Chapter 8

Resource Masters
 Consumable Workbooks
Many of the worksheets contained in the Chapter Resource Masters booklets
are available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-X
Skills Practice Workbook 0-07-828023-0
Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 8 of these workbooks

can be found in the back of this Chapter Resource Masters booklet.

Glencoe/McGraw-Hill

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.

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ISBN: 0-07-828011-7 Algebra 2

Chapter 8 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
 Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii Lesson 8-6
Study Guide and Intervention . . . . . . . . 485–486
Lesson 8-1 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 487
Study Guide and Intervention . . . . . . . . 455–456 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 457 Reading to Learn Mathematics . . . . . . . . . . 489
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 490
Reading to Learn Mathematics . . . . . . . . . . 459
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 460 Lesson 8-7
Study Guide and Intervention . . . . . . . . 491–492
Lesson 8-2 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 493
Study Guide and Intervention . . . . . . . . 461–462 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 463 Reading to Learn Mathematics . . . . . . . . . . 495
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 496
Reading to Learn Mathematics . . . . . . . . . . 465
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 466 Chapter 8 Assessment
Chapter 8 Test, Form 1 . . . . . . . . . . . . 497–498
Lesson 8-3 Chapter 8 Test, Form 2A . . . . . . . . . . . 499–500
Study Guide and Intervention . . . . . . . . 467–468 Chapter 8 Test, Form 2B . . . . . . . . . . . 501–502
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 469 Chapter 8 Test, Form 2C . . . . . . . . . . . 503–504
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 Chapter 8 Test, Form 2D . . . . . . . . . . . 505–506
Reading to Learn Mathematics . . . . . . . . . . 471 Chapter 8 Test, Form 3 . . . . . . . . . . . . 507–508
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 472 Chapter 8 Open-Ended Assessment . . . . . . 509
Chapter 8 Vocabulary Test/Review . . . . . . . 510
Lesson 8-4 Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 511
Study Guide and Intervention . . . . . . . . 473–474 Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 512
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 475 Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 513
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Chapter 8 Cumulative Review . . . . . . . . . . . 514
Reading to Learn Mathematics . . . . . . . . . . 477 Chapter 8 Standardized Test Practice . . 515–516
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 478
Standardized Test Practice
Lesson 8-5 Student Recording Sheet . . . . . . . . . . . . . . A1
Study Guide and Intervention . . . . . . . 479–480
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
Reading to Learn Mathematics . . . . . . . . . . 483
Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

 Teacher’s Guide to Using the
Chapter 8 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resources
you use most often. The Chapter 8 Resource Masters includes the core materials needed
for 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 the
Algebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viii Practice There is one master for each
include a student study tool that presents lesson. These problems more closely follow
up to twenty of the key vocabulary terms the structure of the Practice and Apply
from the chapter. Students are to record section of the Student Edition exercises.
definitions and/or examples for each term. These exercises are of average difficulty.
You may suggest that students highlight or
star the terms with which they are not WHEN TO USE These provide additional
familiar. practice options or may be used as
homework for second day teaching of the
WHEN TO USE Give these pages to lesson.
students before beginning Lesson 8-1.
Encourage them to add these pages to their Reading to Learn Mathematics
Algebra 2 Study Notebook. Remind them One master is included for each lesson. The
to add definitions and examples as they first section of each master asks questions
complete each lesson. about the opening paragraph of the lesson
in the Student Edition. Additional
Study Guide and Intervention questions ask students to interpret the
Each lesson in Algebra 2 addresses two context of and relationships among terms
objectives. There is one Study Guide and in the lesson. Finally, students are asked to
Intervention master for each objective. summarize what they have learned using
various representation techniques.
WHEN TO USE Use these masters as
reteaching activities for students who need WHEN TO USE This master can be used
additional reinforcement. These pages can as a study tool when presenting the lesson
also be used in conjunction with the Student or as an informal reading assessment after
Edition as an instructional tool for students presenting the lesson. It is also a helpful
who have been absent. tool for ELL (English Language Learner)
students.
Skills Practice There is one master for
each lesson. These provide computational Enrichment There is one extension
practice at a basic level. master for each lesson. These activities may
extend the concepts in the lesson, offer an
WHEN TO USE These masters can be historical or multicultural look at the
used with students who have weaker concepts, or widen students’ perspectives on
mathematics backgrounds or need the mathematics they are learning. These
additional reinforcement. are not written exclusively for honors
students, but are accessible for use with all
levels of students.
WHEN TO USE These may be used as
extra credit, short-term projects, or as
activities for days when class periods are
shortened.

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Assessment Options Intermediate Assessment
The assessment masters in the Chapter 8 • Four free-response quizzes are included
Resource Masters offer a wide range of to offer assessment at appropriate
assessment tools for intermediate and final intervals in the chapter.
assessment. The following lists describe each
• A Mid-Chapter Test provides an option
assessment master and its intended use.
to assess the first half of the chapter. It is
composed of both multiple-choice and
Chapter Assessment free-response questions.
CHAPTER TESTS
• Form 1 contains multiple-choice questions Continuing Assessment
and is intended for use with basic level • The Cumulative Review provides
students. students an opportunity to reinforce and
retain skills as they proceed through
• Forms 2A and 2B contain multiple-choice
their study of Algebra 2. It can also be
questions aimed at the average level
used as a test. This master includes
student. These tests are similar in format
free-response questions.
to offer comparable testing situations.
• The Standardized Test Practice offers
• Forms 2C and 2D are composed of free-
continuing review of algebra concepts in
response questions aimed at the average
various formats, which may appear on
level student. These tests are similar in
the standardized tests that they may
format to offer comparable testing
encounter. This practice includes multiple-
situations. Grids with axes are provided
choice, grid-in, and quantitative-
for questions assessing graphing skills.
comparison questions. Bubble-in and
• Form 3 is an advanced level test with grid-in answer sections are provided on
free-response questions. Grids without the master.
axes are provided for questions assessing
graphing skills.
Answers
All of the above tests include a free- • Page A1 is an answer sheet for the
response Bonus question. Standardized Test Practice questions
• The Open-Ended Assessment includes that appear in the Student Edition on
performance assessment tasks that are pages 468–469. This improves students’
suitable for all students. A scoring rubric familiarity with the answer formats they
is included for evaluation guidelines. may encounter in test taking.
Sample answers are provided for • The answers for the lesson-by-lesson
assessment. masters are provided as reduced pages
• A Vocabulary Test, suitable for all with answers appearing in red.
students, includes a list of the vocabulary • Full-size answer keys are provided for
words in the chapter and ten questions the assessment masters in this booklet.
assessing students’ knowledge of those
terms. This can also be used in conjunc-
tion with one of the chapter tests or as a
review worksheet.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8 Reading to Learn Mathematics

Vocabulary Builder

Vocabulary Builder
This 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. Remember
to add the page number where you found the term. Add these pages to your Algebra
Study Notebook to review vocabulary at the end of the chapter.

Found
Vocabulary Term Definition/Description/Example
on Page
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 vii Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8 Reading to Learn Mathematics

Vocabulary Builder (continued)
Found
Vocabulary Term Definition/Description/Example
on Page
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

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Study Guide and Intervention

Midpoint and Distance Formulas
The Midpoint Formula
x
x y
y
Midpoint Formula
1
The midpoint M of a segment with endpoints (x1, y1) and (x 2, y2) is  2 1
,
2
2
.
2 

Example 1 Find the midpoint of the Example 2
B

A diameter A
of a circle
line segment with endpoints at has endpoints A(5,
11) and B(
7, 6).
(4,
7) and (
2, 3). What are the coordinates of the center
x1
x2 y1
y2 of the circle?

 ,   
 ,  
4
(2) 7
3

Lesson 8-1
2 2 2 2 The center of the circle is the midpoint of all

 ,   or (1, 2)
2 4 of its diameters.
2 2 x1
x2 y1
y2
The midpoint of the segment is (1, 2).


2
,   
 ,  
2
5
(7) 11
6
2 2


 ,   or
1, 2  
2 5 1
2 2 2

The circle has center 1, 2  .
1
2 
Exercises
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)
 3

 ,8   1

 ,
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)  33
 ,
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 . If R has coordinates (4, 8) and S has coordinates (1, 4),
ST
what are the coordinates of T? (
9,
20)

15. Segment AD

 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?
 2
B is 5, 8   
1
, D is 15, 13  . 
© Glencoe/McGraw-Hill 455 Glencoe Algebra 2
 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Study Guide and Intervention (continued)

Midpoint and Distance Formulas

The Distance Formula
The distance between two points (x1, y1) and (x2, y2) is given by
Distance Formula
d  
(x2  
x1)2

(y2 
y1)2.

Example 1 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.

Example 2 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 .
PQ
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 210
 Simplify.

Since one side of the square is 210

, the perimeter is 810
 units. The area is (210
 )2, or
40 units2.

Exercises
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)
32

units 10 units 41

units
7. (3, 4) and (6, 11) 8. (13, 9) and (11, 15) 9. (15, 7) and (2, 12)
334

units 210

units 526

units

10. Rectangle ABCD has vertices A(1, 4), B(3, 1), C(3, 2), and D(5, 1). Find the
perimeter and area of ABCD. 2
13  65
units; 365
units2
11. Circle R has diameter  with endpoints S(4, 5) and T(2, 3). What are the
ST
circumference and area of the circle? (Express your answer in terms of .)
10 units; 25 units2

© Glencoe/McGraw-Hill 456 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Skills Practice

Midpoint and Distance Formulas
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, 
1


1 
5. (3, 9), (2, 3)  , 3 
6. (3, 5), (3, 8)
3,

3


Lesson 8-1
7. (3, 2), (5, 0) (
1, 1) 8. (3, 4), (5, 2) (4,
1)


9. (5, 9), (5, 4) 0,

5
 
10. (11, 14), (0, 4)
 , 9
11



11. (3, 6), (8, 3)
 ,

5 9
 12. (0, 10), (2, 5) 
1, 5 

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) 42

units 26. (5, 6), (2, 0) 85

units

© Glencoe/McGraw-Hill 457 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Practice (Average)

Midpoint and Distance Formulas

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,   11

3. (7, 6), (1, 2) (
3,
4) 4. (8, 2), (8, 8) (8,
5)


5. (9, 4), (1, 1) 5,

5
 6. (3, 3), (4, 9)  , 67 
7
7. (4, 2), (3, 7)  ,

9
 8. (6, 7), (4, 4) 5,   11

9. (4, 2), (8, 2) (
6, 0) 10. (5, 2), (3, 7) 4,   5

11. (6, 3), (5, 7)
 ,
2  11
 12. (9, 8), (8, 3)
 ,
  1 5

13. (2.6, 4.7), (8.4, 2.5) (5.5,
1.1)
1

23  1 , 5
14.   , 6 ,  , 4
3

15. (2.5, 4.2), (8.1, 4.2) (2.8, 0)
18 12 
5

16.  ,  ,   ,  
8
1
2  
1 , 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) 265

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) 210

units

25 35 
7
29.  ,  , 1,  1 unit
5  30. (42
, 5
), (52
, 45
) 127

units

31. GEOMETRY Circle O has a diameter  AB. If A is at (6, 2) and B is at (3, 4), find the
center of the circle and the length of its diameter. 9

, 1; 35
units
32. GEOMETRY Find the perimeter of a triangle with vertices at (1, 3), (4, 9), and (2, 1).
18  217

units

© Glencoe/McGraw-Hill 458 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Reading to Learn Mathematics

Midpoint and Distance Formulas
Pre-Activity How are the Midpoint and Distance Formulas used in emergency
medicine?
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 might
also use a ruler.

Lesson 8-1
Reading the Lesson
1. a. Write the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2).
x1  x2 y1  y2

2 ,
2 
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 of
the endpoints.

2. a. Write an expression for the distance between two points with coordinates (x1, y1) and
(x2, y2). (x



x )2 
(y

y )2
2 1 2 1
b. Explain how to find the distance between two points. Do not use subscripts in your
explanation.
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 square
root 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. 65

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. It
is halfway between the endpoints. The coordinates of the midpoint are
found by finding the average of the two x-coordinates (add them and
divide by 2) and the average of the two y-coordinates.

© Glencoe/McGraw-Hill 459 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Enrichment

Quadratic Form
Consider 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  0
y2  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  0
t2  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

© Glencoe/McGraw-Hill 460 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Study Guide and Intervention

Parabolas
Equations of Parabolas A parabola is a curve consisting of all points in the
coordinate 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 xh yk
Vertex (h, k) (h, k )

Focus
h, k

4a 
1

h
1
4a
, k

1 1
Directrix yk  xh 
4a 4a
Direction of Opening upward if a  0, downward if a  0 right if a  0, left if a  0

Length of Latus Rectum a1  units a1  units

Example Identify the coordinates of the vertex and focus, the equations of
the axis of symmetry and directrix, and the direction of opening of the parabola

Lesson 8-2
with 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
7
8 
1
equation of the axis of symmetry is x  3, and the equation of the directrix is y  43  .
8
The parabola opens upward.

Exercises
Identify the coordinates of the vertex and focus, the equations of the axis of
symmetry and directrix, and the direction of opening of the parabola with the
given equation.
1. y  x2
6x  4 2. y  8x  2x2
10 3. x  y2  8y
6
(
3,
13), (2, 18), 2, 17 
1
, (
10, 4), 
9 
3
, 4,


3,
12 3 , x 
3, 1

x  2, y  18  , 1
y  4, x 
10  ,
1
y 
13  , up down right

Write an equation of each parabola described below.

4. focus (2, 3), directrix x  2 

1
12
5. vertex (5, 1), focus 4  , 1
11
12 
1
x  6(y
3)2
2  x 
3(y
1)2  5

© Glencoe/McGraw-Hill 461 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Study Guide and Intervention (continued)

Parabolas
Graph Parabolas To graph an equation for a parabola, first put the given equation in
standard form.

y  a(x  h)2
k for a parabola opening up or down, or

x  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 of
the latus rectum. The vertex and the endpoints of the latus rectum give three points on the
parabola. If you need more points to plot an accurate graph, substitute values for points
near the vertex.

Example Graph y   (x
1)2  2.

1
3
1
In the equation, a   , h  1, k  2.
3
The parabola opens up, since a  0. y
vertex: (1, 2)
axis of symmetry: x  1



1 3
focus: 1, 2
 or 1, 2 
1


4 
3
4


O x
1
length of latus rectum: 
1 or 3 units

3

endpoints of latus rectum: 2  , 2  ,   , 2 
1

2
3
4 
1
2
3
4 

Exercises
The coordinates of the focus and the equation of the directrix of a parabola are
given. 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 y y

O x O x

O x

1 1 1
y (x
3)2  3 y (x
4)2
5 x (y  1)2  4

© Glencoe/McGraw-Hill 462 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Skills Practice

Parabolas
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 of
symmetry and directrix, and the direction of opening of the parabola with the
given 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

y y y

O x O x

Lesson 8-2
O x

vertex: (2, 0); vertex: (3, 2); vertex: (
3, 4);

1
focus: 2, ;  1
focus: 3  ,2;   3
focus:
3, 3  ; 
axis of symmetry: axis of symmetry: axis of symmetry:
x  2; y  2; x 
3;
1 3 1
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,  
1
12  5
focus 5, 
4
 directrix x  
7
8
y 
3x 2 y  (x
5)2  1 x  2(y
3)2 1
y y y

O x

O x O x

© Glencoe/McGraw-Hill 463 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Practice (Average)

Parabolas
Write each equation in standard form.
1 1
1. y  2x2  12x
19 2. y   x2
3x
 3. y  3x2  12x  7
2 2
1
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 of
symmetry and directrix, and the direction of opening of the parabola with the
given equation. Then find the length of the latus rectum and graph the parabola.
1
4. y  (x  4)2
3 5. x    y2
1 6. x  3(y
1)2  3
3
y y

O x

O x

vertex: (4, 3); vertex: (1, 0); vertex: (
3,
1);

1
focus: 4, 3  ;  1
focus:  ,0;    11
focus:
2  ,
1 ; 
axis: x  4; axis: y  0; axis: y 
1;
3 3 1
directrix: y  2  ; directrix: x  1  ; directrix: x 
3  ;
opens up; opens left; opens right;
1
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),

7
focus 0, 3 
8  directrix x  3 axis of symmetry x  1,
latus rectum: 2 units,
a0
1 1
y  2x 2
4 x (y
1)2
2 y 
 (x
1)2  3
y y

O x
O x

10. TELEVISION Write the equation in the form y  ax2 for a satellite dish. Assume that the
bottom of the upward-facing dish passes through (0, 0) and that the distance from the
bottom to the focus point is 8 inches. 1 2
y  x

© Glencoe/McGraw-Hill 464 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Reading to Learn Mathematics

Parabolas
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 a
parabola.
Sample answer: telescope mirror, satellite dish

Reading the Lesson

1. In the parabola shown in the graph, the point (2, 2) is called y

the vertex and the point (2, 0) is called the

(2, 0)
focus . The line y  4 is called the O x

directrix , and the line x  2 is called the (2, –2)

Lesson 8-2
axis of symmetry . y  –4

2. a. Write the standard form of the equation of a parabola that opens upward or
downward. y  a(x
h)2  k

b. The parabola opens downward if a0 and opens upward if a0 . The

equation of the axis of symmetry is xh , and the coordinates of the vertex are
(h, k) .
1 left
3. A parabola has equation x    ( y  2)2
4. This parabola opens to the .
8
It has vertex (4, 2) and focus (2, 2) . The directrix is x6 . The length

of the latus rectum is 8 units.

Helping You Remember

4. How can the way in which you plot points in a rectangular coordinate system help you to
remember 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 to
move to the right and a negative x-coordinate tells you to move to the
left. This is like a parabola whose equation is of the form “x  …”; it
opens 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 you
to 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.

© Glencoe/McGraw-Hill 465 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Enrichment

Tangents to Parabolas
A line that intersects a parabola in exactly one point y
6
without crossing the curve is a tangent to the


parabola. The point where a tangent line touches 5
a parabola is the point of tangency. The line
4
perpendicular to a tangent to a parabola at the point
of tangency is called the normal to the parabola at 3
that point. In the diagram, line
is tangent to the

3–2, 9–4


3 9 2
parabola that is the graph of y  x2 at ,  . The
2 4
1
x-axis is tangent to the parabola at O, and the y-axis y  x2
is the normal to the parabola at O. x
–3 –2 –1 O 1 2 3

Solve each problem.

1. Find an equation for line
in the diagram. Hint: A nonvertical line with an
equation 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 line
that 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 of
y  x2 at (a, a2).

© Glencoe/McGraw-Hill 466 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Study Guide and Intervention

Circles
Equations of Circles The equation of a circle with center (h, k) and radius r units is
(x  h) 2
(y  k) 2  r2.

Example 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.
x
x y
y
1
(h, k)  
2
2 1
, 2
2  Midpoint formula


 ,  
4
6 5
(3)
(x1, y1)  (4, 5), (x2, y2)  (6, 3)
2 2


 ,   or (1, 1)
2 2
Simplify.
2 2
Use the coordinates of the center and one endpoint of the diameter to find the radius.
r  (x
 
2 x1)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.

Exercises

Lesson 8-3
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

© Glencoe/McGraw-Hill 467 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Study Guide and Intervention (continued)

Circles
Graph Circles To graph a circle, write the given equation in the standard form of the
equation 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 that
goes through those four points.

Example Find the center and radius of the circle y

whose equation is x2  2x  y2  4y  11. Then graph x2  2x  y 2  4y  11
the circle.
x2
2x
y2
4y  11
O x
x2
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).

Exercises
Find the center and radius of the circle with the given equation. Then graph the
circle.

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
y y y

O x O x

O x

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
y y y

O x O x
O x

© Glencoe/McGraw-Hill 468 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Skills Practice

Circles
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 units
x 2  (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  y 2  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) x 2  y 2  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 the
circle.

13. x2
y2  9 14. (x  1)2
(y  2)2  4 15. (x
1)2
y2  16

Lesson 8-3
(0, 0), 3 units (1, 2), 2 units (
1, 0), 4 units
y y y

O x O x O x

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
y y y
12 8
O 4 8 12 x
6 4
–4
x
–12 –6 O 6 12 –8 –4 O 4 8x
–8
–6 –4
–12
–12 –8

© Glencoe/McGraw-Hill 469 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Practice (Average)

Circles
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

1
3. center   , 3
4 
 , radius 52
 units 4. center (2.5, 4.2), radius 0.9 unit

x  1 2

)2  50
 (y  3 (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 the
circle.
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
y y y

–8 –4 O 4 8x
O x O x
–4

–8

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 can
affect an area up to 300 miles from the storm’s center. In 1992, Hurricane Andrew devastated
southern Florida. A satellite photo of Andrew’s landfall showed the center of its eye on one
coordinate 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
© Glencoe/McGraw-Hill 470 Glencoe Algebra 2
 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Reading to Learn Mathematics

Circles
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 metropolitan
area and needs to select locations for its stores. Market research has shown
that potential customers are willing to travel up to 12 miles to shop at one
of their stores. How can circles help the managers decide where to place
their store?
Sample answer: A store will draw customers who live inside a
circle with center at the store and a radius of
12 miles. The management should select locations for which
as 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 (
8, 0) and radius
11 .

d. The circle with equation (x  10)2
( y
10)2  1 has center (10,
10) and

Lesson 8-3
radius 1 .

2. a. In order to find center and radius of the circle with equation x2
y2
4x  6y 3  0,

it is necessary to complete the square . Fill in the missing parts of this
process.
x2
y2
4x  6y  3  0
x2
y2
4x  6y  3
x2
4x
4
y2  6y
9  3
4
9
(x
2 )2
( y  3 )2  16
b. This circle has radius 4 and center at (
2, 3) .

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 (x 2, y2) with (x, y). Replace d with r. Square both sides. Now you
have the equation of a circle.

© Glencoe/McGraw-Hill 471 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-3 Enrichment

Tangents to Circles
A line that intersects a circle in exactly one point is y


a tangent to the circle. In the diagram, line
is x 2  y2  25 5
(3, 4)
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 –5 O 5 x
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.
–5

Use the diagram above to solve each problem.

1. What is the slope of the radius to the point with coordinates (3, 4)? What is
the 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 named
for Exercise 3.

5. Find an equation for the tangent at (3, 4).

© Glencoe/McGraw-Hill 472 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Study Guide and Intervention

Ellipses
Equations of Ellipses An ellipse is the set of all points in a plane such that the sum
of the distances from two given points in the plane, called the foci, is constant. An ellipse
has two axes of symmetry which contain the major and minor axes. In the table, the
lengths a, b, and c are related by the formula c2  a2  b2.
(x  h) 2 (y  k)2 (y  k)2 (x  h)2
Standard Form of Equation  2

 1 
 1
a b2 a 2 b2
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

Example Write an equation for the ellipse shown. y

The length of the major axis is the distance between (2, 2)
and (2, 8). This distance is 10 units. F1
2a  10, so a  5
The foci are located at (2, 6) and (2, 0), so c  3.
b2  a2  c2
F2
 25  9 O x
 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.
( y  3)2 (x
2)2
An equation of the ellipse is 
  1.
25 16
Exercises
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)
(x  1)2 (y
2)2
1
36 4

Lesson 8-4
2. major axis 8 units long and parallel to the x-axis, minor axis 2 units long, center at (2, 5)
(x  2)2
  (y  5)2  1
16
3. endpoints of major axis at (8, 4) and (4, 4), foci at (3, 4) and (1, 4)
(x  2)2 (y
4)2
  1
36 35
4. endpoints of major axis at (3, 2) and (3, 14), endpoints of minor axis at (1, 6) and (7, 6)
(y  6)2 (x
3)2
1
64 16
5. minor axis 6 units long and parallel to the x-axis, major axis 12 units long, center at (6, 1)
(y
1)2 (x
6)2
1
36 9

© Glencoe/McGraw-Hill 473 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Study Guide and Intervention (continued)

Ellipses
Graph Ellipses To graph an ellipse, if necessary, write the given equation in the
standard form of an equation for an ellipse.
(x  h)2 ( y  k)2

  1 (for ellipse with major axis horizontal) or
a2 b2
( y  k)2 (x  h)2

  1 (for ellipse with major axis vertical)
a2 b2
Use the center (h, k) and the endpoints of the axes to plot four points of the ellipse. To make
a more accurate graph, use a calculator to find some approximate values for x and y that
satisfy the equation.

Example Graph the ellipse 4x 2  6y 2  8x
36y 
34.

4x2
6y2
8x  36y  34 y
4x2
8x
6y2  36y   34 4x 2  6y 2  8x
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
(x
1)2 ( y  3)2

1
6 4 O x
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 26 , and the length of the minor axis is 4. Since the x-term
has the greater denominator, the major axis is horizontal. Plot the endpoints of the axes.
Then graph the ellipse.

Exercises
Find the coordinates of the center and the lengths of the major and minor axes
for the ellipse with the given equation. Then graph the ellipse.
y2 x2 x2 y2
1. 
  1 (0, 0), 43

, 6 2. 
  1 (0, 0), 10, 4
12 9 25 4
y y

O x O x

3. x2
4y2
24y  32 (0,
3), 4, 2 4. 9x2
6y2  36x
12y  12 (2,
1), 6, 26


y y

O x O x

© Glencoe/McGraw-Hill 474 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Skills Practice

Ellipses
Write an equation for each ellipse.

1. y 2. (0, 5) y 3. (0, 5) y

(0, 2) (0, 3)
(–4, 2) (4, 2)
(–3, 0)
O (3, 0) x O x O x
(0, –1)
(0, –2) (0, –3)

(0, –5)

x2 y2 y2 x2 x2 (y
2)2
1 1 1
9

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)
y2 x2 (x
5)2 (y
6)2 (y
6)2 (x
7)2
1 1 1
9 4 9 4

7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis at
and parallel to x-axis, at (6, 0) and (6, 0), foci (0, 12) and (0, 12), foci at
minor axis 4 units long, at ( 32, 0) and (32, 0) (0, 23 ) and (0, 23 )
center at (0, 0)
x2 y2 x2 y2 y2 x2
1 1 1

Find the coordinates of the center and foci and the lengths of the major and
minor axes for the ellipse with the given equation. Then graph the ellipse.

Lesson 8-4
y2 x2 x2 y2 y2 x2
10. 
  1 11. 
  1 12. 
  1
100 81 81 9 49 25
(0, 0); (0, 19

); (0, 0); (62

, 0); (0, 0), (0, 26

);
20; 18 18; 6 14; 10
y y y
8 8 8

4 4 4

–8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O 4 8x
–4 –4 –4

–8 –8 –8

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Practice (Average)

Ellipses
Write an equation for each ellipse.
1. (0, 3) y 2. (0, 2  5 ) (y0, 5) 3. y
(–6, 3) (4, 3)
2
(–11, 0) (11, 0)
–12 –6 O 6 12 x (–5, 3) (3, 3)
–2 O
x
(0, –3) (0, 2
5 ) (0, –1) O x

x2 y2 (y
2)2 x2 (x  1)2 (y
3)2

1  1 1
9 25 9
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)
x2 y2 (y  3)2 (x
4)2 (x
2)2 (y
1)2
1 1 1
25 9 100 25
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, 215 ) at (4, 0) and (4, 0)
center at (2, 4)
(y  4)2 (x
2)2 y2 x2 x2 y2
1 1 1
25 9
Find the coordinates of the center and foci and the lengths of the major and
minor axes for the ellipse with the given equation. Then graph the ellipse.
y2 x2 ( y  1)2 (x  3)2 (x
4)2 ( y
3)2
10. 
 1 11. 
 1 12. 
1
16 9 36 1 49 25
(0, 0); (0, 7

); 8; 6 (3, 1); (3, 1  35

); (
4,
3);
12; 2 (
4  26
,
3); 14;
10
y y
8

O x –8 –4 O 4 8x
–4

–8

13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that the
center of the first loop is at the origin, with the second loop to its right. Write an equation
to model the first loop if its major axis (along the x-axis) is 12 feet long and its minor
axis is 6 feet long. Write another equation to model the second loop.

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Reading to Learn Mathematics

Ellipses
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 answer
using the words circle and ellipse. No; if the Earth’s orbit were a
circle, it would always be the same distance from the Sun
because every point on a circle is the same distance from the
center. However, the Earth’s orbit is an ellipse, and the points
on an ellipse are not all the same distance from the center.

Reading the Lesson

1. An ellipse is the set of all points in a plane such that the sum of the
distances from two fixed points is constant . The two fixed points are called the
foci of the ellipse.
x2 y2
2. Consider the ellipse with equation 
  1.
9 4
a. For this equation, a  3 and b  2 .
b. Write an equation that relates the values of a, b, and c. c 2  a 2
b 2
c. Find the value of c for this ellipse. 5


y2 x2 x2 y2
3. Consider the ellipses with equations 
  1 and 
  1. Complete the
25 16 9 4
following table to describe characteristics of their graphs.

y2 x2 x2 y2
Standard Form of Equation 
 1 
 1
25 16 9 4

Direction of Major Axis vertical horizontal

Direction of Minor Axis horizontal vertical

Foci (0, 3), (0,
3) (5
, 0), (
5
, 0)

Lesson 8-4
Length of Major Axis 10 units 6 units

Length of Minor Axis 8 units 4 units

Helping You Remember

4. 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 in
these equations? The x-axis is horizontal. If the major axis is horizontal, the
x2
first term is  . The y-axis is vertical. If the major axis is vertical, the
y2
first term is  . a is always the larger of the numbers a and b.

© Glencoe/McGraw-Hill 477 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Enrichment

Eccentricity
c
In an ellipse, the ratio  is called the eccentricity and is denoted by the
d
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
x2 y2 x2 y2
it is. Recall that the equation of an ellipse is 2
2  1 or 2
2  1
a b b a
where a is the length of the major axis, and that c  
a2  b2.

Find the eccentricity of each ellipse rounded to the nearest

hundredth.
x2 y2 x2 y2 x2 y2
1. 
  1 2. 
  1 3. 
  1
9 36 81 9 4 9
0.87 0.94 0.75

x2 y2 x2 y2 x2 y2
4. 
  1 5. 
  1 6. 
  1
16 9 36 16 4 36
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. The
length of the major axis is 186,000,000 miles, and the foci are 3,200,000
miles apart. Find the eccentricity of Earth's orbit.

approximately 0.17

9. An artificial satellite orbiting the earth travels at an altitude that varies

between 132 miles and 583 miles above the surface of the earth. If the
center of the earth is one focus of its elliptical orbit and the radius of the
earth is 3950 miles, what is the eccentricity of the orbit?

approximately 0.052

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Study Guide and Intervention

Hyperbolas
Equations of Hyperbolas A hyperbola is the set of all points in a plane such that
the absolute value of the difference of the distances from any point on the hyperbola to any
two 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.
(x  h)2 (y  k)2 (y  k)2 (x  h)2
Standard Form of Equation   1   1
a2 b2 a2 b2
b a
Equations of the Asymptotes y  k    (x  h) y  k    (x  h)
a b
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)

Example 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 y
the hyperbola is the midpoint of the segment joining the two
2
6
vertices. The center is (  , 1), or (2, 1). The value of a is the
2
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. O x

c2  a2
b2
62  42
b2
b2  36  16  20
Use h, k, a2, and b2 to write an equation of the hyperbola.
(x  2)2 ( y  1)2
1
16 20

Exercises
Write an equation for the hyperbola that satisfies each set of conditions.
x2 y2
1. vertices (7, 0) and (7, 0), conjugate axis of length 10 
  1

(x
1)2 (y  3)2
2. vertices (2, 3) and (4, 3), foci (5, 3) and (7, 3) 
  1
9 27

(y  1)2 (x
4)2
3. vertices (4, 3) and (4, 5), conjugate axis of length 4 
  1
16 4
Lesson 8-5

1 x2 9y 2
4. vertices (8, 0) and (8, 0), equation of asymptotes y    x 
  1
6

(y
2)2 (x  4)2
5. vertices (4, 6) and (4, 2), foci (4, 10) and (4, 6) 
  1
16 48

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Study Guide and Intervention (continued)

Hyperbolas
Graph Hyperbolas To graph a hyperbola, write the given equation in the standard
form of an equation for a hyperbola
(x  h) 2 ( y  k) 2
 2
   1 if the branches of the hyperbola open left and right, or
a b2
( y  k)2 (x  h)2
    1 if the branches of the hyperbola open up and down
a2 b2
Graph the point (h, k), which is the center of the hyperbola. Draw a rectangle with
dimensions 2a and 2b and center (h, k). If the hyperbola opens left and right, the vertices
are (h  a, k) and (h
a, k). If the hyperbola opens up and down, the vertices are (h, k  a)
and (h, k
a).

Example Draw the graph of 6y2
4x2
36y
8x 
26.

Complete the squares to get the equation in standard form. y
6y2  4x2  36y  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
( y  3)2 (x
1)2
1
4 6 O x
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 with
vertical dimension 4 and horizontal dimension 26   4.9. The diagonals of this rectangle
are the asymptotes. The branches of the hyperbola open up and down. Use the vertices and
the asymptotes to sketch the hyperbola.

Exercises
Find the coordinates of the vertices and foci and the equations of the asymptotes
for the hyperbola with the given equation. Then graph the hyperbola.
x2 y2 (x
2)2 y2 x2
1.     1 2. ( y  3)2    1 3.     1
4 16 9 16 9
(2, 0), (
2, 0); (
2, 4), (
2, 2); (0, 4), (0,
4);
(25
, 0), (
25
, 0); (
2, 3  10
), (0, 5), (0,
5);
(
2, 3
10 4
y  2x
); y   x
1 2
y y  x  3 , y

1 1
y 
 x  2
y

O x O x

O x

© Glencoe/McGraw-Hill 480 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Skills Practice

Hyperbolas
Write an equation for each hyperbola.

1. y 2. y 3. y
8 8 8
(0, 

61 ) (0, 6)
(–5, 0) 4 4 4
(–2, 0) (2, 0)
(5, 0)
–8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O 4 8x
–4 –4 (–

29, 0) –4 (

29, 0)
41, 0) 

(–
( 41, 0) (0, –

61 ) (0, –6)
–8 –8 –8

x2 y2 y2 x2 x2 y2

1 
1 
1

Write an equation for the hyperbola that satisfies each set of conditions.
x2 y2
4. vertices (4, 0) and (4, 0), conjugate axis of length 8 
  1

y2 x2
5. vertices (0, 6) and (0, 6), conjugate axis of length 14 
  1

y2 x2
6. vertices (0, 3) and (0, 3), conjugate axis of length 10 
  1

x2 y2
7. vertices (2, 0) and (2, 0), conjugate axis of length 4 
  1

x2 y2
8. vertices (3, 0) and (3, 0), foci (5, 0) 
  1

y2 x2
9. vertices (0, 2) and (0, 2), foci (0, 3) 
  1

(x
3)2 (y  2)2
, 2) 
  1
10. vertices (0, 2) and (6, 2), foci (3  13
9 4

Find the coordinates of the vertices and foci and the equations of the asymptotes
for the hyperbola with the given equation. Then graph the hyperbola.
x2 y2 y2 x2 x2 y2
11.     1 12.     1 13.     1
9 36 49 9 16 1
(3, 0); (35

, 0); (0, 7); (0, 58

); (4, 0); (17

, 0);
7 1
y  2x y   x y   x
y y y
8 8
Lesson 8-5

4 4

O x –8 –4 O 4 8x –8 –4 O 4 8x
–4 –4

–8 –8

© Glencoe/McGraw-Hill 481 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Practice (Average)

Hyperbolas
Write an equation for each hyperbola.

1. y 2. y 3. y
8
(0, 35 ) (–3, 2  

34 )
8
4
O x
(0, 3) (–3, 5) 4 (–1, –2)
–8 –4 O 4 8x (–3, –1) O (3, –2)
–8 –4 4 x
–4 (0, –3)
–4 (1, –2)
(0, –35 ) (–3, 2


34 )
–8

y2 x2 (y
2)2 (x  3)2 (x
1)2 (y  2)2


1 
1 
1
9 25 4 16
Write an equation for the hyperbola that satisfies each set of conditions.
y2 x2
4. vertices (0, 7) and (0, 7), conjugate axis of length 18 units 
  1

(y  5)2 (x
1)2
5. vertices (1, 1) and (1, 9), conjugate axis of length 6 units 
  1
16 9
x2 y2
, 0) 
  1
6. vertices (5, 0) and (5, 0), foci (26

(y  1)2 (x
1)2
) 
  1
7. vertices (1, 1) and (1, 3), foci (1, 1  5
4 1
Find the coordinates of the vertices and foci and the equations of the asymptotes
for the hyperbola with the given equation. Then graph the hyperbola.
y2 x2 ( y  2)2 (x  1)2 ( y
2)2 (x  3)2
8.     1 9.     1 10.     1
16 4 1 4 4 4
(0, 4); (0, 25

); (1, 3), (1, 1); (3, 0), (3,
4);
y  2x (1, 2  5
); (3,
2  22
);
1
y
2   (x
1) y  2  (x
3)
y y
8

–8 –4 O 4 8x
–4
O x
–8

11. ASTRONOMY Astronomers use special X-ray telescopes to observe the sources of
celestial X rays. Some X-ray telescopes are fitted with a metal mirror in the shape of a
hyperbola, which reflects the X rays to a focus. Suppose the vertices of such a mirror are
located at (3, 0) and (3, 0), and one focus is located at (5, 0). Write an equation that
models the hyperbola formed by the mirror. x 2 y2

1

© Glencoe/McGraw-Hill 482 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Reading to Learn Mathematics

Hyperbolas
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. List
three ways in which hyperbolas are different from parabolas.
Sample answer: A hyperbola has two branches, while a
parabola 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 y
x2 y2 y  – 34 x y  34 x
equation in standard form is     1.
16 9
(–5, 0) (5, 0)
The point (0, 0) is the center of the (–4, 0) O (4, 0) x
hyperbola.
The points (4, 0) and (4, 0) are the vertices
of the hyperbola.
The points (5, 0) and (5, 0) are the foci
of the hyperbola.
The segment connecting (4, 0) and (4, 0) is called the transverse axis.
The segment connecting (0, 3) and (0, 3) is called the conjugate axis.
3 3 asymptotes .
The lines y   x and y    x are called the
4 4
2. Study the hyperbola graphed at the right. y

The center is (0, 0) .

The value of a is 2 .
O x
The value of c is 4 .
To find b2, solve the equation c2  a2
b2 .

x2 y2

  1
The equation in standard form for this hyperbola is 4 12 .
Lesson 8-5

Helping You Remember

3. What is an easy way to remember the equation relating the values of a, b, and c for a
hyperbola? This equation looks just like the Pythagorean Theorem,
although the variables represent different lengths in a hyperbola than in
a right triangle.

© Glencoe/McGraw-Hill 483 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-5 Enrichment

Rectangular Hyperbolas
A rectangular hyperbola is a hyperbola with perpendicular asymptotes.
For example, the graph of x2  y2  1 is a rectangular hyperbola. A hyperbola
with asymptotes that are not perpendicular is called a nonrectangular
hyperbola. 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 rectangular

hyperbola below. Be sure to consider negative values for the
variables. See students’ tables.

1. xy  4 2. xy  3
y y

O x O x

3. xy  1 4. xy  8
y y

O x O x

5. Make a conjecture about the asymptotes of rectangular hyperbolas.

The coordinate axes are the asymptotes.

© Glencoe/McGraw-Hill 484 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Study Guide and Intervention

Conic Sections
Standard Form Any conic section in the coordinate plane can be described by an

Lesson 8-6
equation 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 and
complete 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.

Example 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.
(x  5)2 ( y
1)2
 1 Divide each side by 12.
4 3
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 23 units.

Exercises
Write each equation in standard form. State whether the graph of the equation is
a parabola, circle, ellipse, or hyperbola.

1. x2
y2  6x
4y
3  0 2. x2
2y2
6x  20y
53  0
(x  3)2 (y
5)2
(x
3)2  (y  2)2  10; circle     1; ellipse
6 3
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

(x  (y

2)2 2)2

  1; hyperbola x 
3(y
4)2  2; parabola
10 12

7. x2  4y2  16x
24y  36  0 8. x2
2y2
8x
4y
2  0

(x
(y

8)2 3)2 (x  4)2 (y  1)2

  1; hyperbola     1; ellipse
64 16 16 8

9. 4x2
48x
y
158  0 10. 3x2
y2  48x  4y
184  0

(x
8)2 (y
2)2
y 
4(x  6)2
14; parabola     1; ellipse
4 12

11. 3x2
2y2  18x
20y
5  0 12. x2
y2
8x
2y
8  0

(y  (x 
5)2 3)2

  1; hyperbola (x  4)2  (y  1)2  9; circle
9 6

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Study Guide and Intervention (continued)

Conic Sections
Identify Conic Sections If you are given an equation of the form
Ax2
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. Refer
to the following chart.

Relationship of A and C Type of Conic Section

A  0 or C  0, but not both. parabola
AC circle
A and C have the same sign, but A C. ellipse
A and C have opposite signs. hyperbola

Example Without writing the equation in standard form, state whether the
graph of each equation is a parabola, circle, ellipse, or hyperbola.
a. 3x 2
3y 2  5x  12  0 b. y 2  7y
2x  13
A  3 and C  3 have opposite signs, so A  0, so the graph of the equation is
the graph of the equation is a hyperbola. a parabola.

Exercises
Without writing the equation in standard form, state whether the graph of each
equation is a parabola, circle, ellipse, or hyperbola.

1. x2  17x  5y
8 2. 2x2
2y2  3x
4y  5

parabola circle
3. 4x2  8x  4y2  6y
10 4. 8(x  x2)  4(2y2  y)  100
hyperbola circle
5. 6y2  18  24  4x2 6. y  27x  y2
ellipse parabola
7. x2  4( y  y2)
2x  1 8. 10x  x2  2y2  5y
ellipse ellipse
9. x  y2  5y
x2  5 10. 11x2  7y2  77
circle hyperbola
11. 3x2
4y2  50
y2 12. y2  8x  11
circle parabola
13. 9y2  99y  3(3x  3x2) 14. 6x2  4  5y2  3
circle hyperbola
15. 111  11x2
10y2 16. 120x2  119y2
118x  117y  0
ellipse hyperbola
17. 3x2  4y2
12 18. 150  x2  120  y
hyperbola parabola

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 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Skills Practice

Conic Sections
Write each equation in standard form. State whether the graph of the equation is

Lesson 8-6
a 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

x2 y2 x2 y2

1 1 x 2  y 2  16
y y y
4

–8 –4 O 4 8x O x O x
–2

–4

4. x2
8x
y2  9 circle 5. x2
2x  15  y parabola 6. 100x2
25y2  400

x2 y2 ellipse
(x  4)2  y 2  25 y  (x  1)2
16 1
y y y
8
–8 –4 O 4 8x
4 –4

–8
–8 –4 O 4 8x O x
–4 –12

–8 –16

Without writing the equation in standard form, state whether the graph of each
equation 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

© Glencoe/McGraw-Hill 487 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Practice (Average)

Conic Sections
Write each equation in standard form. State whether the graph of the equation is
a parabola, circle, ellipse, or hyperbola. Then graph the equation.

1. y2  3x 2. x2
y2
6x  7 3. 5x2  6y2  30x  12y  9

parabola circle hyperbola
1 2 (x
3)2 (y  1)2
x 
 y (x  3)2  y2  16 
1
6 5
y y y

O x

O x O x

4. 196y2  1225  100x2 5. 3x2  9  3y2  6y 6. 9x2
y2
54x  6y  81

ellipse circle ellipse
y2
x2 (x  3)2 (y
3)2
1 x 2  (y  1)2  4 1
1 9

Without writing the equation in standard form, state whether the graph of each
equation 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  0

circle parabola hyperbola

13. ASTRONOMY A satellite travels in an hyperbolic orbit. It reaches the vertex of its orbit
2
at (5, 0) and then travels along a path that gets closer and closer to the line y   x.
5
Write an equation that describes the path of the satellite if the center of its hyperbolic
orbit is at (0, 0).
x2 y2

1

© Glencoe/McGraw-Hill 488 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Reading to Learn Mathematics

Conic Sections
Pre-Activity How can you use a flashlight to make conic sections?

Lesson 8-6
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 to
form the conic sections. Name the conic section that is formed if the plane
slices the double cone in each of the following ways:
• The plane is parallel to the base of the double cone and slices through
one of the cones that form the double cone. circle
• The plane is perpendicular to the base of the double cone and slices
through 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 the
coordinates of the vertex if the conic section is a parabola and of the center if it is a
circle, an ellipse, or a hyperbola.
(x  3)2 ( y
5)2
a. 
  1 ellipse; (3,
5)
36 15

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 the
values of A and C. Then, without writing the equation in standard form, state whether
the graph of each equation is a parabola, circle, ellipse, or hyperbola.

a. 2x2
y2  6x
8y
12  0 A 2 ;C 1 ; type of graph: ellipse

b. 2x2
3x  2y  5  0 A 2 ;C 0 ; type of graph: parabola

c. 5x2
10x
5y2  20y
1  0 A 5 ;C 5 ; type of graph: circle

d. x2  y2
4x
2y  5  0 A 1 ; C 
1 ; type of graph: hyperbola

Helping You Remember

3. What is an easy way to recognize that an equation represents a parabola rather than
one of the other conic sections?
If the equation has an x 2 term and y term but no y 2 term, then the graph
is a parabola. Likewise, if the equation has a y 2 term and x term but no
x 2 term, then the graph is a parabola.

© Glencoe/McGraw-Hill 489 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Enrichment

Loci
A locus (plural, loci) is the set of all points, and only those points, that satisfy
a given set of conditions. In geometry, figures often are defined as loci. For
example, a circle is the locus of points of a plane that are a given distance
from a given point. The definition leads naturally to an equation whose graph
is the curve described.

Example Write an equation of the locus of points that are the

same 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 units
is the distance from the vertex to both the focus and directrix.
1
Thus, an equation for the parabola is y   (x  3)2.
16
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
1
 (x  3)2  y
16

Describe each locus as a geometric figure. Then write an equation

for 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.

© Glencoe/McGraw-Hill 490 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Study Guide and Intervention

Solving Quadratic Systems
Systems of Quadratic Equations Like systems of linear equations, systems of
quadratic equations can be solved by substitution and elimination. If the graphs are a conic
section and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conic
sections, the system will have 0, 1, 2, 3, or 4 solutions.

Example Solve the system of equations. y  x 2
2x
15

x  y 
3
Rewrite the second equation as y  x  3 and substitute into the first equation.

Lesson 8-7
x  3  x2  2x  15
0  x2  x  12 Add x
3 to each side.
0  (x  4)(x
3) Factor.

Use the Zero Product property to get

x  4 or x  3.
Substitute these values for x in x
y  3:
4
y  3 or 3
y  3
y  7 y0
The solutions are (4, 7) and (3, 0).

Exercises
Find the exact solution(s) of each system of equations.

1. y x2  5 2. x2
( y  5)2  25
y x  3 y  x2
(2,
1), (
1,
4) (0, 0)

3. x2
( y  5)2  25 4. x2
y2  9
y  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  16 x  y2  4


34

, 30
  34

, 
,

30


,

7  29
2


,
1  29
2


,


34

, 
30
  34

,

,

30


 
7
29
2


,
1
29
2




© Glencoe/McGraw-Hill 491 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Study Guide and Intervention (continued)

Solving Quadratic Systems

Systems of Quadratic Inequalities Systems of quadratic inequalities can be solved
by graphing.

Example 1 Solve the system of inequalities by graphing. y

x2  y2 25
5 2
 
25
x
  y2

2 4
The graph of x2
y2
25 consists of all points on or inside O x
the circle with center (0, 0) and radius 5.The graph of

x  25  25
2

y2   consists of all points on or outside the
4

52  5
circle with center  , 0 and radius  . The solution of the
2
system is the set of points in both regions.

Example 2 Solve the system of inequalities by graphing. y

x2  y2 25
y2 x2

 1
4 9
The graph of x2
y2
25 consists of all points on or inside O x
the circle with center (0, 0) and radius 5.The graph of
y2 x2
    1 are the points “inside” but not on the branches of
4 9
the hyperbola shown. The solution of the system is the set of
points in both regions.

Exercises
Solve each system of inequalities below by graphing.
x2 y2
1. 

1 2. x2
y2
169 3. y  (x  2)2
16 4
x2
9y2  225 (x
1)2
( y
1)2
16
1
y  x  2
2
y y y

12

O x –12 –6 O 6 12 x O x
–6

–12

© Glencoe/McGraw-Hill 492 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Skills Practice

Solving Quadratic Systems
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 solution
x
y  4 (
2
2 2

) x
y  25

,
2
2 2 x2
y2  9

Lesson 8-7
7. y  2x
2 (2,
2), 8. x  y
1  0 (1, 2) 9. y  2  x (0, 2), (3,

1)
y2  2x y2  4x y  x2  4x
2
1 , 1
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  144
x2
y2  16 y  2x2
4 x2
8y2  16
y y y
8

O x O x –8 –4 O 4 8x
–4

–8

© Glencoe/McGraw-Hill 493 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Practice (Average)

Solving Quadratic Systems

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  1

xy1 x
y3 y  2x  1 y  x
1
(0,
1), (3, 2) (2, 1), (6.5,
3.5) (
1,
3), (5, 9) 1 2
(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

)
x2 y2
9. x2
y2  25 10. 4x2
9y2  36 11. x  ( y  3)2
2 12.     1
9 16
x  3y  5 2x2  9y2  18 x  ( y  3)2
3
x
y2  9
2

(
5, 0), (4, 3) (3, 0) no solution (3, 0)

13. 25x2
4y2  100 14. x2
y2  4 15. x2  y2  3

5 x2 y2 y2  x2  3
x   
1
2 4 8
no solution (2, 0) no solution
x2 y2
16. 
  1 17. x
2y  3 18. x2
y2  64
7 7
x2
y2  9 x2  y2  8
3x2  y2  9
(2, 3
) (3, 0),
 
9 12
,  (6, 27
)
Solve each system of inequalities by graphing.
( y  3)2 (x
2)2
19. y  x2 20. x2
y2  36 21. 

1
16 4
y  x
2 x2
y2  16
(x
1)2
( y  2)2
4
y y y
8

–8 –4 O 4 8x
–4
O x O x
–8

A B
22. GEOMETRY The top of an iron gate is shaped like half an
ellipse with two congruent segments from the center of the
ellipse to the ellipse as shown. Assume that the center of
(0, 0)
the ellipse is at (0, 0). If the ellipse can be modeled by the
equation x2
4y2  4 for y  0 and the two congruent
3

and 1, 
3
segments can be modeled by y   x and y    x,
2
3
2
 
1,    3

what are the coordinates of points A and B?

© Glencoe/McGraw-Hill 494 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Reading to Learn Mathematics

Solving Quadratic Systems
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 force
field in two points. Is it possible for the spaceship to hit the force field in
either fewer or more than two points? State all possibilities and explain
how these could happen. Sample answer: The spaceship could hit
the force field in zero points if the spaceship missed the force
field all together. The spaceship could also hit the force field

Lesson 8-7
in one point if the spaceship just touched the edge of the
force 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 a

that intersect in that intersect in line that intersect in
2 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 solutions
that this system could have? 0, 1, 2, 3, or 4

Helping You Remember

3. Suppose that the graph of a quadratic inequality is a region whose boundary is a circle.
How can you remember whether to shade the interior or the exterior of the circle?
Sample answer: The solutions of an inequality of the form x 2  y 2  r 2
are all points that are less than r units from the origin, so the graph is
the interior of the circle. The solutions of an inequality of the form
x 2  y 2  r 2 are the points that are more than r units from the origin, so
the graph is the exterior of the circle.

© Glencoe/McGraw-Hill 495 Glencoe Algebra 2

 NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Enrichment

Graphing Quadratic Equations with xy-Terms

You can use a graphing calculator to examine graphs of quadratic equations
that contain xy-terms.

Example Use a graphing calculator to display y

the graph of x2  xy  y2  4. 2

Solve the equation for y in terms of x by using the 1

quadratic formula.
x
y2
xy
(x2  4)  0 –2 –1 O 1 2
–1
To use the formula, let a  1, b  x, and c  (x2  4).
–2
x  
x2  4
(1)(x2 
 4)
y  
2
x  
16  
3x2
y  
2

To graph the equation on the graphing calculator, enter the two equations:
x

16  
3x2 x  
16  
3x2
y   and y  
2 2

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

© Glencoe/McGraw-Hill 496 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Test, Form 1 SCORE

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  41
C. y  (x  5)2  16 D. y  (x  8)(x  2) 5.

Assessment
6. Write an equation for the parabola with vertex (1, 0) if the length of the
latus rectum is 1 and the parabola opens down.
2
1
A. y 
(x
1)2 B. y 
2(x
1)2 C. x 
2(y
1)2 D. x 
1(y
1)2 6.
2 2

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  2
C. 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  9
C. (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  3
C. (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)?
2 2 2 2 2 2
A. y  x  1 B. x2  y2  16 C. x  y  1 D. y  x  1 11.
16 9 16 7 16 7

© Glencoe/McGraw-Hill 497 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Test, Form 1 (continued)

12. Which equation is graphed at the right? y

x2 y2 y2 x2
A.     1 B.     1
16 4 16 4 x
O
x2 y2 y2 x2
C. 
  1 D. 
  1 12.
16 4 16 4

13. Which is the equation of a hyperbola with vertices

(0, 2) and (0,
2) and foci (0, 3) and (0,
3)?
2 2 2 2 2 2 2 2
A. y
x  1 B. y
x  1 C. x
y  1 D. x  y  1 13.
5 4 4 5 4 5 5 4
y
14. Which equation is graphed at the right?
2 2 2 2
A. x
y  1 B. x
y  1
16 4 4 16
2 2 2 2
O x
C. y
x  1 D. y
x  1 14.
16 4 4 16

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? y

A. x2  y2  9 B. x2  y2  9
yx1 yx1
x
C. x2  y2  9 D. x2  y2  9 O

yx1 yx1 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:

© Glencoe/McGraw-Hill 498 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Test, Form 2A SCORE

Write the letter for the correct answer in the blank at y

the right of each question. 9
AMPHITHEATER
8
ZOO
For Questions 1 and 2, refer to the figure at the 7
6
right showing six city locations. The origin is 5 TOWN SQUARE
at the lower left corner of the grid. 4
3
1. What is the location of the point halfway between UNIVERSITY WHARF
2
the wharf and library? 1
LIBRARY

2 2

2 
15 5 O 1 2 3 4 5 6 7 8 9x
A. (7, 2) B.  ,  C. 7, 5 15
D.  , 2 1.
2

2. What is the distance between the library and zoo?

61
A. 11 units  units
B. 61 C. 61 units D.   units 2.
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
15
C. y  2(x
2)2
7 D. y  2(x
4)2
15

Assessment
3.

4. Write an equation for the parabola with focus (4, 0) and vertex (2, 0).
A. x  1y2  2 B. x 
1y2
2 C. y  1x2
2 D. y 
1x2  2 4.
8 8 8 8

5. Which equation is graphed at the right? y

A. y  4x2
16x  16 B. x  4y2
16y  16
C. y  1x2
x  1 D. x  1y2
y  1 5.
4 4 O x

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  36
C. (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  0
C. 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).
(y  1)2 (x
1)2 (x  1)2 (y  1)2
A.      1 B.      1
16 9 16 9
(x
1)2 (y  1)2 (y
1)2 (x  1)2
C.     1 D.     1 8.
16 9 16 9

9. Which is the equation of an ellipse with center (1,
2) and a vertical
major axis?
(y
2)2 (x  1)2 (x  1)2 (y
2)2
A.      1 B.      1
9 4 9 4
(y  2)2 (x
1)2 (x
1)2 (y  2)2
C.     1 D.     1 9.
9 4 9 4
© Glencoe/McGraw-Hill 499 Glencoe Algebra 2
 NAME DATE PERIOD

8 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).
(x  2)2 (y
1)2 (x
2)2 (y  1)2
A.  
  1 B.  
  1
64 36 36 64
(x
2)2 (y  1)2 (x  2)2 (y
1)2
C. 
  1 D. 
  1 11.
64 36 36 64

12. Which equation is graphed at the right? y

A. x2
9y2  9 B. 9y2
x2  9
C. 9x2
y2  9 D. y2
9x2  9 12.
x
13. Write the equation x2
2x  y2  4y  11 in standard form. O

A. (x
1)2  (y  2)2  16 B. (x  1)2  (y
2)2  16

(x  1)2 (y
2)2 (x
1)2 (y
1)2
C.      1 D.  
  1 13.
1 4 4 4

14. Write the equation 4x2  24x
y  34  0 in standard form.

A. y  4(x
3)2  2 B. x  4y2  2
C. 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?
y
A. x2  y2  16 B. x2  y2  16
x
y 
3 x
y 
3
C. x  y  16
2 2 D. x2  y2  16
x
y 
3 x
y 
3 O x 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  2x
A. (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.
© Glencoe/McGraw-Hill 500 Glencoe Algebra 2
 NAME DATE PERIOD

8 Chapter 8 Test, Form 2B SCORE

Write the letter for the correct answer in the blank y

9
at the right of each question. 8
ARENA SAILING CLUB

For Questions 1 and 2, refer to the figure at the 7 CITY CENTER

6
right showing six city locations. The origin is 5
LIBRARY
at the lower left corner of the grid. 4
3
1. What is the location of the point halfway between HOSPITAL
2
the hospital and arena? 1
MUSEUM


2 
2 
O 1 2 3 4 5 6 7 8 9x
A. (2, 6) B. 6, 5 C. 5, 6 D. 1, 3 1.
2

2. What is the distance between the museum and sailing club?

53
 units
A. 53 B. 53 units C.   units D. 9 units 2.
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  5
C. y  4(x
1)2
3 D. y  4(x
4)2
15

Assessment
3.

4. Write an equation for the parabola with focus (1, 3) and vertex (0, 3).

4
2
A. y  4(x
3)2 B. x 
1(y  3)2 C. y 
4 x  3 D. x  1(y
3)2 4.
4 4

5. Which equation is graphed at the right? y

A. y  2x2
8x  7 B. x 
2y2
8y
7 O x
C. 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  25
C. (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  1
C. 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).
(x
1)2 y2 (x  1)2 y2
A.      1 B.      1
36 16 36 16
y2 (x
1)2 y2 x2
C.     1 D.     1 8.
36 16 36 16

9. Which is the equation of an ellipse with center (
4, 2) and a horizontal

major axis?
(x  4)2 (y
2)2 (x
4)2 (y
2)2
A.      1 B.      1
16 4 16 4
(y  2)2 (x  4)2 (y
2)2 (x
4)2
C.     1 D.     1 9.
16 4 16 4

© Glencoe/McGraw-Hill 501 Glencoe Algebra 2

 NAME DATE PERIOD

8 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).
(y  1)2 x2 (x  1)2 y2
A.  
  1 B.  
  1
4 12 12 4
(y
1)2 2 x2 (y
1)2
C. 
x  1 D. 
  1 11.
4 12 4 12
12. Which equation is graphed at the right? y
A. 9x2
4y2  36 B. 4x2
9y2  36
C. 9y2
4x2  36 D. 4y2
9x2  36 O x 12.
13. Write the equation 4x2  8x
y2
4y
4  0
in standard form.
(x  1)2 (y  2)2
A.  
  1 B. (x  1)2  (y
2)2  4
1 4
(x
1)2 (y
2)2 (x
1)2 (y  2)2
C.     1 D.      1 13.
1 4 4 1
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  10
C. 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  1
C. 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? y

A. x2  y2  9 B. x2  y2  9
y x0
2 y2  x  0
O x
C. 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 
4
A. (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
6
A. (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.
© Glencoe/McGraw-Hill 502 Glencoe Algebra 2
 NAME DATE PERIOD

8 Chapter 8 Test, Form 2C SCORE

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.

Assessment
Graph each equation.
7. x2  y2  4x  6y
3  0 7. y

O x

8. 9x2  4y2  36 8. y

O x

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  4
4x2  y2  12

© Glencoe/McGraw-Hill 503 Glencoe Algebra 2

 NAME DATE PERIOD

8 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
y  x2
4x
yx
4
O x

20. Solve the system of inequalities by graphing. 20. y

x2  y2  16
y 
2x2  1

O x

Bonus Write an equation for the circle with the same center as B:
(x
3)2 (y  1)2
the graph of  
  1 and the same radius
4 16
as the graph of x2  y2
4x  10y  9.

© Glencoe/McGraw-Hill 504 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Test, Form 2D SCORE

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.

2 
6. Write an equation for the circle with center 1,
2 that is 6.

Assessment
tangent to the x-axis.

Graph each equation.

7. x2  y2
2x  4y  4 7. y

O x

8. 9x2  16y2  144 8. y

O x

For Questions 9 and 10, write an equation for the ellipse that
satisfies 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  11
3x2  y2  24

© Glencoe/McGraw-Hill 505 Glencoe Algebra 2

 NAME DATE PERIOD

8 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.

, 1)
13. vertices (
10, 1) and (4, 1), foci (
3 70 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. y
y2  9
x2
y 
3x  4
4

O x

20. Solve the system of inequalities by graphing. 20. y

x2  4y2  1
x  4(y  2)2

O x

Bonus Write an equation for the circle with the same center as B:
(x
5)2 (y  2)2
the graph of  
  1 and the same radius
16 9
as the graph of x2  y2
2y  16x  1.

© Glencoe/McGraw-Hill 506 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Test, Form 3 SCORE

1. Find the midpoint of the line segment with endpoints at 1.

(
12, 3.5) and (5.1, 4.8).

,
2) and B(5

2. Find the distance between A(45 , 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 
7.
2

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

Assessment
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. y

O x

9. Graph 5x2
2y2
4y  22. 9. y

O x

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,
1
2 
11. endpoints of major axis at (3,
8) and (3, 4), foci at 11.
(3,
2  25) and (3,
2
25 )

© Glencoe/McGraw-Hill 507 Glencoe Algebra 2

 NAME DATE PERIOD

8 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. 1x
y2  (y
12) 15.

8

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.

x2 y2
    1
25 16
xy

19. Solve the system of equations by graphing. 19.

x2
y2  4x  6y
4  0 y
x2  4x  3y
4  0

O x

20. Solve the system of inequalities by graphing. 20. y

x2
y2  4x  4
10
2  y  (x  1.75)2
O x

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?
© Glencoe/McGraw-Hill 508 Glencoe Algebra 2
 NAME DATE PERIOD

8 Chapter 8 Open-Ended Assessment SCORE

Demonstrate your knowledge by giving a clear, concise solution

to each problem. Be sure to include all relevant drawings and
justify your answers. You may show your solution in more than
one way or investigate beyond the requirements of the problem.
1. Harry was asked to determine whether the graph of the equation
x2  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 your
equation.

2. Do the graphs of any of the conic sections you have studied in

this chapter represent relations that are functions? Explain your
reasoning.

Assessment
3. What do the graphs of the parabolas y  (x
2)2
1 and
x  (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 y

and y  (x
4)2  3 are shown. For parts a and b,
replace each of the
s with one of the inequality 1
3
symbols (, ,  , ) so that the solution of the
2
system is the region indicated. Explain your choices.
a. (x
4)2  (y
3)2
4 4
y
(x
4)2  3
The solution of the system is region 2. O x

b. (x
4)2  (y
3)2
4
y
(x
4)2  3
The 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.
(x
1)2 (y  2)2 y
5. The graph of the equation      1
4 9
is shown. Find values of k for which the given O x

system of equations has the given number of

solutions. Explain the reasoning for your choices.
(x
1)2 (y  2)2
    1
4 9
yk
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.

© Glencoe/McGraw-Hill 509 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Vocabulary Test/Review SCORE

asymptote conjugate axis focus of a parabola parabola

center of a circle directrix hyperbola tangent
center of an ellipse Distance Formula latus rectum transverse axis
center of a hyperbola ellipse major axis vertex of a hyperbola
circle foci of an ellipse Midpoint Formula
conic section foci of a hyperbola minor axis

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

© Glencoe/McGraw-Hill 510 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Quiz SCORE

(Lessons 8–1 and 8–2)

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.

Assessment
of the latus rectum. y
O x

NAME DATE PERIOD

8 Chapter 8 Quiz SCORE

(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. y

O x

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
3)2 y2
    1.
16 4
© Glencoe/McGraw-Hill 511 Glencoe Algebra 2
 NAME DATE PERIOD

8 Chapter 8 Quiz SCORE

(Lessons 8–5 and 8–6)

1. Write an equation for the hyperbola y

whose graph is shown. (1, 2) 1.

2. Write an equation for the hyperbola 2.

with vertices (2,
5) and (2, 3), O x
foci (2,
6) and (2, 4)

3. Find the coordinates of the vertices 3.

and foci and the equations of the y
2 2
asymptotes for the hyperbola x
y  1. Then graph the
4 36
O x
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.

NAME DATE PERIOD

8 Chapter 8 Quiz SCORE

(Lesson 8–7)

Graph each system of equations. Use the graph to solve

the system.
1. y  (x
2)2
3 1.
(x
2)2 (y  2)2
    1 y
1 4
O x

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. y

x2 y2

  1 2
9 4
x2 y2 O 2 x
    1
25 9

© Glencoe/McGraw-Hill 512 Glencoe Algebra 2

 NAME DATE PERIOD

8 Chapter 8 Mid-Chapter Test SCORE

(Lessons 8–1 through 8–4)

Part I 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  3.
4
A. y  (x
2)2  1 B. x  (y
2)2  1
C. y  (x  2)2
1 D. x  (y
1)2
2 3.

4. Which equation is graphed?

y
A. y  4x2  8x  4 B. x  4y2  1
C. y  1x2
1x  1 D. x  1y2
1y  1 4.

Assessment
4 2 4 4 2 4
O x
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  49
C. (x
2)2  (y  7)2  4 D. (x
2)2  (y  7)2  49 5.

Part II
6. Graph x2  y2
4x  12. 6. y

O x

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).

© Glencoe/McGraw-Hill 513 Glencoe Algebra 2

 NAME DATE PERIOD

8 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  5
2x
3y  1. (Lesson 3-2)

4. Perform the indicated matrix operation. If the matrix does 4.

not exist, write impossible. 

3 6 9

2
1 0

4 3 7

2 9 5  
(Lesson 4-2) 
2x2
7x
4
5. Simplify  . Assume that the denominator is not 5.
2 2x  7x  3
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)

f (x )
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) O x

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)

© Glencoe/McGraw-Hill 514 Glencoe Algebra 2

 NAME DATE PERIOD

8 Standardized Test Practice

(Chapters 1–8)

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1. Which of the following is the sum of two consecutive prime numbers?

A. 9 B. 11 C. 17 D. 24 1. A B C D

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. E F G H

3. What is the value of 10m
3 if 2m  9?

A. 2 B. 8 C. 42 D. 48 3. A B C D

4. If 10 pears cost c cents, how many pears will d dollars buy?

1000d
E.   F. d 10d
G.  
10c
H.  4. E F G H
c 10c c d

5. What is the value of x is 2.5x  11y and y

0?

Assessment
y 3
55
A.   B. 2 22
C.   D. 3 5. A B C D
6 3 15 2

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. E F G H

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 only
C. III and IV only D. III only 7. A B C D

8. The table shows the distribution of Number of

Score
quiz scores for a group of students. students
No student scored less than 50 or 90 2
greater than 90. What is the mean 80 6
of the scores?
70 8
E. 70 F. 72.5
60 3
G. 75 H. 74.5 50 1
8. E F G H

9. Square RSTU is inscribed in circle O. R S

If the circumference of circle O is 16,
find the area of triangle ROU. O
A. 32 B. 32
C. 64 D. 16 U T 9. A B C D

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. E F G H

© Glencoe/McGraw-Hill 515 Glencoe Algebra 2

 NAME DATE PERIOD

8 Standardized Test Practice (continued)

Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column box
and then shading in the appropriate oval that corresponds to that entry.

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 0 0 0 0 0 0
1 1 1 1 1 1 1 1
marble
1
? 15
C 2 2 2 2 2 2 2 2
4 B 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
12. In the figure shown, what is 60˚ 5 6 6 6 6 6 6 6 6

D
the length of C
? 7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
A E D 9 9 9 9 9 9 9 9

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? . . . . . . . .
0 0 0 0 0 0

14. Evaluate 7  3  5  22.

1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in
A if the quantity in column A is greater;
B if the quantity in column B is greater;
C if the quantities are equal; or
D if the relationship cannot be determined from the information given.

Column A Column B
15. 2de2 15. A B C D

ed de

16. The average of a, b, and c is x. 16. A B C D

3x 3a

17. 3 17. A B C D
64



4
162

a b
ab
18. 

for all real numbers a, b, and c 18. A B C D
c c

2 3 10 15

4 10

© Glencoe/McGraw-Hill 516 Glencoe Algebra 2

 NAME DATE PERIOD

8 Standardized Test Practice

Student Record Sheet (Use with pages 468–469 of the Student Edition.)

Part 1 Multiple Choice

Select the best answer from the choices given and fill in the corresponding oval.

1 A B C D 4 A B C D 7 A B C D 9 A B C D

2 A B C D 5 A B C D 8 A B C D 10 A B C D

3 A B C D 6 A B C D

Part 2 Short Response/Grid In

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 in
the corresponding oval for that number or symbol.

11 13 15 17

/ / / / / / / /
. . . . . . . . . . . . . . . .
0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7

Answers
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

12 14 16

/ / / / / /
. . . . . . . . . . . .
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9

Part 3 Quantitative Comparison

Select the best answer from the choices given and fill in the corresponding oval.

18 A B C D 20 A B C D

19 A B C D 21 A B C D

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Study Guide and Intervention 8-1 Study Guide and Intervention (continued)

Midpoint and Distance Formulas Midpoint and Distance Formulas

The Midpoint Formula The Distance Formula
x
x
1 2 1 2 y
y The distance between two points (x1, y1) and (x2, y2) is given by
Midpoint Formula The midpoint M of a segment with endpoints (x1, y1) and (x 2, y2) is  , . Distance Formula

2 2  d  
(x2  
x1)2

(y2 
y1)2.

Glencoe/McGraw-Hill
Example 1 Find the midpoint of the Example 2 A diameter A
B

of a circle Example 1 What is the distance between (8,
2) and (
6,
8)?
line segment with endpoints at has endpoints A(5,
11) and B(
7, 6).
(4,
7) and (
2, 3). What are the coordinates of the center d  
(x2  
x1)2
(
y2  
y1)2 Distance Formula
x1
x2 y1
y2 4
(2) 7
3 of the circle?
 ,    ,   
(6 
8)2

[8 
(2)]2 Let (x1, y1)  (8, 2) and (x2, y2)  (6, 8).

2 2 
2 2  The center of the circle is the midpoint of all
2 4 of its diameters.  (14)
2 6)2

( Subtract.
2 2

 ,   or (1, 2)
x1
x2 y1
y2 5
(7) 11
6   196
 36 or 232
 Simplify.
 ,    , 

2 2 
2 2 

Lesson 8-1
The midpoint of the segment is (1, 2). The distance between the points is 232
 or about 15.2 units.
2 5 1
2 2 2

 ,   or
1, 2  
Example 2 Find the perimeter and area of square PQRS with vertices P(
4, 1),
1 Q(
2, 7), R(4, 5), and S(2,
1).
The circle has center 1, 2  .

2 
PQ
Find the length of one side to find the perimeter and the area. Choose .
Exercises d  
(x2  
x1)2
(
y2  
y1)2 Distance Formula
Answers

Find the midpoint of each line segment with endpoints at the given coordinates.  
[4 
(2)]  7)2
2
(1 Let (x1, y1)  (4, 1) and (x2, y2)  (2, 7).

A2
 
(2)2  )2

(6 Subtract.
1. (12, 7) and (2, 11) 2. (8, 3) and (10, 9) 3. (4, 15) and (10, 1)
 40
 or 210
 Simplify.
(5, 9) (1, 3) (7, 8)
Since one side of the square is 210
, the perimeter is 810
 units. The area is (210
 )2, or
4. (3, 3) and (3, 3) 5. (15, 6) and (12, 14) 6. (22, 8) and (10, 6) 40 units2.

(0, 0) (13.5, 10) (6,
1) Exercises

(Lesson 8-1)

7. (3, 5) and (6, 11) 8. (8, 15) and (7, 13) 9. (2.5, 6.1) and (7.9, 13.7) Find the distance between each pair of points with the given coordinates.


32 , 8 21 ,
1 (5.2, 3.8) 1. (3, 7) and (1, 4) 2. (2, 10) and (10, 5) 3. (6, 6) and (2, 0)

10. (7, 6) and (1, 24) 11. (3, 10) and (30, 20) 12. (9, 1.7) and (11, 1.3)
5 units 13 units 10 units
33 4. (7, 2) and (4, 1) 5. (5, 2) and (3, 4) 6. (11, 5) and (16, 9)
(
4, 9)  ,
15 (
10, 1.5)
 2 
32

units 10 units 41

units
13. Segment M N
 has midpoint P. If M has coordinates (14, 3) and P has coordinates 7. (3, 4) and (6, 11) 8. (13, 9) and (11, 15) 9. (15, 7) and (2, 12)
(8, 6), what are the coordinates of N? (
30, 15)
334

units 210

units 526

units
ST
14. Circle R has a diameter . If R has coordinates (4, 8) and S has coordinates (1, 4),
what are the coordinates of T? (
9,
20) 10. Rectangle ABCD has vertices A(1, 4), B(3, 1), C(3, 2), and D(5, 1). Find the
perimeter and area of ABCD. 2
13  65
units; 365
units2
15. Segment AD
 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? ST
11. Circle R has diameter  with endpoints S(4, 5) and T(2, 3). What are the
2 1 circumference and area of the circle? (Express your answer in terms of .)
B is 5, 8 
 , D is 15, 13  . 10 units; 25 units2
3  3

© Glencoe/McGraw-Hill 455 Glencoe Algebra 2 © Glencoe/McGraw-Hill 456 Glencoe Algebra 2

Glencoe Algebra 2
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Skills Practice 8-1 Practice (Average)

Midpoint and Distance Formulas Midpoint and Distance Formulas

Find the midpoint of each line segment with endpoints at the given coordinates. Find the midpoint of each line segment with endpoints at the given coordinates.
11
1. (4, 1), (4, 1) (0, 0) 2. (1, 4), (5, 2) (2, 3) 1. (8, 3), (6, 11) (1,
7) 2. (14, 5), (10, 6)
2,   2 

Glencoe/McGraw-Hill
3. (7, 6), (1, 2) (
3,
4) 4. (8, 2), (8, 8) (8,
5)
1
3. (3, 4), (5, 4) (4, 4) 4. (6, 2), (2, 1) 4, 
 2 
5
5. (9, 4), (1, 1) 5,
 6. (3, 3), (4, 9)  , 6
 2   72 
1 3
5. (3, 9), (2, 3)  , 3 6. (3, 5), (3, 8)
3,
 9 11
2   2
  7. (4, 2), (3, 7)  ,
 8. (6, 7), (4, 4) 5, 
 72 2   2 
5
7. (3, 2), (5, 0) (
1, 1) 8. (3, 4), (5, 2) (4,
1) 9. (4, 2), (8, 2) (
6, 0) 10. (5, 2), (3, 7) 4,   2 

Lesson 8-1
11 1 5
11. (6, 3), (5, 7)
 ,
2 12. (9, 8), (8, 3)
 ,

5 11  2   2 2 
9. (5, 9), (5, 4) 0,
 10. (11, 14), (0, 4)
 , 9
 2   2  1
13. (2.6, 4.7), (8.4, 2.5) (5.5,
1.1) 14.   , 6 ,  , 4

3 
23  16 , 5
5 9 5 1
11. (3, 6), (8, 3)
 ,
 12. (0, 10), (2, 5)
 2 2  
1, 52  15. (2.5, 4.2), (8.1, 4.2) (2.8, 0) 16.  ,  ,   ,  

18 12 
8 2  
14 , 0
Answers

A3
Find the distance between each pair of points with the given coordinates.
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
13. (4, 12), (1, 0) 13 units 14. (7, 7), (5, 2) 15 units

19. (3, 8), (1, 5) 173

units 20. (0, 1), (9, 6) 130

units
15. (1, 4), (1, 4) 2 units 16. (11, 11), (8, 15) 5 units
21. (5, 6), (6, 6) 1 unit 22. (3, 5), (12, 3) 17 units
(Lesson 8-1)

17. (1, 6), (7, 2) 10 units 18. (3, 5), (3, 4) 9 units 23. (2, 3), (9, 3) 157

units 24. (9, 8), (7, 8) 265

units

25. (9, 3), (9, 2) 5 units 26. (1, 7), (0, 6) 170

units
19. (2, 3), (3, 5) 5

units 20. (4, 3), (1, 7) 5 units
27. (10, 3), (2, 8) 13 units 28. (0.5, 6), (1.5, 0) 210

units
21. (5, 5), (3, 10) 17 units 22. (3, 9), (2, 3) 13 units 7
29.  ,  , 1,  1 unit 30. (42
, 5
), (52
, 45
) 127

units

25 35 
5 
23. (6, 2), (1, 3) 74

units 24. (4, 1), (2, 4) 61

units 31. GEOMETRY Circle O has a diameter  AB. If A is at (6, 2) and B is at (3, 4), find the
center of the circle and the length of its diameter. 9

2 , 1; 35
units
25. (0, 3), (4, 1) 42

units 26. (5, 6), (2, 0) 85

units 32. GEOMETRY Find the perimeter of a triangle with vertices at (1, 3), (4, 9), and (2, 1).
18  217

units

© Glencoe/McGraw-Hill 457 Glencoe Algebra 2 © Glencoe/McGraw-Hill 458 Glencoe Algebra 2

Glencoe Algebra 2
Answers
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-1 Reading to Learn Mathematics 8-1 Enrichment

Midpoint and Distance Formulas
Pre-Activity How are the Midpoint and Distance Formulas used in emergency Quadratic Form
medicine?
Consider two methods for solving the following equation.
Read the introduction to Lesson 8-1 at the top of page 412 in your textbook.

Glencoe/McGraw-Hill
How do you find distances on a road map? (y  2)2  5(y  2)
6  0
Sample answer: Use the scale of miles on the map. You might One way to solve the equation is to simplify first, then use factoring.
also use a ruler.
y2  4y
4  5y
10
6  0
y2  9y
20  0
( y  4)( y  5)  0
Reading the Lesson
1. a. Write the coordinates of the midpoint of a segment with endpoints (x1, y1) and (x2, y2). Thus, the solution set is {4, 5}.
x1  x2 y1  y2

Lesson 8-1
 , Another way to solve the equation is first to replace y  2 by a single variable.
 2 2  This will produce an equation that is easier to solve than the original equation.
Let t  y  2 and then solve the new equation.
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. ( y  2)2  5( y  2)
6  0
Sample answer: To find the x-coordinate of the midpoint, add the t2  5t
6  0
x-coordinates of the endpoints and divide by two. To find the (t  2)(t  3)  0
y-coordinate of the midpoint, do the same with the y-coordinates of
Answers

the endpoints. Thus, t is 2 or 3. Since t  y  2, the solution set of the original equation is {4, 5}.

A4
2. a. Write an expression for the distance between two points with coordinates (x1, y1) and
(x2, y2). (x




2x )2 

1 (y


2 y )2
1 Solve each equation using two different methods.
b. Explain how to find the distance between two points. Do not use subscripts in your
explanation. 1. (z
2)2
8(z
2)
7  0 2. (3x  1)2  (3x  1)  20  0

Sample answer: Find the difference between the {
3,
9} {2,
1}
x-coordinates and square it. Find the difference between the
(Lesson 8-1)

y-coordinates and square it. Add the squares. Then find the square
root of the sum.
3. Consider the segment connecting the points (3, 5) and (9, 11).
3. (2t
1)2  4(2t
1)
3  0 4. ( y2  1)2  ( y2  1)  2  0
a. Find the midpoint of this segment. (3, 8)
{0, 1} 0, 3

b. Find the length of the segment. Write your answer in simplified radical form. 65

Helping You Remember

4. How can the “mid” in midpoint help you remember the midpoint formula?
5. (a2  2)2  2(a2  2)  3  0 6. (1
c ) 2
(1
c )  6  0
Sample answer: The midpoint is the point in the middle of a segment. It
is halfway between the endpoints. The coordinates of the midpoint are 1, 5
 {1}
found by finding the average of the two x-coordinates (add them and
divide by 2) and the average of the two y-coordinates.

© Glencoe/McGraw-Hill 459 Glencoe Algebra 2 © Glencoe/McGraw-Hill 460 Glencoe Algebra 2

Glencoe Algebra 2
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-2 Study Guide and Intervention 8-2 Study Guide and Intervention (continued)

Parabolas Parabolas
Equations of Parabolas A parabola is a curve consisting of all points in the Graph Parabolas To graph an equation for a parabola, first put the given equation in
coordinate plane that are the same distance from a given point (the focus) and a given line standard form.
(the directrix). The following chart summarizes important information about parabolas.
y  a(x  h)2
k for a parabola opening up or down, or

Glencoe/McGraw-Hill
Standard Form of Equation y  a(x  h)2
k x  a(y  k)2
h x  a(y  k)2
h for a parabola opening to the left or right
Axis of Symmetry xh yk
Use the values of a, h, and k to determine the vertex, focus, axis of symmetry, and length of
Vertex (h, k) (h, k)
the latus rectum. The vertex and the endpoints of the latus rectum give three points on the
1 1 parabola. If you need more points to plot an accurate graph, substitute values for points
Focus
h, k

4a  4a

h
 , k
near the vertex.
1 1
Directrix yk  xh 
4a 4a
Direction of Opening upward if a  0, downward if a  0 right if a  0, left if a  0 Example 1
Graph y   (x
1)2  2.
3
1 1
Length of Latus Rectum  units  units
a
  a
  1
In the equation, a   , h  1, k  2.
3
Example The parabola opens up, since a  0. y
Identify the coordinates of the vertex and focus, the equations of
the axis of symmetry and directrix, and the direction of opening of the parabola vertex: (1, 2)
with equation y  2x2
12x
25. axis of symmetry: x  1
y  2x2  12x  25 Original equation 1 3
focus: 1, 2
 or 1, 2 
y  2(x2  6x)  25 Factor 2 from the x-terms.

4 

4 

13  
y Complete the square on the right side.
Answers

 2(x2  6x
■)  25  2(■)
O x

Lesson 8-2
y  2(x2  6x
9)  25  2(9) The 9 added to complete the square is multiplied by 2. 1

A5
length of latus rectum: 
1 or 3 units
y  2(x  3)2  43 Write in standard form. 
7
 3
The vertex of this parabola is located at (3, 43), the focus is located at 3, 42  , the
8  1 3 1 3
endpoints of latus rectum: 2  , 2  ,   , 2 
1
2 4 
2 4 
equation of the axis of symmetry is x  3, and the equation of the directrix is y  43  .
8
The parabola opens upward.

Exercises Exercises
(Lesson 8-2)

Identify the coordinates of the vertex and focus, the equations of the axis of The coordinates of the focus and the equation of the directrix of a parabola are
symmetry and directrix, and the direction of opening of the parabola with the given. Write an equation for each parabola and draw its graph.
given equation.
1. (3, 5), y  1 2. (4, 4), y 6 3. (5, 1), x  3
1. y  x2
6x  4 2. y  8x  2x2
10 3. x  y2  8y
6 y y y
1 3
(
3,
13), (2, 18), 2, 17  , (
10, 4), 
9  , 4,
8 4
1 1 O x O x

3,
12 34 , x 
3, x  2, y  18  , y  4, x 
10  ,
8 4
1
y 
13  , up down right
4 O x

Write an equation of each parabola described below.

1 1 1
1 11
y (x
3)2  3 y (x
4)2
5 x (y  1)2  4
4. focus (2, 3), directrix x  2  5. vertex (5, 1), focus 4  , 1 8 4 4
12
12 
1
x  6(y
3)2
2  x 
3(y
1)2  5
24

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8-2 Skills Practice 8-2 Practice (Average)

Parabolas Parabolas
Write each equation in standard form. Write each equation in standard form.
1 1
1. y  x2
2x
2 2. y  x2  2x
4 3. y  x2
4x
1 1. y  2x2  12x
19 2. y   x2
3x
 3. y  3x2  12x  7
2 2
y  [x
(
1)]2 1 y  (x
1)2 3 y  [x
(
2)]2  (
3) 1
(
4)

Glencoe/McGraw-Hill
y  2(x
3)2  1 y [x
(
3)]2  y 
3[x
(
2)]2  5
2
Identify the coordinates of the vertex and focus, the equations of the axis of
Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the
symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola.
given equation. Then find the length of the latus rectum and graph the parabola.
1
4. y  (x  4)2
3 5. x    y2
1 6. x  3(y
1)2  3
3
4. y  (x  2)2 5. x  (y  2)2
3 6. y  (x
3)2
4
y y y
y y y

O x O x
O x O x

O x
O x

vertex: (4, 3); vertex: (1, 0); vertex: (
3,
1);

vertex: (2, 0); vertex: (3, 2); vertex: (
3, 4);
1 1 11
1 1 3 focus: 4, 3  ; focus:  ,0 ; focus:
2  ,
1 ;
Answers

; ,2 ; ;  4 4   12 

Lesson 8-2
focus: 2,  focus: 3  focus:
3, 3 
4  4   4 

A6
axis: x  4; axis: y  0; axis: y 
1;
axis of symmetry: axis of symmetry: axis of symmetry: 3 3 1
x  2; y  2; x 
3; directrix: y  2  ; directrix: x  1  ; directrix: x 
3  ;
4 4 12
1 3 1 opens up; opens left; opens right;
directrix: y 
 ; directrix: x  2  ; directrix: y  4  ;
4 4 4 1
latus rectum: 1 unit latus rectum: 3 units latus rectum:  unit
opens up; opens right; opens down; 3
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, 4), 8. vertex (2, 1), 9. vertex (1, 3),
(Lesson 8-2)

7 directrix x  3 axis of symmetry x  1,

Write an equation for each parabola described below. Then draw the graph. focus 0, 3 

8  latus rectum: 2 units,
7. vertex (0, 0), 8. vertex (5, 1), 9. vertex (1, 3), a0
1 5 7 1 1
focus 0,   focus 5,  directrix x   y  2x 2
4 x (y
1)2
2 y 
 (x
1)2  3

12 
4  8 4 2
2 y y y
y
3x y  (x
5)2  1 x  2(y
3)2 1
y y y

O x
O x O x
O x

O x O x
10. TELEVISION Write the equation in the form y  ax2 for a satellite dish. Assume that the
bottom of the upward-facing dish passes through (0, 0) and that the distance from the
bottom to the focus point is 8 inches. 1 2
y  x
32

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8-2 Reading to Learn Mathematics 8-2 Enrichment

Parabolas
Pre-Activity How are parabolas used in manufacturing? Tangents to Parabolas
Read the introduction to Lesson 8-2 at the top of page 419 in your textbook. A line that intersects a parabola in exactly one point y
6
Name at least two reflective objects that might have the shape of a without crossing the curve is a tangent to the

Glencoe/McGraw-Hill
parabola. parabola. The point where a tangent line touches 5
a parabola is the point of tangency. The line
Sample answer: telescope mirror, satellite dish 4
perpendicular to a tangent to a parabola at the point
of tangency is called the normal to the parabola at 3
that point. In the diagram, line
is tangent to the
3 9 2
–23, –49
Reading the Lesson parabola that is the graph of y  x2 at ,  . The
2 4

1
1. In the parabola shown in the graph, the point (2, 2) is called y
x-axis is tangent to the parabola at O, and the y-axis y  x2
is the normal to the parabola at O. O x
–3 –2 –1 1 2 3
the vertex and the point (2, 0) is called the
(2, 0)
focus . The line y  4 is called the O x

directrix (2, –2) Solve each problem.

, and the line x  2 is called the
axis of symmetry . y  –4 1. Find an equation for line
in the diagram. Hint: A nonvertical line with an
equation of the form y  mx
b will be tangent to the graph of y  x2 at
2. a. Write the standard form of the equation of a parabola that opens upward or
32, 94 if and only if
32, 94 is the only pair of numbers that satisfies both
Answers

downward. y  a(x
h)2  k y  x2 and y  mx
b.

Lesson 8-2

A7
b. The parabola opens downward if a0 and opens upward if a0 . The 9 9
m  3, b 
, y  3x

4 4
equation of the axis of symmetry is xh , and the coordinates of the vertex are
(h, k) .
1 2. If a is any real number, then (a, a2) belongs to the graph of y  x2. Express
3. A parabola has equation x    ( y  2)2
4. This parabola opens to the left . m and b in terms of a to find an equation of the form y  mx
b for the line
8
It has vertex (4, 2) and focus (2, 2) . The directrix is x6 . The length that is tangent to the graph of y  x2 at (a, a2).
(Lesson 8-2)

of the latus rectum is 8 units. m  2a, b  a 2 , y  (2a)x  (
a 2 ) or y  2ax
a 2

Helping You Remember

4. How can the way in which you plot points in a rectangular coordinate system help you to 3. Find an equation for the normal to the graph of y  x2 at ,  .
32 94 
remember what the sign of a tells you about the direction in which a parabola opens? 1 11
y 
x  
Sample answer: In plotting points, a positive x-coordinate tells you to 3 4
move to the right and a negative x-coordinate tells you to move to the
left. This is like a parabola whose equation is of the form “x  …”; it
opens to the right if a  0 and to the left if a  0. Likewise, a positive
4. If a is a nonzero real number, find an equation for the normal to the graph of
y-coordinate tells you to move up and a negative y-coordinate tells you
y  x2 at (a, a2).
to 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. 1 1
y 
 x  a 2  
 2a   2 

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8-3 Study Guide and Intervention 8-3 Study Guide and Intervention (continued)

Circles Circles
Equations of Circles The equation of a circle with center (h, k) and radius r units is Graph Circles To graph a circle, write the given equation in the standard form of the
(x  h) 2
(y  k) 2  r2. equation 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 that

Glencoe/McGraw-Hill
Example Write an equation for a circle if the endpoints of a diameter are at goes through those four points.
(
4, 5) and (6,
3).
Use the midpoint formula to find the center of the circle. Example Find the center and radius of the circle y
1 2 x
x
1 2 y
y whose equation is x2  2x  y2  4y  11. Then graph x 2  2x  y 2  4y  11
(h, k)   , Midpoint formula

2 2  the circle.
4
6 5
(3) x2
2x
y2
4y  11
  ,  (x1, y1)  (4, 5), (x2, y2)  (6, 3)
O x

2 2  x2
2x
■
y2
4y
■  11
■
2 2
Simplify. x2
2x
1
y2
4y
4  11
1
4
2 2

 ,   or (1, 1)
(x
1)2
( y
2)2  16
Use the coordinates of the center and one endpoint of the diameter to find the radius.
Therefore, the circle has its center at (1, 2) and a radius of
r   )2

(x2 x1 ( y2 
y1) 2 Distance formula
16  4. Four points on the circle are (3, 2), (5, 2), (1, 2),
r  
(4 
1) 2
 1) 2
(5   (x1, y1)  (1, 1), (x2, y2)  (4, 5)
and (1, 6).

 
42
(5) 2  41
 Simplify. Exercises
The radius of the circle is 41
, so r2  41.
Find the center and radius of the circle with the given equation. Then graph the
Answers

An equation of the circle is (x  1)2
(y  1) 2  41. circle.

A8
Exercises 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
Write an equation for the circle that satisfies each set of conditions.
y y y

1. center (8, 3), radius 6 (x
8)2  (y  3)2  36 O x O x

2. center (5, 6), radius 4 (x
5)2  (y  6)2  16

Lesson 8-3
O
(Lesson 8-3)

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

4. (x  2)2
(y
4)2  16 5. x2
y2  10x
8y
16  0 6. x2
y2  4x
6y  12
5. center (3, 6), tangent to the x-axis (x
3)2  (y
6)2  36
(2,
4), r  4 (5,
4), r  5 (2,
3), r  5
6. center (4, 7), tangent to x  2 (x  4)2  (y  7)2  36 y y y

O x O x
7. center at (2, 8), tangent to y  4 (x  2)2  (y
8)2  144 O x

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

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8-3 Skills Practice 8-3 Practice (Average)

Circles Circles
Write an equation for the circle that satisfies each set of conditions. 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 units 1. center (4, 2), radius 8 units 2. center (0, 0), radius 4 units
x 2  (y
5)2  1 (x
5)2  (y
12)2  64 (x  4)2  (y
2)2  64 x2  y2  16

Glencoe/McGraw-Hill
1
3. center (4, 0), radius 2 units 4. center (2, 2), radius 3 units 3. center   , 3
 , radius 52
 units 4. center (2.5, 4.2), radius 0.9 unit

4 
(x
4)2  y 2  4 (x
2)2  (y
2)2  9 1 2 2
x  4   (y  3

)  50 (x
2.5)2  (y
4.2)2  0.81
5. center (4, 4), radius 4 units 6. center (6, 4), radius 5 units
5. endpoints of a diameter at (2, 9) and (0, 5) (x  1)2  (y  7)2  5
(x
4)2  (y  4)2  16 (x  6)2  (y
4)2  25
6. center at (9, 12), passes through (4, 5) (x  9)2  (y  12)2  74
7. endpoints of a diameter at (12, 0) and (12, 0) x 2  y 2  144
7. center at (6, 5), tangent to x-axis (x  6)2  (y
5)2  25
8. endpoints of a diameter at (4, 0) and (4, 6) (x  4)2  (y  3)2  9
Find the center and radius of the circle with the given equation. Then graph the
9. center at (7, 3), passes through the origin (x
7)2  (y  3)2  58 circle.
10. center at (4, 4), passes through (4, 1) (x  4)2  (y
4)2  9 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. center at (6, 5), tangent to y-axis (x  6)2  (y  5)2  36 y y y

12. center at (5, 1), tangent to x-axis (x
5)2  (y
1)2  1 4

Answers

–8 –4 O 4 8x
Find the center and radius of the circle with the given equation. Then graph the O x O x
–4

A9
circle.
–8
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
11. (x  1)2
y2
4y  12 12. x2  6x
y2  0 13. x2
y2
2x
6y  1
y y y
(1,
2), 4 units (3, 0), 3 units (
1,
3), 3 units
Lesson 8-3 y y y
(Lesson 8-3)

O x
O x O x O x O x

O x

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 WEATHER For Exercises 14 and 15, use the following information.
y y y On average, the circular eye of a hurricane is about 15 miles in diameter. Gale winds can
12 8
affect an area up to 300 miles from the storm’s center. In 1992, Hurricane Andrew devastated
O 4 8 12 x
6 4 southern Florida. A satellite photo of Andrew’s landfall showed the center of its eye on one
–4 coordinate system could be approximated by the point (80, 26).
x
–12 –6 O 6 12 –8 –4 O 4 8x
–8 14. Write an equation to represent a possible boundary of Andrew’s eye.
–6 –4
–12 (x
80)2  (y
26)2  56.25
–12 –8
15. Write an equation to represent a possible boundary of the area affected by gale winds.
(x
80)2  (y
26)2  90,000
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Answers
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8-3 Reading to Learn Mathematics 8-3 Enrichment

Circles
Pre-Activity Why are circles important in air traffic control? Tangents to Circles
Read the introduction to Lesson 8-3 at the top of page 426 in your textbook. A line that intersects a circle in exactly one point is y


A large home improvement chain is planning to enter a new metropolitan a tangent to the circle. In the diagram, line
is x 2  y2  25 5
(3, 4)

Glencoe/McGraw-Hill
area and needs to select locations for its stores. Market research has shown tangent to the circle with equation x2
y2  25 at
that potential customers are willing to travel up to 12 miles to shop at one the point whose coordinates are (3, 4).
of their stores. How can circles help the managers decide where to place
A line is tangent to a circle at a point P on the circle
their store?
if and only if the line is perpendicular to the radius
Sample answer: A store will draw customers who live inside a from the center of the circle to point P. This fact –5 O 5 x
circle with center at the store and a radius of enables you to find an equation of the tangent to a
12 miles. The management should select locations for which circle at a point P if you know an equation for the
as many people as possible live within a circle of radius circle and the coordinates of P.
12 miles around one of the stores.
–5

Reading the Lesson

1. a. Write the equation of the circle with center (h, k) and radius r. Use the diagram above to solve each problem.
(x
h) 2  (y
k) 2  r 2
1. What is the slope of the radius to the point with coordinates (3, 4)? What is
b. Write the equation of the circle with center (4, 3) and radius 5. the slope of the tangent to that point?
Answers

(x
4)2  (y  3)2  25 4 3
,

c. The circle with equation (x
8)2
y2  121 has center (
8, 0) and radius 3 4

A10
11 .
2. Find an equation of the line
that is tangent to the circle at (3, 4).
d. The circle with equation (x  10)2
( y
10)2  1 has center (10,
10) and
1 3 25
radius . y 
x  
4 4
2. a. In order to find center and radius of the circle with equation x2
y2
4x  6y 3  0,

Lesson 8-3
(Lesson 8-3)

it is necessary to complete the square . Fill in the missing parts of this 3. If k is a real number between 5 and 5, how many points on the circle have
process. x-coordinate k? State the coordinates of these points in terms of k.
x2
y2
4x  6y  3  0
two, (k, 
k2)
25


x2
y2
4x  6y  3
x2
4x
4
y2  6y
9  3
4
9
(x
2 )2
( y  3 )2  16 4. Describe how you can find equations for the tangents to the points you named
for Exercise 3.
b. This circle has radius 4 and center at (
2, 3) . Use the coordinates of (0, 0) and of one of the given points. Find the
slope of the radius to that point. Use the slope of the radius to find what
the slope of the tangent must be. Use the slope of the tangent and the
Helping You Remember coordinates of the point on the circle to find an equation for the tangent.
3. How can the distance formula help you to remember the equation of a circle? 5. Find an equation for the tangent at (3, 4).
Sample answer: Write the distance formula. Replace (x1, y1) with (h, k) 3 25
y  x  
and (x 2, y2) with (x, y). Replace d with r. Square both sides. Now you 4 x
have the equation of a circle.

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8-4 Study Guide and Intervention 8-4 Study Guide and Intervention (continued)

Ellipses Ellipses
Equations of Ellipses An ellipse is the set of all points in a plane such that the sum Graph Ellipses To graph an ellipse, if necessary, write the given equation in the
of the distances from two given points in the plane, called the foci, is constant. An ellipse standard form of an equation for an ellipse.
has two axes of symmetry which contain the major and minor axes. In the table, the
(x  h)2 ( y  k)2
lengths a, b, and c are related by the formula c2  a2  b2. 
  1 (for ellipse with major axis horizontal) or
a2 b2

Glencoe/McGraw-Hill
(x  h) 2 (y  k)2 (y  k)2 (x  h)2 ( y  k)2 (x  h)2
Standard Form of Equation 
 1 
 1 
  1 (for ellipse with major axis vertical)
a2 b2 a2 b2 a2 b2
Center (h, k) (h, k)
Use the center (h, k) and the endpoints of the axes to plot four points of the ellipse. To make
Direction of Major Axis Horizontal Vertical a more accurate graph, use a calculator to find some approximate values for x and y that
Foci (h
c, k ), (h  c, k ) (h, k  c), (h, k
c) satisfy the equation.
Length of Major Axis 2a units 2a units
Example Graph the ellipse 4x 2  6y 2  8x
36y 
34.
Length of Minor Axis 2b units 2b units
4x2
6y2
8x  36y  34 y
4x2
8x
6y2  36y   34 4x 2  6y 2  8x
36y 
34
Example Write an equation for the ellipse shown. y 4(x2
2x
■)
6( y2  6y
■)  34
■
The length of the major axis is the distance between (2, 2) 4(x2
2x
1)
6( y2  6y
9)  34
58
and (2, 8). This distance is 10 units. F1 4(x
1)2
6( y  3)2  24
2a  10, so a  5 (x
1)2 ( y  3)2

1
6 4 O x
The foci are located at (2, 6) and (2, 0), so c  3.
b2  a2  c2 The center of the ellipse is (1, 3). Since a2  6, a  6.
F2 Since b2  4, b  2.
Answers

 25  9 O x
 16 The length of the major axis is 26 , and the length of the minor axis is 4. Since the x-term

A11
The center of the ellipse is at (2, 3), so h  2, k  3, has the greater denominator, the major axis is horizontal. Plot the endpoints of the axes.
a2  25, and b2  16. The major axis is vertical. Then graph the ellipse.
( y  3)2 (x
2)2
An equation of the ellipse is 
  1. Exercises
25 16
Exercises Find the coordinates of the center and the lengths of the major and minor axes
for the ellipse with the given equation. Then graph the ellipse.
Write an equation for the ellipse that satisfies each set of conditions.
y2 x2 x2 y2
(Lesson 8-4)

1. endpoints of major axis at (7, 2) and (5, 2), endpoints of minor axis at (1, 0) and (1, 4) 1. 
  1 (0, 0), 43

, 6 2. 
  1 (0, 0), 10, 4
12 9 25 4
(x  1)2
(y
2)2
1 y y
36 4
2. major axis 8 units long and parallel to the x-axis, minor axis 2 units long, center at (2, 5)
(x  2)2 O x O x
  (y  5)2  1
16
3. endpoints of major axis at (8, 4) and (4, 4), foci at (3, 4) and (1, 4)
Lesson 8-4

(x  2)2 (y
4)2
  1
36 35 3. x2
4y2
24y  32 (0,
3), 4, 2 4. 9x2
6y2  36x
12y  12 (2,
1), 6, 26


4. endpoints of major axis at (3, 2) and (3, 14), endpoints of minor axis at (1, 6) and (7, 6) y y
(y  6)2 (x
3)2
1
64 16 O x O x
5. minor axis 6 units long and parallel to the x-axis, major axis 12 units long, center at (6, 1)
(y
1)2 (x
6)2
1
36 9

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Answers
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-4 Skills Practice 8-4 Practice (Average)

Ellipses Ellipses
Write an equation for each ellipse. Write an equation for each ellipse.
1. (0, 3) y 2. (0, 2  5 ) (y0, 5) 3. y
1. y 2. (0, 5) y 3. (0, 5) y (–6, 3) (4, 3)
2

Glencoe/McGraw-Hill
(–11, 0) (11, 0)
(0, 2) (0, 3)
–12 –6 O 6 12 x (–5, 3) (3, 3)
(–4, 2) (4, 2)
(–3, 0) –2 O
x
O (3, 0) x O x O x (0, –3) (0, 2
5 ) (0, –1) O x
(0, –1)
(0, –2) (0, –3)
x2 y2 (y
2)2 x2 (x  1)2 (y
3)2
1  1 1
121 9 9 4 25 9
(0, –5)
Write an equation for the ellipse that satisfies each set of conditions.
x2 y2 y2 x2 x2 (y
2)2
1 1 1 4. endpoints of major axis 5. endpoints of major axis 6. major axis 20 units long
9 4 25 16 16 9
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,
Write an equation for the ellipse that satisfies each set of conditions. at (0, 3) and (0, 3) at (1, 3) and (7, 3) center at (2, 1)
4. endpoints of major axis 5. endpoints of major axis 6. endpoints of major axis x2 y2 (y  3)2 (x
4)2 (x
2)2 (y
1)2
1 1 1
at (0, 6) and (0, 6), at (2, 6) and (8, 6), at (7, 3) and (7, 9), 81 9 25 9 100 25
endpoints of minor axis endpoints of minor axis endpoints of minor axis 7. major axis 10 units long, 8. major axis 16 units long, 9. endpoints of minor axis
at (3, 0) and (3, 0) at (5, 4) and (5, 8) at (5, 6) and (9, 6) minor axis 6 units long center at (0, 0), foci at at (0, 2) and (0, 2), foci
Answers

y2 x2 (x
5)2
(y
6)2 (y
6)2
(x
7)2 and parallel to x-axis, (0, 2
15 ) and (0, 215 ) at (4, 0) and (4, 0)
1 1 1
36 9 9 4 9 4 center at (2, 4)

A12
(y  4)2 (x
2)2 y2 x2 x2 y2
1 1 1
7. major axis 12 units long 8. endpoints of major axis 9. endpoints of major axis at 25 9 64 4 20 4
and parallel to x-axis, at (6, 0) and (6, 0), foci (0, 12) and (0, 12), foci at
Find the coordinates of the center and foci and the lengths of the major and
minor axis 4 units long, at ( 32, 0) and (32, 0) (0, 23 ) and (0, 23 ) minor axes for the ellipse with the given equation. Then graph the ellipse.
center at (0, 0)
y2 y2 y2 y2 x2 ( y  1)2 (x  3)2 (x
4)2 ( y
3)2
x2 x2 x2 10. 
 1 11. 
 1 12. 
1
1 1 1 16 9 36 1 49 25
36 4 36 4 144 121
(Lesson 8-4)

(0, 0); (0, 7

); 8; 6 (3, 1); (3, 1  35

); (
4,
3);
Find the coordinates of the center and foci and the lengths of the major and 12; 2 (
4  26
,
3); 14; 10
minor axes for the ellipse with the given equation. Then graph the ellipse. y y y
8 4
y2 x2 x2 y2 y2 x2
10. 
  1 11. 
  1 12. 
  1 4
100 81 81 9 49 25 –8 –4 O 4 x
–4
(0, 0); (0, 19

); (0, 0); (62

, 0); (0, 0), (0, 26

); O x –8 –4 O 4 8x
Lesson 8-4

20; 18 18; 6 14; 10 –4 –8

y y y –8 –12
8 8 8

4 4 4
13. SPORTS An ice skater traces two congruent ellipses to form a figure eight. Assume that the
–8 –4 O 4 8x O O
center of the first loop is at the origin, with the second loop to its right. Write an equation
–8 –4 4 8x –8 –4 4 8x
to model the first loop if its major axis (along the x-axis) is 12 feet long and its minor
–4 –4 –4
axis is 6 feet long. Write another equation to model the second loop.
–8 –8 –8 x2 y2 (x
12)2 y2
    1;     1
36 9 36 9
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8-4 Reading to Learn Mathematics 8-4 Enrichment

Ellipses
Pre-Activity Why are ellipses important in the study of the solar system? Eccentricity
Read the introduction to Lesson 8-4 at the top of page 433 in your textbook. c
In an ellipse, the ratio  is called the eccentricity and is denoted by the
d
Is the Earth always the same distance from the Sun? Explain your answer letter e. Eccentricity measures the elongation of an ellipse. The closer e is to 0,

Glencoe/McGraw-Hill
using the words circle and ellipse. No; if the Earth’s orbit were a the more an ellipse looks like a circle. The closer e is to 1, the more elongated
circle, it would always be the same distance from the Sun x2 y2 x2 y2
because every point on a circle is the same distance from the it is. Recall that the equation of an ellipse is 2
2  1 or 2
2  1
a b b a
center. However, the Earth’s orbit is an ellipse, and the points where a is the length of the major axis, and that c  
a2  b2.
on an ellipse are not all the same distance from the center.

Reading the Lesson

1. An ellipse is the set of all points in a plane such that the sum of the Find the eccentricity of each ellipse rounded to the nearest
constant hundredth.
distances from two fixed points is . The two fixed points are called the
x2 y2 x2 y2 x2 y2
foci of the ellipse. 1. 
  1
9 36
2. 
  1
81 9
3. 
  1
4 9
x2 y2 0.87 0.94 0.75
2. Consider the ellipse with equation 
  1.
9 4
a. For this equation, a  3 and b  2 .
b. Write an equation that relates the values of a, b, and c. c 2  a 2
b 2 x2 y2 x2 y2 x2 y2
4. 
  1 5. 
  1 6. 
  1
16 9 36 16 4 36
Answers

c. Find the value of c for this ellipse. 5

0.66 0.75 0.94

A13
y2 x2 x2 y2
3. Consider the ellipses with equations 
  1 and 
  1. Complete the
25 16 9 4
following table to describe characteristics of their graphs.
7. Is a circle an ellipse? Explain your reasoning.
y2 x2 x2 y2
Standard Form of Equation 
 1 
 1
25 16 9 4
Yes; it is an ellipse with eccentricity 0.
Direction of Major Axis vertical horizontal
(Lesson 8-4)

Direction of Minor Axis horizontal vertical

8. The center of the sun is one focus of Earth's orbit around the sun. The
Foci (0, 3), (0,
3) (5
, 0), (
5
, 0) length of the major axis is 186,000,000 miles, and the foci are 3,200,000
miles apart. Find the eccentricity of Earth's orbit.
Length of Major Axis 10 units 6 units
approximately 0.17
Length of Minor Axis 8 units 4 units
Lesson 8-4

Helping You Remember 9. An artificial satellite orbiting the earth travels at an altitude that varies
4. Some students have trouble remembering the two standard forms for the equation of an between 132 miles and 583 miles above the surface of the earth. If the
ellipse. How can you remember which term comes first and where to place a and b in center of the earth is one focus of its elliptical orbit and the radius of the
these equations? The x-axis is horizontal. If the major axis is horizontal, the earth is 3950 miles, what is the eccentricity of the orbit?
x2
first term is 2
. The y-axis is vertical. If the major axis is vertical, the approximately 0.052
a
y2
first term is 2 . a is always the larger of the numbers a and b.
a

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8-5 Study Guide and Intervention 8-5 Study Guide and Intervention (continued)

Hyperbolas Hyperbolas
Equations of Hyperbolas A hyperbola is the set of all points in a plane such that Graph Hyperbolas To graph a hyperbola, write the given equation in the standard
the absolute value of the difference of the distances from any point on the hyperbola to any form of an equation for a hyperbola
two given points in the plane, called the foci, is constant. (x  h) 2 ( y  k) 2
    1 if the branches of the hyperbola open left and right, or
In the table, the lengths a, b, and c are related by the formula c2  a2
b2. a2 b2

Glencoe/McGraw-Hill
( y  k)2 (x  h)2
(x  h)2 (y  k)2 (y  k)2 (x  h)2    1 if the branches of the hyperbola open up and down
Standard Form of Equation   1   1 a2 b2
a2 b2 a2 b2
b a Graph the point (h, k), which is the center of the hyperbola. Draw a rectangle with
Equations of the Asymptotes y  k    (x  h) y  k    (x  h)
a b dimensions 2a and 2b and center (h, k). If the hyperbola opens left and right, the vertices
Transverse Axis Horizontal Vertical are (h  a, k) and (h
a, k). If the hyperbola opens up and down, the vertices are (h, k  a)
and (h, k
a).
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) Example Draw the graph of 6y2
4x2
36y
8x 
26.
Complete the squares to get the equation in standard form. y
Example Write an equation for the hyperbola with vertices (
2, 1) and (6, 1) 6y2  4x2  36y  8x  26
and foci (
4, 1) and (8, 1). 6( y2  6y
■)  4(x2
2x
■)  26
■
Use a sketch to orient the hyperbola correctly. The center of y
6( y2  6y
9)  4(x2
2x
1)  26
50
the hyperbola is the midpoint of the segment joining the two 6( y  3)2  4(x
1)2  24
2
6 ( y  3)2 (x
1)2
vertices. The center is (  , 1), or (2, 1). The value of a is the 1
2 4 6 O x
distance from the center to a vertex, so a  4. The value of c is The center of the hyperbola is (1, 3).
Answers

the distance from the center to a focus, so c  6. O x

According to the equation, a2  4 and b2  6, so a  2 and b  6 .
c2  a2
b2 The transverse axis is vertical, so the vertices are (1, 5) and (1, 1). Draw a rectangle with

A14
62  42
b2 vertical dimension 4 and horizontal dimension 26   4.9. The diagonals of this rectangle
b2  36  16  20 are the asymptotes. The branches of the hyperbola open up and down. Use the vertices and
the asymptotes to sketch the hyperbola.
Use h, k, a2, and b2 to write an equation of the hyperbola.
(x  2)2 ( y  1)2 Exercises
1
16 20
Find the coordinates of the vertices and foci and the equations of the asymptotes
(Lesson 8-5)

for the hyperbola with the given equation. Then graph the hyperbola.
Exercises
x2 y2 (x
2)2 y2 x2
1.     1 2. ( y  3)2    1 3.     1
Write an equation for the hyperbola that satisfies each set of conditions. 4 16 9 16 9
x2 y2 (2, 0), (
2, 0); (
2, 4), (
2, 2); (0, 4), (0,
4);
1. vertices (7, 0) and (7, 0), conjugate axis of length 10 
  1
49 25 (25
, 0), (
25
, 0); (
2, 3  10
), (0, 5), (0,
5);
4
(x
1)2 (y  3)2 y  2x (
2, 3

10 ); y   x
2. vertices (2, 3) and (4, 3), foci (5, 3) and (7, 3) 
  1 3
9 27 y
1 2
y  x  3 , y
3 3
(y  1)2 (x
4)2 1 1
3. vertices (4, 3) and (4, 5), conjugate axis of length 4 
  1 y 
 x  2
16 4 3 3
y
1 x2 9y 2 O x
4. vertices (8, 0) and (8, 0), equation of asymptotes y    x 
  1 O x
6 64 16
(y
2)2 (x  4)2
5. vertices (4, 6) and (4, 2), foci (4, 10) and (4, 6) 
  1 O x
Lesson 8-5

16 48

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8-5 Skills Practice 8-5 Practice (Average)

Hyperbolas Hyperbolas
Write an equation for each hyperbola. Write an equation for each hyperbola.

1. y 2. y 3. y 1. y 2. y 3. y
8 8 8 8
(0, 35 ) 34 )
(–3, 2  

8
61 )
(0, 
(0, 6)

Glencoe/McGraw-Hill
4 4 4
(–5, 0) 4 O x
(–2, 0) (2, 0) (0, 3) (–3, 5) 4
(5, 0) (–1, –2)
–8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O 4 8x –8 –4 O 4 8x (–3, –1) O (3, –2)
–8 –4 4 x
–4 –4 (–

29, 0) –4 29, 0)
(
–4 (0, –3)
(–
( 41, 0) (0, –

61 ) (0, –6) –4 (1, –2)
41, 0) 
(0, –35 )
–8 34 )
(–3, 2


–8 –8 –8

x2 y2 y2 x2 x2 y2 y2 x2 (y
2)2 (x  3)2 (x
1)2 (y  2)2


1 
1 
1 
1 
1 
1
25 16 36 25 4 25 9 36 9 25 4 16
Write an equation for the hyperbola that satisfies each set of conditions.
Write an equation for the hyperbola that satisfies each set of conditions. y2 x2
4. vertices (0, 7) and (0, 7), conjugate axis of length 18 units 
  1
x2 y2 49 81
4. vertices (4, 0) and (4, 0), conjugate axis of length 8 
  1
16 16 (y  5)2 (x
1)2
y2 x2 5. vertices (1, 1) and (1, 9), conjugate axis of length 6 units 
  1
16 9
5. vertices (0, 6) and (0, 6), conjugate axis of length 14 
  1
36 49
y2 x2
y2 x2 6. vertices (5, 0) and (5, 0), foci (26
, 0) 
  1
6. vertices (0, 3) and (0, 3), conjugate axis of length 10 
  1 1 25
9 25
Answers

(y  1)2 (x
1)2
x2 y2 7. vertices (1, 1) and (1, 3), foci (1, 1  
5) 
  1
7. vertices (2, 0) and (2, 0), conjugate axis of length 4 
  1 4 1

A15
4 4
Find the coordinates of the vertices and foci and the equations of the asymptotes
x2 y2
8. vertices (3, 0) and (3, 0), foci (5, 0) 
  1 for the hyperbola with the given equation. Then graph the hyperbola.
9 16
y2 x2 ( y  2)2 (x  1)2 ( y
2)2 (x  3)2
y2 x2 8.     1 9.     1 10.     1
9. vertices (0, 2) and (0, 2), foci (0, 3) 
  1 16 4 1 4 4 4
4 5
(x
3)2 (y  2)2 (0, 4); (0, 25

); (1, 3), (1, 1); (3, 0), (3,
4);
10. vertices (0, 2) and (6, 2), foci (3  13, 2) 
  1
9 4
(Lesson 8-5)

y  2x (1, 2  5
); (3,
2  22
);

1
Find the coordinates of the vertices and foci and the equations of the asymptotes y
2   (x
1) y  2  (x
3)
2
for the hyperbola with the given equation. Then graph the hyperbola. y y y
8
x2 y2 y2 x2 x2 y2
11.     1 12.     1 13.     1
9 36 49 9 16 1 4
O x
(3, 0); (35

, 0); (0, 7); (0, 58

); (4, 0); (17

, 0);
–8 –4 O 4 8x
7 1
y  2x y   x y   x –4
3 4 O x
y y y –8
8 8

4 4
11. ASTRONOMY Astronomers use special X-ray telescopes to observe the sources of
celestial X rays. Some X-ray telescopes are fitted with a metal mirror in the shape of a
O x –8 –4 O 4 8x –8 –4 O 4 8x
hyperbola, which reflects the X rays to a focus. Suppose the vertices of such a mirror are
–4 –4 located at (3, 0) and (3, 0), and one focus is located at (5, 0). Write an equation that
models the hyperbola formed by the mirror. x 2 y2
Lesson 8-5

–8 –8 
1
9 16

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8-5 Reading to Learn Mathematics 8-5 Enrichment

Hyperbolas
Pre-Activity How are hyperbolas different from parabolas? Rectangular Hyperbolas
Read the introduction to Lesson 8-5 at the top of page 441 in your textbook. A rectangular hyperbola is a hyperbola with perpendicular asymptotes.
Look at the sketch of a hyperbola in the introduction to this lesson. List For example, the graph of x2  y2  1 is a rectangular hyperbola. A hyperbola

Glencoe/McGraw-Hill
three ways in which hyperbolas are different from parabolas. with asymptotes that are not perpendicular is called a nonrectangular
Sample answer: A hyperbola has two branches, while a hyperbola. The graphs of equations of the form xy  c, where c is a constant,
parabola is one continuous curve. A hyperbola has two foci, are rectangular hyperbolas.
while a parabola has one focus. A hyperbola has two vertices,
while a parabola has one vertex.
Make a table of values and plot points to graph each rectangular
Reading the Lesson hyperbola below. Be sure to consider negative values for the
variables. See students’ tables.
1. The graph at the right shows the hyperbola whose y
x2 y2 y  – 43 x y  43 x 1. xy  4 2. xy  3
equation in standard form is     1.
16 9 y y
(–5, 0) (5, 0)
The point (0, 0) is the center of the (–4, 0) O (4, 0) x
hyperbola.
The points (4, 0) and (4, 0) are the vertices
O x O x
of the hyperbola.
Answers

The points (5, 0) and (5, 0) are the foci

of the hyperbola.

A16
The segment connecting (4, 0) and (4, 0) is called the transverse axis.
The segment connecting (0, 3) and (0, 3) is called the conjugate axis.
3 3 asymptotes .
The lines y   x and y    x are called the 3. xy  1 4. xy  8
4 4
y y
2. Study the hyperbola graphed at the right. y
(Lesson 8-5)

The center is (0, 0) .

The value of a is 2 .
O x O x O x
The value of c is 4 .
To find b2, solve the equation c2  a2
b2 .

x2 y2

  1
The equation in standard form for this hyperbola is 4 12 .

Helping You Remember 5. Make a conjecture about the asymptotes of rectangular hyperbolas.

3. What is an easy way to remember the equation relating the values of a, b, and c for a
The coordinate axes are the asymptotes.
hyperbola? This equation looks just like the Pythagorean Theorem,
although the variables represent different lengths in a hyperbola than in
Lesson 8-5

a right triangle.

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8-6 Study Guide and Intervention 8-6 Study Guide and Intervention (continued)

Conic Sections Conic Sections

Standard Form Any conic section in the coordinate plane can be described by an Identify Conic Sections If you are given an equation of the form
equation of the form Ax2
Bxy
Cy2
Dx
Ey
F  0, with B  0,
Ax2
Bxy
Cy2
Dx
Ey
F  0, where A, B, and C are not all zero. you can determine the type of conic section just by considering the values of A and C. Refer
One way to tell what kind of conic section an equation represents is to rearrange terms and to the following chart.

Glencoe/McGraw-Hill
complete the square, if necessary, to get one of the standard forms from an earlier lesson.

Lesson 8-6
This method is especially useful if you are going to graph the equation. Relationship of A and C Type of Conic Section
A  0 or C  0, but not both. parabola
Example Write the equation 3x2
4y2
30x
8y  59  0 in standard form.
AC circle
State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.
A and C have the same sign, but A C. ellipse
3x2  4y2  30x  8y
59  0 Original equation
3x2  30x  4y2  8y  59 Isolate terms. A and C have opposite signs. hyperbola
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. Example Without writing the equation in standard form, state whether the
3(x  5)2  4( y
1)2  12 Simplify. graph of each equation is a parabola, circle, ellipse, or hyperbola.
(x  5)2 ( y
1)2
 1 Divide each side by 12. a. 3x 2
3y 2  5x  12  0 b. y 2  7y
2x  13
4 3
A  3 and C  3 have opposite signs, so A  0, so the graph of the equation is
The graph of the equation is a hyperbola with its center at (5, 1). The length of the the graph of the equation is a hyperbola. a parabola.
transverse axis is 4 units and the length of the conjugate axis is 23 units.

Exercises Exercises
Answers

Without writing the equation in standard form, state whether the graph of each
Write each equation in standard form. State whether the graph of the equation is
equation is a parabola, circle, ellipse, or hyperbola.

A17
a parabola, circle, ellipse, or hyperbola.
1. x2  17x  5y
8 2. 2x2
2y2  3x
4y  5
1. x2
y2  6x
4y
3  0 2. x2
2y2
6x  20y
53  0
parabola circle
(x  3)2 (y
5)2
(x
3)2  (y  2)2  10; circle     1; ellipse 3. 4x2  8x  4y2  6y
10 4. 8(x  x2)  4(2y2  y)  100
6 3
hyperbola circle
3. 6x2  60x  y
161  0 4. x2
y2  4x 14y
29  0
5. 6y2  18  24  4x2 6. y  27x  y2
(Lesson 8-6)

y  6(x
5)2  11; parabola (x
2)2  (y
7)2  24; circle ellipse parabola
7. x2  4( y  y2)
2x  1 8. 10x  x2  2y2  5y
5. 6x2  5y2
24x
20y  56  0 6. 3y2
x  24y
46  0
(x  2)2 (y
2)2 ellipse ellipse

  1; hyperbola x 
3(y
4)2  2; parabola
10 12 9. x  y2  5y
x2  5 10. 11x2  7y2  77
circle hyperbola
7. x2  4y2  16x
24y  36  0 8. x2
2y2
8x
4y
2  0
(x
(y
8)2 3)2 (x  (y 4)2 1)2 11. 3x2
4y2  50
y2 12. y2  8x  11

  1; hyperbola     1; ellipse
64 16 16 8 circle parabola
9. 4x2
48x
y
158  0 10. 3x2
y2  48x  4y
184  0 13. 9y2  99y  3(3x  3x2) 14. 6x2  4  5y2  3
(x
8)2 (y
2)2 circle hyperbola
y 
4(x  6)2
14; parabola     1; ellipse
4 12
15. 111  11x2
10y2 16. 120x2  119y2
118x  117y  0
11. 3x2
2y2  18x
20y
5  0 12. x2
y2
8x
2y
8  0 ellipse hyperbola
(y  5)2 (x  3)2 17. 3x2  4y2
12 18. 150  x2  120  y

  1; hyperbola (x  4)2  (y  1)2  9; circle
9 6
hyperbola parabola

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Answers
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8-6 Skills Practice 8-6 Practice (Average)

Conic Sections Conic Sections

Write each equation in standard form. State whether the graph of the equation is Write each equation in standard form. State whether the graph of the equation is
a parabola, circle, ellipse, or hyperbola. Then graph the equation. a 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. y2  3x 2. x2
y2
6x  7 3. 5x2  6y2  30x  12y  9

Glencoe/McGraw-Hill
x2 y2 x2 y2 parabola circle hyperbola

1 1 x2  y2  16
25 1 4 9

Lesson 8-6
1 2 (x
3)2 (y  1)2
y y y x 
 y (x  3)2  y2  16 
1
4 3 6 5
y y y
2

–8 –4 O 4 8x O x O x O x
–2
O x O x
–4

4. x2
8x
y2  9 circle 5. x2
2x  15  y parabola 6. 100x2
25y2  400

x2 y2 ellipse
(x  4)2  y2  25 y  (x  1)2
16 1 4. 196y2  1225  100x2 5. 3x2  9  3y2  6y 6. 9x2
y2
54x  6y  81
4 16
y y y ellipse circle ellipse
8
–8 –4 O 4 8x x2 y2 (x  3)2 (y
3)2
1 x2  (y  1)2 4 1
4 –4 12.25 6.25 1 9
Answers

y y y
–8
–8 –4 O 4 8x O x

A18
–4 –12

–8 –16
O x O x

O x

Without writing the equation in standard form, state whether the graph of each
equation is a parabola, circle, ellipse, or hyperbola.
(Lesson 8-6)

7. 9x2
4y2  36 ellipse 8. x2
y2  25 circle Without writing the equation in standard form, state whether the graph of each
equation is a parabola, circle, ellipse, or hyperbola.
9. y  x2
2x parabola 10. y  2x2  4x  4 parabola
7. 6x2
6y2  36 8. 4x2  y2  16 9. 9x2
16y2  64y  80  0
11. 4y2  25x2  100 hyperbola 12. 16x2
y2  16 ellipse
circle hyperbola ellipse
13. 16x2  4y2  64 hyperbola 14. 5x2
5y2  25 circle
10. 5x2
5y2  45  0 11. x2
2x  y 12. 4y2  36x2
4x  144  0
15. 25y2
9x2  225 ellipse 16. 36y2  4x2  144 hyperbola circle parabola hyperbola

17. y  4x2  36x  144 parabola 18. x2
y2  144  0 circle 13. ASTRONOMY A satellite travels in an hyperbolic orbit. It reaches the vertex of its orbit
2
at (5, 0) and then travels along a path that gets closer and closer to the line y   x.
19. (x
3)2
( y  1)2  4 circle 20. 25y2  50y
4x2  75 ellipse 5
Write an equation that describes the path of the satellite if the center of its hyperbolic
21. x2  6y2
9  0 hyperbola 22. x  y2
5y  6 parabola orbit is at (0, 0).
x2 y2
23. (x
5)2
y2  10 circle 24. 25x2
10y2  250  0 ellipse 
1
25 4

© Glencoe/McGraw-Hill 487 Glencoe Algebra 2 © Glencoe/McGraw-Hill 488 Glencoe Algebra 2

Glencoe Algebra 2
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-6 Reading to Learn Mathematics 8-6 Enrichment

Conic Sections
Pre-Activity How can you use a flashlight to make conic sections? Loci
Read the introduction to Lesson 8-6 at the top of page 449 in your textbook. A locus (plural, loci) is the set of all points, and only those points, that satisfy
The figures in the introduction show how a plane can slice a double cone to a given set of conditions. In geometry, figures often are defined as loci. For

Glencoe/McGraw-Hill
form the conic sections. Name the conic section that is formed if the plane example, a circle is the locus of points of a plane that are a given distance
from a given point. The definition leads naturally to an equation whose graph

Lesson 8-6
slices the double cone in each of the following ways:
is the curve described.
• The plane is parallel to the base of the double cone and slices through
one of the cones that form the double cone. circle
Example Write an equation of the locus of points that are the
• The plane is perpendicular to the base of the double cone and slices
same distance from (3, 4) and y 
4.
through both of the cones that form the double cone. hyperbola
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 units
Reading the Lesson is the distance from the vertex to both the focus and directrix.
1
1. Name the conic section that is the graph of each of the following equations. Give the Thus, an equation for the parabola is y   (x  3)2.
16
coordinates of the vertex if the conic section is a parabola and of the center if it is a The problem also may be approached analytically as follows:
circle, an ellipse, or a hyperbola.
Let (x, y) be a point of the locus.
(x  3)2 ( y
5)2
a. 
  1 ellipse; (3,
5) The distance from (3, 4) to (x, y)  the distance from y  4 to (x, y).
36 15

b. x  2( y
1)2 (x

  3
)2
( 
y  4)2  
(x  x
)2
(  4))2
y  (
Answers

7 parabola; (7,
1)

(x  3)2
y2  8y
16  y2
8y
16

A19
c. (x  5)2  ( y
5)2  1 hyperbola; (5,
5)
(x  3)2  16y
d. (x
6)2
( y  2)2  1 circle; (
6, 2) 1
 (x  3)2  y
16
2. Each of the following is the equation of a conic section. For each equation, identify the
values of A and C. Then, without writing the equation in standard form, state whether Describe each locus as a geometric figure. Then write an equation
the graph of each equation is a parabola, circle, ellipse, or hyperbola. for the locus.
1. All points that are the same distance from (0, 5) and (4, 5).
(Lesson 8-6)

a. 2x2
y2  6x
8y
12  0 A 2 ;C 1 ; type of graph: ellipse

0 ; type of graph: parabola line, x  2
b. 2x2
3x  2y  5  0 A 2 ;C
2. All points that are 4 units from the origin.
c. 5x2
10x
5y2  20y
1  0 A 5 ;C 5 ; type of graph: circle
circle, x 2  y 2  4
d. x2  y2
4x
2y  5  0 A 1 ; C 
1 ; type of graph: hyperbola
3. All points that are the same distance from (2, 1) and x  2.
Helping You Remember
1
parabola, x   (y 2  2y  1)
8
3. What is an easy way to recognize that an equation represents a parabola rather than 4. The locus of points such that the sum of the distances from (2, 0) and (2, 0) is 6.
one of the other conic sections?
x2 y2
If the equation has an x 2 term and y term but no y 2 term, then the graph ellipse,     1
9 5
is a parabola. Likewise, if the equation has a y 2 term and x term but no
5. The locus of points such that the absolute value of the difference of the distances
x 2 term, then the graph is a parabola.
from (3, 0) and (3, 0) is 2.
x2 y2
hyperbola, 
  1
1 8

© Glencoe/McGraw-Hill 489 Glencoe Algebra 2 © Glencoe/McGraw-Hill 490 Glencoe Algebra 2

Glencoe Algebra 2
Answers
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Study Guide and Intervention 8-7 Study Guide and Intervention (continued)

Solving Quadratic Systems Solving Quadratic Systems

Systems of Quadratic Equations Like systems of linear equations, systems of Systems of Quadratic Inequalities Systems of quadratic inequalities can be solved
quadratic equations can be solved by substitution and elimination. If the graphs are a conic by graphing.
section and a line, the system will have 0, 1, or 2 solutions. If the graphs are two conic
sections, the system will have 0, 1, 2, 3, or 4 solutions.
Example 1

Glencoe/McGraw-Hill
Solve the system of inequalities by graphing. y

Example x 2  y 2 25
Solve the system of equations. y  x 2
2x
15 5 2 25
x  y 
3  y2

x
2  4
Rewrite the second equation as y  x  3 and substitute into the first equation. The graph of x2
y2
25 consists of all points on or inside O x
x  3  x2  2x  15 the circle with center (0, 0) and radius 5.The graph of
5 2 25
0  x2  x  12 Add x
3 to each side. x  
y2   consists of all points on or outside the

2  4
0  (x  4)(x
3) Factor. 5 5
circle with center  , 0 and radius  . The solution of the
Use the Zero Product property to get 2
2

Lesson 8-7
x  4 or x  3. system is the set of points in both regions.

Substitute these values for x in x
y  3:

Example 2 Solve the system of inequalities by graphing. y
4
y  3 or 3
y  3
y  7 y0 x2  y2 25
y2 x2
The solutions are (4, 7) and (3, 0). 
 1
4 9
The graph of x2
y2
25 consists of all points on or inside
Answers

O x
the circle with center (0, 0) and radius 5.The graph of
Exercises

A20
y2 x2
    1 are the points “inside” but not on the branches of
4 9
Find the exact solution(s) of each system of equations. the hyperbola shown. The solution of the system is the set of
points in both regions.
1. y x2  5 2. x2
( y  5)2  25
y x  3 y  x2
(2,
1), (
1,
4) (0, 0) Exercises
(Lesson 8-7)

Solve each system of inequalities below by graphing.

x2 y2
3. x2
(y  5)2  25 4. x2
y2
9 1. 

1 2. x2
y2
169 3. y  (x  2)2
16 4
y  x2 x2
y3 1
x2
9y2  225 (x
1)2
( y
1)2
16
y  x  2
(0, 0), (3, 9), (
3, 9) (0, 3), (5

,
2), (
5

,
2) 2
y y y

12

5. x2  y2  1 6. y  x  3 6
x2
y2  16 x  y2  4
O x –12 –6 O 6 12 x O x
34 30 34 30
, 7  29 1  29


, –6

, 
, 
,


2 2   2 2   2 2 ,
–12
34 30 34 30

7
29 1
29


,

, 
,

,



 2 2   2 2   2 2 

© Glencoe/McGraw-Hill 491 Glencoe Algebra 2 © Glencoe/McGraw-Hill 492 Glencoe Algebra 2

Glencoe Algebra 2
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Skills Practice 8-7 Practice (Average)

Solving Quadratic Systems Solving Quadratic Systems

Find the exact solution(s) of each system of equations. 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) 1. (x  2)2
y2  5 2. x  2( y
1)2  6 3. y2  3x2  6 4. x2
2y2  1
y  x2  2 y  2x2 x  y2 xy1 x
y3 y  2x  1 y  x
1
(1.5, 4.5)

Glencoe/McGraw-Hill
1 2
(0,
1), (3, 2) (2, 1), (6.5,
3.5) (
1,
3), (5, 9) (1, 0),  ,
3 3
4. y  x (2
, 2
), 5. x  5 (
5, 0) 6. y  7 no solution
x2
y2  4 (
2 2 2 x2
y2  9 5. 4y2  9x2  36 6. y  x2  3 7. x2
y2  25 8. y2  10  6x2

,
2

) x
y  25 4x2  9y2  36 x2
y2  9 4y  3x 4y2  40  2x2

7. y  2x
2 (2,
2), 8. x  y
1  0 (1, 2) 9. y  2  x (0, 2), (3,
1)

no solution (0,
3), (5

, 2) (4, 3), (
4,
3) (0, 10

)
y2  2x 1 y2  4x y  x2  4x
2 x2 y2
2 , 1 9. x2
y2  25 10. 4x2
9y2  36 11. x  ( y  3)2
2 12.     1
9 16
x  3y  5 2x2  9y2  18 x  ( y  3)2
3
10. y  x  1 no solution 11. y  3x2 (0, 0) 12. y  x2
1 (
1, 2), x2
y2  9

Lesson 8-7
y  x2 y  3x2 y  x2
3 (1, 2) (
5, 0), (4, 3) (3, 0) no solution (3, 0)

13. 25x2
4y2  100 14. x2
y2  4 15. x2  y2  3

13. y  4x (
1,
4), (1, 4) 14. y  1 (0,
1) 15. 4x2
9y2  36 (
3, 0), 5 x2 y2 y2  x2  3
4x2
y2  20 4x2
y2  1 x2  9y2  9 (3, 0) x   
1
2 4 8
no solution (2, 0) no solution
16. 3( y
2)2  4(x  3)2  12 17. x2  4y2  4 (
2, 0), 18. y2  4x2  4 no x2 y2
Answers

y  2x
2 (0, 2), (3,
4) x2
y2  4 (2, 0) y  2x solution 16. 
  1 17. x
2y  3 18. x2
y2  64
7 7
x2
y2  9 x2  y2  8

A21
3x2  y2  9
9 12
(3, 0),
 ,
(2, 3
)  5 5  (6, 27
)
Solve each system of inequalities by graphing.

19. y
3x  2 20. y
x 21. 4y2
9x2  144 Solve each system of inequalities by graphing.
x2
y2  16 y  2x2
4 x2
8y2  16 ( y  3)2 (x
2)2
19. y  x2 20. x2
y2  36 21. 

1
16 4
y y y y  x
2 x2
y2  16
8
(Lesson 8-7)

(x
1)2
( y  2)2
4
y y y
4 8

O x O x O 4
–8 –4 4 8x
–4
–8 –4 O 4 8x
–8 –4
O x O x
–8

22. GARDENING An elliptical garden bed has a path from point A to y B

22. GEOMETRY The top of an iron gate is shaped like half an A
point B. If the bed can be modeled by the equation x2
3y2  12
1 ellipse with two congruent segments from the center of the
and the path can be modeled by the line y    x, what are the A ellipse to the ellipse as shown. Assume that the center of
3 (0, 0)
O
the ellipse is at (0, 0). If the ellipse can be modeled by the
coordinates of points A and B? (
3, 1) and (3,
1) x
B equation x2
4y2  4 for y  0 and the two congruent
3
 3 3

and 1, 
segments can be modeled by y   x and y    x, 
1, 
2   23

2 2
what are the coordinates of points A and B?

© Glencoe/McGraw-Hill 493 Glencoe Algebra 2 © Glencoe/McGraw-Hill 494 Glencoe Algebra 2

Glencoe Algebra 2
Answers
©
NAME ______________________________________________ DATE ____________ PERIOD _____ NAME ______________________________________________ DATE ____________ PERIOD _____

8-7 Reading to Learn Mathematics 8-7 Enrichment

Solving Quadratic Systems
Pre-Activity How do systems of equations apply to video games? Graphing Quadratic Equations with xy-Terms
Read the introduction to Lesson 8-7 at the top of page 455 in your textbook. You can use a graphing calculator to examine graphs of quadratic equations
The figure in your textbook shows that the spaceship hits the circular force that contain xy-terms.

Glencoe/McGraw-Hill
field in two points. Is it possible for the spaceship to hit the force field in
either fewer or more than two points? State all possibilities and explain
how these could happen. Sample answer: The spaceship could hit Example Use a graphing calculator to display y
the force field in zero points if the spaceship missed the force the graph of x2  xy  y2  4. 2
field all together. The spaceship could also hit the force field
in one point if the spaceship just touched the edge of the Solve the equation for y in terms of x by using the 1
force field. quadratic formula.
–2 –1 O 1 2 x
y2
xy
(x2  4)  0
–1
Reading the Lesson

Lesson 8-7
To use the formula, let a  1, b  x, and c  (x2  4).
–2
1. Draw a sketch to illustrate each of the following possibilities. x  
x2  4  4)
(1)(x2 
y  
2
a. a parabola and a line b. an ellipse and a circle c. a hyperbola and a
that intersect in that intersect in line that intersect in x   3x2
16  
y  
2 points 4 points 1 point 2

y y y
To graph the equation on the graphing calculator, enter the two equations:
Answers

x
 3x2
16   x   3x2
16  
y   and y  
O x O x O x 2 2

A22
2. Consider the following system of equations. Use a graphing calculator to graph each equation. State the type
of curve each graph represents.
x23
y2
2x2
3y2  11 1. y2
xy  8 2. x2
y2  2xy  x  0
(Lesson 8-7)

a. What kind of conic section is the graph of the first equation? hyperbola hyperbola parabola
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 solutions
that this system could have? 0, 1, 2, 3, or 4 3. x2  xy
y2  15 4. x2
xy
y2  9
ellipse graph is 
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? 5. 2x2  2xy  y2
4x  20 6. x2  xy  2y2
2x
5y  3  0
Sample answer: The solutions of an inequality of the form x 2  y 2  r 2 hyperbola two intersecting lines
are all points that are less than r units from the origin, so the graph is
the interior of the circle. The solutions of an inequality of the form
x 2  y 2  r 2 are the points that are more than r units from the origin, so
the graph is the exterior of the circle.

© Glencoe/McGraw-Hill 495 Glencoe Algebra 2 © Glencoe/McGraw-Hill 496 Glencoe Algebra 2

Glencoe Algebra 2
 Chapter 8 Assessment Answer Key
Form 1 Form 2A
Page 497 Page 498 Page 499

1. B 12. A 1. D

2. A
2. B
13. B
3. C

D 3. C
4.

14. B
4. A
5. A

15. D
5. B
6. B
16. C

17. A B
6.
7. C

Answers
8. C 7. B
18. D

D 19. C
9.
8. D

10. A
20. A C
9.

11. D a. 4; b. any positive

B: number except 4;
c. any negative
number; d. 0

(continued on the next page)

© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
 Chapter 8 Assessment Answer Key
Form 2A (continued) Form 2B
Page 500 Page 501 Page 502

10. D 1. C 10. A

2. A

11. A 11. C

3. C

12. D 12. B
4. D

13. A
5. C A
13.

14. C
B 14. B
6.
15. D

15. B
7. D
16. B
16. C
17. D
17. D
8. C

18. B
18. D

9. A

19. D 19. C

20. C 20. A

B: no solutions B: no solutions

© Glencoe/McGraw-Hill A24 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Form 2C
Page 503 Page 504
x2 y2
1.
6, 32 12.

  1
81 25
(y  2)2 (x  1)2
2.
61
 units 
  1
36 3
13.
x  1(y
4)2  1
12
3.
14. (5,
1), (1,
1);
(3  2
2 ,
1);
4. y  3(x
1)
1 y  1  (x
3)
2

(2, 4); 9, 4;

5. 4
y  4; x  7; right
4
(x  1)2  (y  1)2  25;
15. circle
(x  3)2 (y
1)2
6. (x  4)  (y
2)  16     1;
2 2
16. 9 4
ellipse

7. y

O x
17. parabola; C  0
hyperbola; A  4,
18. C 
4

19. (4, 0), (1,
3)

Answers
y

8. y
(4, 0)
O x

O x
(1,
3)

20. y

(x  1)2 (y
3)2
    1 O x
9. 100 25

(y
5)2 (x  2)2
    1
10. 36 16

B: (x
3)  (y  1)  38
2 2

,
2), (

2
11. (
2 ,
2)

© Glencoe/McGraw-Hill A25 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Form 2D
Page 505 Page 506
y2 x2
1.
32, 1 12.

  1
144 16

202
 units (x  3)2 (y
1)2
2. 
  1
13. 49 21
y  1(x  1)2
3
3. 16
14. (1, 3), (
3, 3);
(
1  2
2 , 3);
4. y
3  (x  1)

(1, 3); 1, ,

25
5. 8
23
x  1; y   ; y  4(x
2)2  5;
8 15. parabola
upward
(y
3)2 (x  1)2

  1;
 
2
x
1  (y  2)2  4 16. 100 25
2
6. hyperbola

7. y
17.
O x
18. ellipse; A  10; C  3

19. no solution
y

8. y

O x

O x

20. y

(y
2)2 (x
2)2
    1
9. 49 25 O x

(x
1)2 (y  4)2
    1
10. 64 9

B: (x
5)2  (y  2)2  66
,
3), (

5
11. (
5 ,
3)

© Glencoe/McGraw-Hill A26 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Form 3
Page 507 Page 508

1. (
3.45, 4.15) (2, 3); (2, 2),

12. (2, 4); 2
6 ; 2
5 
2.
166
 units (y  2)2 (x
4)2

  1
13. 9 16
3. x 
(y
3)2  2
x 
1 (y
1)2
5
4. 6

2

2(x
1)2  3 y  4  
67
;
3  3
y 
1(x
80)2  40 14. ellipse
5. 160

 2
2
x 
8 y  1
94;
15.
parabola

(10,
1); ,
1,

39
4
41
y 
1; x  ;
6. 4 circle; A  C  3
16. 2

17. parabola; A  0, C  4
(x
3)2  (y
5)2  25
7. 18. no solution

Answers
8. y

19. (
1, 3), (5, 3), (2, 0)

O x y

(–1, 3) (5, 3)

9. y
O (2, 0) x

O x
20. y

x
y  12
2 O
(x
5)2
 
 1
10. 49 25

(y  2)2 (x
3)2 (
2.5, 0.725)

    1 B:
11. 36 16

© Glencoe/McGraw-Hill A27 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Page 509, Open-Ended Assessment
Scoring Rubric

Score General Description Specific Criteria

4 Superior • Shows thorough understanding of the concepts of

A correct solution that identifying, graphing, and writing equations of conic
is supported by well- sections, and solving systems of quadratic equations and
developed, accurate inequalities.
explanations • Uses appropriate strategies to solve problems.
• Computations are correct.
• Written explanations are exemplary.
• Goes beyond requirements of some or all problems.

3 Satisfactory • Shows an understanding of the concepts of identifying,

A generally correct solution, graphing, and writing equations of conic sections, and
but may contain minor flaws solving systems of quadratic equations and inequalities.
in reasoning or computation • Uses appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are effective.
• Satisfies all requirements of problems.

2 Nearly Satisfactory • Shows an understanding of most of the concepts of

A partially correct identifying, graphing, and writing equations of conic
interpretation and/or sections, and solving systems of quadratic equations and
solution to the problem inequalities.
• May not use appropriate strategies to solve problems.
• Computations are mostly correct.
• Written explanations are satisfactory.
• Satisfies the requirements of most of the problems.
1 Nearly Unsatisfactory • Final computation is correct.
A correct solution with no • No written explanations or work is shown to substantiate
supporting evidence or the final computation.
explanation • Satisfies minimal requirements of some of the problems.

0 Unsatisfactory • Shows little or no understanding of most of the concepts of

An incorrect solution identifying, graphing, and writing equations of conic
indicating no mathematical sections, and solving systems of quadratic equations and
understanding of the inequalities.
concept or task, or no • Does not use appropriate strategies to solve problems.
solution is given • Computations are incorrect.
• Written explanations are unsatisfactory.
• Does not satisfy requirements of problems.
• No answer may be given.

© Glencoe/McGraw-Hill A28 Glencoe Algebra 2

 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.

1a. The coefficients of the quadratic terms 4a. (x  4)2  (y  3)2  4

are the same number and have the y  (x  4)2  3
same sign (A  C  1). Region 2 is the intersection of the region
1b. The radius would be the square root of a inside the circle, including its boundary
negative number (5). () and the region above the parabola,
1c. Student responses should indicate that not including its boundary ().
the constant in the original equation 4b. (x  4)2  (y  3)2  4
should be changed to a number less y  (x  4)2  3
than 25, or that the constant obtained Region 3 is the intersection of the region
when the equation is written in outside the circle, including its
standard form should be changed to a boundary () and the region below the
positive number, or that one or both of parabola, not including its boundary
the coefficients of the linear terms, x ().
and y, must be changed to a number 4c. Region 1 is the intersection of the region
sufficiently large to result in a positive outside the circle, including its
number on the right side of the boundary () and the region above the
standard form of the equation. Sample parabola, not including its boundary
answer: Change the constant in the ().
original equation to 24. The center of
the circle is (4, 3) and the radius is 1 5a. Students must select both values such
unit. that 5  k  1 so that the graph of the
horizontal line y  k will intersect the
2. The graphs of circles, ellipses, graph of the ellipse twice.

Answers
hyperbolas, and parabolas that open to 5b. Students may select only k  1 and
the left and right never represent k  5, the equations of the only two
relations that are functions. Of all the horizontal lines that are tangent to the
conic sections studied in this chapter, ellipse, each intersecting the ellipse in
only parabolas that open upward or exactly one point.
downward have graphs which pass the
vertical line test and are therefore 5c. Students must select both values such
functions. that k  1 or k  5 so that the graph
of the horizontal line y  k will not
3. The parabolas share the same vertex. intersect the graph of the ellipse.
Sample answer: The graph of
y  (x  2)2  1 opens upward while the
graph of x  (y  1)2  2 opens to the
right.

© Glencoe/McGraw-Hill A29 Glencoe Algebra 2

 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
1. parabola; focus; (x
1)2 (y
2)2
directrix 1.

1, 72 1.

  1
4 4
(y  1)2 (x
2)2

  1
2. ellipse; 2. 16 9

foci of the ellipse

, 0);
(2, 0); (2
10
3. hyperbola y  3x
2. A 3.
y
4. minor axis; y  1(x
1)2  1
major axis 3. 12
O x
5. transverse axis
4.
39
(
4, 5);
4, ;
8  
41 y  2(x
3)2
23;
6. tangent x 
4; y   ;
8 parabola
downward 4.
7. latus rectum
1
 unit 5. ellipse; A  1, C  4
8. asymptote 5. 6
y
9. conjugate axis O x

10. distance formula

11. Sample answer:

A circle is the set of
all points in a plane
that are the same Quiz (Lesson 8–7)
distance from a Page 512
given point, which Quiz (Lessons 8–3 and 8–4)
is the center. Page 511
1. (1,
2), (3,
2)
12. Sample answer: y
A vertex of a 1. (x  7)2  (y
2)2  81
hyperbola is the O x
point on a branch 2. (x
3)2  (y
1)2  16
(1,
2) (3,
2)
of the hyperbola (1,
1); 3 units
that is closest to 3.
the center of the y

hyperbola.
O x 2.

73,
131, (1, 3)
(1, 2), (1,
2),
3. (
1, 2), (
1,
2)
x2 (y
1)2
    1 4.
4. 25 16 y

2
5. (3, 0); (3  2
3, 0); 8; 4
O 2 x

© Glencoe/McGraw-Hill A30 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Mid-Chapter Test Cumulative Review
Page 513 Page 514

1. A 1.
22.5

D 2.
2.

3. (
1,
1)

3. B

4.

47 9 16
8 5 

4. C
x
4

5. x3

6.
5. A
1, 5
7. 2

6. y 8. {x  x 
3 or x  1}

9. even; 4

Answers
O x

10.
(x
4)2 (y
1)2
    1
7. 25 16

11.

12.

4, 52
8. y 
3(x
3)  32
2

x 
1y2  1
(x
3)2  (y
4)2  50 13. 20
9.

14. (0, 0); (0, 2
2

); 6; 2

(x  3)2 (y
1)2
    1;
9 4
15. ellipse

© Glencoe/McGraw-Hill A31 Glencoe Algebra 2

 Chapter 8 Assessment Answer Key
Standardized Test Practice
Page 515 Page 516
1. A B C D 11. 12.
4 1 2 . 5
/ / / /
. . . . . . . .
0 0 0 0 0 0
2. E F G H
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
3. A B C D
6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9

4. E F G H 13. 14.
9 . 0 1 4 3 / 4
/ / / /
. . . . . . . .
0 0 0 0 0 0
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
5. A B C D
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9
6. E F G H

7. A B C D

15. A B C D

16. A B C D
8. E F G H

17. A B C D

9. A B C D
18. A B C D

10. E F G H

© Glencoe/McGraw-Hill A32 Glencoe Algebra 2