!) ;( 3,
!2
(2$
2!) ;y
!2
%(
(x!
1)y
#2
%(
(x!
3)
11.A
STR
ON
OM
YA
stro
nom
ers
use
spec
ial X
-ray
tel
esco
pes
to o
bser
ve t
he s
ourc
es o
fce
lest
ial X
ray
s.So
me
X-r
ay t
eles
cope
s ar
e fi
tted
wit
h a
met
al m
irro
r in
the
sha
pe o
f a
hype
rbol
a,w
hich
ref
lect
s th
e X
ray
s to
a f
ocus
.Sup
pose
the
ver
tice
s of
suc
h a
mir
ror
are
loca
ted
at ($
3,0)
and
(3,
0),a
nd o
ne f
ocus
is lo
cate
d at
(5,
0).W
rite
an
equa
tion
tha
tm
odel
s th
e hy
perb
ola
form
ed b
y th
e m
irro
r.!
%1
y2" 16
x2" 9
xO
y
xO
y
xO
y
48
8 4 –4 –8
–4–8
1 " 2
(x$
3)2
"4
(y!
2)2
"4
(x $
1)2
"4
(y$
2)2
"1
x2" 4
y2" 16
(x!
1)2
"1
(y#
1)2
"4y2" 1
x2" 25
(x!
1)2
"9
(y#
5)2
"16
x2" 81
y2 " 49
(y#
2)2
"16
(x !
1)2
"4
(x #
3)2
"25
(y !
2)2
"9
x2" 36
y2 " 9
x
y
O(–
1, –
2)
(1, –
2)
(3, –
2)x
y O
( –3,
2 #
$!34
)
( –3,
2 !
$!34
)
( –3,
–1)
(–3,
5)
4
8 4 –4
–4–8
x
y
O
( 0, 3
$%5)
( 0, –
3$%5)
( 0, 3
)
(0, –
3)48
8 4 –4 –8
–4–8
Pra
ctic
e (A
vera
ge)
Hyp
erbo
las
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 8-5)
Rea
din
g t
o L
earn
Math
emati
csH
yper
bola
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-Hi
ll48
3G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Pre-
Act
ivit
yH
ow a
re h
yper
bola
s d
iffe
ren
t fr
om p
arab
olas
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
5 at
the
top
of
page
441
in y
our
text
book
.
Loo
k at
the
ske
tch
of a
hyp
erbo
la in
the
intr
oduc
tion
to
this
less
on.L
ist
thre
e w
ays
in w
hich
hyp
erbo
las
are
diff
eren
t fr
om p
arab
olas
.Sa
mpl
e an
swer
:A h
yper
bola
has
two
bran
ches
,whi
le a
para
bola
is o
ne c
ontin
uous
cur
ve.A
hyp
erbo
la h
as tw
o fo
ci,
whi
le a
par
abol
a ha
s on
e fo
cus.
A hy
perb
ola
has
two
vert
ices
,w
hile
a p
arab
ola
has
one
vert
ex.
Rea
din
g t
he
Less
on
1.T
he g
raph
at
the
righ
t sh
ows
the
hype
rbol
a w
hose
equa
tion
in s
tand
ard
form
is
$#
1.
The
poi
nt (
0,0)
is t
he
of t
he
hype
rbol
a.
The
poi
nts
(4,0
) an
d ($
4,0)
are
the
of
the
hyp
erbo
la.
The
poi
nts
(5,0
) an
d ($
5,0)
are
the
of
the
hyp
erbo
la.
The
seg
men
t co
nnec
ting
(4,
0) a
nd ($
4,0)
is c
alle
d th
e ax
is.
The
seg
men
t co
nnec
ting
(0,
3) a
nd (
0,$
3) is
cal
led
the
axis
.
The
line
s y
#x
and
y#
$x
are
calle
d th
e .
2.St
udy
the
hype
rbol
a gr
aphe
d at
the
rig
ht.
The
cen
ter
is
.
The
val
ue o
f ais
.
The
val
ue o
f cis
.
To fi
nd b
2 ,so
lve
the
equa
tion
#
!.
The
equ
atio
n in
sta
ndar
d fo
rm f
or t
his
hype
rbol
a is
.
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is a
n ea
sy w
ay t
o re
mem
ber
the
equa
tion
rel
atin
g th
e va
lues
of a
,b,a
nd c
for
ahy
perb
ola?
This
equ
atio
n lo
oks
just
like
the
Pyth
agor
ean
Theo
rem
,al
thou
gh th
e va
riabl
es re
pres
ent d
iffer
ent l
engt
hs in
a h
yper
bola
than
ina
right
tria
ngle
.
"x 42 "!
" 1y 22 "%
1
b2a2
c242
(0,0
)
x
y
O
asym
ptot
es3 " 4
3 " 4
conj
ugat
etra
nsve
rse
foci
vert
ices
cent
er
y2" 9
x2" 16
x
y
O( –
4, 0
)( 4
, 0)
( –5,
0)
( 5, 0
)
y % 3 4x
y % –
3 4x
©G
lenc
oe/M
cGra
w-Hi
ll48
4G
lenc
oe A
lgeb
ra 2
Rec
tang
ular
Hyp
erbo
las
A r
ecta
ngu
lar
hyp
erbo
lais
a h
yper
bola
wit
h pe
rpen
dicu
lar
asym
ptot
es.
For
exam
ple,
the
grap
h of
x2
$y2
#1
is a
rec
tang
ular
hyp
erbo
la.A
hyp
erbo
law
ith
asym
ptot
es t
hat
are
not
perp
endi
cula
r is
cal
led
a n
onre
ctan
gula
rh
yper
bola
.The
gra
phs
of e
quat
ions
of
the
form
xy
#c,
whe
re c
is a
con
stan
t,ar
e re
ctan
gula
r hy
perb
olas
.
Mak
e a
tabl
e of
val
ues
an
d p
lot
poi
nts
to
grap
h e
ach
rec
tan
gula
rh
yper
bola
bel
ow.B
e su
re t
o co
nsi
der
neg
ativ
e va
lues
for
th
eva
riab
les.
See
stud
ents
’tab
les.
1.xy
#$
42.
xy#
3
3.xy
#$
14.
xy#
8
5.M
ake
a co
njec
ture
abo
ut t
he a
sym
ptot
es o
f re
ctan
gula
r hy
perb
olas
.
The
coor
dina
te a
xes
are
the
asym
ptot
es.
x
y
Ox
y
O
x
y
Ox
y
O
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-5
8-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-6)
Stu
dy
Gu
ide
and I
nte
rven
tion
Con
ic S
ectio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-Hi
ll48
5G
lenc
oe A
lgeb
ra 2
Lesson 8-6
Stan
dar
d F
orm
Any
con
ic s
ecti
on in
the
coo
rdin
ate
plan
e ca
n be
des
crib
ed b
y an
equa
tion
of
the
form
A
x2!
Bxy
!C
y2!
Dx
!E
y!
F#
0,w
here
A,B
,and
Car
e no
t al
l zer
o.O
ne w
ay t
o te
ll w
hat
kind
of
coni
c se
ctio
n an
equ
atio
n re
pres
ents
is t
o re
arra
nge
term
s an
dco
mpl
ete
the
squa
re,i
f ne
cess
ary,
to g
et o
ne o
f th
e st
anda
rd f
orm
s fr
om a
n ea
rlie
r le
sson
.T
his
met
hod
is e
spec
ially
use
ful i
f yo
u ar
e go
ing
to g
raph
the
equ
atio
n.
Wri
te t
he
equ
atio
n 3
x2!
4y2
!30
x!
8y
#59
%0
in s
tan
dar
d f
orm
.S
tate
wh
eth
er t
he
grap
h o
f th
e eq
uat
ion
is
a pa
rabo
la,c
ircl
e,el
lips
e,or
hyp
erbo
la.
3x2
$4y
2$
30x
$ 8
y!
59#
0O
rigin
al e
quat
ion
3x2
$30
x$
4y2
$8y
#$
59Is
olat
e te
rms.
3(x2
$10
x!
■) $
4(y2
!2y
!■
)#
$59
! ■
!■
Fact
or o
ut c
omm
on m
ultip
les.
3(x2
$10
x!
25)
$4(
y2!
2y!
1)#
$59
!3(
25)
! ($
4)(1
)Co
mpl
ete
the
squa
res.
3(x
$5)
2$
4(y
!1)
2#
12Si
mpl
ify.
$#
1Di
vide
each
sid
e by
12.
The
gra
ph o
f th
e eq
uati
on is
a h
yper
bola
wit
h it
s ce
nter
at
(5,$
1).T
he le
ngth
of
the
tran
sver
se a
xis
is 4
uni
ts a
nd t
he le
ngth
of
the
conj
ugat
e ax
is is
2$
3#un
its.
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Sta
te w
het
her
th
e gr
aph
of
the
equ
atio
n i
sa
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
1.x2
!y2
$6x
!4y
!3
#0
2.x2
!2y
2!
6x$
20y
!53
#0
(x!
3)2
#(y
#2)
2%
10;c
ircle
#%
1;el
lipse
3.6x
2$
60x
$y
!16
1 #
04.
x2!
y2$
4x$
14y
!29
#0
y%
6(x
!5)
2#
11;p
arab
ola
(x!
2)2
#(y
!7)
2%
24;c
ircle
5.6x
2$
5y2
!24
x!
20y
$56
#0
6.3y
2!
x$
24y
!46
#0
!%
1;hy
perb
ola
x%
!3(
y!
4)2
#2;
para
bola
7.x2
$4y
2$
16x
!24
y$
36 #
08.
x2!
2y2
!8x
!4y
!2
#0
!%
1;hy
perb
ola
#%
1;el
lipse
9.4x
2!
48x
!y
!15
8 #
010
.3x2
!y2
$48
x$
4y!
184
#0
y%
!4(
x#
6)2
!14
;par
abol
a#
%1;
ellip
se
11.$
3x2
!2y
2$
18x
!20
y!
5 #
012
.x2
!y2
!8x
!2y
!8
#0
!%
1;hy
perb
ola
(x#
4)2
#(y
#1)
2%
9;ci
rcle
(x#
3)2
"6
(y#
5)2
"9
(y!
2)2
"12
(x!
8)2
"4
(y#
1)2
"8
(x#
4)2
"16
(y!
3)2
"16
(x!
8)2
"64
(y!
2)2
"12
(x#
2)2
"10
(y!
5)2
"3
(x#
3)2
"6
(y!
1)2
"3
(x$
5)2
"4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll48
6G
lenc
oe A
lgeb
ra 2
Iden
tify
Co
nic
Sec
tio
ns
If y
ou a
re g
iven
an
equa
tion
of
the
form
Ax2
!B
xy!
Cy2
!D
x!
Ey
!F
#0,
wit
h B
#0,
you
can
dete
rmin
e th
e ty
pe o
f co
nic
sect
ion
just
by
cons
ider
ing
the
valu
es o
f Aan
d C
.Ref
erto
the
fol
low
ing
char
t.
Rela
tions
hip
of A
and
CTy
pe o
f Con
ic S
ectio
n
A#
0 or
C#
0, b
ut n
ot b
oth.
para
bola
A #
Ccir
cle
Aan
d C
have
the
sam
e sig
n, b
ut A
)C
.el
lipse
Aan
d C
have
opp
osite
sig
ns.
hype
rbol
a
Wit
hou
t w
riti
ng
the
equ
atio
n i
n s
tan
dar
d f
orm
,sta
te w
het
her
th
egr
aph
of
each
equ
atio
n i
s a
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Con
ic S
ectio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
Exam
ple
Exam
ple
a.3x
2!
3y2
#5x
#12
%0
A#
3 an
d C
#$
3 ha
ve o
ppos
ite
sign
s,so
the
grap
h of
the
equ
atio
n is
a h
yper
bola
.
b.y2
%7y
!2x
#13
A#
0,so
the
gra
ph o
f th
e eq
uati
on is
a pa
rabo
la.
Exer
cises
Exer
cises
Wit
hou
t w
riti
ng
the
equ
atio
n i
n s
tan
dar
d f
orm
,sta
te w
het
her
th
e gr
aph
of
each
equ
atio
n i
s a
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
1.x2
#17
x$
5y!
82.
2x2
!2y
2$
3x!
4y#
5pa
rabo
laci
rcle
3.4x
2$
8x#
4y2
$6y
!10
4.8(
x$
x2) #
4(2y
2$
y) $
100
hype
rbol
aci
rcle
5.6y
2$
18 #
24 $
4x2
6.y
#27
x$
y2
ellip
sepa
rabo
la7.
x2#
4(y
$y2
) !2x
$1
8.10
x$
x2$
2y2
#5y
ellip
seel
lipse
9.x
#y2
$5y
!x2
$5
10.1
1x2
$7y
2#
77ci
rcle
hype
rbol
a11
.3x2
!4y
2#
50 !
y212
.y2
#8x
$11
circ
lepa
rabo
la13
.9y2
$99
y#
3(3x
$3x
2 )14
.6x2
$4
#5y
2$
3ci
rcle
hype
rbol
a15
.111
#11
x2!
10y2
16.1
20x2
$11
9y2
!11
8x$
117y
#0
ellip
sehy
perb
ola
17.3
x2#
4y2
!12
18.1
50 $
x2#
120
$y
hype
rbol
apa
rabo
la
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 8-6)
Skil
ls P
ract
ice
Con
ic S
ectio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-Hi
ll48
7G
lenc
oe A
lgeb
ra 2
Lesson 8-6
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Sta
te w
het
her
th
e gr
aph
of
the
equ
atio
n i
sa
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.Th
en g
rap
h t
he
equ
atio
n.
1.x2
$25
y2#
25hy
perb
ola
2.9x
2!
4y2
#36
ellip
se3.
x2!
y2$
16 #
0ci
rcle
!%
1#
%1
x2#
y2%
16
4.x2
!8x
!y2
#9
circ
le5.
x2!
2x$
15 #
ypa
rabo
la6.
100x
2!
25y2
#40
0 ellip
se(x
#4)
2#
y2%
25y
%(x
#1)
2!
16#
%1
Wit
hou
t w
riti
ng
the
equ
atio
n i
n s
tan
dar
d f
orm
,sta
te w
het
her
th
e gr
aph
of
each
equ
atio
n i
s a
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
7.9x
2!
4y2
#36
ellip
se8.
x2!
y2#
25ci
rcle
9.y
#x2
!2x
para
bola
10.y
#2x
2$
4x$
4pa
rabo
la
11.4
y2$
25x2
#10
0hy
perb
ola
12.1
6x2
!y2
#16
ellip
se
13.1
6x2
$4y
2#
64hy
perb
ola
14.5
x2!
5y2
#25
circ
le
15.2
5y2
!9x
2#
225
ellip
se16
.36y
2$
4x2
#14
4hy
perb
ola
17.y
#4x
2$
36x
$14
4pa
rabo
la18
.x2
!y2
$14
4 #
0ci
rcle
19.(
x!
3)2
!(y
$1)
2#
4ci
rcle
20.2
5y2
$50
y!
4x2
#75
ellip
se
21.x
2$
6y2
!9
#0
hype
rbol
a22
.x#
y2!
5y$
6pa
rabo
la
23.(
x!
5)2
!y2
#10
circ
le24
.25x
2!
10y2
$25
0 #
0el
lipse
x
y
O
xy
O4
8
–4 –8 –12
–16
–4–8
x
y
O4
8
8 4 –4 –8
–4–8
y2" 16
x2" 4
x
y
Ox
y
Ox
O
y
48
4 2 –2 –4
–4–8
y2" 9
x2" 4
y2 " 1x2" 25
©G
lenc
oe/M
cGra
w-Hi
ll48
8G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
equ
atio
n i
n s
tan
dar
d f
orm
.Sta
te w
het
her
th
e gr
aph
of
the
equ
atio
n i
sa
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.Th
en g
rap
h t
he
equ
atio
n.
1.y2
#$
3x2.
x2!
y2!
6x#
73.
5x2
$6y
2$
30x
$12
y#
$9
para
bola
circ
lehy
perb
ola
x%
!y
2(x
#3)
2#
y2%
16!
%1
4.19
6y2
#12
25 $
100x
25.
3x2
#9
$3y
2$
6y6.
9x2
!y2
!54
x$
6y#
$81
ellip
seci
rcle
ellip
se#
%1
x2#
(y#
1)2
%4
#%
1
Wit
hou
t w
riti
ng
the
equ
atio
n i
n s
tan
dar
d f
orm
,sta
te w
het
her
th
e gr
aph
of
each
equ
atio
n i
s a
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
7.6x
2!
6y2
#36
8.4x
2$
y2#
169.
9x2
!16
y2$
64y
$80
#0
circ
lehy
perb
ola
ellip
se
10.5
x2!
5y2
$45
#0
11.x
2!
2x#
y12
.4y2
$36
x2!
4x $
144
#0
circ
lepa
rabo
lahy
perb
ola
13.A
STR
ON
OM
YA
sat
ellit
e tr
avel
s in
an
hype
rbol
ic o
rbit
.It
reac
hes
the
vert
ex o
f it
s or
bit
at (
5,0)
and
the
n tr
avel
s al
ong
a pa
th t
hat
gets
clo
ser
and
clos
er t
o th
e lin
e y
#x.
Wri
te a
n eq
uati
on t
hat
desc
ribe
s th
e pa
th o
f th
e sa
telli
te if
the
cen
ter
of it
s hy
perb
olic
orbi
t is
at
(0,0
).
!%
1y2" 4
x2" 25
2 " 5x
y
O
x
y
Ox
y
O
(y!
3)2
"9
(x#
3)2
"1
y2" 6.
25x2
" 12.2
5
xO
y
x
y
Ox
y
O
(y #
1)2
"5
(x !
3)2
"6
1 " 3Pra
ctic
e (A
vera
ge)
Con
ic S
ectio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-6)
Rea
din
g t
o L
earn
Math
emati
csC
onic
Sec
tions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-Hi
ll48
9G
lenc
oe A
lgeb
ra 2
Lesson 8-6
Pre-
Act
ivit
yH
ow c
an y
ou u
se a
fla
shli
ght
to m
ake
con
ic s
ecti
ons?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
6 at
the
top
of
page
449
in y
our
text
book
.
The
fig
ures
in t
he in
trod
ucti
on s
how
how
a p
lane
can
slic
e a
doub
le c
one
tofo
rm t
he c
onic
sec
tion
s.N
ame
the
coni
c se
ctio
n th
at is
for
med
if t
he p
lane
slic
es t
he d
oubl
e co
ne in
eac
h of
the
fol
low
ing
way
s:
•T
he p
lane
is p
aral
lel t
o th
e ba
se o
f th
e do
uble
con
e an
d sl
ices
thr
ough
one
of t
he c
ones
tha
t fo
rm t
he d
oubl
e co
ne.
circ
le•
The
pla
ne is
per
pend
icul
ar t
o th
e ba
se o
f th
e do
uble
con
e an
d sl
ices
thro
ugh
both
of
the
cone
s th
at f
orm
the
dou
ble
cone
.hy
perb
ola
Rea
din
g t
he
Less
on
1.N
ame
the
coni
c se
ctio
n th
at is
the
gra
ph o
f ea
ch o
f th
e fo
llow
ing
equa
tion
s.G
ive
the
coor
dina
tes
of t
he v
erte
x if
the
con
ic s
ecti
on is
a p
arab
ola
and
of t
he c
ente
r if
it is
aci
rcle
,an
ellip
se,o
r a
hype
rbol
a.
a.!
#1
ellip
se;(
3,!
5)
b.x
#$
2(y
!1)
2!
7pa
rabo
la;(
7,!
1)c.
(x$
5)2
$(y
!5)
2#
1hy
perb
ola;
(5,!
5)d.
(x!
6)2
!(y
$2)
2#
1ci
rcle
;(!
6,2)
2.E
ach
of t
he f
ollo
win
g is
the
equ
atio
n of
a c
onic
sec
tion
.For
eac
h eq
uati
on,i
dent
ify
the
valu
es o
f Aan
d C
.The
n,w
itho
ut w
riti
ng t
he e
quat
ion
in s
tand
ard
form
,sta
te w
heth
erth
e gr
aph
of e
ach
equa
tion
is a
par
abol
a,ci
rcle
,ell
ipse
,or
hype
rbol
a.
a.2x
2!
y2$
6x!
8y!
12 #
0A
#;C
#;t
ype
of g
raph
:
b.2x
2!
3x$
2y$
5 #
0A
#;C
#;t
ype
of g
raph
:
c.5x
2!
10x
!5y
2$
20y
!1
#0
A#
;C#
;typ
e of
gra
ph:
d.x2
$y2
!4x
!2y
$5
#0
A#
;C#
;typ
e of
gra
ph:
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is a
n ea
sy w
ay t
o re
cogn
ize
that
an
equa
tion
rep
rese
nts
a pa
rabo
la r
athe
r th
anon
e of
the
oth
er c
onic
sec
tion
s?
If th
e eq
uatio
n ha
s an
x2
term
and
yte
rm b
ut n
o y2
term
,the
n th
e gr
aph
is a
par
abol
a.Li
kew
ise,
if th
e eq
uatio
n ha
s a
y2te
rm a
nd x
term
but
no
x2te
rm,t
hen
the
grap
h is
a p
arab
ola.
hype
rbol
a!
11
circ
le5
5pa
rabo
la0
2el
lipse
12
(y!
5)2
"15
(x$
3)2
"36
©G
lenc
oe/M
cGra
w-Hi
ll49
0G
lenc
oe A
lgeb
ra 2
Loci
A l
ocus
(plu
ral,
loci
) is
the
set
of
all p
oint
s,an
d on
ly t
hose
poi
nts,
that
sat
isfy
a gi
ven
set
of c
ondi
tion
s.In
geo
met
ry,f
igur
es o
ften
are
def
ined
as
loci
.For
exam
ple,
a ci
rcle
is t
he lo
cus
of p
oint
s of
a p
lane
tha
t ar
e a
give
n di
stan
cefr
om a
giv
en p
oint
.The
def
init
ion
lead
s na
tura
lly t
o an
equ
atio
n w
hose
gra
phis
the
cur
ve d
escr
ibed
.
Wri
te a
n e
quat
ion
of
the
locu
s of
poi
nts
th
at a
re t
he
sam
e d
ista
nce
fro
m (
3,4)
an
d y
%!
4.
Rec
ogni
zing
tha
t th
e lo
cus
is a
par
abol
a w
ith
focu
s (3
,4) a
nd d
irec
trix
y#
$4,
you
can
find
that
h#
3,k
#0,
and
a#
4 w
here
(h,k
) is
the
vert
ex a
nd 4
uni
tsis
the
dis
tanc
e fr
om t
he v
erte
x to
bot
h th
e fo
cus
and
dire
ctri
x.
Thu
s,an
equ
atio
n fo
r th
e pa
rabo
la is
y#
" 11 6"(x
$3)
2 .
The
pro
blem
als
o m
ay b
e ap
proa
ched
ana
lyti
cally
as
follo
ws:
Let
(x,
y) b
e a
poin
t of
the
locu
s.
The
dis
tanc
e fr
om (
3,4)
to
(x,y
) #th
e di
stan
ce f
rom
y#
$4
to (x
,y).
$(x
$3
#)2
!(
#y
$4)
#2 ##
$(x
$x
#)2
!(
#y
$($
#4)
)2#
(x$
3)2
!y2
$8y
!16
#y2
!8y
!16
(x$
3)2
#16
y
" 11 6"(x
$3)
2#
y
Des
crib
e ea
ch l
ocu
s as
a g
eom
etri
c fi
gure
.Th
en w
rite
an
equ
atio
nfo
r th
e lo
cus.
1.A
ll po
ints
tha
t ar
e th
e sa
me
dist
ance
fro
m (
0,5)
and
(4,
5).
line,
x%
22.
All
poin
ts t
hat
are
4 un
its
from
the
ori
gin.
circ
le,x
2#
y2%
43.
All
poin
ts t
hat
are
the
sam
e di
stan
ce f
rom
($2,
$1)
and
x#
2.
para
bola
,x%
"! 81 "(y
2#
2y#
1)4.
The
locu
s of
poi
nts
such
tha
t th
e su
m o
f th
e di
stan
ces
from
($2,
0) a
nd (
2,0)
is 6
.
ellip
se,"x 92 "
#"y 52 "
%1
5.T
he lo
cus
of p
oint
s su
ch t
hat
the
abso
lute
val
ue o
f th
e d
iffe
renc
e of
the
dis
tanc
es
from
($
3,0)
and
(3,
0) is
2.
hype
rbol
a,"x 12 "
!"y 82 "
%1
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-6
8-6
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 8-7)
Stu
dy
Gu
ide
and I
nte
rven
tion
Solv
ing
Qua
drat
ic S
yste
ms
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-Hi
ll49
1G
lenc
oe A
lgeb
ra 2
Lesson 8-7
Syst
ems
of
Qu
adra
tic
Equ
atio
ns
Lik
e sy
stem
s of
line
ar e
quat
ions
,sys
tem
s of
quad
rati
c eq
uati
ons
can
be s
olve
d by
sub
stit
utio
n an
d el
imin
atio
n.If
the
gra
phs
are
a co
nic
sect
ion
and
a lin
e,th
e sy
stem
will
hav
e 0,
1,or
2 s
olut
ions
.If
the
grap
hs a
re t
wo
coni
cse
ctio
ns,t
he s
yste
m w
ill h
ave
0,1,
2,3,
or 4
sol
utio
ns.
Sol
ve t
he
syst
em o
f eq
uat
ion
s.y
%x2
!2x
!15
x#
y%
!3
Rew
rite
the
sec
ond
equa
tion
as
y#
$x
$3
and
subs
titu
te in
to t
he f
irst
equ
atio
n.
$x
$3
#x2
$2x
$15
0 #
x2$
x$
12Ad
d x
!3
to e
ach
side.
0 #
(x$
4)(x
!3)
Fact
or.
Use
the
Zer
o P
rodu
ct p
rope
rty
to g
etx
#4
orx
#$
3.
Subs
titu
te t
hese
val
ues
for
xin
x!
y#
$3:
4 !
y#
$3
or$
3 !
y#
$3
y#
$7
y#
0
The
sol
utio
ns a
re (
4,$
7) a
nd (
$3,
0).
Fin
d t
he
exac
t so
luti
on(s
) of
eac
h s
yste
m o
f eq
uat
ion
s.
1.y#
x2$
52.
x2!
(y$
5)2
#25
y#x
$3
y#
$x2
(2,!
1),(
!1,
!4)
(0,0
)
3.x2
!(y
$5)
2#
254.
x2!
y2#
9y
#x2
x2!
y#
3
(0,0
),(3
,9),
(!3,
9)(0
,3),
($5!,
!2)
,(!
$5!,
!2)
5.x2
$y2
#1
6.y
#x
$3
x2!
y2#
16x
#y2
$4
",
#, ",!
#,"
,#,
"!,
#, "!
,!#
",
#1
!$
29!"
" 27
!$
29!"
" 2$
30!"
2$
34!"
2$
30!"
2$
34!"
2
1 #
$29!
"" 2
7 #
$29!
"" 2
$30!
"2
$34!
"2
$30!
"2
$34!
"2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll49
2G
lenc
oe A
lgeb
ra 2
Syst
ems
of
Qu
adra
tic
Ineq
ual
itie
sSy
stem
s of
qua
drat
ic in
equa
litie
s ca
n be
sol
ved
by g
raph
ing.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by g
rap
hin
g.x2
#y2
)25
"x!
#2#
y2*
The
gra
ph o
f x2
!y2
*25
con
sist
s of
all
poin
ts o
n or
insi
de
the
circ
le w
ith
cent
er (
0,0)
and
rad
ius
5.T
he g
raph
of
!x$
"2!
y2+
cons
ists
of
all p
oint
s on
or
outs
ide
the
circ
le w
ith
cent
er !
,0"a
nd r
adiu
s .T
he s
olut
ion
of t
he
syst
em is
the
set
of
poin
ts in
bot
h re
gion
s.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by g
rap
hin
g.x2
#y2
)25
!&
1
The
gra
ph o
f x2
!y2
*25
con
sist
s of
all
poin
ts o
n or
insi
de
the
circ
le w
ith
cent
er (
0,0)
and
rad
ius
5.T
he g
raph
of
$&
1 ar
e th
e po
ints
“in
side
”bu
t no
t on
the
bra
nche
s of
the
hype
rbol
a sh
own.
The
sol
utio
n of
the
sys
tem
is t
he s
et o
fpo
ints
in b
oth
regi
ons.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s be
low
by
grap
hin
g.
1.!
*1
2.x2
!y2
*16
93.
y+
(x$
2)2
y&
x$
2x2
!9y
2+
225
(x!
1)2
!(y
!1)
2*
16 x
y
Ox
y
O6
12
12 6 –6 –12
–6–1
2x
y
O
1 " 2
y2" 4
x2" 16
x2" 9
y2" 4
x2 " 9y2 " 4
x
y
O
5 " 25 " 2
25 " 45 " 2
25 " 45 " 2
x
y
O
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Solv
ing
Qua
drat
ic S
yste
ms
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-7)
Skil
ls P
ract
ice
Solv
ing
Qua
drat
ic S
yste
ms
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-Hi
ll49
3G
lenc
oe A
lgeb
ra 2
Lesson 8-7
Fin
d t
he
exac
t so
luti
on(s
) of
eac
h s
yste
m o
f eq
uat
ion
s.
1.y
#x
$2
(0,!
2),(
1,!
1)2.
y#
x!
3(!
1,2)
,3.
y#
3x(0
,0)
y#
x2$
2y
#2x
2(1
.5,4
.5)
x#
y2
4.y
#x
( $2!,
$2!)
,5.
x#
$5
(!5,
0)6.
y#
7no
sol
utio
nx2
!y2
#4
( !$
2!,!
$2!)
x2!
y2#
25x2
!y2
#9
7.y
#$
2x!
2(2
,!2)
,8.
x$
y!
1 #
0(1
,2)
9.y
#2
$x
(0,2
),(3
,!1)
y2#
2x"
,1#
y2#
4xy
#x2
$4x
!2
10.y
#x
$1
no s
olut
ion
11.y
#3x
2(0
,0)
12.y
#x2
!1
(!1,
2),
y#
x2y
#$
3x2
y#
$x2
!3
(1,2
)
13.y
#4x
(!1,
!4)
,(1,
4)14
.y#
$1
(0,!
1)15
.4x2
!9y
2#
36(!
3,0)
,4x
2!
y2#
204x
2!
y2#
1x2
$9y
2#
9(3
,0)
16.3
(y!
2)2
$4(
x$
3)2
#12
17.x
2$
4y2
#4
(!2,
0),
18.y
2$
4x2
#4
no
y#
$2x
!2
(0,2
),(3
,!4)
x2!
y2#
4(2
,0)
y#
2xso
lutio
n
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
19.y
*3x
$2
20.y
*x
21.4
y2!
9x2
'14
4x2
!y2
'16
y+
$2x
2!
4x2
!8y
2'
16
22.G
AR
DEN
ING
An
ellip
tica
l gar
den
bed
has
a pa
th f
rom
poi
nt A
to
poin
t B
.If
the
bed
can
be m
odel
ed b
y th
e eq
uati
on x
2!
3y2
#12
and
the
path
can
be
mod
eled
by
the
line
y#
$x,
wha
t ar
e th
e
coor
dina
tes
of p
oint
s A
and
B?
(!3,
1) a
nd (3
,!1)
1 " 3x
y
B
A
O
x
y
O4
8
8 4 –4 –8
–4–8
x
y
Ox
y
O
1 " 2
©G
lenc
oe/M
cGra
w-Hi
ll49
4G
lenc
oe A
lgeb
ra 2
Fin
d t
he
exac
t so
luti
on(s
) of
eac
h s
yste
m o
f eq
uat
ion
s.
1.(x
$2)
2!
y2#
52.
x#
2(y
!1)
2$
63.
y2$
3x2
#6
4.x2
!2y
2#
1x
$y
#1
x!
y#
3y
#2x
$1
y#
$x
!1
(0,!
1),(
3,2)
(2,1
),(6
.5,!
3.5)
(!1,
!3)
,(5,
9)(1
,0),
",
#5.
4y2
$9x
2#
366.
y#
x2$
37.
x2!
y2#
258.
y2#
10 $
6x2
4x2
$9y
2#
36x2
!y2
#9
4y#
3x4y
2#
40 $
2x2
no s
olut
ion
(0,!
3),( (
$5!,
2)(4
,3),
(!4,
!3)
( 0,(
$10!
)9.
x2!
y2#
2510
.4x2
!9y
2#
3611
.x#
$(y
$3)
2!
212
.$
#1
x#
3y$
52x
2$
9y2
#18
x#
(y$
3)2
!3
x2!
y2#
9
(!5,
0),(
4,3)
((3,
0)no
sol
utio
n((
3,0)
13.2
5x2
!4y
2#
100
14.x
2!
y2#
415
.x2
$y2
#3
x#
$!
#1
y2$
x2#
3
no s
olut
ion
((2,
0)no
sol
utio
n
16.
!#
117
.x!
2y#
318
.x2
!y2
#64
3x2
$y2
#9
x2!
y2#
9x2
$y2
#8
( (2,
($
3!)(3
,0),
"!,
#( (
6,(
2$7!)
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
19.y
+x2
20.x
2!
y2'
3621
.!
*1
y&
$x
!2
x2!
y2+
16(x
!1)
2!
(y$
2)2
*4
22.G
EOM
ETRY
The
top
of
an ir
on g
ate
is s
hape
d lik
e ha
lf a
n el
lipse
wit
h tw
o co
ngru
ent
segm
ents
fro
m t
he c
ente
r of
the
ellip
se t
o th
e el
lipse
as
show
n.A
ssum
e th
at t
he c
ente
r of
the
ellip
se is
at
(0,0
).If
the
elli
pse
can
be m
odel
ed b
y th
eeq
uati
on x
2!
4y2
#4
for
y+
0 an
d th
e tw
o co
ngru
ent
segm
ents
can
be
mod
eled
by
y#
xan
d y
#$
x,
wha
t ar
e th
e co
ordi
nate
s of
poi
nts
Aan
d B
?
$3#
"2
$3#
"2
BA
(0, 0
)
x
y
O
x
y
O4
8
8 4 –4 –8
–4–8
x
y
O
(x!
2)2
"4
(y$
3)2
"16
12 " 59 " 5
y2" 7
x2" 7
y2" 8
x2" 4
5 " 2
y2" 16
x2" 9
2 " 31 " 3
Pra
ctic
e (A
vera
ge)
Solv
ing
Qua
drat
ic S
yste
ms
NAM
E__
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____
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____
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____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
"!1,
#and
"1,
#$
3!"
2$
3!"
2
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 8-7)
Rea
din
g t
o L
earn
Math
emati
csSo
lvin
g Q
uadr
atic
Sys
tem
s
NAM
E__
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____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-Hi
ll49
5G
lenc
oe A
lgeb
ra 2
Lesson 8-7
Pre-
Act
ivit
yH
ow d
o sy
stem
s of
equ
atio
ns
app
ly t
o vi
deo
gam
es?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 8-
7 at
the
top
of
page
455
in y
our
text
book
.
The
fig
ure
in y
our
text
book
sho
ws
that
the
spa
cesh
ip h
its
the
circ
ular
for
cefi
eld
in t
wo
poin
ts.I
s it
pos
sibl
e fo
r th
e sp
aces
hip
to h
it t
he f
orce
fie
ld in
eith
er f
ewer
or
mor
e th
an t
wo
poin
ts?
Stat
e al
l pos
sibi
litie
s an
d ex
plai
nho
w t
hese
cou
ld h
appe
n.Sa
mpl
e an
swer
:The
spa
cesh
ip c
ould
hit
the
forc
e fie
ld in
zer
o po
ints
if th
e sp
aces
hip
mis
sed
the
forc
efie
ld a
ll to
geth
er.T
he s
pace
ship
cou
ld a
lso
hit t
he fo
rce
field
in o
ne p
oint
if th
e sp
aces
hip
just
touc
hed
the
edge
of t
hefo
rce
field
.
Rea
din
g t
he
Less
on
1.D
raw
a s
ketc
h to
illu
stra
te e
ach
of t
he f
ollo
win
g po
ssib
iliti
es.
a.a
para
bola
and
a li
ne
b.an
elli
pse
and
a ci
rcle
c.
a hy
perb
ola
and
ath
at in
ters
ect
in
that
inte
rsec
t in
lin
e th
at in
ters
ect
in2
poin
ts4
poin
ts1
poin
t
2.C
onsi
der
the
follo
win
g sy
stem
of
equa
tion
s.
x2#
3 !
y2
2x2
!3y
2#
11
a.W
hat
kind
of
coni
c se
ctio
n is
the
gra
ph o
f th
e fi
rst
equa
tion
?hy
perb
ola
b.W
hat
kind
of
coni
c se
ctio
n is
the
gra
ph o
f th
e se
cond
equ
atio
n?el
lipse
c.B
ased
on
your
ans
wer
s to
par
ts a
and
b,w
hat
are
the
poss
ible
num
bers
of
solu
tion
sth
at t
his
syst
em c
ould
hav
e?0,
1,2,
3,or
4
Hel
pin
g Y
ou
Rem
emb
er
3.Su
ppos
e th
at t
he g
raph
of
a qu
adra
tic
ineq
ualit
y is
a r
egio
n w
hose
bou
ndar
y is
a c
ircl
e.H
ow c
an y
ou r
emem
ber
whe
ther
to
shad
e th
e in
teri
or o
r th
e ex
teri
or o
f th
e ci
rcle
?Sa
mpl
e an
swer
:The
sol
utio
ns o
f an
ineq
ualit
y of
the
form
x2
#y2
'r2
are
all p
oint
s th
at a
re le
ss th
an r
units
from
the
orig
in,s
o th
e gr
aph
isth
e in
terio
rof t
he c
ircle
.The
sol
utio
ns o
f an
ineq
ualit
y of
the
form
x2
#y2
&r2
are
the
poin
ts th
at a
re m
ore
than
run
its fr
om th
e or
igin
,so
the
grap
h is
the
exte
rioro
f the
circ
le.
x
y
Ox
y
Ox
y O
©G
lenc
oe/M
cGra
w-Hi
ll49
6G
lenc
oe A
lgeb
ra 2
Gra
phin
g Q
uadr
atic
Equ
atio
ns w
ith x
y-Te
rms
You
can
use
a gr
aphi
ng c
alcu
lato
r to
exa
min
e gr
aphs
of
quad
rati
c eq
uati
ons
that
con
tain
xy-
term
s.
Use
a g
rap
hin
g ca
lcu
lato
r to
dis
pla
y th
e gr
aph
of
x2#
xy#
y2%
4.
Solv
e th
e eq
uati
on f
or y
in t
erm
s of
xby
usi
ng t
he
quad
rati
c fo
rmul
a.
y2!
xy!
(x2
$4)
#0
To u
se t
he f
orm
ula,
let
a#
1,b
#x,
and
c#
(x2
$4)
.
y#
y#
To g
raph
the
equ
atio
n on
the
gra
phin
g ca
lcul
ator
,ent
er t
he t
wo
equa
tion
s:
y#
and
y#
Use
a g
rap
hin
g ca
lcu
lato
r to
gra
ph
eac
h e
quat
ion
.Sta
te t
he
typ
e of
cu
rve
each
gra
ph
rep
rese
nts
.
1.y2
!xy
#8
2.x2
!y2
$2x
y$
x#
0
hype
rbol
apa
rabo
la
3.x2
$xy
!y2
#15
4.x2
!xy
!y2
#$
9
ellip
segr
aph
is +
5.2x
2$
2xy
$y2
!4x
#20
6.x2
$xy
$2y
2!
2x!
5y$
3 #
0
hype
rbol
atw
o in
ters
ectin
g lin
es
$x
$$
16 $
#3x
2#
""
"2
$x
!$
16 $
#3x
2#
""
"2
$x
($
16 $
#3x
2#
""
"2
$x
($
x2$
4#
(1)(
x2#
$4)
#"
""
2
x
y
O1
–1–2
2
2 1 –1 –2
En
rich
men
t
NAM
E__
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____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
8-7
8-7
Exam
ple
Exam
ple
top related