Post on 16-Mar-2018
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�2
x1 � 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 units
4. (7, 2) and (4, �1) 5. (�5, �2) and (3, 4) 6. (11, 5) and (16, 9)
3�2� units 10 units �41� units
7. (�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�2
x1 � 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 equation
y � 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 � 5
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 � 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.
8
x � 6(2, 2)(4, 2)
left1�8
(h, k)
x � h
a � 0a � 0
axis of symmetry
directrix
focus
vertex
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 � 144
8. endpoints of a diameter at (�4, 0) and (�4, �6) (x � 4)2 � (y � 3)2 � 9
9. 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 � 9
11. center at (�6, �5), tangent to y-axis (x � 6)2 � (y � 5)2 � 36
12. 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 � 0.81
5. endpoints of a diameter at (�2, �9) and (0, �5) (x � 1)2 � (y � 7)2 � 5
6. center at (�9, �12), passes through (�4, �5) (x � 9)2 � (y � 12)2 � 74
7. 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)
163294394
3
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�
9x2�
y2�
(y � 2)2�
9x2�
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�
25y2�
x2�
(y � 3)2�
9(x � 1)2�
25x2�
(y � 2)2�
9y2�
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. �x9
2� � �3
y6
2� � 1 2. �8
x1
2� � �
y9
2� � 1 3. �
x4
2� � �
y9
2� � 1
0.87 0.94 0.75
4. �1x6
2� � �
y9
2� � 1 5. �3
x6
2� � �1
y6
2� � 1 6. �
x4
2� � �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�
9x2�
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.
�x4
2� � �
1y2
2� � 1
b2a2c2
4
2
(0, 0)
x
y
O
asymptotes3�4
3�4
conjugate
transverse
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 equation
3x2 � 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 hyperbola
11. 3x2 � 4y2 � 50 � y2 12. y2 � 8x � 11circle parabola
13. 9y2 � 99y � 3(3x � 3x2) 14. 6x2 � 4 � 5y2 � 3circle hyperbola
15. 111 � 11x2 � 10y2 16. 120x2 � 119y2 � 118x � 117y � 0ellipse hyperbola
17. 3x2 � 4y2 � 12 18. 150 � x2 � 120 � yhyperbola 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 hyperbola
x � � 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�
1y2
�x2
�
xO
y
x
y
Ox
y
O
(y � 1)2�
5(x � 3)2�
61�
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�11
circle55
parabola02
ellipse12
( 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���
27 � �29���
2�30���34���30���34��
1 � �29���2
7 � �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. � � 1x � 3y � 5 2x2 � 9y2 � 18 x � ( y � 3)2 � 3
x2 � 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? hyperbola
b. What kind of conic section is the graph of the second equation? ellipse
c. Based on your answers to parts a and b, what are the possible numbers of solutionsthat 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
Chapter 8 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 497 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. What is the midpoint of the line segment with endpoints at (12, 7) and (18, 19)?A. (30, 26) B. (15, 13) C. (�6, �12) D. (3, 6) 1.
2. Choose the midpoint of the line segment with endpoints at (5, 9) and (11, 15).A. (8, 12) B. (16, 24) C. (6, 6) D. (�6, �6) 2.
3. Find the distance between A(12, 8) and B(4, 2).A. 14 units B. 100 units C. 10 units D. �10 units 3.
4. What is the distance between C(4, 3) and D(7, 7)?A. �5 units B. 7 units C. 25 units D. 5 units 4.
5. Write the equation of the parabola y � x2 � 10x � 16 in standard form.A. y � (x � 5)2 � 9 B. y � (x � 5)2 � 41C. y � (x � 5)2 � 16 D. y � (x � 8)(x � 2) 5.
6. Write an equation for the parabola with vertex (1, 0) if the length of the
latus rectum is �12� and the parabola opens down.
A. y � ��12�(x � 1)2 B. y � �2(x � 1)2 C. x � �2(y � 1)2 D. x � ��
12�(y � 1)2 6.
7. Which is the equation of a parabola that opens downward and has axis of symmetry x � �1?A. y � (x � 1)2 � 2 B. y � (x � 1)2 � 2C. y � �(x � 1)2 � 2 D. y � �(x � 1)2 � 2 7.
8. Find the center and radius of the circle with equation (x � 2)2 � y2 � 9.A. (�2, 0); 9 B. (0, 2); 9 C. (2, 0); 3 D. (0, �2); 3 8.
9. Write an equation for the circle with center (2, �3) that is tangent to the y-axis.A. (x � 2)2 � (y � 3)2 � 9 B. (x � 2)2 � (y � 3)2 � 9C. (x � 2)2 � (y � 3)2 � 4 D. (x � 2)2 � (y � 3)2 � 4 9.
10. Which is the equation of a circle with center (2, 1) that passes through (2, 4)?A. (x � 2)2 � (y � 1)2 � 9 B. (x � 2)2 � (y � 1)2 � 3C. (x � 2)2 � (y � 1)2 � 9 D. (x � 2)2 � (y � 1)2 � 3 10.
11. Which is the equation of an ellipse with foci at (0, 3) and (0, �3) that has the endpoints of its major axis at (0, 4) and (0, �4)?
A. �1y62� � �
x92� � 1 B. x2 � y2 � 16 C. �1
x62� � �
y72� � 1 D. �1
y62� � �
x72� � 1 11.
88
© Glencoe/McGraw-Hill 498 Glencoe Algebra 2
Chapter 8 Test, Form 1 (continued)
12. Which equation is graphed at the right?
A. �1x62� � �
y42� � 1 B. �1
y62� � �
x42� � 1
C. �1x62� � �
y42� � 1 D. �1
y62� � �
x42� � 1 12.
13. Which is the equation of a hyperbola with vertices (0, 2) and (0, �2) and foci (0, 3) and (0, �3)?
A. �y52� � �
x42� � 1 B. �
y42� � �
x52� � 1 C. �
x42� � �
y52� � 1 D. �
x52� � �
y42� � 1 13.
14. Which equation is graphed at the right?
A. �1x62� � �
y42� � 1 B. �
x42� � �1
y62� � 1
C. �1y62� � �
x42� � 1 D. �
y42� � �1
x62� � 1 14.
15. What is the standard form of the equation 5x2 � 5y2 � 20 � 0?A. 5x2 � 5y2 � 20 B. y2 � �x2 � 4 C. x2 � y2 � 4 � 0 D. x2 � y2 � 4 15.
16. What is the graph of x2 � 4y2 � 2y � 8?A. parabola B. circle C. ellipse D. hyperbola 16.
17. Which equation has a hyperbola as its graph?A. 4x2 � 4y2 � 16 B. 4x2 � 4y � 16 C. 4x2 � 4y2 � 16 D. x2 � 4y2 � 16 17.
18. Find the exact solution(s) of the system of equations x2 � y2 � 16 and x � y � 4.A. (�4, 0) and (0, 4) B. (4, 0) and (�4, 0)C. (0, 4) and (0, �4) D. (4, 0) and (0, �4) 18.
19. Solve the system of equations by graphing y � x2 and y � 2x.A. (0, 0) and (4, �2) B. (0, 0) and (�2, 4)C. (0, 0) and (2, 4) D. (0, �1) and (2, 2) 19.
20. Which system of inequalities is graphed at the right?A. x2 � y2 � 9 B. x2 � y2 � 9
y � x � 1 y � x � 1
C. x2 � y2 � 9 D. x2 � y2 � 9y � x � 1 y � x � 1 20.
Bonus For the equation 4x2 � ky2 � 8x � 17y � 3, find a value of k so that the graph of the equation is a. a circle b. an ellipse c. a hyperbola d. a parabola B:
NAME DATE PERIOD
88
y
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y
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y
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Chapter 8 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 499 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1 and 2, refer to the figure at the right showing six city locations. The origin is at the lower left corner of the grid.
1. What is the location of the point halfway between the wharf and library?
A. (7, 2) B. ��125�, �
52�� C. �7, �
52�� D. ��
125�, 2� 1.
2. What is the distance between the library and zoo?
A. 11 units B. �61� units C. 61 units D. �621� units 2.
3. Write the equation of the parabola y � 2x2 � 8x � 1 in standard form.A. y � 2(x � 2)2 � 9 B. y � (x � 4)2 � 15C. y � 2(x � 2)2 � 7 D. y � 2(x � 4)2 � 15 3.
4. Write an equation for the parabola with focus (4, 0) and vertex (2, 0).
A. x � �18�y2 � 2 B. x � ��
18�y2 � 2 C. y � �
18�x2 � 2 D. y � ��
18�x2 � 2 4.
5. Which equation is graphed at the right?A. y � 4x2 � 16x � 16 B. x � 4y2 � 16y � 16
C. y � �14�x2 � x � 1 D. x � �
14�y2 � y � 1 5.
6. Write an equation for a circle if the endpoints of a diameter are at (�7, 1) and (5, 1).A. x2 � (y � 1)2 � 6 B. (x � 1)2 � (y � 1)2 � 36C. (x � 1)2 � y2 � 6 D. (x � 1)2 � (y � 1)2 � 36 6.
7. Which is the equation of a circle with center (2, 0) and radius 2 units?A. x2 � y2 � 4x � 0 B. x2 � y2 � 4x � 0C. x2 � y2 � 4y � 0 D. x2 � y2 � 4y � 0 7.
8. Write an equation for an ellipse if the endpoints of the major axis are at (�1, 5) and (�1, �3) and the endpoints of the minor axis are at (�4, 1) and (2, 1).
A. �(y �
161)2� � �
(x �9
1)2� � 1 B. �
(x �16
1)2� � �
(y �9
1)2� � 1
C. �(x �
161)2� � �
(y �9
1)2� � 1 D. �
(y �16
1)2� � �
(x �9
1)2� � 1 8.
9. Which is the equation of an ellipse with center (1, �2) and a vertical major axis?
A. �(y �
92)2� � �
(x �4
1)2� � 1 B. �
(x �9
1)2� � �
(y �4
2)2� � 1
C. �(y �
92)2� � �
(x �4
1)2� � 1 D. �
(x �9
1)2� � �
(y �4
2)2� � 1 9.
88
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AMPHITHEATER
TOWN SQUARE
UNIVERSITY
LIBRARY
WHARF
ZOO
y
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© Glencoe/McGraw-Hill 500 Glencoe Algebra 2
Chapter 8 Test, Form 2A (continued)
10. Find the center and radius of the circle with equation x2 � (y � 4)2 � 9.A. (0, 4); 9 B. (4, 0); 3 C. (�4, 0); 9 D. (0, 4); 3 10.
11. Write an equation for the hyperbola with vertices (�10, 1) and (6, 1) and foci (�12, 1) and (8, 1).
A. �(x �
642)2� � �
(y �36
1)2� � 1 B. �
(x �36
2)2� � �
(y �64
1)2� � 1
C. �(x �
642)2� � �
(y �36
1)2� � 1 D. �
(x �36
2)2� � �
(y �64
1)2� � 1 11.
12. Which equation is graphed at the right?A. x2 � 9y2 � 9 B. 9y2 � x2 � 9C. 9x2 � y2 � 9 D. y2 � 9x2 � 9 12.
13. Write the equation x2 � 2x � y2 � 4y � 11 in standard form.A. (x � 1)2 � (y � 2)2 � 16 B. (x � 1)2 � (y � 2)2 � 16
C. �(x �
11)2� � �
(y �4
2)2� � 1 D. �
(x �4
1)2� � �
(y �4
1)2� � 1 13.
14. Write the equation 4x2 � 24x � y � 34 � 0 in standard form.A. y � 4(x � 3)2 � 2 B. x � 4y2 � 2C. y � 4(x � 3)2 � 2 D. x � 4(y � 3)2 � 2 14.
15. What is the graph of 4x2 � y2 � 8y � 32?A. parabola B. circle C. ellipse D. hyperbola 15.
16. The graph of which equation is a circle?A. 5x2 � 10x � 9 � 5y2 B. 5x2 � 10x � 9 � 5y2
C. 5x2 � 5x � y2 � 9 D. 5x2 � 10x � 5y � 9 16.
17. Solve the system of equations by graphing x2 � y2 � 16 and y � �x � 4.A. (4, 0), (0, �4) B. (0, �4), (�4, 0) C. (�4, 0), (0, �4) D. (0, 4), (4, 0) 17.
18. Which system of inequalities is graphed at the right?A. x2 � y2 � 16 B. x2 � y2 � 16
x � y � �3 x � y � �3C. x2 � y2 � 16 D. x2 � y2 � 16
x � y � �3 x � y � �3 18.
Find the exact solution(s) of each system of equations.
19. x2 � y2 � 25 and 9y � 4x2
A. (4, 3), (�4, 3) B. (3, 4), (3, �4) C. (4, 3), (4, �3) D. (3, 4), (�3, 4) 19.
20. y � x2 � 1 and y � 2xA. (1, 2), (�1, 2) B. (�1, 2) C. (1, 2) D. (�1, 2), (0, 2) 20.
Bonus Solve the system of equations (x � 2)2 � y2 � 1 and B:(x � 2)2 � y2 � 1.
NAME DATE PERIOD
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y
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Chapter 8 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 501 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1 and 2, refer to the figure at the right showing six city locations. The origin is at the lower left corner of the grid.
1. What is the location of the point halfway between the hospital and arena?
A. (2, 6) B. �6, �52�� C. ��
52�, 6� D. ��
12�, 3� 1.
2. What is the distance between the museum and sailing club?
A. �53� units B. 53 units C. �523� units D. 9 units 2.
3. Write the equation of the parabola y � 4x2 � 8x � 1 in standard form.A. y � (x � 4)2 � 15 B. y � 4(x � 1)2 � 5C. y � 4(x � 1)2 � 3 D. y � 4(x � 4)2 � 15 3.
4. Write an equation for the parabola with focus (1, 3) and vertex (0, 3).
A. y � 4(x � 3)2 B. x � ��14�(y � 3)2 C. y � �4�x � �
34��2
D. x � �14�(y � 3)2 4.
5. Which equation is graphed at the right?A. y � 2x2 � 8x � 7 B. x � �2y2 � 8y � 7C. y � �2x2 � 8x � 7 D. y � �2x2 � 8x � 7 5.
6. Write an equation for a circle if the endpoints of a diameter are at (1, 1) and (1, �9).A. (x � 1)2 � (y � 4)2 � 5 B. (x � 1)2 � (y � 4)2 � 25C. (x � 1)2 � (y � 4)2 � 5 D. (x � 1)2 � (y � 4)2 � 25 6.
7. Which is the equation of a circle with center (0, 1) and radius 2 units?A. x2 � y2 � 2y � 3 B. x2 � y2 � 2y � 1C. x2 � y2 � 2y � 4 D. x2 � y2 � 2y � 3 7.
8. Write an equation for an ellipse if the endpoints of the major axis are at (1, 6) and (1, �6) and the endpoints of the minor axis are at (5, 0) and (�3, 0).
A. �(x �
361)2� � �1
y62� � 1 B. �
(x �36
1)2� � �1
y62� � 1
C. �3y62� � �
(x �16
1)2� � 1 D. �3
y62� � �1
x62� � 1 8.
9. Which is the equation of an ellipse with center (�4, 2) and a horizontal major axis?
A. �(x �
164)2� � �
(y �4
2)2� � 1 B. �
(x �16
4)2� � �
(y �4
2)2� � 1
C. �(y �
162)2� � �
(x �4
4)2� � 1 D. �
(y �16
2)2� � �
(x �4
4)2� � 1 9.
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ARENA SAILING CLUB
CITY CENTER
HOSPITAL
LIBRARY
MUSEUM
y
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© Glencoe/McGraw-Hill 502 Glencoe Algebra 2
Chapter 8 Test, Form 2B (continued)
10. Find the center and radius of the circle with equation (x � 1)2 � y2 � 16.A. (1, 0); 4 B. (�1, 0); 16 C. (0, 1); 4 D. (0, �1); 16 10.
11. Write an equation for the hyperbola with vertices (0, �1) and (0, 3) and foci (0, �3) and (0, 5).
A. �(y �
41)2� � �1
x22� � 1 B. �
(x �12
1)2� � �
y42� � 1
C. �(y �
41)2� � �1
x22� � 1 D. �
x42� � �
(y �12
1)2� � 1 11.
12. Which equation is graphed at the right?A. 9x2 � 4y2 � 36 B. 4x2 � 9y2 � 36C. 9y2 � 4x2 � 36 D. 4y2 � 9x2 � 36 12.
13. Write the equation 4x2 � 8x � y2 � 4y � 4 � 0 in standard form.
A. �(x �
11)2� � �
(y �4
2)2� � 1 B. (x � 1)2 � (y � 2)2 � 4
C. �(x �
11)2� � �
(y �4
2)2� � 1 D. �
(x �4
1)2� � �
(y �1
2)2� � 1 13.
14. Write the equation 2y2 � 4y � x � 12 � 0 in standard form.A. y � 2(x � 1)2 � 6 B. x � 2(y � 1)2 � 10C. y � (x � 1)2 � 10 D. x � 2(y � 1)2 � 6 14.
15. The graph of which equation is a circle?A. 6x2 � 12x � 6y2 � 1 B. 6x2 � 12x � 6y2 � 1C. 6x2 � 6y2 � 12x � 1 D. 6x2 � 6y � 12x � 1 15.
16. What is the graph of x2 � 25y2 � 50?A. parabola B. circle C. ellipse D. hyperbola 16.
17. Solve the system of equations by graphing y � x2 � 2 and y � 2x � 2.A. (�2, 0), (2, 2) B. (2, 0), (0, �2) C. (�2, 0), (�2, 2) D. (0, �2), (2, 2) 17.
18. Which system of inequalities is graphed at the right?A. x2 � y2 � 9 B. x2 � y2 � 9
y2 � x � 0 y2 � x � 0C. x2 � y2 � 9 D. x2 � y2 � 9
y2 � x � 0 y2 � x � 0 18.
Find the exact solution(s) of each system of equations.
19. x2 � 4y2 � 16 and x � 2y � �4A. (0, �2), (0, 2) B. (0, �2), (4, 0) C. (0, �2), (�4, 0) D. (0, 2), (4, 0) 19.
20. x2 � y2 � 36 and y � x � 6A. (0, �6), (6, 0) B. (0, 6), (6, 0) C. (6, 0), (�6, 0) D. (�6, 0), (0, 6) 20.
Bonus Solve the system of equations x2 � (y � 3)2 � 4 and B:x2 � (y � 3)2 � 4.
NAME DATE PERIOD
88
y
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y
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Chapter 8 Test, Form 2C
© Glencoe/McGraw-Hill 503 Glencoe Algebra 2
1. Find the midpoint of the line segment with endpoints at 1.(�2, 3) and (14, 6).
2. Find the distance between A(4, �2) and B(10, �7). 2.
3. Write an equation for the parabola with focus (4, 4) and 3.directrix x � �2.
4. Write the y � 3x2 � 6x � 2 in standard form. 4.
5. Identify the coordinates of the vertex and focus, the 5.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y2 � 8y � 18 � x.
6. Write an equation for the circle with center (�4, 2) that is 6.tangent to the y-axis.
Graph each equation.
7. x2 � y2 � 4x � 6y � 3 � 0 7.
8. 9x2 � 4y2 � 36 8.
For Questions 9 and 10, write an equation for the ellipse that satisfies each set of conditions.
9. endpoints of major axis at (9, 3) and (�11, 3), 9.endpoints of minor axis at (�1, 8) and (�1, �2)
10. major axis 12 units long and parallel to the y-axis, 10.minor axis 8 units long, center at (�2, 5)
11. Find the exact solution(s) of the system of equations. 11.x2 � y � 44x2 � y2 � 12
y
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y
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NAME DATE PERIOD
SCORE 88
Ass
essm
ent
© Glencoe/McGraw-Hill 504 Glencoe Algebra 2
Chapter 8 Test, Form 2C (continued)
For Questions 12 and 13, write an equation for the hyperbola that satisfies each set of conditions.
12. vertices (9, 0) and (�9, 0), conjugate axis of length 10 units 12.
13. vertices (�1, 4) and (�1, �8), foci ( � 1, � 2 �39�) 13.
14. Find the coordinates of the vertices and foci and the 14.equations of the asymptotes for the hyperbola (x � 3)2 � (y � 1)2 � 4.
Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
15. x2 � y2 � 2x � 2y � 23 15.
16. 4x2 � 9y2 � 24x � 18y � 9 � 0 16.
For Questions 17 and 18, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. State the values used to identify each conic section without writing each equation in standard form.
17. 3(x � 5)2 � 3y � 15 � 0 17.
18. 4x2 � 8x � 4(y2 � 2y) � 7 18.
19. Graph the system of equations. Use the graph to solve the 19.system.y � x2 � 4xy � x � 4
20. Solve the system of inequalities by graphing. 20.x2 � y2 � 16y � �2x2 � 1
Bonus Write an equation for the circle with the same center as B:
the graph of �(x �4
3)2� � �
(y �16
1)2� � 1 and the same radius
as the graph of x2 � y2 � 4x � 10y � 9.
y
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y
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NAME DATE PERIOD
88
Chapter 8 Test, Form 2D
© Glencoe/McGraw-Hill 505 Glencoe Algebra 2
1. Find the midpoint of the line segment with endpoints at 1.(�4, 5) and (7, �3).
2. Find the distance between A(�7, 3) and B(4, �6). 2.
3. Write an equation for the parabola with focus (�1, 1) and 3.directrix y � �7.
4. Write the equation of the parabola x � 5y2 � 10y � 2 in 4.standard form.
5. Identify the coordinates of the vertex and focus, the 5.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y � 2x2 � 4x � 5.
6. Write an equation for the circle with center ��12�, �2� that is 6.
tangent to the x-axis.
Graph each equation.
7. x2 � y2 � 2x � 4y � 4 7.
8. 9x2 � 16y2 � 144 8.
For Questions 9 and 10, write an equation for the ellipse thatsatisfies each set of conditions.
9. endpoints of major axis at (2, �5) and (2, 9), 9.endpoints of minor axis at (�4, 2) and (6, 2)
10. major axis 16 units long and parallel to the x-axis, 10.minor axis 6 units long, center at (1, �4)
11. Find the exact solution(s) of the system of equations. 11.x2 � 2y � 113x2 � y2 � 24
y
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y
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NAME DATE PERIOD
SCORE 88
Ass
essm
ent
© Glencoe/McGraw-Hill 506 Glencoe Algebra 2
Chapter 8 Test, Form 2D (continued)
For Questions 12 and 13, write an equation for the hyperbola that satisfies each set of conditions.
12. vertices (0, 12) and (0, �12), conjugate axis of length 8 units 12.
13. vertices (�10, 1) and (4, 1), foci (�3 �70�, 1) 13.
14. Find the coordinates of the vertices and foci and the 14.equations of the asymptotes for the hyperbola (x � 1)2 � (y � 3)2 � 4.
Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
15. 4x2 � 16x � y � 21 � 0 15.
16. y2 � 6y � 4x2 � 8x � 95 16.
For Questions 17 and 18, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. State the values used to identify each conic section without writing each equation in standard form.
17. 2x2 � 10x � 8y � 2y2 � 5 17.
18. 3(y � 2)2 � 8 � 9x � 10x2 18.
19. Graph the system of equations. Use the graph to solve the 19.system.y2 � 9 � x2
y � ��34�x � 4
20. Solve the system of inequalities by graphing. 20.x2 � 4y2 � 1x � 4(y � 2)2
Bonus Write an equation for the circle with the same center as B:
the graph of �(x �16
5)2� � �
(y �9
2)2� � 1 and the same radius
as the graph of x2 � y2 � 2y � 16x � 1.
y
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y
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NAME DATE PERIOD
88
Chapter 8 Test, Form 3
© Glencoe/McGraw-Hill 507 Glencoe Algebra 2
1. Find the midpoint of the line segment with endpoints at 1.(�12, 3.5) and (5.1, 4.8).
2. Find the distance between A(4�5�, �2) and B(�5�, 9). 2.
3. Write the equation x � �y2 � 6y � 7 in standard form. 3.
4. Write an equation for the parabola with vertex (�5, 1) and 4.
directrix x � ��72�.
5. The path traveled by Pati’s remote-controlled model 5.airplane is shaped like a parabola. It took off from the ground and landed on the ground 160 feet away from where it took off. If the airplane reached a maximum height of 40 feet, write an equation for the parabola that models the path of the plane. Assume that the point of take-off is the origin.
6. Identify the coordinates of the vertex and focus, the 6.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation x � �y2 � 2y � 9.
7. Write an equation for a circle if its center is in the first 7.quadrant, and it is tangent to x � �2, x � 8 and the x-axis.
8. Graph x2 � y2 � 4x � 2y � 3 � 0. 8.
9. Graph 5x2 � 2y2 � 4y � 22. 9.
For Questions 10 and 11, write an equation for the ellipse that satisfies each set of conditions.
10. major axis 14 units long and parallel to the x-axis, minor 10.
axis 10 units long, center at �5, ��12��
11. endpoints of major axis at (3, �8) and (3, 4), foci at 11.(3, �2 � 2�5�) and (3, �2 � 2�5�)
y
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y
xO
NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 508 Glencoe Algebra 2
Chapter 8 Test, Form 3 (continued)
12. Find the coordinates of the center and foci and the lengths 12.of the major and minor axes for the ellipse with equation 6x2 � 5y2 � 24x � 30y � �39.
13. Write an equation for the hyperbola with vertices (4, �5) 13.and (4, 1) and foci (4, 3) and (4, �7).
Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
14. 2x2 � 3y2 � 15 � 4(x � 2y) 14.
15. �18�x � y2 � �(y � 12) 15.
For Questions 16 and 17, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. State the values used to identify each conic section without writing each equation in standard form.
16. 3x2 � 9x � y2 � 2(24y � y2 � 27) 16.
17. 34x2 � 40y2 � 18x � 25y � 17(2x2 � 1) 17.
18. Find the exact solution(s) of the system of equations. 18.
�2x52� � �1
y62� � 1
x � y
19. Solve the system of equations by graphing. 19.x2 � y2 � 4x � 6y � 4 � 0x2 � 4x � 3y � 4 � 0
20. Solve the system of inequalities by graphing. 20.x2 � y2 � 4x � �4 � 102 � y � (x � 1.75)2
Bonus The parabolic curve of a certain camera lens can be B:represented by the equation y � 10x2 � 50x � 63.2.What are the coordinates of the focus?
y
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NAME DATE PERIOD
88
Chapter 8 Open-Ended Assessment
© Glencoe/McGraw-Hill 509 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.
1. Harry was asked to determine whether the graph of the equationx2 � y2 � 8x � 6y � 30 � 0 was a parabola, circle, ellipse, or hyperbola.At first glance, he identified the equation as that of a circle.a. What made Harry think he was looking at the equation of a circle?b. When Harry attempted to find the center and radius of the circle,
he ran into a problem. What was the problem?c. Change the equation so that Harry’s problem no longer exists,
then find the center and radius of the circle represented by yourequation.
2. Do the graphs of any of the conic sections you have studied in this chapter represent relations that are functions? Explain your reasoning.
3. What do the graphs of the parabolas y � (x � 2)2 � 1 andx � (y � 1)2 � 2 have in common? How are the graphs different?
4. The graphs of the equations (x � 4)2 � (y � 3)2 � 4 and y � (x � 4)2 � 3 are shown. For parts a and b,replace each of the �s with one of the inequality symbols (�, �, � , �) so that the solution of the system is the region indicated. Explain your choices.a. (x � 4)2 � (y � 3)2 � 4
y � (x � 4)2 � 3The solution of the system is region 2.
b. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3The solution of the system is region 3.
c. What region is represented by the system (x � 4)2 � (y � 3)2 � 4 and y � (x � 4)2 � 3? Explain.
5. The graph of the equation �(x �4
1)2� � �
(y �9
2)2� � 1
is shown. Find values of k for which the given system of equations has the given number of solutions. Explain the reasoning for your choices.
�(x �
41)2� � �
(y �9
2)2� � 1
y � k
a. For k � ____ and k � ____, the system has two solutions.b. For k � ____ and k � ____, the system has one solution.c. For k � ____ and k � ____, the system has no solutions.
NAME DATE PERIOD
SCORE 88
Ass
essm
ent
y
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1
4
2
3
y
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© Glencoe/McGraw-Hill 510 Glencoe Algebra 2
Chapter 8 Vocabulary Test/Review
Choose from the terms above to complete each sentence.
1. A is the set of all points in a plane that are the same distance from a given point and a given line. The given point is called the
and the given line is called the .
2. The set of all points in a plane the sum of whose distances from two fixed points is
constant is a(n) . The two fixed points are called the
.
3. The set of all points in a plane such that the absolute value of the difference of their
distances from the two given points is constant is a(n) .
4. The points at which an ellipse intersects its axes of symmetry determine two segments
on the ellipse. The shorter of these segments is called the
and the longer one is called the .
5. The segment that connects the two vertices of a hyperbola is called the
.
6. A line that intersects a circle in exactly one point is to the circle.
7. The line segment through the focus of a parabola and perpendicular
to the line of symmetry is called the .
8. A line that the branches of a hyperbola approach but do not intersect
is called a(n) .
9. The segment of length 2b units that is perpendicular to the transverse axis
of a hyperbola at its center is called the .
10. The formula that can be used to find the length of a line segment if you know
the coordinates of its endpoints is called the .
In your own words—Define each term.
11. circle
12. vertex of a hyperbola
asymptotecenter of a circlecenter of an ellipsecenter of a hyperbolacircleconic section
conjugate axisdirectrixDistance Formulaellipsefoci of an ellipsefoci of a hyperbola
focus of a parabolahyperbolalatus rectummajor axisMidpoint Formulaminor axis
parabolatangenttransverse axisvertex of a hyperbola
NAME DATE PERIOD
SCORE 88
Chapter 8 Quiz (Lessons 8–1 and 8–2)
88
© Glencoe/McGraw-Hill 511 Glencoe Algebra 2
1. Find the midpoint of the line segment with endpoints at 1.(�7, �3) and (5, 10).
2. Standardized Test Practice Which point is farthest from (2, �1)?A. (3, 3) B. (�2, �1) C. (4, 0) D. (�1, 0) 2.
3. Write an equation for the parabola with focus (1, 4) and 3.directrix y � �2.
4. Identify the coordinates of the vertex and focus, the 4.equations of the axis of symmetry and directrix, and the direction of opening of the parabola with equation y � �2x2 � 16x � 27.
5. Graph the parabola x � 6y2 � 24y � 25 and find the length 5.of the latus rectum. y
xO
NAME DATE PERIOD
SCORE
Chapter 8 Quiz (Lessons 8–3 and 8–4)
For Questions 1 and 2, write an equation for the circle that satisfies each set of conditions.
1. center (�7, 2), radius 9 units 1.
2. endpoints of a diameter at (�1, 1) and (7, 1) 2.
3. Find the center and radius of the circle with equation 3.x2 � y2 � 2x � 2y � 7. Then graph the circle.
4. Write an equation for an ellipse if the endpoints of the major axis are at (5, 1) and (�5, 1) and the endpoints of the minor axis are at (0, 5) and (0, �3). 4.
5. Find the coordinates of the center and foci and the lengths 5.of the major and minor axes for the ellipse with equation
�(x �
163)2� � �
y42� � 1.
y
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NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 512 Glencoe Algebra 2
1. Write an equation for the hyperbola whose graph is shown. 1.
2. Write an equation for the hyperbola 2.with vertices (2, �5) and (2, 3),foci (2, �6) and (2, 4)
3. Find the coordinates of the vertices 3.and foci and the equations of the
asymptotes for the hyperbola �x42� � �3
y62� � 1. Then graph the
hyperbola.
4. Write 2x2 � 12x � y � 5 in standard form. Then state whether the graph of the equation is a parabola, circle,ellipse, or hyperbola. 4.
5. State whether the graph of x2 � 2x � 4y2 � 24y � 37 � 0 is 5.a parabola, circle, ellipse, or hyperbola. State the values used to identify the conic section without writing the equation in standard form.
y
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Chapter 8 Quiz (Lesson 8–7)
Graph each system of equations. Use the graph to solve the system.
1. y � (x � 2)2 � 3 1.
�(x �
12)2� � �
(y �4
2)2� � 1
For Questions 3 and 4, find the exact solution(s) of each system of equations.
2. x2 � y2 � � 8 2.y � 2x � 1
3. 2x2 � 5y2 � 22 3.y2 � 3x2 � 1
4. Solve the system of inequalities by graphing. 4.
�x92� � �
y42� � 1
�2x52� � �
y92� � 1
y
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2
2
y
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NAME DATE PERIOD
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Chapter 8 Quiz (Lessons 8–5 and 8–6)
88
NAME DATE PERIOD
SCORE
88
y
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(1, 2)
Chapter 8 Mid-Chapter Test (Lessons 8–1 through 8–4)
© Glencoe/McGraw-Hill 513 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. What is the midpoint of the line segment with endpoints at (�6, 3) and (�10, 7)?A. (�8, 5) B. (�16, 10) C. (2, �2) D. (4, �4) 1.
2. Find the distance between A(�3, 1) and B(5, �5).A. 100 B. 32 C. 4 D. 10 2.
3. Write an equation for the parabola with vertex (1, 2) and directrix x � �34�.
A. y � (x � 2)2 � 1 B. x � (y � 2)2 � 1C. y � (x � 2)2 � 1 D. x � (y � 1)2 � 2 3.
4. Which equation is graphed?A. y � 4x2 � 8x � 4 B. x � 4y2 � 1
C. y � �14�x2 � �
12�x � �
14� D. x � �
14�y2 � �
12�y � �
14� 4.
5. Write an equation of the circle with center (�2, 7) that is tangent to the y-axis.A. (x � 2)2 � (y � 7)2 � 4 B. (x � 2)2 � (y � 7)2 � 49C. (x � 2)2 � (y � 7)2 � 4 D. (x � 2)2 � (y � 7)2 � 49 5.
6. Graph x2 � y2 � 4x � 12. 6.
7. Write an equation of the ellipse centered at (4, 1) if its minor 7.axis is 8 units long and its major axis is 10 units long and parallel to the x-axis.
8. Write the equation of the parabola y � �3x2 � 18x � 5 in 8.standard form.
9. Write an equation for a circle if the endpoints of a diameter 9.are at (�2, �1) and (8, 9).
y
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Part II
Part I
NAME DATE PERIOD
SCORE 88
Ass
essm
ent
y
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© Glencoe/McGraw-Hill 514 Glencoe Algebra 2
Chapter 8 Cumulative Review (Chapters 1–8)
1. Evaluate �3� 5a � b � if a � �3.5 and b � 10. 1.(Lesson 1-4)
2. Write an equation in slope-intercept form for the line that 2.has a slope of 4 and passes through (2, 5). (Lesson 2-4)
3. Solve the system of equations by using substitution. 3.y � 6x � 52x � 3y � 1. (Lesson 3-2)
4. Perform the indicated matrix operation. If the matrix does 4.
not exist, write impossible. � � � � � (Lesson 4-2)
5. Simplify �22xx
2
2��
77xx
��
43�. Assume that the denominator is not 5.
equal to 0. (Lesson 5-4)
6. Simplify (3 � 4i) � (2 � 5i). (Lesson 5-9) 6.
7. Find the exact solutions to 2x2 � 7x � 5 � 0 by using the 7.Quadratic Formula. (Lesson 6-5)
8. Solve the inequality � 2x � 3 � x2 algebraically. 8.(Lesson 6-7)
9. Determine whether the graph 9.represents an odd degree or an even degree polynomial function. Then state the number of real zeros. (Lesson 7-1)
10. One factor of 2x3 � 7x2 � 2x � 3 is x � 3. 10.Find the remaining factors. (Lesson 7-4)
11. If g(x) � 4x and h(x) � 3x � 5, find [gh](x). (Lesson 7-7) 11.
12. Find the midpoint of the line segment with end points at 12.(�10, 8) and (2, �3). (Lesson 8-1)
13. Write an equation for the parabola with focus (�4, 0) and 13.directrix x � 6. (Lesson 8-2)
14. Find the coordinates of the center and foci and the lengths 14.of the major and minor axes for the ellipse with equation 9x2 � y2 � 9. Then graph the ellipse. (Lesson 8-4)
15. Write the equation 4x2 � 9y2 � 24x � 18y � 9 � 0 in 15.standard form. Then state whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.(Lesson 8-6)
4 3 7�2 9 5
3 6 9�2 �1 0
NAME DATE PERIOD
88
xO
f(x )
Standardized Test Practice (Chapters 1–8)
© Glencoe/McGraw-Hill 515 Glencoe Algebra 2
1. Which of the following is the sum of two consecutive prime numbers? A. 9 B. 11 C. 17 D. 24 1.
2. If (r � 2)(r � 1) � (r � 2)(r � 6), which of the following is true?E. r � 14 F. r � �14 G. r � 14 H. r � 19 2.
3. What is the value of 10m � 3 if 2m � 9?A. 2 B. 8 C. 42 D. 48 3.
4. If 10 pears cost c cents, how many pears will d dollars buy?
E. �100
c0d� F. �1
d0c�
G. �10
cd
� H. �1d0c� 4.
5. What is the value of �xy� is 2.5x � �
131�y and y 0?
A. �565� B. �
23� C. �
2125�
D. �32� 5.
6. The sum of five integers is what percent of the average of the same five integers?E. 5 F. 50 G. 500 H. 5000 6.
7. Which of the following are always true statements?I. x2 � 0 II. x2 � x III. x � 1 � x IV. x � �x
A. I and II only B. I and IV onlyC. III and IV only D. III only 7.
8. The table shows the distribution of quiz scores for a group of students.No student scored less than 50 or greater than 90. What is the mean of the scores?E. 70 F. 72.5G. 75 H. 74.5 8.
9. Square RSTU is inscribed in circle O.If the circumference of circle O is 16�,find the area of triangle ROU.A. 32 B. 32�
C. 64 D. 16� 9.
10. What is the value of t if r � �1 and t � (r � 1)(r � 2)(r � 3)?E. 0 F. 2 G. 6 H. 24 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
88
Ass
essm
ent
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
Score Number ofstudents
90 2
80 6
70 8
60 3
50 1
R S
O
U T
© Glencoe/McGraw-Hill 516 Glencoe Algebra 2
Standardized Test Practice (continued)
11. A jar contains 2 white marbles, 5 red marbles, 11. 12.and 13 blue marbles. How many white marbles must be added to the jar to make the probability of randomly selecting a white
marble �14�?
12. In the figure shown, what is the length of C�D�?
13. If the sales tax on a $22.00 purchase is 13. 14.$1.32, what is the total cost of an item priced at $8.50?
14. Evaluate 7 � 3 � 5 � 22.
Column A Column B
15. 2 � d � e � 2 15.
16. The average of a, b, and c is x. 16.
17. 17.
18. � �a �
cb
� for all real numbers a, b, and c 18.
10 15
10
2 3
4
DCBA
a b
c
DCBA�4
162��3
64�
3a3x
DCBA
d � ee � d
DCBA
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater;if the quantity in column B is greater;if the quantities are equal; or if the relationship cannot be determined from the information given.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
NAME DATE PERIOD
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NAME DATE PERIOD
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
A
D
C
B
A E D
CB
15
560˚
Standardized Test PracticeStudent Record Sheet (Use with pages 468–469 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
88
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 9
2 5 8 10
3 6
Solve the problem and write your answer in the blank.
Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.
11 13 15 17
12 14 16
Select the best answer from the choices given and fill in the corresponding oval.
18 20
19 21 DCBADCBA
DCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
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0 0 0
.. ./ /
.
99 9 987654321
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87654321
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DCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 8-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Mid
po
int
and
Dis
tan
ce F
orm
ula
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
5G
lenc
oe A
lgeb
ra 2
Lesson 8-1
The
Mid
po
int
Form
ula
Mid
po
int
Fo
rmu
laT
he m
idpo
int
Mof
a s
egm
ent
with
end
poin
ts (
x 1,
y 1)
and
(x2,
y2)
is �
, �.
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)
Th
e m
idpo
int
of t
he
segm
ent
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
?
Th
e ce
nte
r of
th
e ci
rcle
is
the
mid
poin
t of
all
of i
ts d
iam
eter
s.
�,
���
,�
��
,�o
r ��
1,�
2�
Th
e ci
rcle
has
cen
ter ��
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)
an
d (�
2,11
)2.
(�8,
�3)
an
d (1
0,9)
3.(4
,15)
an
d (1
0,1)
(5,9
)(1
,3)
(7,8
)
4.(�
3,�
3) a
nd
(3,3
)5.
(15,
6) a
nd
(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)
an
d (�
1,24
) 11
.(3,
�10
) an
d (3
0,�
20)
12.(
�9,
1.7)
an
d (�
11,1
.3)
(�4,
9)�
,�15
�(�
10,1
.5)
13.S
egm
ent
M�N�
has
mid
poin
t P
.If
Mh
as c
oord
inat
es (
14,�
3) a
nd
Ph
as c
oord
inat
es
(�8,
6),w
hat
are
th
e co
ordi
nat
es o
f N
?(�
30,1
5)
14.C
ircl
e R
has
a d
iam
eter
S�T�
.If
Rh
as c
oord
inat
es (
�4,
�8)
an
d S
has
coo
rdin
ates
(1,
4),
wh
at a
re t
he
coor
din
ates
of
T?
(�9,
�20
)
15.S
egm
ent
A�D�
has
mid
poin
t B
,an
d B�
D�h
as m
idpo
int
C.I
f A
has
coo
rdin
ates
(�
5,4)
an
d C
has
coo
rdin
ates
(10
,11)
,wh
at a
re t
he
coor
din
ates
of
Ban
d D
?
Bis
�5,8
�,Dis
�15,
13�.
1 � 32 � 3
33 � 21 � 23 � 2
©G
lenc
oe/M
cGra
w-H
ill45
6G
lenc
oe A
lgeb
ra 2
The
Dis
tan
ce F
orm
ula
Dis
tan
ce F
orm
ula
The
dis
tanc
e be
twee
n tw
o po
ints
(x 1
, y 1
) an
d (x
2, y
2) is
giv
en b
y
d�
�(x
2�
�x 1
)2�
�(y
2�
�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�
Dis
tanc
e F
orm
ula
��
(�6
��
8)2
��
[�8
��
(�2)
]�
2 �Le
t (x
1, y
1) �
(8,
�2)
and
(x 2
, y 2
) �
(�6,
�8)
.
��
(�14
)�
2�
(��
6)2
�S
ubtr
act.
��
196
��
36�or
�23
2�
Sim
plify
.
Th
e di
stan
ce b
etw
een
th
e po
ints
is
�23
2�
or a
bou
t 15
.2 u
nit
s.
Fin
d t
he
per
imet
er a
nd
are
a of
sq
uar
e P
QR
Sw
ith
ver
tice
s P
(�4,
1),
Q(�
2,7)
,R(4
,5),
and
S(2
,�1)
.
Fin
d th
e le
ngt
h o
f on
e si
de t
o fi
nd
the
peri
met
er a
nd
the
area
.Ch
oose
P�Q�
.
d�
�(x
2�
�x 1
)2�
�(y
2�
�y 1
)2�
Dis
tanc
e F
orm
ula
��
[�4
��
(�2)
]�
2�
(1�
� 7
)2�
Let
(x1,
y1)
�(�
4, 1
) an
d (x
2, y
2) �
(�2,
7).
��
(�2)
2�
�(�
6�
)2 �S
ubtr
act.
��
40�or
2�
10�S
impl
ify.
Sin
ce o
ne
side
of
the
squ
are
is 2
�10�
,th
e pe
rim
eter
is
8�10�
un
its.
Th
e ar
ea i
s (2
�10�
)2 ,or
40 u
nit
s2.
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 u
nit
s13
un
its
10 u
nit
s
4.(7
,2)
and
(4,�
1)
5.(�
5,�
2) a
nd
(3,4
) 6.
(11,
5) a
nd
(16,
9)
3�2�
un
its
10 u
nit
s�
41�u
nit
s
7.(�
3,4)
an
d (6
,�11
) 8.
(13,
9) a
nd
(11,
15)
9.(�
15,�
7) a
nd
(2,1
2)
3�34�
un
its
2�10�
un
its
5�26�
un
its
10.R
ecta
ngl
e A
BC
Dh
as v
erti
ces
A(1
,4),
B(3
,1),
C(�
3,�
2),a
nd
D(�
5,1)
.Fin
d th
epe
rim
eter
an
d ar
ea o
f A
BC
D.
2 �13 �
�6 �
5 �u
nit
s;3 �
65 �u
nit
s2
11.C
ircl
e R
has
dia
met
er S�
T�w
ith
en
dpoi
nts
S(4
,5)
and
T(�
2,�
3).W
hat
are
th
eci
rcu
mfe
ren
ce a
nd
area
of
the
circ
le?
(Exp
ress
you
r an
swer
in
ter
ms
of �
.)10
�u
nit
s;25
�u
nit
s2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Mid
po
int
and
Dis
tan
ce F
orm
ula
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
po
int
and
Dis
tan
ce F
orm
ula
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
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
nit
s14
.(7,
7),(
�5,
�2)
15 u
nit
s
15.(
�1,
4),(
1,4)
2 u
nit
s16
.(11
,11)
,(8,
15)
5 u
nit
s
17.(
1,�
6),(
7,2)
10 u
nit
s18
.(3,
�5)
,(3,
4)9
un
its
19.(
2,3)
,(3,
5)�
5�u
nit
s20
.(�
4,3)
,(�
1,7)
5 u
nit
s
21.(
�5,
�5)
,(3,
10)
17 u
nit
s22
.(3,
9),(
�2,
�3)
13 u
nit
s
23.(
6,�
2),(
�1,
3)�
74�u
nit
s24
.(�
4,1)
,(2,
�4)
�61�
un
its
25.(
0,�
3),(
4,1)
4�2�
un
its
26.(
�5,
�6)
,(2,
0)�
85�u
nit
s
5 � 29 � 2
5 � 2
11 � 25 � 2
3 � 21 � 2
1 � 2
©G
lenc
oe/M
cGra
w-H
ill45
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 u
nit
s18
.(�
2,�
4),(
4,4)
10 u
nit
s
19.(
�3,
8),(
�1,
�5)
�17
3�
un
its
20.(
0,1)
,(9,
�6)
�13
0�
un
its
21.(
�5,
6),(
�6,
6)1
un
it22
.(�
3,5)
,(12
,�3)
17 u
nit
s
23.(
�2,
�3)
,(9,
3)�
157
�u
nit
s24
.(�
9,�
8),(
�7,
8)2�
65�u
nit
s
25.(
9,3)
,(9,
�2)
5 u
nit
s26
.(�
1,�
7),(
0,6)
�17
0�
un
its
27.(
10,�
3),(
�2,
�8)
13 u
nit
s28
.(�
0.5,
�6)
,(1.
5,0)
2�10�
un
its
29. �
,�, �
1,�1
un
it30
.(�
4�2�,
��
5�),(
�5�
2�,4�
5�)�
127
�u
nit
s
31.G
EOM
ETRY
Cir
cle
Oh
as a
dia
met
er A�
B�.I
f A
is a
t (�
6,�
2) a
nd
Bis
at
(�3,
4),f
ind
the
cen
ter
of t
he
circ
le a
nd
the
len
gth
of
its
diam
eter
.��
,1�;3
�5�
un
its
32.G
EOM
ETRY
Fin
d th
e pe
rim
eter
of
a tr
iang
le 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 (
Ave
rag
e)
Mid
po
int
and
Dis
tan
ce F
orm
ula
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 8-1)
Readin
g t
o L
earn
Math
em
ati
csM
idp
oin
t an
d D
ista
nce
Fo
rmu
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill45
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
uct
ion
to
Les
son
8-1
at
the
top
of p
age
412
in y
our
text
book
.
How
do
you
fin
d di
stan
ces
on a
roa
d m
ap?
Sam
ple
an
swer
:U
se t
he
scal
e o
f m
iles
on
th
e m
ap.Y
ou
mig
ht
also
use
a r
ule
r.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he
coor
din
ates
of
the
mid
poin
t of
a s
egm
ent
wit
h e
ndp
oin
ts (
x 1,y
1) a
nd
(x2,
y 2).
�,
�b
.E
xpla
in h
ow t
o fi
nd
the
mid
poin
t of
a s
egm
ent
if y
ou k
now
th
e co
ordi
nat
es o
f th
een
dpoi
nts
.Do
not
use
su
bscr
ipts
in
you
r ex
plan
atio
n.
Sam
ple
an
swer
:To
fin
d t
he
x-co
ord
inat
e o
f th
e m
idp
oin
t,ad
d t
he
x-co
ord
inat
es o
f th
e en
dp
oin
ts a
nd
div
ide
by t
wo
.To
fin
d t
he
y-co
ord
inat
e o
f th
e m
idp
oin
t,d
o t
he
sam
e w
ith
th
e y-
coo
rdin
ates
of
the
end
po
ints
.
2.a.
Wri
te a
n e
xpre
ssio
n f
or t
he
dist
ance
bet
wee
n t
wo
poin
ts w
ith
coo
rdin
ates
(x 1
,y1)
an
d(x
2,y 2
).�
(x2
��
x 1)2
��
(y2
��
y 1)2
�b
.E
xpla
in h
ow t
o fi
nd
the
dist
ance
bet
wee
n t
wo
poin
ts.D
o n
ot u
se s
ubs
crip
ts i
n y
our
expl
anat
ion
.
Sam
ple
an
swer
:F
ind
th
e d
iffe
ren
ce b
etw
een
th
e x-
coo
rdin
ates
an
d s
qu
are
it.F
ind
th
e d
iffe
ren
ce b
etw
een
th
e y-
coo
rdin
ates
an
d s
qu
are
it.A
dd
th
e sq
uar
es.T
hen
fin
d t
he
squ
are
roo
t o
f th
e su
m.
3.C
onsi
der
the
segm
ent
con
nec
tin
g th
e po
ints
(�
3,5)
an
d (9
,11)
.
a.F
ind
the
mid
poin
t of
th
is s
egm
ent.
(3,8
)
b.
Fin
d th
e le
ngt
h o
f th
e se
gmen
t.W
rite
you
r an
swer
in
sim
plif
ied
radi
cal
form
.6�
5�
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an t
he
“mid
”in
mid
poin
t h
elp
you
rem
embe
r th
e m
idpo
int
form
ula
?
Sam
ple
an
swer
:Th
e m
idp
oin
tis
th
e p
oin
t in
th
e m
idd
leo
f a
seg
men
t.It
is h
alfw
ay b
etw
een
th
e en
dp
oin
ts.T
he
coo
rdin
ates
of
the
mid
po
int
are
fou
nd
by
fin
din
g t
he
aver
age
of
the
two
x-c
oo
rdin
ates
(ad
d t
hem
an
dd
ivid
e by
2)
and
th
e av
erag
e o
f th
e tw
o y
-co
ord
inat
es.
y 1�
y 2�
2x 1
�x 2
�2
©G
lenc
oe/M
cGra
w-H
ill46
0G
lenc
oe A
lgeb
ra 2
Qu
adra
tic
Fo
rmC
onsi
der
two
met
hod
s fo
r so
lvin
g th
e fo
llow
ing
equ
atio
n.
(y�
2)2
�5(
y�
2) �
6�
0
On
e w
ay t
o so
lve
the
equ
atio
n i
s 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
Th
us,
the
solu
tion
set
is
{4,5
}.
An
oth
er w
ay t
o so
lve
the
equ
atio
n i
s fi
rst
to r
epla
ce y
�2
by a
sin
gle
vari
able
.T
his
wil
l pr
odu
ce a
n e
quat
ion
th
at i
s ea
sier
to
solv
e th
an t
he
orig
inal
equ
atio
n.
Let
t�
y�
2 an
d th
en s
olve
th
e n
ew e
quat
ion
.
(y�
2)2
�5(
y�
2) �
6�
0t2
�5t
�6
�0
(t�
2)(t
�3)
�0
Th
us,
tis
2 o
r 3.
Sin
ce t
�y
�2,
the
solu
tion
set
of
the
orig
inal
equ
atio
n i
s {4
,5}.
Sol
ve e
ach
eq
uat
ion
usi
ng
two
dif
fere
nt
met
hod
s.
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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Par
abo
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
1G
lenc
oe A
lgeb
ra 2
Lesson 8-2
Equ
atio
ns
of
Para
bo
las
A p
arab
ola
is a
cu
rve
con
sist
ing
of a
ll p
oin
ts i
n t
he
coor
din
ate
plan
e th
at a
re t
he
sam
e di
stan
ce f
rom
a g
iven
poi
nt
(th
e fo
cus)
an
d a
give
n l
ine
(th
e d
irec
trix
).T
he
foll
owin
g ch
art
sum
mar
izes
im
port
ant
info
rmat
ion
abo
ut
para
bola
s.
Sta
nd
ard
Fo
rm o
f E
qu
atio
ny
�a
(x�
h)2
�k
x�
a(y
�k)
2�
h
Axi
s o
f S
ymm
etry
x�
hy
�k
Ver
tex
(h,
k)
(h,
k)
Fo
cus
�h, k
��
�h�
, k �
Dir
ectr
ixy
�k
�x
�h
�
Dir
ecti
on
of
Op
enin
gup
war
d if
a�
0, d
ownw
ard
if a
�0
right
if a
�0,
left
if a
�0
Len
gth
of
Lat
us
Rec
tum
un
its
units
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
eq
uat
ion
y�
2x2
�12
x�
25.
y�
2x2
�12
x�
25O
rigin
al e
quat
ion
y�
2(x2
�6x
) �
25F
acto
r 2
from
the
x-t
erm
s.
y�
2(x2
�6x
�■
) �
25 �
2(■
)C
ompl
ete
the
squa
re o
n th
e rig
ht s
ide.
y�
2(x2
�6x
�9)
�25
�2(
9)T
he 9
add
ed t
o co
mpl
ete
the
squa
re is
mul
tiplie
d by
2.
y�
2(x
�3)
2�
43W
rite
in s
tand
ard
form
.
Th
e ve
rtex
of
this
par
abol
a is
loc
ated
at
(3,�
43),
the
focu
s is
loc
ated
at �3,
�42
�,th
e
equ
atio
n o
f th
e ax
is o
f sy
mm
etry
is
x�
3,an
d th
e eq
uat
ion
of
the
dire
ctri
x is
y�
�43
.T
he
para
bola
ope
ns
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
qu
atio
n.
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
do
wn
rig
ht
Wri
te a
n e
qu
atio
n o
f ea
ch p
arab
ola
des
crib
ed b
elow
.
4.fo
cus
(�2,
3),d
irec
trix
x�
�2
5.ve
rtex
(5,
1),f
ocu
s �4
,1�
x�
6(y
�3)
2�
2x
��
3(y
�1)
2�
51 � 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-H
ill46
2G
lenc
oe A
lgeb
ra 2
Gra
ph
Par
abo
las
To
grap
h a
n e
quat
ion
for
a p
arab
ola,
firs
t pu
t th
e gi
ven
equ
atio
n i
nst
anda
rd f
orm
.
y�
a(x
�h
)2�
kfo
r a
para
bola
ope
nin
g u
p or
dow
n,o
rx
�a(
y�
k)2
�h
for
a pa
rabo
la o
pen
ing
to t
he
left
or
righ
t
Use
th
e va
lues
of
a,h
,an
d k
to d
eter
min
e th
e ve
rtex
,foc
us,
axis
of
sym
met
ry,a
nd
len
gth
of
the
latu
s re
ctu
m.T
he
vert
ex a
nd
the
endp
oin
ts o
f th
e la
tus
rect
um
giv
e th
ree
poin
ts o
n t
he
para
bola
.If
you
nee
d m
ore
poin
ts t
o pl
ot a
n a
ccu
rate
gra
ph,s
ubs
titu
te v
alu
es f
or p
oin
tsn
ear
the
vert
ex. G
rap
h y
�(x
�1)
2�
2.
In t
he
equ
atio
n,a
�,h
�1,
k�
2.
Th
e pa
rabo
la o
pen
s u
p,si
nce
a�
0.ve
rtex
:(1,
2)ax
is o
f sy
mm
etry
:x�
1
focu
s:�1,
2 �
�or �1
,2�
len
gth
of
latu
s re
ctu
m:
or 3
un
its
endp
oin
ts o
f la
tus
rect
um
: �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
qu
atio
n f
or e
ach
par
abol
a an
d d
raw
its
gra
ph
.
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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Par
abo
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
Par
abo
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
3G
lenc
oe A
lgeb
ra 2
Lesson 8-2
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.
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
qu
atio
n.T
hen
fin
d t
he
len
gth
of
the
latu
s re
ctu
m a
nd
gra
ph
th
e p
arab
ola.
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�;
axis
of
sym
met
ry:
axis
of
sym
met
ry:
axis
of
sym
met
ry:
x�
2;y
�2;
x�
�3;
dir
ectr
ix:
y�
�;
dir
ectr
ix:
x�
2;
dir
ectr
ix:
y�
4;
op
ens
up
;o
pen
s ri
gh
t;o
pen
s d
ow
n;
latu
s re
ctu
m:
1 u
nit
latu
s re
ctu
m:
1 u
nit
latu
s re
ctu
m:
1 u
nit
Wri
te a
n e
qu
atio
n f
or e
ach
par
abol
a d
escr
ibed
bel
ow.T
hen
dra
w t
he
grap
h.
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-H
ill46
4G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.
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
qu
atio
n.T
hen
fin
d t
he
len
gth
of
the
latu
s re
ctu
m a
nd
gra
ph
th
e p
arab
ola.
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)
;
focu
s:�4,
3�;
focu
s:�
,0�;
focu
s:��
2,�
1 �;ax
is:
x�
4;ax
is:
y�
0;ax
is:
y�
�1;
dir
ectr
ix:
y�
2;
dir
ectr
ix:
x�
1;
dir
ectr
ix:
x�
�3
;
op
ens
up
;o
pen
s le
ft;
op
ens
rig
ht;
latu
s re
ctu
m:
1 u
nit
latu
s re
ctu
m:
3 u
nit
sla
tus
rect
um
:u
nit
Wri
te a
n e
qu
atio
n f
or e
ach
par
abol
a d
escr
ibed
bel
ow.T
hen
dra
w t
he
grap
h.
7.ve
rtex
(0,
�4)
,8.
vert
ex (
�2,
1),
9.ve
rtex
(1,
3),
focu
s �0,
�3
�di
rect
rix
x�
�3
axis
of
sym
met
ry x
�1,
latu
s re
ctu
m:2
un
its,
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
for
m y
�ax
2fo
r a
sate
llit
e di
sh.A
ssum
e th
at t
hebo
ttom
of
the
upw
ard-
faci
ng
dish
pas
ses
thro
ugh
(0,
0) a
nd
that
th
e di
stan
ce f
rom
th
ebo
ttom
to
the
focu
s po
int
is 8
in
ches
.y
�x
21 � 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 (
Ave
rag
e)
Par
abo
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-2)
Readin
g t
o L
earn
Math
em
ati
csP
arab
ola
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill46
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
uct
ion
to
Les
son
8-2
at
the
top
of p
age
419
in y
our
text
book
.
Nam
e at
lea
st t
wo
refl
ecti
ve o
bjec
ts t
hat
mig
ht
hav
e th
e sh
ape
of a
para
bola
.
Sam
ple
an
swer
:te
lesc
op
e m
irro
r,sa
telli
te d
ish
Rea
din
g t
he
Less
on
1.In
th
e pa
rabo
la s
how
n i
n t
he
grap
h,t
he
poin
t (2
,�2)
is
call
ed
the
and
the
poin
t (2
,0)
is c
alle
d th
e
.Th
e li
ne
y�
�4
is c
alle
d th
e
,an
d th
e li
ne
x�
2 is
cal
led
the
.
2.a.
Wri
te t
he
stan
dard
for
m o
f th
e eq
uat
ion
of
a pa
rabo
la t
hat
ope
ns
upw
ard
ordo
wn
war
d.y
�a
(x�
h)2
�k
b.T
he
para
bola
ope
ns
dow
nw
ard
if
and
open
s u
pwar
d if
.T
he
equ
atio
n o
f th
e ax
is o
f sy
mm
etry
is
,an
d th
e co
ordi
nat
es o
f th
e ve
rtex
are
.
3.A
par
abol
a h
as e
quat
ion
x�
�(y
�2)
2�
4.T
his
par
abol
a op
ens
to t
he
.
It h
as v
erte
x an
d fo
cus
.Th
e di
rect
rix
is
.Th
e le
ngt
h
of t
he
latu
s re
ctu
m i
s u
nit
s.
Hel
pin
g Y
ou
Rem
emb
er
4.H
ow c
an t
he
way
in
wh
ich
you
plo
t po
ints
in
a r
ecta
ngu
lar
coor
din
ate
syst
em h
elp
you
to
rem
embe
r w
hat
th
e si
gn o
f a
tell
s yo
u a
bou
t th
e di
rect
ion
in
wh
ich
a p
arab
ola
open
s?S
amp
le a
nsw
er:
In p
lott
ing
po
ints
,a p
osi
tive
x-c
oo
rdin
ate
tells
yo
u t
om
ove
to t
he
rig
ht
and
a n
egat
ive
x-co
ord
inat
e te
lls y
ou
to
mov
e to
th
ele
ft.T
his
is li
ke a
par
abo
la w
ho
se e
qu
atio
n is
of
the
form
“x
�…
”;it
op
ens
to t
he
rig
ht
if a
�0
and
to
th
e le
ftif
a�
0.L
ikew
ise,
a p
osi
tive
y-
coo
rdin
ate
tells
yo
u t
o m
ove
up
and
a n
egat
ive
y-co
ord
inat
e te
lls y
ou
to m
ove
do
wn
.Th
is is
like
a p
arab
ola
wh
ose
eq
uat
ion
is o
f th
e fo
rm
“y�
…”;
it o
pen
s u
pw
ard
if a
�0
and
do
wn
war
dif
a�
0.
8
x�
6(2
,2)
(4,2
)
left
1 � 8
(h,k
)
x�
h
a�
0a
�0
axis
of
sym
met
ry
dir
ectr
ix
focu
s
vert
ex
x
y O
( 2, –
2)
( 2, 0
)
y �
–4
©G
lenc
oe/M
cGra
w-H
ill46
6G
lenc
oe A
lgeb
ra 2
Tan
gen
ts t
o P
arab
ola
sA
lin
e th
at i
nte
rsec
ts a
par
abol
a in
exa
ctly
on
e po
int
wit
hou
t cr
ossi
ng
the
curv
e is
a t
ange
nt
to t
he
para
bola
.Th
e po
int
wh
ere
a ta
nge
nt
lin
e to
uch
es
a pa
rabo
la i
s th
e p
oin
t of
tan
gen
cy.T
he
lin
e pe
rpen
dicu
lar
to a
tan
gent
to
a pa
rabo
la a
t th
e po
int
of t
ange
ncy
is c
alle
d th
e n
orm
alto
th
e pa
rabo
la a
t th
at p
oin
t.In
th
e di
agra
m,l
ine
�is
tan
gen
t to
th
e
para
bola
th
at i
s th
e gr
aph
of
y�
x2at
��3 2� ,�9 4� �.
Th
e
x-ax
is i
s ta
nge
nt
to t
he
para
bola
at
O,a
nd
the
y-ax
is
is t
he
nor
mal
to
the
para
bola
at
O.
Sol
ve e
ach
pro
ble
m.
1.F
ind
an e
quat
ion
for
line
�in
the
dia
gram
.Hin
t:A
non
vert
ical
line
wit
h a
neq
uat
ion
of
the
form
y�
mx
�b
wil
l be
tan
gen
t to
th
e gr
aph
of
y�
x2at
��3 2� ,�9 4� �i
f an
d on
ly i
f ��3 2� ,
�9 4� �is
the
only
pai
r of
nu
mbe
rs t
hat
sat
isfi
es b
oth
y�
x2an
d y
�m
x�
b.
m�
3,b
��
�9 4� ,y
�3
x�
�9 4�
2.If
ais
an
y re
al n
um
ber,
then
(a,
a2)
belo
ngs
to
the
grap
h o
f y
�x2
.Exp
ress
m
and
bin
ter
ms
of a
to f
ind
an e
quat
ion
of
the
form
y�
mx
�b
for
the
lin
eth
at i
s ta
nge
nt
to t
he
grap
h o
f y
�x2
at (
a,a2
).
m�
2a
,b�
a2,y
�(2
a)x
�(�
a2)
or
y�
2a
x�
a2
3.F
ind
an e
quat
ion
for
th
e n
orm
al t
o th
e gr
aph
of
y�
x2at
��3 2� ,�9 4� �.
y�
��1 3� x
��1 41 �
4.If
ais
a n
onze
ro r
eal
nu
mbe
r,fi
nd
an e
quat
ion
for
th
e n
orm
al t
o th
e gr
aph
of
y�
x2at
(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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 8-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Cir
cles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill46
7G
lenc
oe A
lgeb
ra 2
Lesson 8-3
Equ
atio
ns
of
Cir
cles
Th
e eq
uat
ion
of
a ci
rcle
wit
h c
ente
r (h
,k)
and
radi
us
ru
nit
s is
(x
�h
)2�
(y�
k)2
�r2
.
Wri
te a
n e
qu
atio
n f
or a
cir
cle
if t
he
end
poi
nts
of
a d
iam
eter
are
at
(�4,
5) a
nd
(6,
�3)
.
Use
th
e m
idpo
int
form
ula
to
fin
d th
e ce
nte
r of
th
e ci
rcle
.
(h,k
) �
�,
�M
idpo
int
form
ula
��
,�
(x1,
y1)
�(�
4, 5
), (
x 2,
y 2)
�(6
, �
3)
��
,�o
r (1
,1)
Sim
plify
.
Use
th
e co
ordi
nat
es o
f th
e ce
nte
r an
d on
e en
dpoi
nt
of t
he
diam
eter
to
fin
d th
e ra
diu
s.
r�
�(x
2�
x�
1)2
��
(y2
��
y 1)2
�D
ista
nce
form
ula
r�
�(�
4 �
�1)
2�
�(5
��
1)2
�(x
1, y
1) �
(1,
1),
(x2,
y2)
�(�
4, 5
)
��
(�5)
2�
�42
��
�41�
Sim
plify
.
Th
e ra
diu
s of
th
e ci
rcle
is
�41�
,so
r2�
41.
An
equ
atio
n o
f th
e ci
rcle
is
(x�
1)2
�(y
�1)
2�
41.
Wri
te a
n e
qu
atio
n f
or t
he
circ
le t
hat
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.ce
nte
r (8
,�3)
,rad
ius
6(x
�8)
2�
(y�
3)2
�36
2.ce
nte
r (5
,�6)
,rad
ius
4(x
�5)
2�
(y�
6)2
�16
3.ce
nte
r (�
5,2)
,pas
ses
thro
ugh
(�
9,6)
(x�
5)2
�(y
�2)
2�
32
4.en
dpoi
nts
of
a di
amet
er a
t (6
,6)
and
(10,
12)
(x�
8)2
�(y
�9)
2�
13
5.ce
nte
r (3
,6),
tan
gen
t to
th
e x-
axis
(x�
3)2
�(y
�6)
2�
36
6.ce
nte
r (�
4,�
7),t
ange
nt
to x
�2
(x�
4)2
�(y
�7)
2�
36
7.ce
nte
r at
(�
2,8)
,tan
gen
t to
y�
�4
(x�
2)2
�(y
�8)
2�
144
8.ce
nte
r (7
,7),
pass
es t
hro
ugh
(12
,9)
(x�
7)2
�(y
�7)
2�
29
9.en
dpoi
nts
of
a di
amet
er a
re (
�4,
�2)
an
d (8
,4)
(x�
2)2
�(y
�1)
2�
45
10.e
ndp
oin
ts o
f a
diam
eter
are
(�
4,3)
an
d (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-H
ill46
8G
lenc
oe A
lgeb
ra 2
Gra
ph
Cir
cles
To
grap
h a
cir
cle,
wri
te t
he
give
n e
quat
ion
in
th
e st
anda
rd f
orm
of
the
equ
atio
n o
f a
circ
le,(
x�
h)2
�(y
�k)
2�
r2.
Plo
t th
e ce
nte
r (h
,k)
of t
he
circ
le.T
hen
use
rto
cal
cula
te a
nd
plot
th
e fo
ur
poin
ts (
h�
r,k)
,(h
�r,
k),(
h,k
�r)
,an
d (h
,k�
r),w
hic
h a
re a
ll p
oin
ts o
n t
he
circ
le.S
ketc
h t
he
circ
le t
hat
goes
th
rou
gh t
hos
e fo
ur
poin
ts.
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
Th
eref
ore,
the
circ
le h
as i
ts c
ente
r at
(�
1,�
2) a
nd
a ra
diu
s of
�
16��
4.F
our
poin
ts o
n t
he
circ
le a
re (
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
eq
uat
ion
.Th
en g
rap
h t
he
circ
le.
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
� y
2 �
4y
� 1
1
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Cir
cles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
Cir
cles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill46
9G
lenc
oe A
lgeb
ra 2
Lesson 8-3
Wri
te a
n e
qu
atio
n f
or t
he
circ
le t
hat
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.ce
nte
r (0
,5),
radi
us
1 u
nit
2.ce
nte
r (5
,12)
,rad
ius
8 u
nit
sx
2�
(y�
5)2
�1
(x�
5)2
�(y
�12
)2�
64
3.ce
nte
r (4
,0),
radi
us
2 u
nit
s4.
cen
ter
(2,2
),ra
diu
s 3
un
its
(x�
4)2
�y
2�
4(x
�2)
2�
(y�
2)2
�9
5.ce
nte
r (4
,�4)
,rad
ius
4 u
nit
s6.
cen
ter
(�6,
4),r
adiu
s 5
un
its
(x�
4)2
�(y
�4)
2�
16(x
�6)
2�
(y�
4)2
�25
7.en
dpoi
nts
of
a di
amet
er a
t (�
12,0
) an
d (1
2,0)
x2
�y
2�
144
8.en
dpoi
nts
of
a di
amet
er a
t (�
4,0)
an
d (�
4,�
6)(x
�4)
2�
(y�
3)2
�9
9.ce
nte
r at
(7,
�3)
,pas
ses
thro
ugh
th
e or
igin
(x�
7)2
�(y
�3)
2�
58
10.c
ente
r at
(�
4,4)
,pas
ses
thro
ugh
(�
4,1)
(x�
4)2
�(y
�4)
2�
9
11.c
ente
r at
(�
6,�
5),t
ange
nt
to y
-axi
s(x
�6)
2�
(y�
5)2
�36
12.c
ente
r at
(5,
1),t
ange
nt
to x
-axi
s(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
eq
uat
ion
.Th
en g
rap
h t
he
circ
le.
13.x
2�
y2�
914
.(x
�1)
2�
(y�
2)2
�4
15.(
x�
1)2
�y2
�16
(0,0
),3
un
its
(1,2
),2
un
its
(�1,
0),4
un
its
16.x
2�
(y�
3)2
�81
17.(
x�
5)2
�(y
�8)
2�
4918
.x2
�y2
�4y
�32
�0
(0,�
3),9
un
its
(5,�
8),7
un
its
(0,2
),6
un
its
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-H
ill47
0G
lenc
oe A
lgeb
ra 2
Wri
te a
n e
qu
atio
n f
or t
he
circ
le t
hat
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.ce
nte
r (�
4,2)
,rad
ius
8 u
nit
s2.
cen
ter
(0,0
),ra
diu
s 4
un
its
(x�
4)2
�(y
�2)
2�
64x2
�y2
�16
3.ce
nte
r��
,��
3� �,ra
diu
s 5�
2�u
nit
s4.
cen
ter
(2.5
,4.2
),ra
diu
s 0.
9 u
nit
�x�
�2�
( y�
�3�)
2�
50(x
�2.
5)2
�(y
�4.
2)2
�0.
81
5.en
dpoi
nts
of
a di
amet
er a
t (�
2,�
9) a
nd
(0,�
5)(x
�1)
2�
(y�
7)2
�5
6.ce
nte
r at
(�
9,�
12),
pass
es t
hro
ugh
(�
4,�
5)(x
�9)
2�
(y�
12)2
�74
7.ce
nte
r at
(�
6,5)
,tan
gen
t to
x-a
xis
(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
eq
uat
ion
.Th
en g
rap
h t
he
circ
le.
8.(x
�3)
2�
y2�
169.
3x2
�3y
2�
1210
.x2
�y2
�2x
�6y
�26
(�3,
0),4
un
its
(0,0
),2
un
its
(�1,
�3)
,6 u
nit
s
11.(
x �
1)2
�y2
�4y
�12
12.x
2�
6x�
y2�
013
.x2
�y2
�2x
�6y
��
1(1
,�2)
,4 u
nit
s(3
,0),
3 u
nit
s(�
1,�
3),3
un
its
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 hu
rric
ane
is a
bout
15
mil
es i
n di
amet
er.G
ale
win
ds c
anaf
fect
an
area
up
to 3
00 m
iles
fro
m t
he s
torm
’s c
ente
r.In
199
2,H
urri
cane
And
rew
dev
asta
ted
sout
hern
Flo
rida
.A s
atel
lite
pho
to o
f And
rew
’s l
andf
all
show
ed t
he c
ente
r of
its
eye
on
one
coor
dina
te s
yste
m c
ould
be
appr
oxim
ated
by
the
poin
t (8
0,26
).
14.W
rite
an
equ
atio
n t
o re
pres
ent
a po
ssib
le b
oun
dary
of A
ndr
ew’s
eye
.(x
�80
)2�
(y�
26)2
�56
.25
15.W
rite
an
equ
atio
n t
o re
pres
ent
a po
ssib
le b
oun
dary
of
the
area
aff
ecte
d by
gal
e w
inds
.(x
�80
)2�
(y�
26)2
�90
,000
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 (
Ave
rag
e)
Cir
cles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 8-3)
Readin
g t
o L
earn
Math
em
ati
csC
ircl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill47
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
uct
ion
to
Les
son
8-3
at
the
top
of p
age
426
in y
our
text
book
.
A l
arge
hom
e im
prov
emen
t ch
ain
is
plan
nin
g to
en
ter
a n
ew m
etro
poli
tan
area
an
d n
eeds
to
sele
ct l
ocat
ion
s fo
r it
s st
ores
.Mar
ket
rese
arch
has
sh
own
that
pot
enti
al c
ust
omer
s ar
e w
illi
ng
to t
rave
l u
p to
12
mil
es t
o sh
op a
t on
eof
th
eir
stor
es.H
ow c
an c
ircl
es h
elp
the
man
ager
s de
cide
wh
ere
to p
lace
thei
r st
ore?
Sam
ple
an
swer
:A
sto
re w
ill d
raw
cu
sto
mer
s w
ho
live
insi
de
aci
rcle
wit
h c
ente
r at
th
e st
ore
an
d a
rad
ius
of
12 m
iles.
Th
e m
anag
emen
t sh
ou
ld s
elec
t lo
cati
on
s fo
r w
hic
has
man
y p
eop
le a
s p
oss
ible
live
wit
hin
a c
ircl
e o
f ra
diu
s 12
mile
s ar
ou
nd
on
e o
f th
e st
ore
s.
Rea
din
g t
he
Less
on
1.a.
Wri
te t
he
equ
atio
n o
f th
e ci
rcle
wit
h c
ente
r (h
,k)
and
radi
us
r.(x
�h
)2�
(y�
k)2
�r2
b.
Wri
te t
he
equ
atio
n o
f th
e ci
rcle
wit
h c
ente
r (4
,�3)
an
d ra
diu
s 5.
(x�
4)2
�(y
�3)
2�
25
c.T
he
circ
le w
ith
equ
atio
n (
x �
8)2
�y2
�12
1 h
as c
ente
r an
d ra
diu
s
.
d.
Th
e ci
rcle
wit
h e
quat
ion
(x
�10
)2�
(y�
10)2
�1
has
cen
ter
and
radi
us
.
2.a.
In o
rder
to
find
cen
ter
and
radi
us o
f th
e ci
rcle
wit
h eq
uati
on x
2�
y2�
4x�
6y�
3 �
0,
it i
s n
eces
sary
to
.Fil
l in
th
e m
issi
ng
part
s of
th
ispr
oces
s.
x2�
y2�
4x�
6y�
3 �
0
x2�
y2�
4x�
6y�
x2�
4x�
�y2
�6y
��
��
(x�
)2�
(y�
)2�
b.
Th
is c
ircl
e h
as r
adiu
s 4
and
cen
ter
at
.
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an t
he
dist
ance
for
mu
la h
elp
you
to
rem
embe
r th
e eq
uat
ion
of
a ci
rcle
?S
amp
le a
nsw
er:W
rite
th
e d
ista
nce
fo
rmu
la.R
epla
ce (
x 1,y
1) w
ith
(h
,k)
and
(x
2,y 2
) w
ith
(x,
y).
Rep
lace
dw
ith
r.S
qu
are
bo
th s
ides
.No
w y
ou
hav
e th
e eq
uat
ion
of
a ci
rcle
.
(�2,
3)
163
29
43
94
3
com
ple
te t
he
squ
are
1(1
0,�
10)
11(�
8,0)
©G
lenc
oe/M
cGra
w-H
ill47
2G
lenc
oe A
lgeb
ra 2
Tan
gen
ts t
o C
ircl
esA
lin
e th
at i
nte
rsec
ts a
cir
cle
in e
xact
ly o
ne
poin
t is
a
tan
gen
tto
th
e ci
rcle
.In
th
e di
agra
m,l
ine
�is
ta
nge
nt
to t
he
circ
le w
ith
equ
atio
n x
2�
y2�
25 a
t th
e po
int
wh
ose
coor
din
ates
are
(3,
4).
A l
ine
is t
ange
nt
to a
cir
cle
at a
poi
nt
Pon
th
e ci
rcle
if
an
d on
ly i
f th
e li
ne
is p
erpe
ndi
cula
r to
th
e ra
diu
s fr
om t
he
cen
ter
of t
he
circ
le t
o po
int
P.T
his
fac
t en
able
s yo
u t
o fi
nd
an e
quat
ion
of
the
tan
gen
t to
a
circ
le a
t a
poin
t P
if y
ou k
now
an
equ
atio
n f
or t
he
circ
le a
nd
the
coor
din
ates
of
P.
Use
th
e d
iagr
am a
bov
e to
sol
ve e
ach
pro
ble
m.
1.W
hat
is
the
slop
e of
th
e ra
diu
s to
th
e po
int
wit
h c
oord
inat
es (
3,4)
? W
hat
is
the
slop
e of
th
e ta
nge
nt
to t
hat
poi
nt?
�4 3� ,�
�3 4�
2.F
ind
an e
quat
ion
of
the
lin
e �
that
is
tan
gen
t to
th
e ci
rcle
at
(3,4
).
y�
��3 4� x
��2 45 �
3.If
kis
a r
eal
nu
mbe
r be
twee
n �
5 an
d 5,
how
man
y po
ints
on
th
e ci
rcle
hav
e x-
coor
din
ate
k? S
tate
th
e co
ordi
nat
es o
f th
ese
poin
ts i
n t
erm
s of
k.
two
,(k,
��
25 �
�k
2 �)
4.D
escr
ibe
how
you
can
fin
d eq
uat
ion
s fo
r th
e ta
nge
nts
to
the
poin
ts y
ou n
amed
for
Exe
rcis
e 3.
Use
th
e co
ord
inat
es o
f (0
,0)
and
of
on
e o
f th
e g
iven
po
ints
.Fin
d t
he
slo
pe
of
the
rad
ius
to t
hat
po
int.
Use
th
e sl
op
e o
f th
e ra
diu
s to
fin
d w
hat
the
slo
pe
of
the
tan
gen
t m
ust
be.
Use
th
e sl
op
e o
f th
e ta
ng
ent
and
th
eco
ord
inat
es o
f th
e p
oin
t o
n t
he
circ
le t
o f
ind
an
eq
uat
ion
fo
r th
e ta
ng
ent.
5.F
ind
an e
quat
ion
for
th
e ta
nge
nt
at (
�3,
4).
y�
�3 4� x�
�2 x5 �
5
–5
–5
5
(3, 4
)
y
xO
�x2
� y
2 �
25
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Elli
pse
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
3G
lenc
oe A
lgeb
ra 2
Lesson 8-4
Equ
atio
ns
of
Ellip
ses
An
ell
ipse
is t
he
set
of a
ll p
oin
ts i
n a
pla
ne
such
th
at t
he
sum
of t
he
dist
ance
s fr
om t
wo
give
n p
oin
ts i
n t
he
plan
e,ca
lled
th
e fo
ci,i
s co
nst
ant.
An
ell
ipse
has
tw
o ax
es o
f sy
mm
etry
wh
ich
con
tain
th
e m
ajor
and
min
or a
xes.
In t
he
tabl
e,th
ele
ngt
hs
a,b,
and
car
e re
late
d by
th
e fo
rmu
la c
2�
a2�
b2 .
Sta
nd
ard
Fo
rm o
f E
qu
atio
n�
�1
��
1
Cen
ter
(h,
k)(h
, k)
Dir
ecti
on
of
Maj
or
Axi
sH
oriz
onta
lV
ertic
al
Fo
ci(h
�c,
k),
(h
�c,
k)
(h,
k�
c),
(h,
k�
c)
Len
gth
of
Maj
or
Axi
s2a
units
2aun
its
Len
gth
of
Min
or
Axi
s2b
units
2bun
its
Wri
te a
n e
qu
atio
n f
or t
he
elli
pse
sh
own
.
Th
e le
ngt
h o
f th
e m
ajor
axi
s is
th
e di
stan
ce b
etw
een
(�
2,�
2)
and
(�2,
8).T
his
dis
tan
ce i
s 10
un
its.
2a�
10,s
o a
�5
Th
e fo
ci a
re l
ocat
ed a
t (�
2,6)
an
d (�
2,0)
,so
c�
3.b2
�a2
�c2
�25
�9
�16
Th
e ce
nte
r of
th
e el
lips
e is
at
(�2,
3),s
o h
��
2,k
�3,
a2�
25,a
nd
b2�
16.T
he
maj
or a
xis
is v
erti
cal.
An
equ
atio
n o
f th
e el
lips
e is
�
�1.
Wri
te a
n e
qu
atio
n f
or t
he
elli
pse
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
1.en
dpoi
nts
of m
ajor
axi
s at
(�
7,2)
and
(5,
2),e
ndpo
ints
of
min
or a
xis
at (
�1,
0) a
nd (
�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
maj
or a
xis
at (
�8,
4) a
nd
(4,4
),fo
ci a
t (�
3,4)
an
d (�
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
to t
he x
-axi
s,m
ajor
axi
s 12
uni
ts l
ong,
cent
er a
t (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
�b
2
(y�
k)2
�a
2(y
�k)
2�
b2
(x�
h)2
�a
2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill47
4G
lenc
oe A
lgeb
ra 2
Gra
ph
Elli
pse
sT
o gr
aph
an
ell
ipse
,if
nec
essa
ry,w
rite
th
e gi
ven
equ
atio
n i
n t
he
stan
dard
for
m o
f an
equ
atio
n f
or a
n e
llip
se.
��
1 (f
or e
llip
se w
ith
maj
or a
xis
hor
izon
tal)
or
��
1 (f
or e
llip
se w
ith
maj
or a
xis
vert
ical
)
Use
th
e ce
nte
r (h
,k)
and
the
endp
oin
ts o
f th
e ax
es t
o pl
ot f
our
poin
ts o
f th
e el
lips
e.T
o m
ake
a m
ore
accu
rate
gra
ph,u
se a
cal
cula
tor
to f
ind
som
e ap
prox
imat
e va
lues
for
xan
d y
that
sati
sfy
the
equ
atio
n.
Gra
ph
th
e el
lip
se 4
x2
�6y
2�
8x�
36y
��
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
Th
e ce
nte
r of
th
e el
lips
e is
(�
1,3)
.Sin
ce a
2�
6,a
��
6�.S
ince
b2
�4,
b�
2.T
he
len
gth
of
the
maj
or a
xis
is 2
�6�,
and
the
len
gth
of
the
min
or a
xis
is 4
.Sin
ce t
he
x-te
rmh
as t
he
grea
ter
den
omin
ator
,th
e m
ajor
axi
s is
hor
izon
tal.
Plo
t th
e en
dpoi
nts
of
the
axes
.T
hen
gra
ph t
he
elli
pse.
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
qu
atio
n.T
hen
gra
ph
th
e el
lip
se.
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
� 6
y2 �
8x
� 3
6y �
�34
(x�
h)2
�b2
(y�
k)2
�a2
(y�
k)2
�b2
(x�
h)2
�a2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Elli
pse
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 8-4)
Skil
ls P
ract
ice
Elli
pse
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
5G
lenc
oe A
lgeb
ra 2
Lesson 8-4
Wri
te a
n e
qu
atio
n f
or e
ach
ell
ipse
.
1.2.
3.
��
1�
�1
��
1
Wri
te a
n e
qu
atio
n f
or t
he
elli
pse
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
4.en
dpoi
nts
of
maj
or a
xis
5.
endp
oin
ts o
f m
ajor
axi
s6.
endp
oin
ts o
f m
ajor
axi
s at
(0,
6) a
nd
(0,�
6),
at (
2,6)
an
d (8
,6),
at (
7,3)
an
d (7
,9),
endp
oin
ts o
f m
inor
axi
s en
dpoi
nts
of
min
or a
xis
endp
oin
ts o
f m
inor
axi
s
at (
�3,
0) a
nd
(3,0
)at
(5,
4) a
nd
(5,8
)at
(5,
6) a
nd
(9,6
)
��
1�
�1
��
1
7.m
ajor
axi
s 12
un
its
lon
g 8.
endp
oin
ts o
f m
ajor
axi
s 9.
endp
oin
ts o
f m
ajor
axi
s at
and
para
llel
to
x-ax
is,
at (
�6,
0) a
nd
(6,0
),fo
ci
(0,1
2) a
nd
(0,�
12),
foci
at
min
or a
xis
4 u
nit
s lo
ng,
at ( �
�32�
,0) a
nd
(�32�
,0)
( 0,�
23�) a
nd
( 0,�
�23�
)ce
nte
r at
(0,
0)
��
1�
�1
��
1
Fin
d t
he
coor
din
ates
of
the
cen
ter
and
foc
i an
d t
he
len
gth
s of
th
e m
ajor
an
dm
inor
axe
s fo
r th
e el
lip
se w
ith
th
e gi
ven
eq
uat
ion
.Th
en g
rap
h t
he
elli
pse
.
10.
��
111
.�
�1
12.
��
1
(0,0
);( 0
,��
19�) ;
(0,0
);( �
6�2�,
0);
(0,0
),( 0
,�2�
6�);
20;
1818
;6
14;
10
x
y
O4
8
8 4 –4 –8
–4–8
x
y
O4
8
8 4 –4 –8
–4–8
x
y
O4
8
8 4 –4 –8
–4–8
x2� 25
y2� 49
y2� 9
x2� 81
x2� 81
y2� 10
0
x2
� 121
y2
� 144
y2
� 4x
2� 36
y2
� 4x
2� 36
(x�
7)2
�4
(y�
6)2
�9
(y �
6)2
�4
(x �
5)2
�9
x2
� 9y
2� 36
(y�
2)2
�9
x2
� 16x
2� 16
y2
� 25y
2� 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
)
©G
lenc
oe/M
cGra
w-H
ill47
6G
lenc
oe A
lgeb
ra 2
Wri
te a
n e
qu
atio
n f
or e
ach
ell
ipse
.
1.2.
3.
��
1�
�1
��
1
Wri
te a
n e
qu
atio
n f
or t
he
elli
pse
th
at s
atis
fies
eac
h s
et o
f co
nd
itio
ns.
4.en
dpoi
nts
of
maj
or a
xis
5.
endp
oint
s of
maj
or a
xis
6.
maj
or a
xis
20 u
nit
s lo
ng
at (
�9,
0) a
nd
(9,0
),at
(4,
2) a
nd
(4,�
8),
and
para
llel
to
x-ax
is,
endp
oin
ts o
f m
inor
axi
s
endp
oint
s of
min
or a
xis
m
inor
axi
s 10
un
its
lon
g,at
(0,
3) a
nd
(0,�
3)at
(1,
�3)
and
(7,
�3)
cen
ter
at (
2,1)
��
1�
�1
��
1
7.m
ajor
axi
s 10
un
its
lon
g,8.
maj
or a
xis
16 u
nit
s lo
ng,
9.en
dpoi
nts
of
min
or a
xis
min
or a
xis
6 u
nit
s lo
ng
ce
nte
r at
(0,
0),f
oci
at
at (
0,2)
an
d (0
,�2)
,foc
i an
d pa
rall
el t
o x-
axis
,( 0
,2�
15�) a
nd
( 0,�
2�15�
)at
(�
4,0)
an
d (4
,0)
cen
ter
at (
2,�
4)
��
1�
�1
��
1
Fin
d t
he
coor
din
ates
of
the
cen
ter
and
foc
i an
d t
he
len
gth
s of
th
e m
ajor
an
dm
inor
axe
s fo
r th
e el
lip
se w
ith
th
e gi
ven
eq
uat
ion
.Th
en g
rap
h t
he
elli
pse
.
10.
��
111
.�
�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.S
PORT
SA
n ic
e sk
ater
tra
ces
two
cong
ruen
t el
lipse
s to
for
m a
fig
ure
eigh
t.A
ssum
e th
at t
hece
nter
of
the
firs
t lo
op i
s at
the
ori
gin,
wit
h th
e se
cond
loo
p to
its
rig
ht.W
rite
an
equa
tion
to m
odel
th
e fi
rst
loop
if
its
maj
or a
xis
(alo
ng
the
x-ax
is)
is 1
2 fe
et l
ong
and
its
min
orax
is i
s 6
feet
lon
g.W
rite
an
oth
er e
quat
ion
to
mod
el t
he
seco
nd
loop
.
��
1;�
�1
y2
� 9(x
�12
)2�
� 36y
2� 9
x2
� 36
4
4 –4 –8 –12
–4–8
x
y
O
x
y
O4
8
8 4 –4 –8
–4–8
x
y
O
(y�
3)2
�25
(x�
4)2
�49
(x�
3)2
�1
(y�
1)2
�36
x2� 9
y2� 16
y2
� 4x
2� 20
x2
� 4y
2� 64
(x�
2)2
�9
(y�
4)2
�25
(y �
1)2
�25
(x �
2)2
�10
0(x
�4)
2�
9(y
�3)
2�
25y
2� 9
x2
� 81
(y�
3)2
�9
(x �
1)2
�25
x2
� 4(y
�2)
2�
9y
2� 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)
612
2 –2
–6–1
2
Pra
ctic
e (
Ave
rag
e)
Elli
pse
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-4)
Readin
g t
o L
earn
Math
em
ati
csE
llip
ses
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill47
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
uct
ion
to
Les
son
8-4
at
the
top
of p
age
433
in y
our
text
book
.
Is t
he
Ear
th a
lway
s th
e sa
me
dist
ance
fro
m t
he
Su
n?
Exp
lain
you
r an
swer
usi
ng
the
wor
ds c
ircl
ean
d el
lips
e.N
o;
if t
he
Ear
th’s
orb
it w
ere
aci
rcle
,it
wo
uld
alw
ays
be
the
sam
e d
ista
nce
fro
m t
he
Su
nb
ecau
se e
very
po
int
on
a c
ircl
e is
th
e sa
me
dis
tan
ce f
rom
th
ece
nte
r.H
ow
ever
,th
e E
arth
’s o
rbit
is a
n e
llip
se,a
nd
th
e p
oin
tso
n a
n e
llip
se a
re n
ot
all t
he
sam
e d
ista
nce
fro
m t
he
cen
ter.
Rea
din
g t
he
Less
on
1.A
n e
llip
se i
s th
e se
t of
all
poi
nts
in
a p
lan
e su
ch t
hat
th
e of
th
e
dist
ance
s fr
om t
wo
fixe
d po
ints
is
.Th
e tw
o fi
xed
poin
ts a
re c
alle
d th
e
of t
he
elli
pse.
2.C
onsi
der
the
elli
pse
wit
h e
quat
ion
�
�1.
a.F
or t
his
equ
atio
n,a
�an
d b
�.
b.
Wri
te a
n e
quat
ion
th
at r
elat
es t
he
valu
es o
f a,
b,an
d c.
c2
�a
2�
b2
c.F
ind
the
valu
e of
cfo
r th
is e
llip
se.
�5�
3.C
onsi
der
the
elli
pses
wit
h e
quat
ion
s �
�1
and
��
1.C
ompl
ete
the
foll
owin
g ta
ble
to d
escr
ibe
char
acte
rist
ics
of t
hei
r gr
aph
s.
Sta
nd
ard
Fo
rm o
f E
qu
atio
n�
�1
��
1
Dir
ecti
on
of
Maj
or
Axi
sve
rtic
alh
ori
zon
tal
Dir
ecti
on
of
Min
or
Axi
sh
ori
zon
tal
vert
ical
Fo
ci(0
,3),
(0,�
3)( �
5�,0)
,(�
�5�,
0)
Len
gth
of
Maj
or
Axi
s10
un
its
6 u
nit
s
Len
gth
of
Min
or
Axi
s8
un
its
4 u
nit
s
Hel
pin
g Y
ou
Rem
emb
er4.
Som
e st
ude
nts
hav
e tr
oubl
e re
mem
beri
ng
the
two
stan
dard
for
ms
for
the
equ
atio
n o
f an
elli
pse.
How
can
you
rem
embe
r w
hic
h t
erm
com
es f
irst
an
d w
her
e to
pla
ce a
an
d b
inth
ese
equ
atio
ns?
Th
e x-
axis
is h
ori
zon
tal.
If t
he
maj
or
axis
is h
ori
zon
tal,
the
firs
t te
rm is
.T
he
y-ax
is is
ver
tica
l.If
th
e m
ajo
r ax
is is
ver
tica
l,th
e
firs
t te
rm is
.a
is a
lway
s th
e la
rger
of
the
nu
mb
ers
aan
d b
.y
2� a
2
x2
� a2
y2
� 4x2� 9
x2� 16
y2
� 25
y2� 4
x2� 9
x2� 16
y2� 25
23
y2� 4
x2� 9
foci
con
stan
tsu
m
©G
lenc
oe/M
cGra
w-H
ill47
8G
lenc
oe A
lgeb
ra 2
Ecc
entr
icit
y In
an
ell
ipse
,th
e ra
tio
� dc �is
cal
led
the
ecce
ntr
icit
yan
d is
den
oted
by
the
lett
er e
.Ecc
entr
icit
y m
easu
res
the
elon
gati
on o
f an
ell
ipse
.The
clo
ser
eis
to
0,th
e m
ore
an e
llip
se l
ooks
lik
e a
circ
le.T
he
clos
er e
is t
o 1,
the
mor
e el
onga
ted
it i
s.R
ecal
l th
at t
he
equ
atio
n o
f an
ell
ipse
is
� ax2 2��
� by2 2��
1 or
� bx2 2��
� ay2 2��
1
wh
ere
ais
th
e le
ngt
h o
f th
e m
ajor
axi
s,an
d th
at 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
ell
ipse
? E
xpla
in y
our
reas
onin
g.
Yes;
it is
an
elli
pse
wit
h e
ccen
tric
ity
0.
8.T
he
cen
ter
of t
he
sun
is
one
focu
s of
Ear
th's
orb
it a
rou
nd
the
sun
.Th
ele
ngt
h o
f th
e m
ajor
axi
s is
186
,000
,000
mil
es,a
nd
the
foci
are
3,2
00,0
00m
iles
apa
rt.F
ind
the
ecce
ntr
icit
y of
Ear
th's
orb
it.
app
roxi
mat
ely
0.17
9.A
n a
rtif
icia
l sa
tell
ite
orbi
tin
g th
e ea
rth
tra
vels
at
an a
ltit
ude
th
at v
arie
sbe
twee
n 1
32 m
iles
an
d 58
3 m
iles
abo
ve t
he
surf
ace
of t
he
eart
h.I
f th
ece
nte
r of
th
e ea
rth
is
one
focu
s of
its
ell
ipti
cal
orbi
t an
d th
e ra
diu
s of
th
eea
rth
is
3950
mil
es,w
hat
is
the
ecce
ntr
icit
y of
th
e or
bit?
app
roxi
mat
ely
0.05
2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 8-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Hyp
erb
ola
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill47
9G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Equ
atio
ns
of
Hyp
erb
ola
sA
hyp
erb
ola
is t
he
set
of a
ll p
oin
ts i
n a
pla
ne
such
th
atth
e ab
solu
te v
alu
e of
th
e d
iffe
ren
ceof
th
e di
stan
ces
from
an
y po
int
on t
he
hyp
erbo
la t
o an
ytw
o gi
ven
poi
nts
in
th
e pl
ane,
call
ed t
he
foci
,is
con
stan
t.
In t
he
tabl
e,th
e le
ngt
hs
a,b,
and
car
e re
late
d by
th
e fo
rmu
la c
2�
a2�
b2.
Sta
nd
ard
Fo
rm o
f E
qu
atio
n�
�1
��
1
Eq
uat
ion
s o
f th
e A
sym
pto
tes
y �
k�
�(x
�h)
y �
k�
�(x
�h)
Tran
sver
se A
xis
Hor
izon
tal
Ver
tical
Fo
ci(h
�c,
k),
(h
�c,
k)
(h,
k�
c),
(h,
k�
c)
Ver
tice
s(h
�a,
k),
(h
�a,
k)
(h,
k�
a),
(h,
k�
a)
Wri
te a
n e
qu
atio
n f
or t
he
hyp
erb
ola
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 t
o or
ien
t th
e h
yper
bola
cor
rect
ly.T
he
cen
ter
of
the
hyp
erbo
la i
s th
e m
idpo
int
of t
he
segm
ent
join
ing
the
two
vert
ices
.Th
e ce
nte
r is
(,1
),or
(2,
1).T
he
valu
e of
ais
th
e
dist
ance
fro
m t
he
cen
ter
to a
ver
tex,
so a
�4.
Th
e va
lue
of c
is
the
dist
ance
fro
m t
he
cen
ter
to a
foc
us,
so c
�6.
c2�
a2�
b2
62�
42�
b2
b2�
36 �
16 �
20
Use
h,k
,a2 ,
and
b2to
wri
te a
n e
quat
ion
of
the
hyp
erbo
la.
��
1
Wri
te a
n e
qu
atio
n f
or t
he
hyp
erb
ola
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
1.ve
rtic
es (
�7,
0) a
nd
(7,0
),co
nju
gate
axi
s of
len
gth
10
��
1
2.ve
rtic
es (
�2,
�3)
an
d (4
,�3)
,foc
i (�
5,�
3) a
nd
(7,�
3)�
�1
3.ve
rtic
es (
4,3)
an
d (4
,�5)
,con
juga
te a
xis
of l
engt
h 4
��
1
4.ve
rtic
es (
�8,
0) a
nd
(8,0
),eq
uat
ion
of
asym
ptot
es y
��
x�
�1
5.ve
rtic
es (
�4,
6) a
nd
(�4,
�2)
,foc
i (�
4,10
) an
d (�
4,�
6)�
�1
(x�
4)2
�48
(y�
2)2
�16
9y2
� 16x
2� 64
1 � 6
(x�
4)2
�4
(y�
1)2
�16
(y�
3)2
�27
(x�
1)2
�9
y2
� 25x
2� 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
�b
2(x
�h)
2�
a2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill48
0G
lenc
oe A
lgeb
ra 2
Gra
ph
Hyp
erb
ola
sT
o gr
aph
a h
yper
bola
,wri
te t
he
give
n e
quat
ion
in
th
e st
anda
rdfo
rm o
f an
equ
atio
n f
or a
hyp
erbo
la
��
1 if
th
e br
anch
es o
f th
e h
yper
bola
ope
n l
eft
and
righ
t,or
��
1 if
th
e br
anch
es o
f th
e h
yper
bola
ope
n u
p an
d do
wn
Gra
ph t
he
poin
t (h
,k),
wh
ich
is
the
cen
ter
of t
he
hyp
erbo
la.D
raw
a r
ecta
ngl
e w
ith
dim
ensi
ons
2aan
d 2b
and
cen
ter
(h,k
).If
th
e h
yper
bola
ope
ns
left
an
d ri
ght,
the
vert
ices
are
(h�
a,k)
an
d (h
�a,
k).I
f th
e h
yper
bola
ope
ns
up
and
dow
n,t
he
vert
ices
are
(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
uar
es t
o ge
t th
e eq
uat
ion
in
sta
nda
rd f
orm
.6y
2�
4x2
�36
y�
8x�
�26
6(y2
�6y
�■
) �
4(x2
�2x
�■
) �
�26
�■
6(y2
�6y
�9)
�4(
x2�
2x�
1) �
�26
�50
6(y
�3)
2�
4(x
�1)
2�
24
��
1
Th
e ce
nte
r of
th
e h
yper
bola
is
(�1,
3).
Acc
ordi
ng
to t
he
equ
atio
n,a
2�
4 an
d b2
�6,
so a
�2
and
b�
�6�.
Th
e tr
ansv
erse
axi
s is
ver
tica
l,so
th
e ve
rtic
es a
re (
�1,
5) a
nd
(�1,
1).D
raw
a r
ecta
ngl
e w
ith
vert
ical
dim
ensi
on 4
an
d h
oriz
onta
l di
men
sion
2�
6��
4.9.
Th
e di
agon
als
of t
his
rec
tan
gle
are
the
asym
ptot
es.T
he
bran
ches
of
the
hyp
erbo
la o
pen
up
and
dow
n.U
se t
he
vert
ices
an
dth
e as
ympt
otes
to
sket
ch t
he
hyp
erbo
la.
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
bol
a w
ith
th
e gi
ven
eq
uat
ion
.Th
en g
rap
h t
he
hyp
erb
ola.
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 � 31 � 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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Hyp
erb
ola
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
erb
ola
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill48
1G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Wri
te a
n e
qu
atio
n f
or e
ach
hyp
erb
ola.
1.2.
3.
��
1�
�1
��
1
Wri
te a
n e
qu
atio
n f
or t
he
hyp
erb
ola
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
4.ve
rtic
es (
�4,
0) a
nd
(4,0
),co
nju
gate
axi
s of
len
gth
8�
�1
5.ve
rtic
es (
0,6)
an
d (0
,�6)
,con
juga
te a
xis
of l
engt
h 1
4�
�1
6.ve
rtic
es (
0,3)
an
d (0
,�3)
,con
juga
te a
xis
of l
engt
h 1
0�
�1
7.ve
rtic
es (
�2,
0) a
nd
(2,0
),co
nju
gate
axi
s of
len
gth
4�
�1
8.ve
rtic
es (
�3,
0) a
nd
(3,0
),fo
ci (
�5,
0)�
�1
9.ve
rtic
es (
0,2)
an
d (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
bol
a w
ith
th
e gi
ven
eq
uat
ion
.Th
en g
rap
h t
he
hyp
erb
ola.
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
� 4x
2� 4
x2
� 25y
2� 9
x2
� 49y
2� 36
y2
� 16x
2� 16
y2
� 25x
2� 4
x2
� 25y
2� 36
y2
� 16x
2� 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-H
ill48
2G
lenc
oe A
lgeb
ra 2
Wri
te a
n e
qu
atio
n f
or e
ach
hyp
erb
ola.
1.2.
3.
��
1�
�1
��
1
Wri
te a
n e
qu
atio
n f
or t
he
hyp
erb
ola
that
sat
isfi
es e
ach
set
of
con
dit
ion
s.
4.ve
rtic
es (
0,7)
an
d (0
,�7)
,con
juga
te a
xis
of l
engt
h 1
8 u
nit
s�
�1
5.ve
rtic
es (
1,�
1) a
nd
(1,�
9),c
onju
gate
axi
s of
len
gth
6 u
nit
s�
�1
6.ve
rtic
es (
�5,
0) a
nd
(5,0
),fo
ci (�
�26�
,0)
��
1
7.ve
rtic
es (
1,1)
an
d (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
bol
a w
ith
th
e gi
ven
eq
uat
ion
.Th
en g
rap
h t
he
hyp
erb
ola.
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
spe
cial
X-r
ay t
eles
cope
s to
obs
erve
th
e so
urc
es o
fce
lest
ial
X r
ays.
Som
e X
-ray
tel
esco
pes
are
fitt
ed w
ith
a m
etal
mir
ror
in t
he
shap
e of
ah
yper
bola
,wh
ich
ref
lect
s th
e X
ray
s to
a f
ocu
s.S
upp
ose
the
vert
ices
of
such
a m
irro
r ar
elo
cate
d at
(�
3,0)
an
d (3
,0),
and
one
focu
s is
loc
ated
at
(5,0
).W
rite
an
equ
atio
n t
hat
mod
els
the
hyp
erbo
la f
orm
ed b
y th
e m
irro
r.�
�1
y2
� 16x
2� 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
�4y
2� 1
x2
� 25
(x�
1)2
�9
(y�
5)2
�16
x2
� 81y
2� 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 (
Ave
rag
e)
Hyp
erb
ola
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 8-5)
Readin
g t
o L
earn
Math
em
ati
csH
yper
bo
las
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill48
3G
lenc
oe A
lgeb
ra 2
Lesson 8-5
Pre-
Act
ivit
yH
ow a
re h
yper
bol
as d
iffe
ren
t fr
om p
arab
olas
?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-5
at
the
top
of p
age
441
in y
our
text
book
.
Loo
k at
th
e sk
etch
of
a h
yper
bola
in
th
e in
trod
uct
ion
to
this
les
son
.Lis
tth
ree
way
s in
wh
ich
hyp
erbo
las
are
diff
eren
t fr
om p
arab
olas
.S
amp
le a
nsw
er:
A h
yper
bo
la h
as t
wo
bra
nch
es,w
hile
ap
arab
ola
is o
ne
con
tin
uo
us
curv
e.A
hyp
erb
ola
has
tw
o f
oci
,w
hile
a p
arab
ola
has
on
e fo
cus.
A h
yper
bo
la h
as t
wo
ver
tice
s,w
hile
a p
arab
ola
has
on
e ve
rtex
.
Rea
din
g t
he
Less
on
1.T
he
grap
h a
t th
e ri
ght
show
s th
e h
yper
bola
wh
ose
equ
atio
n i
n s
tan
dard
for
m i
s �
�1.
Th
e po
int
(0,0
) is
th
e of
th
e h
yper
bola
.
Th
e po
ints
(4,
0) a
nd
(�4,
0) a
re t
he
of t
he
hyp
erbo
la.
Th
e po
ints
(5,
0) a
nd
(�5,
0) a
re t
he
of t
he
hyp
erbo
la.
Th
e se
gmen
t co
nn
ecti
ng
(4,0
) an
d (�
4,0)
is
call
ed t
he
axis
.
Th
e se
gmen
t co
nn
ecti
ng
(0,3
) an
d (0
,�3)
is
call
ed t
he
axis
.
Th
e li
nes
y�
xan
d y
��
xar
e ca
lled
th
e .
2.S
tudy
th
e h
yper
bola
gra
phed
at
the
righ
t.
Th
e ce
nte
r is
.
Th
e va
lue
of a
is
.
Th
e va
lue
of c
is
.
To
fin
d b2
,sol
ve t
he
equ
atio
n
��
.
Th
e eq
uat
ion
in
sta
nda
rd f
orm
for
th
is h
yper
bola
is
.
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
rem
embe
r th
e eq
uat
ion
rel
atin
g th
e va
lues
of
a,b,
and
cfo
r a
hyp
erbo
la?
Th
is e
qu
atio
n lo
oks
just
like
th
e P
yth
ago
rean
Th
eore
m,
alth
ou
gh
th
e va
riab
les
rep
rese
nt
dif
fere
nt
len
gth
s in
a h
yper
bo
la t
han
ina
rig
ht
tria
ng
le.
�x 42 ��
� 1y 22 ��
1
b2
a2
c2
42
(0,0
)
x
y
O
asym
pto
tes
3 � 43 � 4
con
jug
ate
tran
sver
se
foci
vert
ices
cen
tery2� 9
x2� 16
x
y
O( –
4, 0
)( 4
, 0)
( –5,
0)
( 5, 0
)
y �
3 4xy
� –
3 4x
©G
lenc
oe/M
cGra
w-H
ill48
4G
lenc
oe A
lgeb
ra 2
Rec
tan
gu
lar
Hyp
erb
ola
s A
rec
tan
gula
r h
yper
bol
ais
a h
yper
bola
wit
h p
erpe
ndi
cula
r as
ympt
otes
.F
or e
xam
ple,
the
grap
h o
f x2
�y2
�1
is a
rec
tan
gula
r h
yper
bola
.A h
yper
bola
wit
h a
sym
ptot
es t
hat
are
not
per
pen
dicu
lar
is c
alle
d a
non
rect
angu
lar
hyp
erb
ola.
Th
e gr
aph
s of
equ
atio
ns
of t
he
form
xy
�c,
wh
ere
cis
a c
onst
ant,
are
rect
angu
lar
hyp
erbo
las.
Mak
e a
tab
le o
f va
lues
an
d p
lot
poi
nts
to
grap
h e
ach
rec
tan
gula
rh
yper
bol
a b
elow
.Be
sure
to
con
sid
er n
egat
ive
valu
es f
or t
he
vari
able
s.S
ee s
tud
ents
’tab
les.
1.xy
��
42.
xy�
3
3.xy
��
14.
xy�
8
5.M
ake
a co
nje
ctu
re a
bou
t th
e as
ympt
otes
of
rect
angu
lar
hyp
erbo
las.
Th
e co
ord
inat
e ax
es a
re t
he
asym
pto
tes.
x
y
Ox
y
O
x
y
Ox
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Co
nic
Sec
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
5G
lenc
oe A
lgeb
ra 2
Lesson 8-6
Stan
dar
d F
orm
An
y co
nic
sec
tion
in
th
e co
ordi
nat
e pl
ane
can
be
desc
ribe
d by
an
equ
atio
n o
f th
e fo
rm
Ax2
�B
xy�
Cy2
�D
x�
Ey
�F
�0,
wh
ere
A,B
,an
d C
are
not
all
zer
o.O
ne
way
to
tell
wh
at k
ind
of c
onic
sec
tion
an
equ
atio
n r
epre
sen
ts i
s to
rea
rran
ge t
erm
s an
dco
mpl
ete
the
squ
are,
if n
eces
sary
,to
get
one
of t
he
stan
dard
for
ms
from
an
ear
lier
les
son
.T
his
met
hod
is
espe
cial
ly u
sefu
l if
you
are
goi
ng
to g
raph
th
e eq
uat
ion
.
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 p
ara
bola
,cir
cle,
elli
pse
,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 �
■�
■F
acto
r ou
t co
mm
on m
ultip
les.
3(x2
�10
x�
25)
�4(
y2�
2y�
1)�
�59
�3(
25)
� (
�4)
(1)
Com
plet
e th
e sq
uare
s.
3(x
�5)
2�
4(y
�1)
2�
12S
impl
ify.
��
1D
ivid
e ea
ch s
ide
by 1
2.
Th
e gr
aph
of
the
equ
atio
n i
s a
hyp
erbo
la w
ith
its
cen
ter
at (
5,�
1).T
he
len
gth
of
the
tran
sver
se a
xis
is 4
un
its
and
the
len
gth
of
the
con
juga
te a
xis
is 2
�3�
un
its.
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.S
tate
wh
eth
er t
he
grap
h o
f th
e eq
uat
ion
is
a p
ara
bola
,cir
cle,
elli
pse
,or
hyp
erbo
la.
1.x2
�y2
�6x
�4y
�3
�0
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2�
6x�
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10;
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le�
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se
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60x
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1 �
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5 �
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per
bo
la(x
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2�
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circ
le(x
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2�
6(y
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9
(y�
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8)2
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(y�
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(y�
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8)2
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(y�
5)2
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(x�
5)2
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Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill48
6G
lenc
oe A
lgeb
ra 2
Iden
tify
Co
nic
Sec
tio
ns
If y
ou a
re g
iven
an
equ
atio
n o
f th
e fo
rmA
x2�
Bxy
�C
y2�
Dx
�E
y�
F�
0,w
ith
B�
0,yo
u c
an d
eter
min
e th
e ty
pe o
f co
nic
sec
tion
just
by
con
side
rin
g th
e va
lues
of
Aan
d C
.Ref
erto
th
e fo
llow
ing
char
t.
Rel
atio
nsh
ip o
f A
and
CTy
pe
of
Co
nic
Sec
tio
n
A�
0 or
C�
0, b
ut n
ot b
oth.
para
bola
A �
Cci
rcle
Aan
d C
have
the
sam
e si
gn,
but
A
C.
ellip
se
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
eq
uat
ion
is
a p
ara
bola
,cir
cle,
elli
pse
,or
hyp
erbo
la.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Co
nic
Sec
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
Exam
ple
Exam
ple
a.3x
2�
3y2
�5x
�12
�0
A�
3 an
d C
��
3 h
ave
oppo
site
sig
ns,
soth
e gr
aph
of
the
equ
atio
n i
s a
hyp
erbo
la.
b.y
2�
7y�
2x�
13A
�0,
so t
he
grap
h o
f th
e eq
uat
ion
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
pa
rabo
la,c
ircl
e,el
lip
se,o
r h
yper
bola
.
1.x2
�17
x�
5y�
82.
2x2
�2y
2�
3x�
4y�
5p
arab
ola
circ
le3.
4x2
�8x
�4y
2�
6y�
104.
8(x
�x2
) �
4(2y
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100
hyp
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ola
circ
le5.
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26.
y�
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y�
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8.10
x�
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seel
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9.x
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10.1
1x2
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rcle
hyp
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11.3
x2�
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abo
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4 �
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15.1
11 �
11x2
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y216
.120
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119y
2�
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hyp
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17.3
x2�
4y2
�12
18.1
50 �
x2�
120
�y
hyp
erb
ola
par
abo
la
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 8-6)
Skil
ls P
ract
ice
Co
nic
Sec
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
7G
lenc
oe A
lgeb
ra 2
Lesson 8-6
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.S
tate
wh
eth
er t
he
grap
h o
f th
e eq
uat
ion
is
a p
ara
bola
,cir
cle,
elli
pse
,or
hyp
erbo
la.T
hen
gra
ph
th
e eq
uat
ion
.
1.x2
�25
y2�
25hy
per
bo
la2.
9x2
�4y
2�
36el
lipse
3.x2
�y2
�16
�0
circ
le�
�1
��
1x
2�
y2
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4.x2
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circ
le5.
x2�
2x�
15 �
yp
arab
ola
6.10
0x2
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y2�
400 ellip
se(x
�4)
2�
y2
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y�
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1)2
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��
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
pa
rabo
la,c
ircl
e,el
lip
se,o
r h
yper
bola
.
7.9x
2�
4y2
�36
ellip
se8.
x2�
y2�
25ci
rcle
9.y
�x2
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par
abo
la10
.y�
2x2
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par
abo
la
11.4
y2�
25x2
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0hy
per
bo
la12
.16x
2�
y2�
16el
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13.1
6x2
�4y
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64hy
per
bo
la14
.5x2
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2�
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15.2
5y2
�9x
2�
225
ellip
se16
.36y
2�
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�14
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per
bo
la
17.y
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18.x
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144
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circ
le
19.(
x�
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2�
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y�
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ellip
se
21.x
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22.x
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23.(
x�
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le24
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0 �
0el
lipse
x
y
O
xy
O4
8
–4 –8 –12
–16
–4–8
x
y
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8
8 4 –4 –8
–4–8
y2
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2� 4
x
y
Ox
y
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y
48
4 2 –2 –4
–4–8
y2� 9
x2� 4
y2
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2� 25
©G
lenc
oe/M
cGra
w-H
ill48
8G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
eq
uat
ion
in
sta
nd
ard
for
m.S
tate
wh
eth
er t
he
grap
h o
f th
e eq
uat
ion
is
a p
ara
bola
,cir
cle,
elli
pse
,or
hyp
erbo
la.T
hen
gra
ph
th
e eq
uat
ion
.
1.y2
��
3x2.
x2�
y2�
6x�
73.
5x2
�6y
2�
30x
�12
y�
�9
par
abo
laci
rcle
hyp
erb
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
��
1x
2�
(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
pa
rabo
la,c
ircl
e,el
lip
se,o
r h
yper
bola
.
7.6x
2�
6y2
�36
8.4x
2�
y2�
169.
9x2
�16
y2�
64y
�80
�0
circ
lehy
per
bo
lael
lipse
10.5
x2�
5y2
�45
�0
11.x
2�
2x�
y12
.4y2
�36
x2�
4x �
144
�0
circ
lep
arab
ola
hyp
erb
ola
13.A
STR
ON
OM
YA
sat
elli
te t
rave
ls i
n a
n h
yper
boli
c or
bit.
It r
each
es t
he
vert
ex o
f it
s or
bit
at (
5,0)
an
d th
en t
rave
ls a
lon
g a
path
th
at g
ets
clos
er a
nd
clos
er t
o th
e li
ne
y�
x.
Wri
te a
n e
quat
ion
th
at d
escr
ibes
th
e pa
th o
f th
e sa
tell
ite
if t
he
cen
ter
of i
ts h
yper
boli
cor
bit
is a
t (0
,0).
��
1y
2� 4
x2
� 25
2 � 5x
y
O
x
y
Ox
y
O
(y�
3)2
�9
(x�
3)2
�1
y2
� 6.25
x2
� 12.2
5
xO
y
x
y
Ox
y
O
(y �
1)2
�5
(x �
3)2
�6
1 � 3Pra
ctic
e (
Ave
rag
e)
Co
nic
Sec
tio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 8-6)
Readin
g t
o L
earn
Math
em
ati
csC
on
ic S
ecti
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill48
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
uct
ion
to
Les
son
8-6
at
the
top
of p
age
449
in y
our
text
book
.
Th
e fi
gure
s in
th
e in
trod
uct
ion
sh
ow h
ow a
pla
ne
can
sli
ce a
dou
ble
con
e to
form
th
e co
nic
sec
tion
s.N
ame
the
con
ic s
ecti
on t
hat
is
form
ed i
f th
e pl
ane
slic
es t
he
dou
ble
con
e in
eac
h o
f th
e fo
llow
ing
way
s:
•T
he
plan
e is
par
alle
l to
th
e ba
se o
f th
e do
ubl
e co
ne
and
slic
es t
hro
ugh
one
of t
he
con
es t
hat
for
m t
he
dou
ble
con
e.ci
rcle
•T
he
plan
e is
per
pen
dicu
lar
to t
he
base
of
the
dou
ble
con
e an
d sl
ices
thro
ugh
bot
h o
f th
e co
nes
th
at f
orm
th
e do
ubl
e co
ne.
hyp
erb
ola
Rea
din
g t
he
Less
on
1.N
ame
the
con
ic s
ecti
on t
hat
is
the
grap
h o
f ea
ch o
f th
e fo
llow
ing
equ
atio
ns.
Giv
e th
eco
ordi
nat
es o
f th
e ve
rtex
if
the
con
ic s
ecti
on i
s a
para
bola
an
d of
th
e ce
nte
r if
it
is a
circ
le,a
n e
llip
se,o
r a
hyp
erbo
la.
a.�
�1
ellip
se;
(3,�
5)
b.
x�
�2(
y�
1)2
�7
par
abo
la;
(7,�
1)
c.(x
�5)
2�
(y�
5)2
�1
hyp
erb
ola
;(5
,�5)
d.
(x�
6)2
�(y
�2)
2�
1ci
rcle
;(�
6,2)
2.E
ach
of
the
foll
owin
g is
th
e eq
uat
ion
of
a co
nic
sec
tion
.For
eac
h e
quat
ion
,ide
nti
fy t
he
valu
es o
f A
and
C.T
hen
,wit
hou
t w
riti
ng
the
equ
atio
n i
n s
tan
dard
for
m,s
tate
wh
eth
erth
e gr
aph
of
each
equ
atio
n i
s a
para
bola
,cir
cle,
elli
pse,
or h
yper
bola
.
a.2x
2�
y2�
6x�
8y�
12 �
0A
�;C
�;t
ype
of g
raph
:
b.
2x2
�3x
�2y
�5
�0
A�
;C�
;typ
e of
gra
ph:
c.5x
2�
10x
�5y
2�
20y
�1
�0
A�
;C�
;typ
e of
gra
ph:
d.
x2�
y2�
4x�
2y�
5 �
0A
�;C
�;t
ype
of g
raph
:
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
reco
gniz
e th
at a
n e
quat
ion
rep
rese
nts
a p
arab
ola
rath
er t
han
one
of t
he
oth
er c
onic
sec
tion
s?
If t
he
equ
atio
n h
as a
n x
2te
rm a
nd
yte
rm b
ut
no
y2
term
,th
en t
he
gra
ph
is a
par
abo
la.L
ikew
ise,
if t
he
equ
atio
n h
as a
y2
term
an
d x
term
bu
t n
ox
2te
rm,t
hen
th
e g
rap
h is
a p
arab
ola
.
hyp
erb
ola
�1
1
circ
le5
5
par
abo
la0
2
ellip
se1
2
(y�
5)2
�15
(x�
3)2
�36
©G
lenc
oe/M
cGra
w-H
ill49
0G
lenc
oe A
lgeb
ra 2
Lo
ciA
loc
us
(plu
ral,
loci
) is
th
e se
t of
all
poi
nts
,an
d on
ly t
hos
e po
ints
,th
at s
atis
fya
give
n s
et o
f co
ndi
tion
s.In
geo
met
ry,f
igu
res
ofte
n a
re d
efin
ed a
s lo
ci.F
orex
ampl
e,a
circ
le is
the
locu
s of
poi
nts
of a
pla
ne t
hat
are
a gi
ven
dist
ance
from
a g
iven
poi
nt.T
he
defi
nit
ion
lea
ds n
atu
rall
y to
an
equ
atio
n w
hos
e gr
aph
is t
he
curv
e de
scri
bed.
Wri
te a
n e
qu
atio
n o
f th
e lo
cus
of p
oin
ts t
hat
are
th
esa
me
dis
tan
ce f
rom
(3,
4) a
nd
y�
�4.
Rec
ogni
zing
tha
t th
e lo
cus
is a
par
abol
a w
ith
focu
s (3
,4)
and
dire
ctri
x y
��
4,yo
u ca
n fi
nd t
hat
h�
3,k
�0,
and
a�
4 w
here
(h,
k) is
the
ver
tex
and
4 un
its
is t
he
dist
ance
fro
m t
he
vert
ex t
o bo
th t
he
focu
s an
d di
rect
rix.
Th
us,
an e
quat
ion
for
th
e pa
rabo
la i
s y
�� 11 6�
(x�
3)2 .
Th
e pr
oble
m a
lso
may
be
appr
oach
ed a
nal
ytic
ally
as
foll
ows:
Let
(x,
y) b
e a
poin
t of
th
e lo
cus.
Th
e di
stan
ce f
rom
(3,
4) t
o (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
eq
uat
ion
for
the
locu
s.
1.A
ll p
oin
ts t
hat
are
th
e sa
me
dist
ance
fro
m (
0,5)
an
d (4
,5).
line,
x�
2
2.A
ll p
oin
ts t
hat
are
4 u
nit
s fr
om t
he
orig
in.
circ
le,x
2�
y2
�4
3.A
ll p
oin
ts t
hat
are
th
e sa
me
dist
ance
fro
m (
�2,
�1)
an
d x
�2.
par
abo
la,x
��� 81 �
(y2
�2y
�1)
4.T
he
locu
s of
poi
nts
su
ch t
hat
th
e su
m o
f th
e di
stan
ces
from
(�
2,0)
an
d (2
,0)
is 6
.
ellip
se,�
x 92 ��
�y 52 ��
1
5.T
he
locu
s of
poi
nts
su
ch t
hat
th
e ab
solu
te v
alu
e of
th
e d
iffe
ren
ce o
f th
e di
stan
ces
from
(�
3,0)
an
d (3
,0)
is 2
.
hyp
erb
ola
,�x 12 �
��y 82 �
�1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 8-7)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g Q
uad
rati
c S
yste
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
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
lin
ear
equ
atio
ns,
syst
ems
ofqu
adra
tic
equ
atio
ns
can
be
solv
ed b
y su
bsti
tuti
on a
nd
elim
inat
ion
.If
the
grap
hs
are
a co
nic
sect
ion
an
d a
lin
e,th
e sy
stem
wil
l h
ave
0,1,
or 2
sol
uti
ons.
If t
he
grap
hs
are
two
con
icse
ctio
ns,
the
syst
em w
ill
hav
e 0,
1,2,
3,or
4 s
olu
tion
s.
Sol
ve t
he
syst
em o
f eq
uat
ion
s.y
�x
2�
2x�
15x
�y
��
3
Rew
rite
th
e se
con
d eq
uat
ion
as
y�
�x
�3
and
subs
titu
te i
nto
th
e fi
rst
equ
atio
n.
�x
�3
�x2
�2x
�15
0 �
x2�
x�
12A
dd x
�3
to e
ach
side
.
0 �
(x�
4)(x
�3)
Fac
tor.
Use
th
e Z
ero
Pro
duct
pro
pert
y to
get
x�
4 or
x�
�3.
Su
bsti
tute
th
ese
valu
es f
or x
in x
�y
��
3:
4 �
y�
�3
or�
3 �
y�
�3
y�
�7
y�
0
Th
e so
luti
ons
are
(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-H
ill49
2G
lenc
oe A
lgeb
ra 2
Syst
ems
of
Qu
adra
tic
Ineq
ual
itie
sS
yste
ms
of q
uad
rati
c in
equ
alit
ies
can
be
solv
edby
gra
phin
g.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by
grap
hin
g.x
2�
y2
25
�x�
�2�
y2
Th
e gr
aph
of
x2�
y2
25 c
onsi
sts
of a
ll p
oin
ts o
n o
r in
side
th
e ci
rcle
wit
h c
ente
r (0
,0)
and
radi
us
5.T
he
grap
h o
f
�x�
�2�
y2�
con
sist
s of
all
poi
nts
on
or
outs
ide
the
circ
le w
ith
cen
ter �
,0�a
nd
radi
us
.Th
e so
luti
on o
f th
e
syst
em i
s th
e se
t of
poi
nts
in
bot
h r
egio
ns.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by
grap
hin
g.x2
�y2
25
��
1
Th
e gr
aph
of
x2�
y2
25 c
onsi
sts
of a
ll p
oin
ts o
n o
r in
side
th
e ci
rcle
wit
h c
ente
r (0
,0)
and
radi
us
5.T
he
grap
h o
f
��
1 ar
e th
e po
ints
“in
side
”bu
t n
ot o
n t
he
bran
ches
of
the
hyp
erbo
la s
how
n.T
he
solu
tion
of
the
syst
em i
s th
e se
t of
poin
ts i
n b
oth
reg
ion
s.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
elow
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 G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g Q
uad
rati
c S
yste
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
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
So
lvin
g Q
uad
rati
c S
yste
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
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�
7n
o s
olu
tio
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
so
luti
on
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
�2x
solu
tio
n
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s b
y gr
aph
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
ell
ipti
cal
gard
en b
ed h
as a
pat
h f
rom
poi
nt
Ato
po
int
B.I
f th
e be
d ca
n b
e m
odel
ed b
y th
e eq
uat
ion
x2
�3y
2�
12
and
the
path
can
be
mod
eled
by
the
lin
e y
��
x,w
hat
are
th
e
coor
din
ates
of
poin
ts 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-H
ill49
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
so
luti
on
(0,�
3),(
��
5�,2)
(4,3
),(�
4,�
3)( 0
,��
10�)
9.x2
�y2
�25
10.4
x2�
9y2
�36
11.x
��
(y�
3)2
�2
12.
��
1x
�3y
�5
2x2
�9y
2�
18x
�(y
�3)
2�
3x2
�y2
�9
(�5,
0),(
4,3)
(�3,
0)n
o s
olu
tio
n(�
3,0)
13.2
5x2
�4y
2�
100
14.x
2�
y2�
415
.x2
�y2
�3
x�
��
�1
y2�
x2�
3
no
so
luti
on
(�2,
0)n
o s
olu
tio
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 b
y gr
aph
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
Th
e to
p of
an
iro
n g
ate
is s
hap
ed l
ike
hal
f an
el
lips
e w
ith
tw
o co
ngr
uen
t se
gmen
ts f
rom
th
e ce
nte
r of
th
eel
lips
e to
th
e el
lips
e as
sh
own
.Ass
um
e th
at t
he
cen
ter
ofth
e el
lips
e is
at
(0,0
).If
th
e el
lips
e ca
n b
e m
odel
ed b
y th
eeq
uat
ion
x2
�4y
2�
4 fo
r y
�0
and
the
two
con
gru
ent
segm
ents
can
be
mod
eled
by
y�
xan
d y
��
x,
wh
at a
re t
he
coor
din
ates
of
poin
ts A
and
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 (
Ave
rag
e)
So
lvin
g Q
uad
rati
c S
yste
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
��1,
�an
d �1
,�
�3�
�2
�3�
�2
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 8-7)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Qu
adra
tic
Sys
tem
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill49
5G
lenc
oe A
lgeb
ra 2
Lesson 8-7
Pre-
Act
ivit
yH
ow d
o sy
stem
s of
eq
uat
ion
s ap
ply
to
vid
eo g
ames
?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-7
at
the
top
of p
age
455
in y
our
text
book
.
Th
e fi
gure
in
you
r te
xtbo
ok s
how
s th
at t
he
spac
esh
ip h
its
the
circ
ula
r fo
rce
fiel
d in
tw
o po
ints
.Is
it p
ossi
ble
for
the
spac
esh
ip t
o h
it t
he
forc
e fi
eld
inei
ther
few
er o
r m
ore
than
tw
o po
ints
? S
tate
all
pos
sibi
liti
es a
nd
expl
ain
how
th
ese
cou
ld h
appe
n.
Sam
ple
an
swer
:Th
e sp
aces
hip
co
uld
hit
the
forc
e fi
eld
in z
ero
po
ints
if t
he
spac
esh
ip m
isse
d t
he
forc
efi
eld
all
tog
eth
er.T
he
spac
esh
ip c
ou
ld a
lso
hit
th
e fo
rce
fiel
din
on
e p
oin
t if
th
e sp
aces
hip
just
to
uch
ed t
he
edg
e o
f th
efo
rce
fiel
d.
Rea
din
g t
he
Less
on
1.D
raw
a s
ketc
h t
o il
lust
rate
eac
h o
f th
e fo
llow
ing
poss
ibil
itie
s.
a.a
para
bola
an
d a
lin
e b
.an
ell
ipse
an
d a
circ
le
c.a
hyp
erbo
la a
nd
ath
at i
nte
rsec
t in
th
at i
nte
rsec
t in
li
ne
that
in
ters
ect
in2
poin
ts4
poin
ts1
poin
t
2.C
onsi
der
the
foll
owin
g sy
stem
of
equ
atio
ns.
x2�
3 �
y2
2x2
�3y
2�
11
a.W
hat
kin
d of
con
ic s
ecti
on i
s th
e gr
aph
of
the
firs
t eq
uat
ion
?hy
per
bo
la
b.
Wh
at k
ind
of c
onic
sec
tion
is
the
grap
h o
f th
e se
con
d eq
uat
ion
?el
lipse
c.B
ased
on
you
r an
swer
s to
par
ts a
an
d b,
wh
at a
re t
he
poss
ible
nu
mbe
rs o
f so
luti
ons
that
th
is s
yste
m c
ould
hav
e?0,
1,2,
3,o
r 4
Hel
pin
g Y
ou
Rem
emb
er
3.S
upp
ose
that
th
e gr
aph
of
a qu
adra
tic
ineq
ual
ity
is a
reg
ion
wh
ose
bou
nda
ry i
s a
circ
le.
How
can
you
rem
embe
r w
het
her
to
shad
e th
e in
teri
or o
r th
e ex
teri
or o
f th
e ci
rcle
?S
amp
le a
nsw
er:T
he
solu
tio
ns
of
an in
equ
alit
y o
f th
e fo
rm x
2�
y2
�r2
are
all p
oin
ts t
hat
are
less
th
an r
un
its
fro
m t
he
ori
gin
,so
th
e g
rap
h is
the
inte
rio
ro
f th
e ci
rcle
.Th
e so
luti
on
s o
f an
ineq
ual
ity
of
the
form
x
2�
y2
�r2
are
the
po
ints
th
at a
re m
ore
th
an r
un
its
fro
m t
he
ori
gin
,so
the
gra
ph
is t
he
exte
rio
ro
f th
e ci
rcle
.
x
y
Ox
y
Ox
y O
©G
lenc
oe/M
cGra
w-H
ill49
6G
lenc
oe A
lgeb
ra 2
Gra
ph
ing
Qu
adra
tic
Eq
uat
ion
s w
ith
xy-
Term
sYo
u c
an u
se a
gra
phin
g ca
lcu
lato
r to
exa
min
e gr
aph
s of
qu
adra
tic
equ
atio
ns
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.
Sol
ve t
he
equ
atio
n f
or y
in t
erm
s of
xby
usi
ng
the
quad
rati
c fo
rmu
la.
y2�
xy�
(x2
�4)
�0
To
use
th
e fo
rmu
la,l
et a
�1,
b�
x,an
d c
�(x
2�
4).
y�
y�
To
grap
h t
he
equ
atio
n o
n t
he
grap
hin
g ca
lcu
lato
r,en
ter
the
two
equ
atio
ns:
y�
and
y�
Use
a g
rap
hin
g ca
lcu
lato
r to
gra
ph
eac
h e
qu
atio
n.S
tate
th
e ty
pe
of c
urv
e ea
ch g
rap
h r
epre
sen
ts.
1.y2
�xy
�8
2.x2
�y2
�2x
y�
x�
0
hyp
erb
ola
par
abo
la
3.x2
�xy
�y2
�15
4.x2
�xy
�y2
��
9
ellip
seg
rap
h is
�
5.2x
2�
2xy
�y2
�4x
�20
6.x2
�xy
�2y
2�
2x�
5y�
3 �
0
hyp
erb
ola
two
inte
rsec
tin
g li
nes
�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
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9. C
D
B
B
B
A
C
B
D
a. 4; b. any positivenumber except 4; c. any negativenumber; d. 0
A
C
D
A
C
D
B
B
A
D
A
D
C
C
B
A
D
C
A
B
An
swer
s
(continued on the next page)
Chapter 8 Assessment Answer Key Form 1 Form 2APage 497 Page 498 Page 499
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: no solutions
A
C
D
D
C
B
B
A
B
C
A
A
C
D
B
C
D
C
A
C
no solutions
C
D
B
D
B
D
C
A
D
A
D
Chapter 8 Assessment Answer Key Form 2A (continued) Form 2BPage 500 Page 501 Page 502
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: (x � 3)2 � (y � 1)2 � 38
y
xO
y
xO(4, 0)
(1, �3)
(4, 0), (1, �3)
hyperbola; A � 4,C � �4
parabola; C � 0
�(x �
93)2� � �
(y �4
1)2� � 1;
ellipse
(x � 1)2 � (y � 1)2 � 25;circle
(5, �1), (1, �1); (3 � 2�2�, �1);
y � 1 � �(x � 3)
�(y �
362)2� � �
(x �3
1)2� � 1
�8x1
2� � �
2y5
2� � 1
(�2�, �2), (��2�, �2)
�(y �
365)2� � �
(x �16
2)2� � 1
�(x
1�00
1)2� � �
(y �25
3)2� � 1
y
xO
y
xO
(x � 4)2 � (y � 2)2 � 16
(2, 4); ��94
�, 4�;y � 4; x � �
74
�; right
y � 3(x � 1)2 � 1
x � �112�(y � 4)2 � 1
�61� units
�6, �32
��
An
swer
s
Chapter 8 Assessment Answer Key Form 2CPage 503 Page 504
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: (x � 5)2 � (y � 2)2 � 66
y
xO
y
xO
no solution
ellipse; A � 10; C � 3
�(y
1�00
3)2� � �
(x �25
1)2� � 1;
hyperbola
y � 4(x � 2)2 � 5;parabola
(1, 3), (�3, 3); (�1 � 2�2�, 3);
y � 3 � �(x � 1)
�(x �
493)2� � �
(y �21
1)2� � 1
�1y4
2
4� � �
1x6
2� � 1
(�5�, �3), (��5�, �3)
�(x �
641)2� � �
(y �9
4)2� � 1
�(y �
492)2� � �
(x �25
2)2� � 1
y
xO
y
xO
�x � �12
��2� (y � 2)2 � 4
(1, 3); �1, �285��,
x � 1; y � �283�;
upward
y � �116�(x � 1)2 � 3
�202� units
��32
�, 1�
Chapter 8 Assessment Answer Key Form 2DPage 505 Page 506
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: (�2.5, 0.725)
y
xO
y
xO
(5, 3)(–1, 3)
(2, 0)
(�1, 3), (5, 3), (2, 0)
no solution
parabola; A � 0, C � 4
circle; A � C � �32
�
x � �8�y � �12
��2� 94;
parabola
2(x � 1)2 � 3�y � �43
��2� 67
�3
;
ellipse
�(y �
92)2� � �
(x �16
4)2� � 1
(2, 3); (2, 2), (2, 4); 2�6�; 2�5�
�(y �
362)2� � �
(x �16
3)2� � 1
y
xO
y
xO
(x � 3)2 � (y � 5)2 � 25
(10, �1); ��349�, �1�,
y � �1; x � �441�;
y � ��1160�(x � 80)2 � 40
x � ��16
�(y � 1)2 � 5
x � �(y � 3)2 � 2
�166� units
(�3.45, 4.15)
An
swer
s
Chapter 8 Assessment Answer Key Form 3Page 507 Page 508
�(x �
495)2� � � 1
�y � �12
��2
�25
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
Chapter 8 Assessment Answer KeyPage 509, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of identifying,graphing, and writing equations of conic sections, andsolving systems of quadratic equations and inequalities.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofidentifying, graphing, and writing equations of conicsections, and solving systems of quadratic equations andinequalities.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1a. The coefficients of the quadratic termsare the same number and have thesame sign (A � C � 1).
1b. The radius would be the square root of anegative number (�5).
1c. Student responses should indicate thatthe constant in the original equationshould be changed to a number lessthan 25, or that the constant obtainedwhen the equation is written instandard form should be changed to apositive number, or that one or both ofthe coefficients of the linear terms, xand y, must be changed to a numbersufficiently large to result in a positivenumber on the right side of thestandard form of the equation. Sampleanswer: Change the constant in theoriginal equation to 24. The center ofthe circle is (�4, 3) and the radius is 1unit.
2. The graphs of circles, ellipses,hyperbolas, and parabolas that open tothe left and right never representrelations that are functions. Of all theconic sections studied in this chapter,only parabolas that open upward ordownward have graphs which pass thevertical line test and are thereforefunctions.
3. The parabolas share the same vertex.Sample answer: The graph ofy � (x � 2)2 � 1 opens upward while thegraph of x � (y � 1)2 � 2 opens to theright.
4a. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3 Region 2 is the intersection of the regioninside the circle, including its boundary(�) and the region above the parabola,not including its boundary (�).
4b. (x � 4)2 � (y � 3)2 � 4y � (x � 4)2 � 3Region 3 is the intersection of the regionoutside the circle, including itsboundary (�) and the region below theparabola, not including its boundary(�).
4c. Region 1 is the intersection of the regionoutside the circle, including itsboundary (�) and the region above theparabola, not including its boundary(�).
5a. Students must select both values suchthat �5 � k � 1 so that the graph of thehorizontal line y � k will intersect thegraph of the ellipse twice.
5b. Students may select only k � 1 andk � �5, the equations of the only twohorizontal lines that are tangent to theellipse, each intersecting the ellipse inexactly one point.
5c. Students must select both values suchthat k � 1 or k � �5 so that the graphof the horizontal line y � k will notintersect the graph of the ellipse.
An
swer
s
Chapter 8 Assessment Answer Key Page 509, Open-Ended Assessment
Sample Answers
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
1. parabola; focus;directrix
2. ellipse; foci of the ellipse
3. hyperbola
4. minor axis;major axis
5. transverse axis
6. tangent
7. latus rectum
8. asymptote
9. conjugate axis
10. distance formula
11. Sample answer: A circle is the set ofall points in a planethat are the samedistance from agiven point, whichis the center.
12. Sample answer: A vertex of ahyperbola is thepoint on a branch of the hyperbolathat is closest to the center of thehyperbola.
1.
2.
3.
4.
5.
Quiz (Lessons 8–3 and 8–4)
Page 511
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
Quiz (Lesson 8–7)
Page 512
1.
2.
3.
4. y
xO
2
2
(1, 2), (1, �2), (�1, 2), (�1, �2)
���73
�, ��131��, (1, 3)
y
xO
(1, �2) (3, �2)
(1, �2), (3, �2)
ellipse; A � 1, C � 4
y � 2(x � 3)2 � 23;parabola
y
xO
(�2, 0); (�2�10�, 0);y � �3x
�(y �
161)2� � �
(x �9
2)2� � 1
�(x �
41)2� � �
(y �4
2)2� � 1
(3, 0); (3 � 2�3�, 0); 8; 4
�2x5
2� � �
(y �16
1)2� � 1
y
xO
(1, �1); 3 units
(x � 3)2 � (y � 1)2 � 16
(x � 7)2 � (y � 2)2 � 81
yxO
�16
� unit
(�4, 5); ��4, �389��;
x � �4; y � �481�;
downward
y � �112�(x � 1)2 � 1
A
��1, �72
��
Chapter 8 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 8–1 and 8–2) Quiz (Lessons 8–5 and 8–6)
Page 510 Page 511 Page 512
© Glencoe/McGraw-Hill A31 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.�(x �
93)2� � �
(y �4
1)2� � 1;
ellipse
(0, 0); (0, �2�2�); 6; 2
x � ��210�y2 � 1
��4, �52
��
even; 4
{x � x � �3 or x � 1}
1, �52
�
�xx
��
43
�
(�1, �1)
�22.5
(x � 3)2 � (y � 4)2 � 50
y � �3(x � 3)2 � 32
�(x �
254)2� � �
(y �16
1)2� � 1
y
xO
A
C
B
D
A
An
swer
s
Chapter 8 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 513 Page 514
� �7 9 16�4 8 5
© Glencoe/McGraw-Hill A32 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13. 14.
15.
16.
17.
18. DCBA
DCBA
DCBA
DCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
4 3 / 4
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
9 . 0 1
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
1 2 . 5
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
4
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
Chapter 8 Assessment Answer KeyStandardized Test Practice
Page 515 Page 516