Chapter 8 A1 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Anticipation Guide and Lesson 8-1)

8-1

Less

on R

eadi

ng G

uide

Mu

ltip

lyin

g a

nd

Div

idin

g R

atio

nal

Exp

ress

ion

s

Cha

pter

85

Gle

ncoe

Alg

ebra

2

Lesson 8-1

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-1 i

n y

our

text

boo

k.

•S

upp

ose

that

th

e G

oodi

e S

hop

pe a

lso

sell

s a

can

dy m

ixtu

re m

ade

wit

h

4 po

un

ds o

f ch

ocol

ate

min

ts a

nd

3 po

un

ds o

f ca

ram

els,

then

of t

he

mix

ture

is

min

ts a

nd

of t

he

mix

ture

is

cara

mel

s.

•If

th

e st

ore

man

ager

add

s an

oth

er y

pou

nds

of

min

ts t

o th

e m

ixtu

re,w

hat

fra

ctio

n o

f th

em

ixtu

re w

ill

be m

ints

?

Rea

d t

he

Less

on

1.a.

In o

rder

to

sim

plif

y a

rati

onal

nu

mbe

r or

rat

ion

al e

xpre

ssio

n,

the

nu

mer

ator

an

d an

d di

vide

bot

h o

f th

em b

y th

eir

.

b.

A r

atio

nal e

xpre

ssio

n is

und

efin

ed w

hen

its

is e

qual

to

.

To

fin

d th

e va

lues

th

at m

ake

the

expr

essi

on u

nde

fin

ed,c

ompl

etel

y

the

orig

inal

an

d se

t ea

ch f

acto

r eq

ual

to

.

2.a.

To

mu

ltip

ly t

wo

rati

onal

exp

ress

ion

s,th

e an

dm

ult

iply

th

e de

nom

inat

ors.

b.

To

divi

de t

wo

rati

onal

exp

ress

ion

s,by

th

e of

the

.

3.a.

Wh

ich

of

the

foll

owin

g ex

pres

sion

s ar

e co

mpl

ex f

ract

ion

s?ii,

iv,v

i.ii

.ii

i.iv

.v.

b.

Doe

s a

com

plex

fra

ctio

n e

xpre

ss a

mu

ltip

lica

tion

or

divi

sion

pro

blem

?d

ivis

ion

How

is

mu

ltip

lica

tion

use

d in

sim

plif

yin

g a

com

plex

fra

ctio

n?

Sam

ple

an

swer

:To

div

ide

the

nu

mer

ato

r o

f th

e co

mp

lex

frac

tio

n b

y th

e d

eno

min

ato

r,m

ult

iply

th

e n

um

erat

or

by t

he

reci

pro

cal o

f th

e d

eno

min

ato

r.

Rem

emb

er W

hat

Yo

u L

earn

ed

4.O

ne

way

to

rem

embe

r so

met

hin

g n

ew i

s to

see

how

it

is s

imil

ar t

o so

met

hin

g yo

ual

read

y kn

ow.H

ow c

an y

our

know

ledg

e of

div

isio

n o

f fr

acti

ons

in a

rith

met

ic h

elp

you

to

un

ders

tan

d h

ow t

o di

vide

rat

ion

al e

xpre

ssio

ns?

Sam

ple

an

swer

:To

div

ide

rati

on

alex

pre

ssio

ns,

mu

ltip

ly t

he

firs

t ex

pre

ssio

n b

y th

e re

cip

roca

l of

the

seco

nd

.Th

is is

th

e sa

me

“inv

ert

and

mu

ltip

ly”

pro

cess

th

at is

use

d w

hen

div

idin

g a

rith

met

ic f

ract

ion

s.

�r2� 9

25�

� �r� 3

5�

�z� z

1�

�z

r�

5� r

�5

�3 8�

� � 15 6�

7 � 12

div

iso

rre

cip

roca

lm

ult

iply

nu

mer

ato

rsm

ult

iply

0d

eno

min

ato

rfa

cto

r0d

eno

min

ato

rg

reat

est

com

mo

n f

acto

rd

eno

min

ato

rfa

cto

r

4 �

y� 7

�y

�3 7�

�4 7�

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

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____

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__D

ATE

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__P

ER

IOD

____

_

8A

ntic

ipat

ion

Gui

deR

atio

nal

Exp

ress

ion

s an

d E

qu

atio

ns

Cha

pter

83

Gle

ncoe

Alg

ebra

2

Bef

ore

you

beg

in C

ha

pte

r 8

•R

ead

each

sta

tem

ent.

•D

ecid

e w

het

her

you

Agr

ee (

A)

or D

isag

ree

(D)

wit

h t

he

stat

emen

t.

•W

rite

A o

r D

in

th

e fi

rst

colu

mn

OR

if

you

are

not

su

re w

het

her

you

agr

ee o

r di

sagr

ee,

wri

te N

S(N

ot S

ure

).

Aft

er y

ou c

omp

lete

Ch

ap

ter

8

•R

erea

d ea

ch s

tate

men

t an

d co

mpl

ete

the

last

col

um

n b

y en

teri

ng

an A

or

a D

.

•D

id a

ny

of y

our

opin

ion

s ab

out

the

stat

emen

ts c

han

ge f

rom

th

e fi

rst

colu

mn

?

•F

or t

hos

e st

atem

ents

th

at y

ou m

ark

wit

h a

D,u

se a

pie

ce o

f pa

per

to w

rite

an

exa

mpl

e of

wh

y yo

u d

isag

ree.

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Step

1

ST

EP

1S

TE

P 2

A,D

,or

NS

Sta

tem

ent

A o

r D

1.S

ince

a d

enom

inat

or c

ann

ot e

qual

0,t

he

expr

essi

on

Ais

un

defi

ned

for

x�

�5.

2.T

o di

vide

tw

o ra

tion

al e

xpre

ssio

ns,

mu

ltip

ly b

y th

e re

cipr

ocal

of

th

e di

viso

r.A

3.T

he

leas

t co

mm

on m

ult

iple

of

thre

e m

onom

ials

is

fou

nd

by

mu

ltip

lyin

g th

e m

onom

ials

tog

eth

er.

D

4.B

efor

e ad

din

g tw

o ra

tion

al e

xpre

ssio

ns,

a co

mm

on

den

omin

ator

mu

st b

e fo

un

d.A

5.T

he

grap

h o

f a

rati

onal

fu

nct

ion

con

tain

ing

an a

sym

ptot

e w

ill

be s

ymm

etri

c ov

er t

he

asym

ptot

e.D

6.S

ince

f(x

) �

can

be

sim

plifi

ed t

o f(

x) �

m�

2,

the

grap

hof

f(x)

wil

lbe

the

stra

igh

tli

ne

defi

ned

byy

�m

�2.

D

7.y

�kx

yzis

an

exa

mpl

e of

a jo

int

vari

atio

n i

f k,

x,y,

and

zar

e al

l n

ot e

qual

to

0.A

8.T

he

shap

e of

th

e gr

aph

of

y�

�3x

2�

2x�

4 ca

n o

nly

be

dete

rmin

ed b

y gr

aph

ing

the

fun

ctio

n.

D

9.B

ecau

se t

he

grap

h o

f an

abs

olu

te v

alu

e fu

nct

ion

is

in t

he

shap

e of

a V

,th

e gr

aph

of

y�

�x��

4 w

ill

also

be

in t

he

shap

e A

of a

V.

10.

Wh

en s

olvi

ng

rati

onal

equ

atio

ns,

solu

tion

s th

at r

esu

lt i

n a

ze

ro i

n t

he

den

omin

ator

mu

st b

e ex

clu

ded.

A

(m�

4)(m

�2)

��

m�

4

3x2(x

�1)

��

x�

5

Step

2

Chapter Resources

Chapter 8 A2 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Mu

ltip

lyin

g a

nd

Div

idin

g R

atio

nal

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Cha

pter

87

Gle

ncoe

Alg

ebra

2

Lesson 8-1

Sim

plif

y C

om

ple

x Fr

acti

on

sA

com

ple

x fr

acti

onis

a r

atio

nal

exp

ress

ion

wh

ose

nu

mer

ator

an

d/or

den

omin

ator

con

tain

s a

rati

onal

exp

ress

ion

.To

sim

plif

y a

com

plex

frac

tion

,fir

st r

ewri

te i

t as

a d

ivis

ion

pro

blem

.

Sim

pli

fy

.

��

Exp

ress

as

a di

visi

on p

robl

em.

��

Mul

tiply

by

the

reci

proc

al o

f th

e di

viso

r.

�Fa

ctor

.

�S

impl

ify.

Sim

pli

fy.

1.2.

3.(b

�1)

2

4.5.

6.a

�4

7.x

�3

8.9.

1� x

�5

x2�

x�

2�

��

x3�

6x2

�x

�30

��

�x

�1

� x�

3

b�

4�

�(b

�1)

(b�

2)

� b2�b

� 6b2 �

8�

��

�b2

b2��b

1� 62

�

�2x2

x��9x

1�9

�

��

�10 5x x2 2

� �1 79 xx �

�26

�

�a a2� �

1 26�

��

�a a2 2� �3 aa ��

24�

1�

�(x

�3)

(x�

2)

� x2�x

� 6x4 �

9�

��

�x2� 3

�2xx�

8�

2(b

�10

)�

�b

(3b

�1)

�b2� b210

0�

��

�3b2

�3 21 bb

�10

�

� 3b b2� �

1 2�

��

� 3b2b �

�b1 �

2�

ac7

� by

�a x2 2b yc 23�

� � ca 4 xb 22 y�

xyz

� a5

�x a3 2y b2 2z�

� �a3

bx 22 y �

s3� s

�3(3s

�1)

s4�

�s(

3s�

1)(s

�3)s4

��

3s2

�8s

�3

3s�

1�

s

3s2

�8s

�3

�� s4

3s�

1�

s

�3ss�

1�

��

�3s2

�s8 4

s�

3�

�3ss�

1�

��

�3s2

�

s8 4s

�3

�

1

11

s3

8-1

Exer

cise

s

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

8-1

Cha

pter

86

Gle

ncoe

Alg

ebra

2

Sim

plif

y R

atio

nal

Exp

ress

ion

sA

rat

io o

f tw

o po

lyn

omia

l ex

pres

sion

s is

a r

atio

nal

exp

ress

ion

.To

sim

plif

y a

rati

onal

exp

ress

ion

,div

ide

both

th

e n

um

erat

or a

nd

the

den

omin

ator

by

thei

r gr

eate

st c

omm

on f

acto

r (G

CF

).

Mu

ltip

lyin

g R

atio

nal

Exp

ress

ion

sFo

r al

l rat

iona

l exp

ress

ions

an

d ,

��

, if b

�0

and

d�

0.

Div

idin

g R

atio

nal

Exp

ress

ion

sFo

r al

l rat

iona

l exp

ress

ions

an

d ,

��

, if b

�0,

c�

0, a

nd d

�0.

Sim

pli

fy e

ach

exp

ress

ion

.

a.

��

b.

� ��

��

c.� �

��

�

�

Sim

pli

fy e

ach

exp

ress

ion

.

1.�(�

22 0a ab b2 4)3�

�2.

3.

4.�

2m2 (

m�

1)5.

�

6.�

m7.

�

8.�

9.�

4�

�(2

m�

1)(m

�5)

p(4

p�

1)�

�2(

p�

2)

4m2

�1

��

4m�

82m

�1

��

m2

�3m

�10

4p2

�7p

�2

��

7p5

16p2

�8p

�1

��

14p4

y5

� 15z5

18xz

2�

5y6x

y4� 25

z3m

3�

9m�

�m

2�

9(m

�3)

2�

�m

2�

6m�

9

c� c

�5

c2�

4c�

5�

�c2

�4c

�3

c2�

3c� c2

�25

4m5

� m�

13m

3�

3m�

�6m

4

x�

2� x

�9

x2�

x�

6�

�x2

�6x

�27

3 �

2x�

34x

2�

12x

�9

��

9 �

6x2a

2 b2

�5

x�

4� 2(

x�

2)(x

�4)

(x�

4)(x

�1)

��

�2(

x�

1)(x

�2)

(x�

4)x�

1�

�x2

�2x

�8

x2�

8x�

16�

�2x

�2

x2�

2x�

8�

�x

�1

x2�

8x�

16�

�2x

�2

x2�

2x�

8�

�x

�1

x2�

8x�

16�

�2x

�2

4s2

� 3rt2

2 �

2 �

s�

s�

�3

�r

�t

�t

3 �

r�

r�

s�

s�

s�

2 �

2 �

5 �

t�

t�

��

�5

�t

�t

�t

�t

�3

�3

�r

�r

�r

�s

20t2

� 9r3 s

3r2 s

3�

5t4

20t2

� 9r3 s

3r2 s

3�

5t4

3a � 2b2

2 �

2 �

2 �

3 �

a�

a�

a�

a�

a�

b�

b�

��

��

2 �

2 �

2 �

2 �

a�

a�

a�

a�

b�

b�

b�

b24

a5 b2

� (2ab

)4

24a

5 b2

� (2a

b)4

ad � bcc � d

a � bc � d

a � b

ac � bdc � d

a � bc � d

a � b

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Stud

y G

uide

and

Inte

rven

tion

Mu

ltip

lyin

g a

nd

Div

idin

g R

atio

nal

Exp

ress

ion

s

11

11

11

11

11

1

11

11

11

1

11

11

11

11

1

11

11

11

11

1

Exer

cise

s

Exam

ple

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-1)

Chapter 8 A3 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-1)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Prac

tice

Mu

ltip

lyin

g a

nd

Div

idin

g R

atio

nal

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-1

Cha

pter

89

Gle

ncoe

Alg

ebra

2

Lesson 8-1

Sim

pli

fy e

ach

exp

ress

ion

.

1.2.

�3.

4.5.

�

6.7.

��

8.�

9.�

10.

�n

�w

11.

��

12.

�13

.�

14. �

�3�

15.

�

16.

�17

.�

18.

��

19.

20.

�2(

x�

3)21

.

22.G

EOM

ETRY

A r

igh

t tr

ian

gle

wit

h a

n a

rea

of x

2�

4 sq

uar

e u

nit

s h

as a

leg

th

atm

easu

res

2x�

4 u

nit

s.D

eter

min

e th

e le

ngt

h o

f th

e ot

her

leg

of

the

tria

ngl

e.x

�2

un

its

23.G

EOM

ETRY

A r

ecta

ngu

lar

pyra

mid

has

a b

ase

area

of

squ

are

cen

tim

eter

s

and

a h

eigh

t of

ce

nti

met

ers.

Wri

te a

rat

ion

al e

xpre

ssio

n t

o de

scri

be t

he

volu

me

of t

he

rect

angu

lar

pyra

mid

.cm

3x

� 5

�6

x2�

3x�

�x2

�5x

�6

x2�

3x�

10�

� 2x

x2

�2x

�4

��

x(x

�2)

� xx 23

��22 x3

�

��

� x2( �x

� 4x2 �)3

4�

�x2

4�9

�

� �3� 8

x�

2x�

1� 4

�x

�2xx�

1�

� �4� x

x�

5 � 22a

�6

� 5a�

109

�a2

��

a2�

5a�

6

2s�

3�

�(s

�4)

(s�

5)s2

�10

s�

25�

�s

�4

2s2

�7s

�15

��

(s�

4)2

2�

�x(

x�

3)6x

2�

12x

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8-1

Chapter 8 A4 Glencoe Algebra 2

8-2

8-1

Enri

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Dim

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on

al A

nal

ysis

Cha

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811

Gle

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2

Lesson 8-1

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the

un

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r so

luti

on.I

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ambi

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sta

te a

sol

uti

on r

egar

din

g di

stan

ce a

s 17

;S

even

teen

wh

at,f

eet,

mil

es,m

eter

s? O

ften

it

is h

elpf

ul

to a

nal

yze

the

un

its

of t

he

quan

titi

es i

n a

for

mu

la t

o de

term

ine

the

desi

red

un

its

of a

n o

utp

ut.

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exa

mpl

e,it

is

know

n t

hat

tor

que

is t

he

prod

uct

of

forc

e an

d di

stan

ce,b

ut

wh

at a

re t

he

un

its

of f

orce

?

Th

e u

nit

s al

so d

epen

d on

th

e m

easu

rin

g sy

stem

.Th

e tw

o m

ost

com

mon

lyu

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syst

ems

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the

Bri

tish

sys

tem

an

d th

e in

tern

atio

nal

sys

tem

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its

(SI)

.Som

e co

mm

on u

nit

s of

th

e B

riti

sh s

yste

m a

re i

nch

es,f

eet,

mil

es,

and

pou

nds

.Com

mon

SI

un

its

incl

ude

met

ers,

kilo

met

ers,

New

ton

s,an

dgr

ams.

Fre

quen

tly

con

vers

ion

fro

m o

ne

syst

em t

o an

oth

er i

s n

eces

sary

an

d ac

com

plis

hed

by

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ltip

lica

tion

by

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vers

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fac

tors

.

Con

side

r ch

angi

ng

un

its

from

mil

es p

er h

our

to k

ilom

eter

s pe

r h

our.

Wh

at i

s60

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es p

er h

our

in k

ilom

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s pe

r h

our?

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th

e co

nve

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n 1

ft

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e B

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sin

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1.JE

LLY

BEA

NS

A l

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atio

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inth

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t?

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GE

Bet

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car

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mil

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in

th

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ty a

nd

26 m

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hig

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eth

use

s C

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ons

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th

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ty a

nd

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llon

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gas

on

the

hig

hw

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um

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gal

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Bet

h g

ets

wit

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ar i

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s of

C

and

H.

3.H

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Th

e fr

ont

face

of

a N

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ouse

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lar.

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rfac

e ar

ea

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face

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x2�

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NN

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an

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the

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atio

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old

run

s to

th

e lo

cal

food

mar

t to

bu

y a

gall

on o

f so

y m

ilk.

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ause

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eigh

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wn

on

his

ret

urn

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p,h

e ru

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er o

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ay b

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sec

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on t

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the

food

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t an

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2fe

et

per

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nd

on t

he

way

bac

k.L

et d

be t

he

dist

ance

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to t

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mar

t.R

emem

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dist

ance

�ra

te �

tim

e.

5.W

rite

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atio

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tota

lti

me

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t ru

nn

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for

this

erra

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t�

�

6.W

hat

spe

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ould

Har

old

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ru

n

if h

e w

ante

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mai

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in a

con

stan

tsp

eed

for

the

enti

re t

rip

yet

take

th

esa

me

amou

nt

of t

ime

run

nin

g?

2S1S

2� S

1�

S2

d � S2

d � S1V � �r2

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-1)

Chapter 8 A5 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

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, a

divi

sion

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cGra

w-H

ill C

ompa

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, In

c.

Answers (Lesson 8-2)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exam

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Stud

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

812

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

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ER

IOD

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Less

on R

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xpre

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for

the

Less

on

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d t

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intr

odu

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n t

o L

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n 8

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n y

our

text

boo

k.

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n i

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andi

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5 fe

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len

s w

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gth

of

3 fe

et.

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te a

n e

quat

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at y

ou c

ould

sol

ve t

o fi

nd

how

far

th

e fi

lm s

hou

ld b

e fr

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len

s to

get

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on

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In w

ork

wit

h r

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s,L

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nds

for

and

LC

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tan

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or

.Th

e L

CD

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b.

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fin

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e L

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2.T

o ad

d an

d ,y

ou s

hou

ld f

irst

fac

tor

the

of

each

fra

ctio

n.T

hen

use

th

e fa

ctor

izat

ion

s to

fin

d th

e of

x2

�5x

�6

and

x3�

4x2

�4x

.Th

is i

s th

e fo

r th

e tw

o fr

acti

ons.

3.W

hen

you

add

or

subt

ract

fra

ctio

ns,

you

oft

en n

eed

to r

ewri

te t

he

frac

tion

s as

equ

ival

ent

frac

tion

s.Yo

u d

o th

is s

o th

at t

he

resu

ltin

g eq

uiv

alen

t fr

acti

ons

wil

l ea

ch h

ave

a

den

omin

ator

equ

al t

o th

e of

th

e or

igin

al f

ract

ion

s.

4.T

o ad

d or

su

btra

ct t

wo

frac

tion

s th

at h

ave

the

sam

e de

nom

inat

or,y

ou a

dd o

r su

btra

ct

thei

r an

d ke

ep t

he

sam

e .

5.T

he

sum

or

diff

eren

ce o

f tw

o ra

tion

al e

xpre

ssio

ns

shou

ld b

e w

ritt

en a

s a

poly

nom

ial

or

as a

fra

ctio

n i

n

.

Rem

emb

er W

hat

Yo

u L

earn

ed

6.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g w

het

her

a c

omm

on d

enom

inat

or i

s n

eede

d to

add

and

subt

ract

rat

ion

al e

xpre

ssio

ns

or t

o m

ult

iply

an

d di

vide

th

em.H

ow c

an y

our

know

ledg

e of

wor

kin

g w

ith

fra

ctio

ns

in a

rith

met

ic h

elp

you

rem

embe

r th

is?

Sam

ple

an

swer

:In

ari

thm

etic

,a c

om

mo

n d

eno

min

ato

r is

nee

ded

to

ad

dan

d s

ub

trac

t fr

acti

on

s,bu

t n

ot

to m

ult

iply

an

d d

ivid

e th

em.T

he

situ

atio

nis

th

e sa

me

for

rati

on

al e

xpre

ssio

ns.

sim

ple

st f

orm

den

om

inat

or

nu

mer

ato

rs

LC

D

LC

D

LC

M

den

om

inat

or

x�

4�

�x3

�4x

2�

4xx2

�3

��

x2�

5x�

6

fact

or

gre

ates

tfa

cto

rp

oly

no

mia

lfa

cto

r

LC

Mle

ast

com

mo

n m

ult

iple

leas

t co

mm

on

den

om

inat

or

1 � 51 � 3

1 � q

Chapter 8 A6 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Skill

s Pr

acti

ceA

dd

ing

an

d S

ub

trac

tin

g R

atio

nal

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Cha

pter

815

Gle

ncoe

Alg

ebra

2

Lesson 8-2

Fin

d t

he

LC

M o

f ea

ch s

et o

f p

olyn

omia

ls.

1.12

c,6c

2 d12

c2 d

2.18

a3bc

2 ,24

b2c2

72a

3 b2 c

2

3.2x

�6,

x�

32(

x�

3)4.

5a,a

�1

5a(a

�1)

5.t2

�25

,t�

5(t

�5)

(t�

5)6.

x2�

3x�

4,x

�1

(x�

4)(x

�1)

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pli

fy e

ach

exp

ress

ion

.

7.�

8.�

9.�

410

.�

11.

�12

.�

13.

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.�

15.

�16

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17.

�18

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19.

�20

.�

21.

�22

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y�

12�

��

(y�

4)(y

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(y�

2)n

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3

2�

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3�

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2�

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1)2

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4)(z

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5 �

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h2

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27

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��

5y2 z

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2

2 �

5m2

��

m2 n

5 � n2

� m2 n

2c�

5�

32c

�7

�3

13� 8p

2 q5

� 4p2 q

3� 8p

2 q5x

�3y

�xy

5 � y3 � x8-2

Lesson 8-2

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Ad

din

g a

nd

Su

btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

Cha

pter

814

Gle

ncoe

Alg

ebra

2

Ad

d a

nd

Su

btr

act

Rat

ion

al E

xpre

ssio

ns

To

add

or s

ubt

ract

rat

ion

al e

xpre

ssio

ns,

foll

ow t

hes

e st

eps.

Ste

p 1

If ne

cess

ary,

fin

d eq

uiva

lent

fra

ctio

ns t

hat

have

the

sam

e de

nom

inat

or.

Ste

p 2

Add

or

subt

ract

the

num

erat

ors.

Ste

p 3

Com

bine

any

like

ter

ms

in t

he n

umer

ator

.S

tep

4Fa

ctor

if p

ossi

ble.

Ste

p 5

Sim

plify

if p

ossi

ble.

Sim

pli

fy

�.

�

��

Fact

or t

he d

enom

inat

ors.

��

The

LC

D is

2(x

�3)

(x�

2)(x

�2)

.

�S

ubtr

act

the

num

erat

ors.

�D

istr

ibut

ive

Pro

pert

y

�C

ombi

ne li

ke t

erm

s.

�S

impl

ify.

Sim

pli

fy e

ach

exp

ress

ion

.

1.�

�2.

�

3.�

4.�

5.�

6.�

�2x

2�

9x�

4�

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x�

1)(2

x�

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x2�

54

��

4x2

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4� x

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x�

1� x2

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3x�

3�

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4x�

14� 3x

�6

4x�

5� 3x

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3� x

�2

4a2

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2�

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bc

15b

� 5ac

4a � 3bc

x�

1�

�(x

�1)

(x�

3)1

� x�

12

� x�

3y � 3

4y2

� 2y�

7xy

�3x

x�

��

(x�

3)(x

�2)

(x�

2)

2x�

��

2(x

�3)

(x�

2)(x

�2)

6x�

12 �

4x�

12�

��

2(x

�3)

(x�

2)(x

�2)

6(x

�2)

�4(

x�

3)�

��

2(x

�3)

(x�

2)(x

�2)

2 �

2(x

�3)

��

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x�

3)(x

�2)

(x�

2)6(

x�

2)�

��

2(x

�3)

(x�

2)(x

�2)

2�

�(x

�2)

(x�

2)6

��

2(x

�3)

(x�

2)

2� x2

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6�

�2x

2�

2x�

12

2� x2

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6�

�2x

2�

2x�

12

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-2

Exer

cise

s

Exam

ple

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-2)

Chapter 8 A7 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-2)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Wor

d Pr

oble

m P

ract

ice

Ad

din

g a

nd

Su

btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-2

Cha

pter

817

Gle

ncoe

Alg

ebra

2

Lesson 8-2

1.SQ

UA

RES

Su

san

’s f

avor

ite

perf

ect

squ

are

is s

2an

d T

ravi

s’ i

s t2

,wh

ere

san

d t

are

wh

ole

nu

mbe

rs.W

hat

per

fect

squ

are

is g

uar

ante

ed t

o be

div

isib

le b

ybo

th S

usa

n’s

an

d T

ravi

s’ f

avor

ite

perf

ect

squ

ares

reg

ardl

ess

of t

hei

r sp

ecifi

cva

lue?

s2 t

2

2.EL

ECTR

IC P

OTE

NTI

AL

Th

e el

ectr

ical

pote

nti

al f

un

ctio

n b

etw

een

tw

o el

ectr

ons

is g

iven

by

a fo

rmu

la t

hat

has

th

e fo

rm

�.S

impl

ify

this

exp

ress

ion

.

3.TR

APE

ZOID

ST

he

cros

s se

ctio

n o

f a

stan

d co

nsi

sts

of t

wo

trap

ezoi

ds s

tack

edon

e on

top

of

the

oth

er.

Th

e to

tal

area

of

the

cros

s se

ctio

n i

s x2

squ

are

un

its.

Ass

um

ing

the

trap

ezoi

dsh

ave

the

sam

e h

eigh

t,w

rite

an

expr

essi

on f

or t

he

hei

ght

of t

he

stan

d in

term

s of

x.P

ut

you

r an

swer

in

sim

ples

tfo

rm.(

Rec

all

that

th

e ar

ea o

f a

trap

ezoi

dw

ith

hei

ght

han

d ba

ses

b 1an

d b 2

is

give

n b

y h

(b1

�b 2

).)

x2

� 2x�

3

1 � 2

x �

4

x �

2

x

1� r(

1 �

r)1� 1

�r

1 � r

4.FR

AC

TIO

NS

In t

he

seve

nte

enth

cen

tury

,Lor

d B

rou

nck

er w

rote

dow

n a

mos

t pe

culi

ar m

ath

emat

ical

equ

atio

n:

� �4 ��

1 �

12

2 �

32

2 �

52

2 �

�7 ∞2 �

Th

is i

s an

exa

mpl

e of

a c

onti

nu

edfr

acti

on.S

impl

ify

the

con

tin

ued

fra

ctio

n

n�

.

REL

AY

RA

CE

For

Exe

rcis

es 5

-7,u

se t

he

foll

owin

g in

form

atio

n.

Mar

k,C

onn

ell,

Zac

k,an

d M

oses

ru

n t

he

4by

400

met

er r

elay

tog

eth

er.T

hei

r av

erag

esp

eeds

wer

e s,

s�

0.5,

s�

0.5,

and

s�

1m

eter

s pe

r se

con

d,re

spec

tive

ly.

5.W

hat

wer

e th

eir

indi

vidu

al t

imes

for

thei

r ow

n l

egs

of t

he

race

?

,,

,

6.W

rite

an

exp

ress

ion

for

th

eir

tim

e as

ate

am.W

rite

you

r an

swer

as

a ra

tio

oftw

o po

lyn

omia

ls.

400

7.If

sw

as 6

met

ers

per

seco

nd,

wh

at w

asth

e te

am’s

tim

e? R

oun

d yo

ur

answ

er t

oth

e n

eare

st s

econ

d.

281

seco

nd

s

16s

3�

12s

2�

2s�

1�

��

4s4

�4s

3�

s2

�s

400

� s�

140

0� s

��1 2�

400

� s�

�1 2�

400

�sn3

�2n

� n2

�11

� n�

� n1 �

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Prac

tice

Ad

din

g a

nd

Su

btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

Cha

pter

816

Gle

ncoe

Alg

ebra

2

Fin

d t

he

LC

M o

f ea

ch s

et o

f p

olyn

omia

ls.

1.x2

y,xy

32.

a2b3

c,ab

c43.

x�

1,x

�3

x2 y

3a

2 b3 c

4(x

�1)

(x�

3)

4.g

�1,

g2�

3g�

45.

2r�

2,r2

�r,

r�

16.

3,4w

�2,

4w2

�1

(g�

1)(g

�4)

2r(r

�1)

6(2w

�1)

(2w

�1)

7.x2

�2x

�8,

x�

48.

x2�

x�

6,x2

�6x

�8

9.d

2�

6d�

9,2(

d2

�9)

(x�

4)(x

�2)

(x�

2)(x

�4)

(x�

3)2(

d�

3)(d

�3)

2

Sim

pli

fy e

ach

exp

ress

ion

.

10.

�11

.�

12.

�

13.

�2

14.2

x�

5 �

15.

�

16.

�17

.�

18.

�

19.

�20

.�

21.

��

22.

��

23.

24.

25. G

EOM

ETRY

The

exp

ress

ions

,

,and

re

pres

ent

the

leng

ths

of t

he s

ides

of

a

tria

ngle

.Wri

te a

sim

plif

ied

expr

essi

on f

or t

he p

erim

eter

of

the

tria

ngle

.

26.K

AYA

KIN

GM

ai i

s ka

yaki

ng

on a

riv

er t

hat

has

a c

urr

ent

of 2

mil

es p

er h

our.

If r

repr

esen

ts h

er r

ate

in c

alm

wat

er,t

hen

r�

2 re

pres

ents

her

rat

e w

ith

th

e cu

rren

t,an

d r

�2

repr

esen

ts h

er r

ate

agai

nst

th

e cu

rren

t.M

ai k

ayak

s 2

mil

es d

own

stre

am a

nd

then

back

to

her

sta

rtin

g po

int.

Use

th

e fo

rmu

la f

or t

ime,

t�

,wh

ere

dis

th

e di

stan

ce,t

o

wri

te a

sim

plif

ied

expr

essi

on f

or t

he

tota

l ti

me

it t

akes

Mai

to

com

plet

e th

e tr

ip.

h4r

��

(r�

2)(r

�2)

d � r

5(x3

�4x

�16

)�

�2(

x�

4)(x

�4)

10� x

�4

20� x

�4

5x � 2

r�

4� r

�1

3x�

y� x

�y

12� a

�3

�r� r

6�

�� r

�12

�

��

�r2

r� 2�4r

2� r3

�

� x�2

y�

�� x

�1y

�

��

� x�1

y�

36� a2

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3(6

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n2p

2�

2p�

1�

��

(p�

2)(p

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(p�

3)5

��

2(x

�2)

7� 10

n3 � 4

1 � 5n5

� p2�

92p

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��

p2�

5p�

620

��

x2�

4x�

125

� 2x�

12

2y�

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(y�

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92

� x�

4

y�

�y2

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y�

5�

�y2

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�10

4m�

5� 9

�m

2 �

5m� m

�9

2� x

�4

16� x2

�16

13a

�47

��

(a�

3)(a

�5)

2(x

�3)

(x�

2)�

�x

�4

2(2

�3n

)�

�3n

9� a

�5

4� a

�3

x�

8� x

�4

4m � 3mn

2d2

�9c

��

12c

2 d3

25y

2�

12x

2�

�60

x4 y

320

�21

b�

�24

ab

3� 4c

d3

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2 d1

� 5x2 y

35

� 12x4 y

7 � 8a5

� 6ab

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-2

Chapter 8 A8 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Less

on R

eadi

ng G

uide

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

819

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-3 i

n y

our

text

boo

k.

•If

15

stu

den

ts c

ontr

ibu

te t

o th

e gi

ft,h

ow m

uch

wou

ld e

ach

of

them

pay

?$1

0•

If e

ach

stu

den

t pa

ys $

5,h

ow m

any

stu

den

ts c

ontr

ibu

ted?

30 s

tud

ents

Rea

d t

he

Less

on

1.W

hic

h o

f th

e fo

llow

ing

are

rati

onal

fu

nct

ion

s?A

an

d C

A.

f(x)

�B

.g(x

) �

�x�

C.h

(x)

�

2.a.

Gra

phs

of r

atio

nal

fu

nct

ion

s m

ay h

ave

brea

ks i

n

.Th

ese

may

occ

ur

as v

erti

cal

or a

s po

int

.Th

e of

a r

atio

nal

fu

nct

ion

is

lim

ited

to

valu

es f

or w

hic

h t

he

fun

ctio

n i

s de

fin

ed.

b.

Th

e gr

aph

s of

tw

o ra

tion

al f

un

ctio

ns

are

show

n b

elow

.

I.II

.

Gra

ph I

has

a

at x

�.

Gra

ph I

I h

as a

at

x�

.

Mat

ch e

ach

fu

nct

ion

wit

h i

ts g

raph

abo

ve.

f(x)

�II

g(x)

�I

Rem

emb

er W

hat

Yo

u L

earn

ed

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng n

ew i

s to

see

how

it

is r

elat

ed t

o so

met

hing

you

alr

eady

know

.How

can

kn

owin

g th

at d

ivis

ion

by

zero

is

un

defi

ned

hel

p yo

u t

o re

mem

ber

how

to

fin

d th

e pl

aces

wh

ere

a ra

tion

al f

un

ctio

n h

as a

poi

nt

disc

onti

nu

ity

or a

n a

sym

ptot

e?

Sam

ple

an

swer

:A

po

int

dis

con

tin

uit

y o

r ve

rtic

al a

sym

pto

te o

ccu

rsw

her

e th

e fu

nct

ion

is u

nd

efin

ed,t

hat

is,w

her

e th

e d

eno

min

ato

r o

f th

ere

late

d r

atio

nal

exp

ress

ion

is e

qu

al t

o 0

.Th

eref

ore

,set

th

e d

eno

min

ato

req

ual

to

zer

o a

nd

so

lve

for

the

vari

able

.

x2�

4� x

�2

x� x

�2

�2

vert

ical

asy

mp

tote

�2

po

int

dis

con

tin

uit

y

x

y Ox

y

O

do

mai

nd

isco

nti

nu

itie

sas

ymp

tote

sco

nti

nu

ity

x2�

25�

�x2

�6x

�9

1� x

�5

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

818

Gle

ncoe

Alg

ebra

2

Enri

chm

ent

Z

eno

’s P

arad

ox

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-2

Th

e G

reek

ph

ilos

oph

er Z

eno

of E

lea

(bor

n s

omet

ime

betw

een

495

an

d 48

0 B

.C.)

pro

pose

d fo

ur

para

doxe

s to

ch

alle

nge

th

e n

otio

ns

of s

pace

an

d ti

me.

Zen

o’s

firs

t pa

rado

x w

orks

lik

e th

is:

Su

ppos

e yo

u a

re o

n y

our

way

to

sch

ool.

Ass

um

e yo

u a

re a

ble

to c

over

hal

f of

the

rem

ain

ing

dist

ance

eac

h m

inu

te t

hat

you

wal

k.Yo

u l

eave

you

r h

ouse

at

7:45

A.M

.Aft

er t

he

firs

t m

inu

te,y

ou a

re h

alf

of t

he

way

to

sch

ool.

In t

he

nex

tm

inu

teyo

uco

ver

hal

fof

the

rem

ain

ing

dist

ance

tosc

hoo

l,an

dat

7:47

A.M

.you

are

thre

e-qu

arte

rs o

f th

e w

ay t

o sc

hoo

l.T

his

pat

tern

con

tin

ues

eac

h m

inu

te.

At

wh

at t

ime

wil

l yo

u a

rriv

e at

sch

ool?

Bef

ore

8:00

A.M

.? B

efor

e lu

nch

?

Sin

ce s

pace

is

infi

nit

ely

divi

sibl

e,w

e ca

n r

epea

t th

is p

atte

rn f

orev

er.T

hu

s,on

th

e w

ay t

o sc

hoo

l yo

u m

ust

rea

ch a

n i

nfi

nit

e n

um

ber

of ‘m

idpo

ints

’ in

afi

nit

e ti

me.

Th

is i

s im

poss

ible

,so

you

can

nev

er r

each

you

r go

al.I

n g

ener

al,

acco

rdin

g to

Zen

o an

yon

e w

ho

wan

ts t

o m

ove

from

on

e po

int

to a

not

her

m

ust

mee

t th

ese

requ

irem

ents

,an

d m

otio

n i

s im

poss

ible

.Th

eref

ore,

wh

at

we

perc

eive

as

mot

ion

is

mer

ely

an i

llu

sion

.

Add

itio

n o

f fr

acti

ons

can

be

defi

ned

by

��

,sim

ilar

ly f

orsu

btra

ctio

n.

Ass

um

e yo

ur

hou

se i

s on

e m

ile

from

sch

ool.

At

7:46

A.M

.,yo

u h

ave

wal

ked

hal

f of

a m

ile,

so y

ou h

ave

left

1 �

,or

a m

ile.

At

7:47

A.M

.you

on

ly h

ave

��

of a

mil

e to

go.

To

dete

rmin

e h

ow f

ar y

ou h

ave

wal

ked

and

how

far

aw

ay f

rom

th

e sc

hoo

l yo

u

are

at 7

:48

A.M

.,ad

d th

e di

stan

ces

wal

ked

each

min

ute

,�

��

of

a m

ile

so f

ar a

nd

you

sti

ll h

ave

1 �

�of

a m

ile

to g

o.

1.D

eter

min

e h

ow f

ar y

ou h

ave

wal

ked

and

how

far

aw

ay f

rom

th

e sc

hoo

lyo

u a

re a

t 7:

50 A

.M.

You

hav

e w

alke

d

of

a m

ile,a

nd

will

be

of

a m

ile a

way

fro

m s

cho

ol.

2.S

upp

ose

inst

ead

of c

over

ing

one-

hal

f th

e di

stan

ce t

o sc

hoo

l ea

ch m

inu

te,

you

cov

er t

hre

e-qu

arte

rs o

f th

e di

stan

ce r

emai

nin

g to

sch

ool

each

min

ute

,n

ow w

ill

you

be

able

to

mak

e it

to

sch

ool

on t

ime?

Det

erm

ine

how

far

you

stil

l h

ave

left

to

go a

t 7:

47 A

.M.

No

.Yo

u w

ill h

ave

of

a m

ile r

emai

nin

g a

t 7:

47 A

.M.

3.S

upp

ose

that

in

stea

d of

cov

erin

g on

e-h

alf

or t

hre

e-qu

arte

rs o

f th

e

dist

ance

to

sch

ool

each

min

ute

,you

cov

er

of t

he

dist

ance

rem

ain

ing,

wh

ere

xis

a w

hol

e n

um

ber

grea

ter

than

2.W

hat

is

you

r di

stan

ce f

rom

sch

ool

at 7

:46

A.M

.?

You

are

o

f a

mile

fro

m s

cho

ol a

t 7:

46 A

.M.

x2

� (x �

1)2

1� x

�1

1 � 16

1 � 3231 � 32

1 � 87 � 8

7 � 81 � 8

1 � 41 � 2

1 � 41 � 4

1 � 2

1 � 21 � 2

ad�

bc�

bdc � d

a � b

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lessons 8-2 and 8-3)

Chapter 8 A9 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-3)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

821

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Gra

ph

Rat

ion

al F

un

ctio

ns

Use

th

e fo

llow

ing

step

s to

gra

ph a

rat

ion

al f

un

ctio

n.

Ste

p 1

Firs

t se

e if

the

func

tion

has

any

vert

ical

asy

mpt

otes

or

poin

t di

scon

tinui

ties.

Ste

p 2

Dra

w a

ny v

ertic

al a

sym

ptot

es.

Ste

p 3

Mak

e a

tabl

e of

val

ues.

Ste

p 4

Plo

t th

e po

ints

and

dra

w t

he g

raph

.

Gra

ph

f(x

) �

.

�or

Th

eref

ore

the

grap

h o

f f(

x) h

as a

n a

sym

ptot

e at

x�

�3

and

a po

int

disc

onti

nu

ity

at x

�1.

Mak

e a

tabl

e of

val

ues

.Plo

t th

e po

ints

an

d dr

aw t

he

grap

h.

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

4.f(

x) �

5.f(

x) �

6.f(

x) �

xO

f(x)

xO

f(x)

xO

f(x)

x2�

6x�

8�

�x2

�x

�2

x2�

x�

6�

�x

�3

2� (x

�3)

2

xO

f(x)

48

8 4 –4 –8

–4–8

xO

f(x)

xO

f(x)

2x�

1� x

�3

2 � x3

� x�

1

x�

2.5

�2

�1

�3.

5�

4�

5

f(x

)2

10.

5�

2�

1�

0.51� x

�3

x�

1�

�(x

�1)

(x�

3)x

�1

��

x2�

2x�

3

x

f(x)

O

x�

1�

�x2

�2x

�3

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

Exer

cise

s

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exer

cise

s

Stud

y G

uide

and

Inte

rven

tion

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

Cha

pter

820

Gle

ncoe

Alg

ebra

2

Do

mai

n a

nd

Ran

ge

Rat

ion

al F

un

ctio

nan

equ

atio

n of

the

form

f(x

) �

, w

here

p(x

) an

d q

(x)

are

poly

nom

ial e

xpre

ssio

ns a

nd

q(x

) �

0

Do

mai

nT

he d

omai

n of

a r

atio

nal f

unct

ion

is li

mite

d to

val

ues

for

whi

ch t

he f

unct

ion

is d

efin

ed.

Ver

tica

l Asy

mp

tote

A

n as

ympt

ote

is a

line

tha

t th

e gr

aph

of a

fun

ctio

n ap

proa

ches

.If

the

sim

plifi

ed fo

rm o

f th

ere

late

d ra

tiona

l exp

ress

ion

is u

ndef

ined

for

x�

a, t

hen

x�

ais

a v

ertic

al a

sym

ptot

e.

Po

int

Dis

con

tin

uit

y P

oint

dis

cont

inui

ty is

like

a h

ole

in a

gra

ph.I

f th

e or

igin

al r

elat

ed e

xpre

ssio

n is

und

efin

ed

for

x�

abu

t th

e si

mpl

ified

exp

ress

ion

is d

efin

ed fo

r x

�a,

the

n th

ere

is a

hol

e in

the

gr

aph

at x

�a.

Ho

rizo

nta

l O

ften

a ho

rizon

tal a

sym

ptot

e oc

curs

in t

he g

raph

of

a ra

tiona

l fun

ctio

n w

here

a v

alue

isA

sym

pto

teex

clud

ed f

rom

the

ran

ge.

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es

of x

for

any

hol

es i

n t

he

grap

h o

f f(

x) �

.

Fir

st f

acto

r th

e n

um

erat

or a

nd

the

den

omin

ator

of

the

rati

onal

exp

ress

ion

.

f(x)

��

Th

e fu

nct

ion

is

un

defi

ned

for

x�

1 an

d x

��

1.

Sin

ce

�,x

�1

is a

ver

tica

l as

ympt

ote.

Th

e si

mpl

ifie

d ex

pres

sion

is

defi

ned

for

x�

�1,

so t

his

val

ue

repr

esen

ts a

hol

e in

th

e gr

aph

.

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

asym

pto

tes:

x�

2,h

ole

:x

�as

ymp

tote

:x

�0;

x�

�5

ho

le x

�4

4.f(

x) �

5.f(

x) �

6.f(

x) �

asym

pto

te:

x�

�2;

asym

pto

tes:

x�

1,as

ymp

tote

:x

��

3

ho

le:

x�

x�

�7

7.f(

x) �

8.f(

x) �

9.f(

x) �

asym

pto

tes:

x�

1,as

ymp

tote

:x

��

3;h

ole

s:x

�1,

x�

3 x

�5

ho

le:

x�

3 � 2

x3�

2x2

�5x

�6

��

�x2

�4x

�3

2x2

�x

�3

��

2x2

�3x

�9

x�

1�

�x2

�6x

�5

1 � 3

3x2

�5x

�2

��

x�

3x2

�6x

�7

��

x2�

6x�

73x

�1

��

3x2

�5x

�2

5 � 2

x2�

x�

12�

�x2

�4x

2x2

�x

�10

��

2x�

54

��

x2�

3x�

10

4x�

3� x

�1

(4x

�3)

(x�

1)�

�(x

�1)

(x�

1)

(4x

�3)

(x�

1)�

�(x

�1)

(x�

1)4x

2�

x�

3�

�x2

�1

4x2

�x

�3

��

x2�

1

p(x

)� q

(x)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Exam

ple

Chapter 8 A10 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Prac

tice

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

823

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

3.f(

x) �

asym

pto

tes:

x�

2,as

ymp

tote

:x

�3;

asym

pto

te:

x�

�2

x�

�5

ho

le:

x�

74.

f(x)

�5.

f(x)

�6.

f(x)

�

ho

le:

x�

�10

ho

le:

x�

6h

ole

:x

��

5

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

7.f(

x) �

8.f(

x) �

9.f(

x) �

10.P

AIN

TIN

GW

orki

ng

alon

e,T

awa

can

giv

e th

e sh

ed a

coa

t of

pai

nt

in 6

hou

rs.I

t ta

kes

her

fat

her

xh

ours

wor

kin

g al

one

to g

ive

the

shed

a c

oat

of p

ain

t.T

he

equ

atio

n f

(x)

�de

scri

bes

the

port

ion

of

the

job

Taw

a an

d h

er f

ath

er w

orki

ng

toge

ther

can

com

plet

e in

1 h

our.

Gra

ph f

(x)

�fo

r x

0,

y

0.If

Taw

a’s

fath

er c

an c

ompl

ete

the

job

in 4

hou

rs a

lon

e,w

hat

por

tion

of

the

job

can

th

ey c

ompl

ete

toge

ther

in

1 h

our?

Wh

at d

omai

n a

nd

ran

geva

lues

are

mea

nin

gfu

l in

th

e co

nte

xt o

f th

e pr

oble

m?

;S

amp

le a

nsw

er:T

he

nu

mb

er o

f h

ou

rs it

tak

es h

er f

ath

er t

o g

ive

the

shed

a c

oat

of

pai

nt

sho

uld

be

po

siti

ve.T

her

efo

re,o

nly

val

ues

of

x

gre

ater

th

an 0

an

d v

alu

es o

f f(

x) g

reat

er t

han

ar

e m

ean

ing

ful.

11.L

IGH

TT

he

rela

tion

ship

bet

wee

n t

he

illu

min

atio

n a

n o

bjec

t re

ceiv

es f

rom

a l

igh

t so

urc

e of

Ifo

ot-c

andl

es a

nd

the

squ

are

of

the

dist

ance

din

fee

t of

th

e ob

ject

fro

m t

he

sou

rce

can

be

mod

eled

by

I(d

) �

.Gra

ph t

he

fun

ctio

n I

(d)

�fo

r

0 �

I�

80 a

nd

0 �

d�

80.W

hat

is

the

illu

min

atio

n i

n

foot

-can

dles

th

at t

he

obje

ct r

ecei

ves

at a

dis

tan

ce o

f 20

fee

t fr

om t

he

ligh

t so

urc

e?W

hat

dom

ain

an

d ra

nge

val

ues

are

m

ean

ingf

ul

in t

he

con

text

of

the

prob

lem

? 11

.25

foo

t-ca

nd

les;

Sam

ple

an

swer

:T

he

dis

tan

ce o

f th

e o

bje

ct f

rom

th

e so

urc

e sh

ou

ld b

e p

osi

tive

.Th

eref

ore

,o

nly

val

ues

of

dg

reat

er t

han

0 a

nd

val

ues

of

I(d

) g

reat

er t

han

0 a

rem

ean

ing

ful.

4500

�d

245

00�

d2

1 � 16

5 � 12

6 �

x�

6x

6 �

x�

6x

xO

f(x)

xOf(

x)

xO

f(x)

xO

f(x)

3x� (x

�3)

2x

�3

� x�

2�

4� x

�2

x2�

9x�

20�

�x

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x2�

2x�

24�

�x

�6

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100

��

x�

10

x�

2�

�x2

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�4

x�

7�

�x2

�10

x�

216

��

x2�

3x�

10

2040

Dist

ance

(ft)

Illu

min

atio

n

Illumination (foot-candles)

60

60 40 20

dOIII

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

822

Gle

ncoe

Alg

ebra

2

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es o

f x

for

any

hol

es i

n t

he

grap

h o

f ea

ch r

atio

nal

fu

nct

ion

.

1.f(

x) �

2.f(

x) �

asym

pto

tes:

x�

4,x

��

2as

ymp

tote

s:x

�4,

x�

9

3.f(

x) �

4.f(

x) �

asym

pto

te:

x�

2;h

ole

:x

��

12as

ymp

tote

:x

�3;

ho

le:

x�

1

5.f(

x) �

6.f(

x) �

ho

le:

x�

�2

ho

le:

x�

3

Gra

ph

eac

h r

atio

nal

fu

nct

ion

.

7.f(

x) �

8.f(

x) �

9.f(

x) �

10.f

(x)

�11

.f(x

) �

12.f

(x)

�

xO

f(x)

xO

f(x)

xO

f(x)

x2�

4� x

�2

x� x

�2

2� x

�1

xO

f(x)

xO

f(x) 2

2

xO

f(x)

�4

�x

10 � x�

3�

x

x2�

x�

12�

�x

�3

x2�

8x�

12�

�x

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x�

1�

�x2

�4x

�3

x�

12�

�x2

�10

x�

24

10�

�x2

�13

x�

363

��

x2�

2x�

8

Skill

s Pr

acti

ceG

rap

hin

g R

atio

nal

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-3)

Chapter 8 A11 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-3)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Enri

chm

ent

Ch

arac

teri

stic

s o

f R

atio

nal

Fu

nct

ion

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Cha

pter

825

Gle

ncoe

Alg

ebra

2

Lesson 8-3

Use

th

e in

form

atio

n i

n t

he

tabl

e to

gra

ph r

atio

nal

fu

nct

ion

s

A s

ign

ch

art

use

s an

xva

lue

from

th

e le

ft a

nd

righ

t of

ea

ch c

riti

cal

valu

e to

det

erm

ine

if t

he

grap

h i

s po

siti

ve o

r n

egat

ive

on t

hat

inte

rval

.A s

ign

ch

art

for

y�

is s

how

n b

elow

.

Th

e gr

aph

of

is s

how

n

to t

he

righ

t.

Cre

ate

a si

gn c

har

t fo

r y

�.U

se a

n x

-val

ue

from

th

e le

ft a

nd

righ

t of

eac

h c

riti

cal

valu

e to

det

erm

ine

if t

he

grap

h i

s p

osit

ive

orn

egat

ive

on t

hat

in

terv

al.T

hen

gra

ph

th

e fu

nct

ion

.

y

x�

22

�3

�2

�1

��

��

01

23

x�

1� x

2�

4

x�

1�

�x2

�x

�6

y

x�

23

�3

�2

�1

��

��

01

23

4

x�

1�

�x2

�x

�6

CH

AR

AC

TE

RIS

TIC

ME

AN

ING

HO

W T

O F

IND

IT

Ver

tica

l asy

mp

tote

sA

ver

tical

line

at

an x

valu

e w

here

the

S

et t

he d

enom

inat

or e

qual

to

zero

and

ra

tiona

l fun

ctio

n is

und

efin

edso

lve

for

x.

Ho

rizo

nta

l asy

mp

tote

sA

hor

izon

tal l

ine

that

the

rat

iona

l S

tudy

the

end

-beh

avio

rs.

func

tion

Rig

ht

end

-beh

avio

rH

ow t

he g

raph

beh

aves

at

larg

e E

valu

ate

the

ratio

nal e

xpre

ssio

n at

po

sitiv

e va

lues

of

xin

crea

sing

pos

itive

val

ues

of x

.

Lef

t en

d-b

ehav

ior

How

the

gra

ph b

ehav

es a

t la

rge

Eva

luat

e th

e ra

tiona

l exp

ress

ion

at

nega

tive

valu

es o

f x

incr

easi

ng n

egat

ive

valu

es o

f x.

Ro

ots

,zer

os,

or

x-i

nte

rcep

tsP

oint

(s)

whe

re t

he g

raph

cro

sses

the

S

et t

he n

umer

ator

equ

al t

o ze

ro a

nd

x-ax

isso

lve

for

x.

y-i

nte

rcep

tsP

oint

whe

re t

he g

raph

cro

sses

the

S

et x

= 0

to

dete

rmin

e th

e y-

inte

rcep

t.y-

axis

Exam

ple

Exer

cise

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

824

Gle

ncoe

Alg

ebra

2

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

1.R

OA

D T

RIP

Rob

ert

and

Sar

ah s

tart

of

f on

a r

oad

trip

fro

m t

he

sam

e h

ouse

.D

uri

ng

the

trip

,Rob

ert’s

an

d S

arah

’sca

rs r

emai

n s

epar

ated

by

a co

nst

ant

dist

ance

.Th

e gr

aph

sh

ows

the

rati

o of

the

dist

ance

Sar

ah h

as t

rave

led

to t

he

dist

ance

Rob

ert

has

tra

vele

d.T

he

dott

edli

ne

show

s h

ow t

his

gra

ph w

ould

be

exte

nde

d to

hyp

oth

etic

al n

egat

ive

valu

esof

x.W

hat

doe

s th

e x-

coor

din

ate

of t

he

vert

ical

asy

mpt

ote

repr

esen

t?

the

dis

tan

ce b

y w

hic

h S

arah

trai

ls R

ob

ert

2.G

RA

PHS

Alm

a gr

aph

ed t

he

fun

ctio

n

f(x)

�be

low

.

Th

ere

is a

pro

blem

wit

h h

er g

raph

.E

xpla

in h

ow t

o co

rrec

t it

.

Th

e p

oin

t (4

,4)

nee

ds

to b

eer

ased

an

d a

sm

all c

ircl

e p

ut

aro

un

d it

.

y

xO

x2�

4x�x

�4

y

xO

3.FI

NA

NC

EA

qu

ick

way

to

get

an i

dea

of h

ow m

any

year

s be

fore

a s

avin

gsac

cou

nt

wil

l do

ubl

e at

an

in

tere

st r

ate

of I

perc

ent

com

pou

nde

d an

nu

ally

,is

todi

vide

Iin

to 7

2.S

ketc

h a

gra

ph o

f th

e

fun

ctio

n f

(I)

�.

4.N

EWTO

NS

ir I

saac

New

ton

stu

died

th

e ra

tion

al f

un

ctio

n

f(x)

�.

Ass

um

ing

that

d�

0,w

her

e w

ill

ther

ebe

a v

erti

cal

asym

ptot

e to

th

e gr

aph

of

this

fu

nct

ion

?

x�

0

BA

TTIN

G A

VER

AG

ESF

or E

xerc

ises

5an

d 6

,use

th

e fo

llow

ing

info

rmat

ion

.

Josh

has

mad

e 26

hit

s in

80

at b

ats

for

a ba

ttin

g av

erag

e of

.325

.Jos

h g

oes

on a

hit

tin

g st

reak

an

d m

akes

xh

its

in t

he

nex

t2x

at b

ats.

5.W

hat

fu

nct

ion

des

crib

es J

osh

’s b

atti

ng

aver

age

duri

ng

this

str

eak?

f(x)

�

6.W

hat

is

the

equ

atio

n o

f th

e h

oriz

onta

las

ympt

ote

to t

he

grap

h o

f th

e fu

nct

ion

you

wro

te f

or E

xerc

ise

5? W

hat

is

its

mea

nin

g?

y�

0.5;

0.5

rep

rese

nts

an

up

per

bo

un

d o

n J

osh

’s b

atti

ng

ave

rag

eif

his

hit

rat

e d

oes

no

t ch

ang

e

26 �

x� 80

�2x

ax3

�bx

2�

cx�

d�

��

x

I

50

5O

f(I)

72 � I

Chapter 8 A12 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

827

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-4 i

n y

our

text

boo

k.

•F

or e

ach

add

itio

nal

stu

den

t w

ho

enro

lls

in a

pu

blic

col

lege

,th

e to

tal

hig

h-t

ech

spe

ndi

ng

wil

l (i

ncr

ease

/dec

reas

e) b

y .

•F

or e

ach

dec

reas

e in

en

roll

men

t of

100

stu

den

ts i

n a

pu

blic

col

lege

,th

e to

tal

hig

h-t

ech

spe

ndi

ng

wil

l (i

ncr

ease

/dec

reas

e) b

y .

Rea

d t

he

Less

on

1.W

rite

an

equ

atio

n t

o re

pres

ent

each

of

the

foll

owin

g va

riat

ion

sta

tem

ents

.Use

kas

th

eco

nst

ant

of v

aria

tion

.

a.m

vari

es i

nve

rsel

y as

n.

m�

b.

sva

ries

dir

ectl

y as

r.

s�

kr

c.t

vari

es jo

intl

y as

pan

d q.

t�

kpq

2.W

hic

h t

ype

of v

aria

tion

,dir

ect

or i

nve

rse,

is r

epre

sen

ted

by e

ach

gra

ph?

a.in

vers

eb

.d

irec

t

Rem

emb

er W

hat

Yo

u L

earn

ed

3.H

ow c

an y

our

know

ledg

e of

th

e eq

uat

ion

of

the

slop

e-in

terc

ept

form

of

the

equ

atio

n o

f a

lin

e h

elp

you

rem

embe

r th

e eq

uat

ion

for

dir

ect

vari

atio

n?

Sam

ple

an

swer

:Th

e g

rap

h o

f an

eq

uat

ion

exp

ress

ing

dir

ect

vari

atio

n is

alin

e.T

he

slo

pe-

inte

rcep

t fo

rm o

f th

e eq

uat

ion

of

a lin

e is

y�

mx

�b

.In

dir

ect

vari

atio

n,i

f o

ne

of

the

qu

anti

ties

is 0

,th

e o

ther

qu

anti

ty is

als

o 0

,so

b�

0 an

d t

he

line

go

es t

hro

ug

h t

he

ori

gin

.Th

e eq

uat

ion

of

a lin

eth

rou

gh

th

e o

rig

in is

y�

mx,

wh

ere

mis

th

e sl

op

e.T

his

is t

he

sam

e as

the

equ

atio

n f

or

dir

ect

vari

atio

n w

ith

k�

m.

x

y Ox

y

O

k � n

$20,

300

dec

reas

e

$203

incr

ease

Less

on R

eadi

ng G

uide

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Lesson 8-4

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

826

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-3

Th

e li

ne

y�

bis

a h

oriz

onta

l as

ympt

ote

for

the

rati

onal

fu

nct

ion

f(x

) if

f(

x)→

bas

x→

or

as

x→

�

.Th

e h

oriz

onta

l as

ympt

ote

can

be

fou

nd

by

usi

ng

the

TA

BL

Efe

atu

re o

f th

e gr

aph

ing

calc

ula

tor.

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

a.f(

x)�� x2

�41 x

�5

�

En

ter

the

fun

ctio

n in

to Y

1.P

lace

[T

blS

et]

in t

he

Ask

mod

e.E

nte

r th

en

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

inth

e x-

list

.K

eyst

roke

s:1

4 5

[TB

LS

ET

] [T

AB

LE

].T

hen

en

ter

the

valu

es f

or x

.

Not

ice

that

as

xin

crea

ses,

yap

proa

ches

0.T

hu

s,y

�0

is t

he

hor

izon

tal a

sym

ptot

e.

b.f

(x)

�� 2x

2�3x

52 x�

6�

En

ter

the

equ

atio

n in

to Y

1.E

nte

r th

e n

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

in t

he

x-li

st.N

ote

the

patt

ern

.As

xin

crea

ses,

yap

proa

ches

1.5

.Th

us,

y�

1.5

is t

he

hor

izon

tal a

sym

ptot

e.

2nd

EN

TER

2nd

)—

+x

2(

�Y

=

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

1.f(

x)�

� x2 �x

1�

y�

22.

f(x)

�� 2x

2x �2

7� x1 �

12�

y�

�1 2�3.

f(x)

�� 2x

3�

6 2x x3

2�

2�

y�

3

4.f(

x)�� 3x

2�

2 5x x�

1�

y�

05.

f(x)

��15

x2� x33x

�7

�y

�0

6.f(

x)�

y�

0

7.f(

x)�

�5 xx2 ��23

�n

on

e8.

f(x)

�� 2x

2�6x

33 x�

6�

no

ne

9.f(

x)�

�2x2�

4�

no

ne

x3�

8x2

�4x

�11

��

�x4

�3x

3 �4x

�6

Gra

phin

g Ca

lcul

ator

Act

ivit

yH

ori

zon

tal A

sym

pto

tes

and

Tab

les

Exer

cise

s

Exam

ple

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lessons 8-3 and 8-4)

Chapter 8 A13 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-4)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

829

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Inve

rse

Var

iati

on

Inve

rse

Var

iati

on

yva

ries

inve

rsel

y as

xif

ther

e is

som

e no

nzer

o co

nsta

nt k

such

tha

t xy

�k

or y

�.

If a

vari

es i

nve

rsel

y as

ban

d a

�8

wh

en b

�12

,fin

d a

wh

en b

�4.

�In

vers

e va

riatio

n

�a 1

�8,

b1

�12

, b 2

�4

8(12

) �

4a2

Cro

ss m

ultip

ly.

96 �

4a2

Sim

plify

.

24 �

a 2D

ivid

e ea

ch s

ide

by 4

.

Wh

en b

�4,

the

valu

e of

ais

24.

Fin

d e

ach

val

ue.

1.If

yva

ries

in

vers

ely

as x

and

y�

12 w

hen

x�

10,f

ind

yw

hen

x�

15.

8

2.If

yva

ries

in

vers

ely

as x

and

y�

100

wh

en x

�38

,fin

d y

wh

en x

�76

.50

3.If

yva

ries

in

vers

ely

as x

and

y�

32 w

hen

x�

42,f

ind

yw

hen

x�

24.

56

4.If

yva

ries

in

vers

ely

as x

and

y�

36 w

hen

x�

10,f

ind

yw

hen

x�

30.

12

5.If

yva

ries

in

vers

ely

as x

and

y�

18 w

hen

x�

124,

fin

d y

wh

en x

�93

.24

6.If

yva

ries

in

vers

ely

as x

and

y�

90 w

hen

x�

35,f

ind

yw

hen

x�

50.

63

7.If

yva

ries

in

vers

ely

as x

and

y�

42 w

hen

x�

48,f

ind

yw

hen

x�

36.

56

8.If

yva

ries

in

vers

ely

as x

and

y�

44 w

hen

x�

20,f

ind

yw

hen

x�

55.

16

9.If

yva

ries

in

vers

ely

as x

and

y�

80 w

hen

x�

14,f

ind

yw

hen

x�

35.

32

10.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�8,

fin

d y

wh

en x

�40

.0.

6

11.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�42

,fin

d y

wh

en x

�14

.48

12.I

f y

vari

es i

nve

rsel

y as

xan

d y

�23

wh

en x

�12

,fin

d y

wh

en x

�15

.18

.4

a 2� 12

8 � 4

a 2� b 1

a 1� b 2

k � x

Lesson 8-4

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

Exer

cise

s

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

828

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Stud

y G

uide

and

Inte

rven

tion

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

nD

irec

t V

aria

tio

n a

nd

Jo

int

Var

iati

on

Dir

ect V

aria

tio

ny

varie

s di

rect

ly a

s x

if th

ere

is s

ome

nonz

ero

cons

tant

ksu

ch t

hat

y�

kx.k

is c

alle

d th

eco

nsta

nt o

f va

riatio

n.

Join

t Var

iati

on

yva

ries

join

tly a

s x

and

zif

ther

e is

som

e nu

mbe

r k

such

tha

t y

�kx

z, w

here

x�

0 an

d z

�0.

Fin

d e

ach

val

ue.

a.If

yva

ries

dir

ectl

y as

xan

d y

�16

wh

en x

�4,

fin

d x

wh

en y

�20

.

�D

irect

pro

port

ion

�y 1

�16

, x 1

�4,

and

y2

�20

16x 2

�(2

0)(4

)C

ross

mul

tiply

.

x 2�

5S

impl

ify.

Th

e va

lue

of x

is 5

wh

en y

is 2

0.

20 � x 2

16 � 4

y 2� x 2

y 1� x 1

b.

If y

vari

es j

oin

tly

as x

and

zan

d y

�10

wh

en x

�2

and

z �

4,fi

nd

yw

hen

x

�4

and

z�

3.

�Jo

int

varia

tion

�y 1

�10

, x 1

�2,

z 1

�4,

x2

�4,

an

d z 2

�3

120

�8y

2S

impl

ify.

y 2�

15D

ivid

e ea

ch s

ide

by 8

.

Th

e va

lue

of y

is 1

5 w

hen

x�

4 an

d z

�3.

y 2� 4

�3

10� 2

�4

y 2� x 2

z 2

y 1� x 1z

1

Fin

d e

ach

val

ue.

1.If

yva

ries

dir

ectl

y as

xan

d y

�9

wh

en

2.If

yva

ries

dir

ectl

y as

xan

d y

�16

wh

en

x�

6,fi

nd

yw

hen

x�

8.12

x�

36,f

ind

yw

hen

x�

54.

24

3.If

yva

ries

dir

ectl

y as

xan

d x

�15

4.

If y

vari

es d

irec

tly

as x

and

x�

33 w

hen

w

hen

y�

5,fi

nd

xw

hen

y�

9.27

y�

22,f

ind

xw

hen

y�

32.

48

5.S

upp

ose

yva

ries

join

tly

as x

and

z.6.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�5

and

z�

3,if

y�

18

wh

en x

�6

and

z�

8,if

y�

6 w

hen

x�

4w

hen

x�

3 an

d z

�2.

45an

d z

�2.

36

7.S

upp

ose

yva

ries

join

tly

as x

and

z.8.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�4

and

z�

11,i

f y

�60

w

hen

x�

5 an

d z

�2,

if y

�84

wh

en

wh

en x

�3

and

z�

5.17

6x

�4

and

z�

7.30

9.If

yva

ries

dir

ectl

y as

xan

d y

�39

10

.If

yva

ries

dir

ectl

y as

xan

d x

�60

wh

enw

hen

x�

52,f

ind

yw

hen

x�

22.

16.5

y�

75,f

ind

xw

hen

y�

42.

33.6

11.S

upp

ose

yva

ries

join

tly

as x

and

z.12

.Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�7

and

z�

18,i

f w

hen

x�

5 an

d z

�27

,if

y�

480

wh

en

y�

351

wh

en x

�6

and

z�

13.

567

x�

9 an

d z

�20

.36

0

Exer

cise

s

Exam

ple

Chapter 8 A14 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Cha

pter

831

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Sta

te w

het

her

eac

h e

qu

atio

n r

epre

sen

ts a

dir

ect,

join

t,or

in

vers

eva

riat

ion

.Th

enn

ame

the

con

stan

t of

var

iati

on.

1.u

�8w

zjo

int;

82.

p�

4sd

irec

t;4

3.L

�

inve

rse;

54.

xy�

4.5

inve

rse;

4.5

5.�

6.

2d�

mn

7.�

h8.

y�

dir

ect;

�jo

int;

inve

rse;

1.25

inve

rse;

Fin

d e

ach

val

ue.

9.If

yva

ries

dir

ectl

y as

xan

d y

�8

wh

en x

�2,

fin

d y

wh

en x

�6.

24

10.I

f y

vari

es d

irec

tly

as x

and

y�

�16

wh

en x

�6,

fin

d x

wh

en y

��

4.1.

5

11.I

f y

vari

es d

irec

tly

as x

and

y�

132

wh

en x

�11

,fin

d y

wh

en x

�33

.39

6

12.I

f y

vari

es d

irec

tly

as x

and

y�

7 w

hen

x�

1.5,

fin

d y

wh

en x

�4.

13.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

24 w

hen

x�

2 an

d z

�1,

fin

d y

wh

en x

�12

an

d z

�2.

288

14.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

60 w

hen

x�

3 an

d z

�4,

fin

d y

wh

en x

�6

and

z�

8.24

0

15.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

12 w

hen

x�

�2

and

z�

3,fi

nd

yw

hen

x�

4 an

d z

��

1.8

16.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�4,

fin

d y

wh

en x

�3.

17.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�5,

fin

d x

wh

en y

�2.

5.6

18.I

f y

vari

es i

nve

rsel

y as

xan

d y

��

18 w

hen

x�

6,fi

nd

yw

hen

x�

5.�

21.6

19.I

f y

vari

es d

irec

tly

as x

and

y�

5 w

hen

x�

0.4,

fin

d x

wh

en y

�37

.5.

3

20.G

ASE

ST

he

volu

me

Vof

a g

as v

arie

s in

vers

ely

as i

ts p

ress

ure

P.I

f V

�80

cu

bic

cen

tim

eter

s w

hen

P�

2000

mil

lim

eter

s of

mer

cury

,fin

d V

wh

en P

�32

0 m

illi

met

ers

ofm

ercu

ry.

500

cm3

21.S

PRIN

GS

Th

e le

ngt

h S

that

a s

prin

g w

ill

stre

tch

var

ies

dire

ctly

wit

h t

he

wei

ght

Fth

atis

att

ach

ed t

o th

e sp

rin

g.If

a s

prin

g st

retc

hes

20

inch

es w

ith

25

pou

nds

att

ach

ed,h

owfa

r w

ill

it s

tret

ch w

ith

15

pou

nds

att

ach

ed?

12 in

.

22.G

EOM

ETRY

Th

e ar

ea A

of a

tra

pezo

id v

arie

s jo

intl

y as

its

hei

ght

and

the

sum

of

its

base

s.If

th

e ar

ea i

s 48

0 sq

uar

e m

eter

s w

hen

th

e h

eigh

t is

20

met

ers

and

the

base

s ar

e28

met

ers

and

20 m

eter

s,w

hat

is

the

area

of

a tr

apez

oid

wh

en i

ts h

eigh

t is

8 m

eter

s an

dit

s ba

ses

are

10 m

eter

s an

d 15

met

ers?

100

m2

64 � 3

56 � 3

3 � 41 � 2

3 � 4x1.

25�

gC � d

5 � k

Lesson 8-4

Prac

tice

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

830

Gle

ncoe

Alg

ebra

2

Sta

te w

het

her

eac

h e

qu

atio

n r

epre

sen

ts a

dir

ect,

join

t,or

in

vers

eva

riat

ion

.Th

enn

ame

the

con

stan

t of

var

iati

on.

1.c

�12

md

irec

t;12

2.p

�in

vers

e;4

3.A

�bh

join

t;

4.rw

�15

inve

rse;

155.

y�

2rst

join

t;2

6.f

�52

80m

dir

ect;

5280

7.y

�0.

2sd

irec

t;0.

28.

vz�

�25

inve

rse;

�25

9.t

�16

rhjo

int;

16

10.R

�in

vers

e;8

11.

�d

irec

t;12

.C�

2r

dir

ect;

2�

Fin

d e

ach

val

ue.

13.I

f y

vari

es d

irec

tly

as x

and

y�

35 w

hen

x�

7,fi

nd

yw

hen

x�

11.

55

14.I

f y

vari

es d

irec

tly

as x

and

y�

360

wh

en x

�18

0,fi

nd

yw

hen

x�

270.

540

15.I

f y

vari

es d

irec

tly

as x

and

y�

540

wh

en x

�10

,fin

d x

wh

en y

�10

80.

20

16.I

f y

vari

es d

irec

tly

as x

and

y�

12 w

hen

x�

72,f

ind

xw

hen

y�

9.54

17.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

18 w

hen

x�

2 an

d z

�3,

fin

d y

wh

en x

�5

and

z�

6.90

18.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

�16

wh

en x

�4

and

z�

2,fi

nd

yw

hen

x�

�1

and

z�

7.14

19.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

120

wh

en x

�4

and

z�

6,fi

nd

yw

hen

x�

3 an

d z

�2.

30

20.I

f y

vari

es i

nve

rsel

y as

xan

d y

�2

wh

en x

�2,

fin

d y

wh

en x

�1.

4

21.I

f y

vari

es i

nve

rsel

y as

xan

d y

�6

wh

en x

�5,

fin

d y

wh

en x

�10

.3

22.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�14

,fin

d x

wh

en y

�6.

7

23.I

f y

vari

es i

nve

rsel

y as

xan

d y

�27

wh

en x

�2,

fin

d x

wh

en y

�9.

6

24.I

f y

vari

es d

irec

tly

as x

and

y�

�15

wh

en x

�5,

fin

d x

wh

en y

��

36.

12

1 � 31 � 3

a � b8 � w

1 � 21 � 2

4 � q

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Skill

s Pr

acti

ceD

irec

t,Jo

int,

and

Inve

rse

Var

iati

on

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-4)

Chapter 8 A15 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-4)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

8-4

Enri

chm

ent

Geo

syn

chro

no

us

Sat

ellit

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Cha

pter

833

Gle

ncoe

Alg

ebra

2

Lesson 8-4

Sat

elli

tes

circ

lin

g th

e E

arth

are

alm

ost

as c

omm

on a

s th

e ce

ll p

hon

es t

hat

depe

nd

on t

hem

.A g

eosy

nch

ron

ous

sate

llit

e is

on

e th

at m

ain

tain

s th

e sa

me

posi

tion

abo

ve t

he

Ear

th a

t al

l ti

mes

.Geo

syn

chro

nou

s sa

tell

ites

are

use

d in

cell

ph

one

com

mu

nic

atio

ns,

tran

smit

tin

g si

gnal

s fr

om t

ower

s on

Ear

th a

nd

to e

ach

oth

er.

Th

e sp

eed

at w

hic

h t

hey

tra

vel

is v

ery

impo

rtan

t.If

th

e sp

eed

is t

oo l

ow,

the

sate

llit

e w

ill

be f

orce

d ba

ck d

own

to

Ear

th d

ue

to t

he

Ear

th’s

gra

vity

.H

owev

er,i

f it

is

too

fast

,it

wil

l ov

erco

me

grav

ity’

s fo

rce

and

esca

pe i

nto

spac

e,n

ever

to

retu

rn.N

ewto

n’s

sec

ond

law

of

mot

ion

say

s th

at f

orce

on

an

obje

ct i

s eq

ual

to

mas

s ti

mes

acc

eler

atio

n o

r F

�m

a.It

is

also

wel

l kn

own

that

th

e n

et g

ravi

tati

onal

for

ce b

etw

een

tw

o ob

ject

s is

in

vers

ely

prop

orti

onal

to t

he

squ

are

of t

he

dist

ance

bet

wee

n t

hem

.Th

eref

ore,

ther

e ar

e tw

ova

riab

les

on w

hic

h t

he

forc

e de

pen

ds:s

peed

an

d h

eigh

t ab

ove

the

Ear

th.

In p

arti

cula

r,N

ewto

n’s

sec

ond

law

,F�

ma,

show

s th

at f

orce

var

ies

dire

ctly

wit

h a

ccel

erat

ion

,wh

ere

mis

th

e co

nst

ant

taki

ng

the

plac

e of

“k.

”

1.S

how

th

at t

he

net

gra

vita

tion

al f

orce

pro

vidi

ng

a sa

tell

ite

wit

h a

ccel

era-

tion

is

inve

rsel

y pr

opor

tion

al t

o th

e sq

uar

e of

th

e di

stan

ce b

etw

een

th

emby

exp

ress

ing

this

var

iati

on a

s an

equ

atio

n.

F�

,wh

ere

his

th

e h

eig

ht

of

the

sate

llite

ab

ove

the

surf

ace

of

the

Ear

th.

2.U

se y

our

equ

atio

n f

rom

Nu

mbe

r 1

and

equ

ate

it w

ith

New

ton

’s f

orm

ula

abov

e to

det

erm

ine

how

th

e sa

tell

ite’

s ac

cele

rati

on v

arie

s w

ith

its

hei

ght

abov

e th

e E

arth

.

ma

�⇒

a�

��

,th

eref

ore

it v

arie

s in

vers

ely

wit

h

the

squ

are

of

the

hei

gh

t.

3.D

eter

min

e h

ow t

he

spee

d of

a g

eosy

nch

ron

ous

sate

llit

e va

ries

wit

h i

tsh

eigh

t ab

ove

the

Ear

th b

y u

sin

g th

e fa

ct t

hat

spe

ed i

s eq

ual

to

dist

ance

divi

ded

by t

ime

and

the

path

of

the

sate

llit

e is

cir

cula

r.

Dir

ect

vari

atio

n.s

pee

d�

⇒sp

eed

�,w

her

e

r�

h�

Rad

ius

of

the

Ear

th.

2�r

� day

dis

tan

ce�

�ti

me

K � h2

1 � h2

k � mk � h

2

k � h2

Exer

cise

s

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

832

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

Wor

d Pr

oble

m P

ract

ice

D

irec

t,Jo

int,

and

Inve

rse

Var

iati

on

1.D

IVIN

GT

he

hei

ght

that

a d

iver

lea

psab

ove

a di

vin

g bo

ard

vari

es d

irec

tly

wit

hth

e am

oun

t th

at t

he

tip

of t

he

divi

ng

boar

d di

ps b

elow

its

nor

mal

lev

el.I

f a

dive

r le

aps

44 i

nch

es a

bove

th

e di

vin

gbo

ard

wh

en t

he

divi

ng

boar

d ti

p di

ps 1

2in

ches

,how

hig

h w

ill

the

dive

r le

apab

ove

the

divi

ng

boar

d if

th

e ti

p di

ps 1

8in

ches

?

66 in

ches

2.PA

RK

ING

LO

T D

ESIG

NA

s a

gen

eral

rule

,th

e n

um

ber

of p

arki

ng

spac

es i

n

a pa

rkin

g lo

t fo

r a

mov

ie t

hea

ter

com

plex

var

ies

dire

ctly

wit

h t

he

nu

mbe

rof

th

eate

rs i

n t

he

com

plex

.A t

ypic

alth

eate

r h

as 3

0 pa

rkin

g sp

aces

for

eac

hth

eate

r.A

bu

sin

essm

an w

ants

to

buil

d a

new

cin

ema

com

plex

on

a l

ot t

hat

h

as e

nou

gh s

pace

for

210

par

kin

gsp

aces

.How

man

y th

eate

rs s

hou

ld t

he

busi

nes

sman

bu

ild

in h

is c

ompl

ex?

7

3.R

ENT

An

apa

rtm

ent

ren

ts f

or m

doll

ars

per

mon

th.I

f n

stu

den

ts s

har

e th

e re

nt

equ

ally

,how

mu

ch w

ould

eac

h s

tude

nt

hav

e to

pay

? H

ow d

oes

the

cost

per

stu

den

t va

ry w

ith

th

e n

um

ber

ofst

ude

nts

? If

2 s

tude

nts

hav

e to

pay

$7

00 e

ach

,how

mu

ch m

oney

wou

ld

each

stu

den

t h

ave

to p

ay i

f th

ere

wer

e 5

stu

den

ts s

har

ing

the

ren

t?

Eac

h s

tud

ent

pay

s d

olla

rs.

Th

e co

st p

er s

tud

ent

vari

esin

vers

ely

wit

h t

he

nu

mb

er o

fst

ud

ents

,so

eac

h s

tud

ent

wo

uld

pay

$28

0.

m � n

4.PA

INTI

NG

Th

e co

st o

f pa

inti

ng

a w

all

vari

es d

irec

tly

wit

h t

he

area

of

the

wal

l.W

rite

a f

orm

ula

for

th

e co

st o

f pa

inti

ng

a re

ctan

gula

r w

all

wit

h d

imen

sion

s �

byw

.Wit

h r

espe

ct t

o �

and

w,d

oes

the

cost

vary

dir

ectl

y,jo

intl

y,or

in

vers

ely?

C�

k�w

,wh

ere

Cis

th

e co

st a

nd

kis

a c

on

stan

t.C

vari

es jo

intl

yw

ith

�an

d w

.

HY

DR

OG

ENF

or E

xerc

ises

5-7

,use

th

efo

llow

ing

info

rmat

ion

.

Th

e co

st o

f a

hyd

roge

n s

tora

ge t

ank

vari

esdi

rect

ly w

ith

th

e vo

lum

e of

th

e ta

nk.

Ala

bora

tory

wan

ts t

o pu

rch

ase

a st

orag

e ta

nk

shap

ed l

ike

a bl

ock

wit

h d

imen

sion

s L

by W

by H

.

5.F

ill

in t

he

mis

sin

g sp

aces

in

th

efo

llow

ing

tabl

e fr

om a

bro

chu

re o

fva

riou

s ta

nk

size

s.

6.T

he

hyd

roge

n t

ank

mu

st fi

t in

a s

hel

fth

at h

as a

fixe

d h

eigh

t an

d de

pth

.How

does

th

e co

st o

f th

e h

ydro

gen

sto

rage

tan

k va

ry w

ith

th

e w

idth

of

tan

k w

ith

fixe

d de

pth

an

d h

eigh

t?

Th

e co

st v

arie

s d

irec

tly

wit

h t

he

wid

th.

7.H

ow m

uch

wou

ld a

sph

eric

al t

ank

ofra

diu

s 24

in

ches

cos

t? (

Rec

all

that

th

e

volu

me

of a

sph

ere

is g

iven

by

�r3

,

wh

ere

ris

th

e ra

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s.)

$1,1

17.0

1

4 � 3

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rog

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ank

Dim

ensi

on

s (i

nch

es)

Co

st

LW

H

3636

36$9

0018

1824

$150

2424

72$8

00

Chapter 8 A16 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Lesson 8-5

Less

on R

eadi

ng G

uide

Cla

sses

of

Fu

nct

ion

s

Cha

pter

835

Gle

ncoe

Alg

ebra

2

Lesson 8-5

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-5 i

n y

our

text

boo

k.

•B

ased

on

th

e gr

aph

,est

imat

e th

e w

eigh

t on

Mar

s of

a c

hil

d w

ho

wei

ghs

40 p

oun

ds o

n E

arth

.ab

ou

t 15

po

un

ds

•A

lth

ough

th

e gr

aph

doe

s n

ot e

xten

d fa

r en

ough

to

the

righ

t to

rea

d it

dir

ectl

y fr

om t

he

grap

h,u

se t

he

wei

ght

you

fou

nd

abov

e an

d yo

ur

know

ledg

e th

at t

his

gr

aph

rep

rese

nts

dir

ect

vari

atio

n t

o es

tim

ate

the

wei

ght

on M

ars

of a

wom

an

wh

o w

eigh

s 12

0 po

un

ds o

n E

arth

.ab

ou

t 45

po

un

ds

Rea

d t

he

Less

on

1.M

atch

eac

h g

raph

bel

ow w

ith

th

e ty

pe o

f fu

nct

ion

it

repr

esen

ts.S

ome

type

s m

ay b

e u

sed

mor

e th

an o

nce

an

d ot

her

s n

ot a

t al

l.I.

squ

are

root

II.

quad

rati

cII

I.ab

solu

te v

alu

eIV

.ra

tion

alV.

grea

test

in

tege

rV

I.co

nst

ant

VII

.ide

nti

ty

a.III

b.

Ic.

VI

d.

IIe.

IVf.

V

Rem

emb

er W

hat

Yo

u L

earn

ed

2.H

ow c

an t

he

sym

boli

c de

fin

itio

n o

f ab

solu

te v

alu

e th

at y

ou l

earn

ed i

n L

esso

n 1

-4 h

elp

you

to

rem

embe

r th

e gr

aph

of

the

fun

ctio

n f

(x)

�|x

|?S

amp

le a

nsw

er:

Usi

ng

th

ed

efin

itio

n o

f ab

solu

te v

alu

e,f(

x)

�x

if x

�0

and

f(x

) �

�x

if x

�0.

Th

eref

ore

,th

e g

rap

h is

mad

e u

p o

f p

iece

s o

f tw

o li

nes

,on

e w

ith

slo

pe

1an

d o

ne

wit

h s

lop

e �

1,m

eeti

ng

at

the

ori

gin

.Th

is f

orm

s a

V-sh

aped

gra

ph

wit

h “

vert

ex”

at t

he

ori

gin

.

x

y

Ox

y Ox

y O

x

y

Ox

y

Ox

y

O

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

834

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-4

You

have

lear

ned

to s

olve

pro

blem

s in

volv

ing

dire

ct,i

nver

se,a

nd jo

int

vari

atio

n.M

any

phys

ical

sit

uat

ion

s in

volv

e at

lea

st o

ne

of t

hes

e ty

pes

of v

aria

tion

.For

exam

ple,

acco

rdin

g to

New

ton

’s l

aw o

f u

niv

ersa

l gr

avit

atio

n,t

he

wei

ght

of a

mas

s n

ear

Ear

th d

epen

ds o

n t

he

dist

ance

bet

wee

n t

he

mas

s an

d th

e ce

nte

r of

Ear

th.S

tudy

th

e sp

read

shee

t be

low

to

dete

rmin

e th

e ty

pe o

f va

riat

ion

th

atex

ists

bet

wee

n t

he

quan

tity

of

an a

stro

nau

t’s w

eigh

t an

d th

e di

stan

ce o

f th

eas

tron

aut

from

th

e ce

nte

r of

Ear

th.

In t

he

spre

adsh

eet,

the

valu

es f

or t

he

astr

onau

t’s w

eigh

t in

new

ton

s ar

e en

tere

din

th

e ce

lls

in c

olu

mn

A,a

nd

the

valu

es f

or t

he

astr

onau

t’s d

ista

nce

in

met

ers

from

the

cen

ter

of E

arth

are

ent

ered

in c

ells

in c

olum

n B

.Col

umn

C c

onta

ins

the

astr

onau

t’s d

ista

nce

fro

m E

arth

’s s

urf

ace.

Spre

adsh

eet

Act

ivit

yV

aria

tio

n

1.U

se t

he

valu

es i

n t

he

spre

adsh

eet

to m

ake

a gr

aph

of

the

astr

onau

t’s w

eigh

t pl

otte

d ag

ain

st t

he

astr

onau

t’s

dist

ance

fro

m E

arth

’s c

ente

r.

2.B

ased

on

you

r gr

aph

,is

this

an

in

vers

e or

dir

ect

vari

atio

n?

inve

rse

3.W

rite

an

equ

atio

n t

hat

rep

rese

nts

th

is s

itua

tion

.Let

Wre

pres

ent

the

astr

onau

t’sw

eigh

t,k

the

con

stan

t of

vari

atio

n,a

nd

R t

he

dist

ance

fro

m E

arth

’s c

ente

r.

W�

� RK2�

4.U

se t

he

equ

atio

n t

o fi

nd

the

wei

ght

of t

he

astr

onau

t at

th

ese

dist

ance

s fr

om E

arth

’s s

urf

ace.

(Hin

t:R

emem

ber

to a

dd t

hes

e va

lues

to

the

valu

e in

cel

l B

2 to

fin

d th

e di

stan

ce f

rom

Ear

th’s

cen

ter.

)a.

145,

300,

000

mb

.65

mc.

25,6

00 m

1.29

9615

N73

4.54

94 N

728.

7047

N

d.

300,

800,

700

me.

6580

mf.

180,

560

m0.

3168

72 N

733.

0515

N69

4.68

73 N

A1 32 4 5 6 7

BC

Gra

vita

tio

n.x

ls 734.

5843

712.

0675

548.

9825

111.

4406

2.64

2112

6,38

0,00

06,

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7,38

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,380

,000

106,

380,

000

010

010

0010

,000

100,

000

Ast

rona

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Wei

ght (

N)

Dis

tanc

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om E

arth

’s C

ente

r (m

)D

ista

nce

from

Ear

th’s

Sur

face

(km

)

Sh

eet

1S

hee

t 2

Sh

eet

3

Exer

cise

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Weight (N)

200

300

100 0

400

500

600

700

800

Dis

tan

ce (

mill

ion

s o

f m

eter

s)20

4060

100

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lessons 8-4 and 8-5)

Chapter 8 A17 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-5)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Exer

cise

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

837

Gle

ncoe

Alg

ebra

2

Lesson 8-5 Lesson 8-5

Iden

tify

Eq

uat

ion

sYo

u sh

ould

be

able

to

grap

h th

e eq

uati

ons

of t

he f

ollo

win

g fu

ncti

ons.

Fu

nct

ion

Gen

eral

Eq

uat

ion

Co

nst

ant

y�

a

Dir

ect V

aria

tio

ny

�ax

Gre

ates

t In

teg

ereq

uatio

n in

clud

es a

var

iabl

e w

ithin

the

gre

ates

t in

tege

r sy

mbo

l, ��

Ab

solu

te V

alu

eeq

uatio

n in

clud

es a

var

iabl

e w

ithin

the

abs

olut

e va

lue

sym

bol,

||

Qu

adra

tic

y�

ax2

�bx

�c,

whe

re a

�0

Sq

uar

e R

oo

teq

uatio

n in

clud

es a

var

iabl

e be

neat

h th

e ra

dica

l sig

n, �

�

Rat

ion

aly

�

Inve

rse

Var

iati

on

y�

Iden

tify

th

e fu

nct

ion

rep

rese

nte

d b

y ea

ch e

qu

atio

n.T

hen

gra

ph

th

e eq

uat

ion

.

1.y

�in

vers

e va

riat

ion

2.y

�x

dir

ect

vari

atio

n3.

y�

�q

uad

rati

c

4.y

�| 3

x|�

1ab

solu

teva

lue

5.y

��

inve

rse

vari

atio

n6.

y�

gre

ates

tin

teg

er

7.y

��

x�

2�

squ

are

roo

t8.

y�

3.2

con

stan

t9.

y�

rati

on

al

x

y

Ox

y

Ox

y O

x2�

5x�

6�

�x

�2

x

y

Ox

y

Ox

y

O

x � 22 � x

x

y Ox

y

Ox

y

O

x2� 2

4 � 36 � x

a � xp(x

)� q

(x)

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

Cla

sses

of

Fu

nct

ion

s

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

836

Gle

ncoe

Alg

ebra

2

Iden

tify

Gra

ph

sYo

u s

hou

ld b

e fa

mil

iar

wit

h t

he

grap

hs

of t

he

foll

owin

g fu

nct

ion

s.

Fu

nct

ion

Des

crip

tio

n o

f G

rap

h

Co

nst

ant

a ho

rizon

tal l

ine

that

cro

sses

the

y-a

xis

at a

Dir

ect V

aria

tio

na

line

that

pas

ses

thro

ugh

the

orig

in a

nd is

nei

ther

hor

izon

tal n

or v

ertic

al

Iden

tity

a lin

e th

at p

asse

s th

roug

h th

e po

int

(a,

a),

whe

re a

is a

ny r

eal n

umbe

r

Gre

ates

t In

teg

era

step

fun

ctio

n

Ab

solu

te V

alu

eV

-sha

ped

grap

h

Qu

adra

tic

a pa

rabo

la

Sq

uar

e R

oo

ta

curv

e th

at s

tart

s at

a p

oint

and

cur

ves

in o

nly

one

dire

ctio

n

Rat

ion

ala

grap

h w

ith o

ne o

r m

ore

asym

ptot

es a

nd/o

r ho

les

Inve

rse

Var

iati

on

a gr

aph

with

2 c

urve

d br

anch

es a

nd 2

asy

mpt

otes

, x

�0

and

y�

0 (s

peci

al c

ase

of r

atio

nal f

unct

ion)

Iden

tify

th

e fu

nct

ion

rep

rese

nte

d b

y ea

ch g

rap

h.

1.2.

3.

qu

adra

tic

rati

on

ald

irec

t va

riat

ion

4.5.

6.

con

stan

tab

solu

te v

alu

eg

reat

est

inte

ger

7.8.

9.

iden

tity

squ

are

roo

tin

vers

e va

riat

ionx

y

O

x

y O

x

y O

x

y

Ox

y

Ox

y

O

x

y

Ox

y O

x

y ONA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Stud

y G

uide

and

Inte

rven

tion

Cla

sses

of

Fu

nct

ion

s

Exer

cise

s

Chapter 8 A18 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

839

Gle

ncoe

Alg

ebra

2

Lesson 8-5

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

gra

ph

.

1.2.

3.

rati

on

alsq

uar

e ro

ot

abso

lute

val

ue

Mat

ch e

ach

gra

ph

wit

h a

n e

qu

atio

n b

elow

.

A.y

�| 2

x�

1|

B.y

��2

x�

1�C

.y�

D.y

��

�x

�

4.D

5.C

6.A

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

eq

uat

ion

.Th

en g

rap

h t

he

equ

atio

n.

7.y

��

38.

y�

2x2

�1

9.y

�

con

stan

tq

uad

rati

cra

tio

nal

10.B

USI

NES

SA

sta

rtup

com

pany

use

s th

e fu

ncti

on P

�1.

3x2

�3x

�7

to p

redi

ct it

s pr

ofit

or

loss

du

rin

g it

s fi

rst

7 ye

ars

of o

pera

tion

.Des

crib

e th

e sh

ape

of t

he

grap

h o

f th

e fu

nct

ion

.T

he

gra

ph

is U

-sh

aped

;it

is a

par

abo

la.

11.P

AR

KIN

GA

par

kin

g lo

t ch

arge

s $1

0 to

par

k fo

r th

e fi

rst

day

or p

art

of a

day

.Aft

er t

hat

,it

ch

arge

s an

add

itio

nal

$8

per

day

or p

art

of a

day

.Des

crib

e th

e gr

aph

an

d fi

nd

the

cost

of p

arki

ng

for

6da

ys.

Th

e g

rap

h lo

oks

like

a s

erie

s o

f st

eps,

sim

ilar

to a

g

reat

est

inte

ger

fu

nct

ion

,bu

t w

ith

op

en c

ircl

es o

n t

he

left

an

d c

lose

dci

rcle

s o

n t

he

rig

ht;

$58.

1 � 2

x

y

Ox

y O

x

y

O

x2�

5x�

6�

�x

�2

x

y

O

x

y

Ox

y O

x�

3�

2

x

y

O

x

y O

x

y

O

Lesson 8-5

Prac

tice

Cla

sses

of

Fu

nct

ion

s

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

838

Gle

ncoe

Alg

ebra

2

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

gra

ph

.

1.2.

3.

con

stan

td

irec

t va

riat

ion

qu

adra

tic

Mat

ch e

ach

gra

ph

wit

h a

n e

qu

atio

n b

elow

.

A.

y�

|x�

1|B

.y�

C.y

��

1 �

x�

D.y

��x

��

1

4.B

5.C

6.A

Iden

tify

th

e ty

pe

of f

un

ctio

n r

epre

sen

ted

by

each

eq

uat

ion

.Th

en g

rap

h t

he

equ

atio

n.

7.y

�8.

y�

2�x�

9.y

��

3x

inve

rse

vari

atio

n

gre

ates

t in

teg

erd

irec

t va

riat

ion

or

rati

on

al

x

y

Ox

y

Ox

O

y

2 � x

x

y

O

x

y Ox

y

O

1� x

�1

x

y O

x

y Ox

y O

8-5

Skill

s Pr

acti

ceC

lass

es o

f F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_ Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-5)

Chapter 8 A19 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-5)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Enri

chm

ent

Phy

sica

l Pro

per

ties

of

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

Cha

pter

841

Gle

ncoe

Alg

ebra

2

Lesson 8-5

Mat

hem

atic

al f

un

ctio

ns

are

clas

sifi

ed b

ased

on

pro

pert

ies

sim

ilar

to

how

bio

logi

sts

clas

sify

anim

al s

peci

es.F

un

ctio

ns

can

be

clas

sifi

ed a

s co

nti

nu

ous

or n

on-c

onti

nu

ous,

incr

easi

ng

orde

crea

sin

g,po

lyn

omia

l or

non

-pol

ynom

ial

for

exam

ple.

Th

e cl

ass

of p

olyn

omia

ls f

un

ctio

ns

can

be

furt

her

cla

ssifi

ed a

s li

nea

r,qu

adra

tic,

cubi

c,et

c.,b

ased

on

its

deg

ree.

Ch

arac

teri

stic

s of

fu

nct

ion

s in

clu

de:

•A

fu

nct

ion

is

bou

nd

ed b

elow

if t

her

e ex

ists

a n

um

ber

that

is

less

th

an a

ny

fun

ctio

nva

lue.

•A

fu

nct

ion

is

bou

nd

ed a

bov

eif

a n

um

ber

exis

ts t

hat

is

grea

ter

than

an

y fu

nct

ion

valu

e.•

A f

un

ctio

n i

s sy

mm

etri

c(a

bou

t a

vert

ical

axi

s) i

f it

is

a m

irro

r im

age

abou

t th

atve

rtic

al a

xis.

•A

fu

nct

ion

is

con

tin

uou

sif

it

can

be

draw

n w

ith

out

lift

ing

you

r pe

nci

l.•

A f

un

ctio

n i

s in

crea

sin

gif

f(x

)

f(y)

wh

en x

y.

Con

tin

ual

gro

wth

fro

m l

eft

to r

igh

t.•

A f

un

ctio

n i

s d

ecre

asin

gif

f(x

) �

f(y)

wh

en x

�y.

Con

tin

ual

dec

ay f

rom

lef

t to

rig

ht.

1.S

ketc

h t

he

grap

h o

f y

�x2

�5x

�6.

Lis

t th

e ch

arac

teri

stic

s of

fu

nct

ion

s di

spla

yed

by t

his

gra

ph.

So

me

pro

per

ties

incl

ud

e:sy

mm

etri

c,co

nti

nu

ou

s,b

ou

nd

ed b

elo

w.

2.W

hat

ch

arac

teri

stic

s do

abs

olu

te v

alu

e fu

nct

ion

s an

d qu

adra

tic

fun

ctio

ns

hav

e in

com

mon

? H

ow d

o th

ey d

iffe

r?

Co

mm

on

:S

ymm

etri

c,co

nti

nu

ou

s,b

ou

nd

ed

bel

ow

(o

r ab

ove)

.Dif

fer:

On

e is

U s

hap

ed

and

th

e o

ther

V s

hap

ed,o

ne

is s

mo

oth

an

d o

ne

has

a r

igid

co

rner

,an

d o

ne

incr

ease

s m

ore

rap

idly

th

an t

he

oth

er.

3.G

raph

y�

⏐x�

3 ⏐.

4.G

raph

y�

x2�

8x�

7.

y

x

y

x

y

x

Exer

cise

s

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

840

Gle

ncoe

Alg

ebra

2

Wor

d Pr

oble

m P

ract

ice

C

lass

es o

f F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-5

1.ST

AIR

SW

hat

typ

e of

a f

un

ctio

n h

as

a gr

aph

th

at c

ould

be

use

d to

mod

el

a st

airc

ase?

the

gre

ates

t in

teg

er f

un

ctio

n

2.G

OLF

BA

LLS

Th

e tr

ajec

tory

of

a go

lfba

ll h

it b

y an

ast

ron

aut

on t

he

moo

n

is d

escr

ibed

by

the

fun

ctio

n

f(x)

��

0.01

25(x

�40

)2�

20.

Des

crib

e th

e sh

ape

of t

his

tra

ject

ory.

a p

arab

ola

3.R

AV

INE

Th

e gr

aph

sh

ows

the

cros

s-se

ctio

n o

f a

ravi

ne.

Wh

at k

ind

of f

un

ctio

n i

s re

pres

ente

d by

the

grap

h?

Wri

te t

he

fun

ctio

n.

an a

bso

lute

val

ue

fun

ctio

n;

f(x)

�⏐x

�2 ⏐

�2

4.LE

AK

Y F

AU

CET

SA

lea

ky f

auce

t le

aks

1 m

illi

lite

r of

wat

er e

very

sec

ond.

Wri

te a

fu

nct

ion

th

at g

ives

th

e n

um

ber

of m

illi

lite

rs l

eake

d in

tse

con

ds a

s a

fun

ctio

n o

f t.

Wh

at t

ype

of f

un

ctio

n i

s it

?

f(t)

�t;

an id

enti

ty f

un

ctio

n

y

xO

y

x80

21 O

PUB

LISH

ING

For

Exe

rcis

es 5

-8,u

se t

he

foll

owin

g in

form

atio

n.

Kat

e h

as ju

st fi

nis

hed

wri

tin

g a

book

th

atex

plai

ns

how

to

sew

you

r ow

n s

tuff

edan

imal

s.S

he

hop

es t

o m

ake

$100

0 fr

omsa

les

of t

he

book

bec

ause

th

at i

s h

ow m

uch

it w

ould

cos

t h

er t

o go

to

the

Eu

rope

anS

ewin

g C

onve

nti

on.E

ach

boo

k co

sts

$2 t

opr

int

and

asse

mbl

e.L

et P

be t

he

sell

ing

pric

e of

th

e bo

ok.L

et N

be t

he

nu

mbe

r of

peop

le w

ho

wil

l bu

y th

e bo

ok.

5.W

rite

an

equ

atio

n t

hat

rel

ates

Pan

d N

if s

he

earn

s ex

actl

y $1

,000

fro

m s

ales

of

the

book

.

1000

�N

(P �

2)

6.S

olve

th

e eq

uat

ion

you

wro

te f

orE

xerc

ise

5 fo

r P

in t

erm

s of

N.

P�

7.W

hat

kin

d of

fu

nct

ion

is

Pin

ter

ms

ofN

? S

ketc

h a

gra

ph o

f P

as a

fu

nct

ion

of

N.

rati

on

al;

8.If

Kat

e th

inks

th

at 1

25 p

eopl

e w

ill

buy

her

boo

k,h

ow m

uch

sh

ould

sh

e ch

arge

for

the

book

?

$10Sale Price

050100

Nu

mb

er o

f B

uye

rs10

0

2N�

1000

�� N

Chapter 8 A20 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Cha

pter

843

Gle

ncoe

Alg

ebra

2

Lesson 8-6

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

8-6

Lesson 8-6

Solv

e R

atio

nal

Eq

uat

ion

sA

rat

ion

al e

qu

atio

nco

nta

ins

one

or m

ore

rati

onal

expr

essi

ons.

To

solv

e a

rati

onal

equ

atio

n,f

irst

mu

ltip

ly e

ach

sid

e by

th

e le

ast

com

mon

den

omin

ator

of

all

of t

he

den

omin

ator

s.B

e su

re t

o ex

clu

de a

ny

solu

tion

th

at w

ould

pro

duce

a de

nom

inat

or o

f ze

ro.

Sol

ve

��

.

��

Orig

inal

equ

atio

n

10(x

�1)

��

��10

(x�

1)�

�Mul

tiply

eac

h si

de b

y 10

(x�

1).

9(x

�1)

�2(

10)

�4(

x�

1)M

ultip

ly.

9x�

9 �

20 �

4x�

4D

istr

ibut

ive

Pro

pert

y

5x�

�25

Sub

trac

t 4x

and

29

from

eac

h si

de.

x�

�5

Div

ide

each

sid

e by

5.

Ch

eck

��

Orig

inal

equ

atio

n

��

x�

�5

��

Sim

plify

.

�

Sol

ve e

ach

eq

uat

ion

.

1.�

�2

52.

��

12

3.�

��

4.�

�4

�5.

�

76.

��

10

7.N

AV

IGA

TIO

NT

he

curr

ent

in a

riv

er i

s 6

mil

es p

er h

our.

In h

er m

otor

boat

Mar

issa

can

trav

el 1

2 m

iles

ups

trea

m o

r 16

mil

es d

own

stre

am i

n t

he

sam

e am

oun

t of

tim

e.W

hat

is

the

spee

dof

her

mot

orbo

atin

stil

lwat

er?

Isth

isa

reas

onab

lean

swer

?E

xpla

in.

42m

ph

;S

amp

le a

nsw

er:T

he

answ

er is

rea

son

able

.Th

e b

oat

will

tra

vel

48 m

ph

on

e w

ay a

nd

36

mp

h t

he

oth

er w

ay.T

her

efo

re it

will

tak

e o

f an

ho

ur

to t

rave

l 16

mile

s an

d 1

2 m

iles,

resp

ecti

vely

.

8.W

OR

KA

dam

,Bet

han

y,an

d C

arlo

s ow

n a

pai

nti

ng

com

pan

y.T

o pa

int

a pa

rtic

ula

r h

ouse

alon

e,A

dam

est

imat

es t

hat

it

wou

ld t

ake

him

4 d

ays,

Bet

han

y es

tim

ates

5da

ys,a

nd

Car

los

6 da

ys.I

f th

ese

esti

mat

es a

re a

ccu

rate

,how

lon

g sh

ould

it

take

th

e th

ree

of t

hem

to p

ain

t th

e h

ouse

if

they

wor

k to

geth

er?

Isth

isa

reas

onab

lean

swer

?ab

ou

t 1

day

s;S

amp

le a

nsw

er:

It is

a r

easo

nab

le a

nsw

er.I

t w

ill t

ake

each

per

son

abo

ut

5 d

ays

to p

ain

t th

e h

ou

se a

lon

e,so

it s

ho

uld

tak

e ab

ou

t o

f th

eti

me

to p

ain

t th

e h

ou

se t

og

eth

er.

1 � 3

2 � 3

1 � 2

1 � 3

8 � 34

� x�

2x

� x�

2x

�1

�12

4� x

�1

1 � 242m

�1

�2m

3m�

2�

5m

13 � 51 � 2

x�

5�

42x

�1

�3

4 �

2t�

34t

�3

�5

y�

3�

62y � 3

2 � 52 � 5

2 � 510 � 20

18 � 20

2 � 52

� �5

�1

9 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

Exer

cise

s

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

842

Gle

ncoe

Alg

ebra

2

Less

on R

eadi

ng G

uide

S

olv

ing

Rat

ion

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Get

Rea

dy

for

the

Less

on

Rea

d t

he

intr

odu

ctio

n t

o L

esso

n 8

-6 i

n y

our

text

boo

k.

•If

you

in

crea

se t

he

nu

mbe

r of

son

gs t

hat

you

dow

nlo

ad,w

ill

you

r to

tal

bill

in

crea

se o

rde

crea

se?

incr

ease

•W

ill

you

r ac

tual

cos

t pe

r so

ng

incr

ease

or

decr

ease

?d

ecre

ase

Rea

d t

he

Less

on

1.W

hen

sol

vin

g a

rati

onal

equ

atio

n,a

ny

poss

ible

sol

uti

on t

hat

res

ult

s in

in

th

e de

nom

inat

or m

ust

be

excl

ude

d fr

om t

he

list

of

solu

tion

s.

2.S

upp

ose

that

on

a q

uiz

you

are

ask

ed t

o so

lve

the

rati

onal

in

equ

alit

y �

0.

Com

plet

e th

e st

eps

of t

he

solu

tion

.

Ste

p 1

Th

e ex

clu

ded

valu

es a

re

and

.

Ste

p 2

Th

e re

late

d eq

uat

ion

is

��

0 .

To s

olve

thi

s eq

uati

on,m

ulti

ply

both

sid

es b

y th

e L

CD

,whi

ch i

s .

Sol

vin

g th

is e

quat

ion

wil

l sh

ow t

hat

th

e on

ly s

olu

tion

is

�4.

Ste

p 3

Div

ide

a n

um

ber

lin

e in

to

regi

ons

usi

ng

the

excl

ude

d va

lues

an

d th

eso

luti

on o

f th

e re

late

d eq

uat

ion

.Dra

w d

ash

ed v

erti

cal

lin

es o

n t

he

nu

mbe

r li

ne

belo

w t

o sh

ow t

hes

e re

gion

s.

Con

side

r th

e fo

llow

ing

valu

es o

f �

for

vari

ous

test

val

ues

of

z.

If z

��

5,�

�0.

2.If

z�

�3,

��

�1.

If z

��

1,�

�9.

If z

�1,

��

�5.

Usi

ng

this

in

form

atio

n a

nd

you

r n

um

ber

lin

e,w

rite

th

e so

luti

on o

f th

e in

equ

alit

y.

z�

�4

or

�2

�z

�0

Rem

emb

er W

hat

Yo

u L

earn

ed3.

How

are

th

e pr

oces

ses

of a

ddin

g ra

tion

al e

xpre

ssio

ns

wit

h d

iffe

ren

t de

nom

inat

ors

and

ofso

lvin

g ra

tion

al e

xpre

ssio

ns

alik

e,an

d h

ow a

re t

hey

dif

fere

nt?

Sam

ple

an

swer

:Th

eyar

e al

ike

bec

ause

bo

th u

se t

he

LC

D o

f al

l th

e ra

tio

nal

exp

ress

ion

s in

th

ep

rob

lem

.Th

ey a

re d

iffe

ren

t b

ecau

se in

an

ad

dit

ion

pro

ble

m,t

he

LC

Dre

mai

ns

afte

r th

e fr

acti

on

s ar

e ad

ded

,wh

ile in

so

lvin

g a

rat

ion

aleq

uat

ion

,th

e L

CD

is e

limin

ated

.

6 � z3

� z�

26 � z

3� z

�2

6 � z3

� z�

26 � z

3� z

�2

6 � z3

� z�

2

�3

�4

�5

�6

�2

�1

01

23

45

64

z(z

�2)

6 � z3

� z�

2

0�

2

6 � z3

� z�

2

0

Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-6)

Chapter 8 A21 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

-Hill

, a

divi

sion

of T

he M

cGra

w-H

ill C

ompa

nies

, In

c.

Answers (Lesson 8-6)

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Cha

pter

845

Gle

ncoe

Alg

ebra

2

Lesson 8-6

Skill

s Pr

acti

ceS

olv

ing

Rat

ion

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

Lesson 8-6

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

1.�

�1

2.2

��

3.�

�1

4.3

�z

�1,

2

5.�

56.

��

5,8

7.�

�3

8.�

�y

�7

3,4

9.�

810

.�

0

k�

0

11.2

��

0 �

v�

412

.n�

�n

��

3 o

r 0

�n

�3

13.

��

�0

�m

�1

14.

��

10

�x

�

15.

��

93

16.

�4

�4

17.2

��

�5

18.8

��

19.

��

�4

20.

��

�

21.

��

�22

.�

�2

23.

��

�6

24.

��

52

� t�

34

� t�

38

� t2�

92

� e�

21

� e�

22e

� e2�

4

5� s

�4

3� s

�3

12s

�19

��

s2�

7s�

122x

�3

� x�

1x

� 2x�

2x

�8

� 2x�

2

4� w

2�

41

� w�

21

� w�

22

� n�

35

� n2

�9

1� n

�3

2 � 58z

�8

� z�

24 � z

2q� q

�1

5 � 2q

b�

2� b

�1

3b�

2� b

�1

9x�

7� x

�2

15 � x

3 � 22 � x

1 � 2x5 � 2

3 � m1

� 2m

12 � n3 � n

5 � v3 � v

4 � 3k3 � k

x�

1� x

�10

x�

2� x

�4

12 � y3 � 2

2x�

3� x

�1

8 � ss

�3

�5

1� d

�2

2� d

� 1

2 � z�

6�2

9 � 3x

12 � 51 � 3

4 � n1 � 2

x� x

�1

Lesson 8-6

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

844

Gle

ncoe

Alg

ebra

2

Solv

e R

atio

nal

Ineq

ual

itie

sTo

sol

ve a

rat

iona

l ine

qual

ity,

com

plet

e th

e fo

llow

ing

step

s.

Ste

p 1

Sta

te t

he e

xclu

ded

valu

es.

Ste

p 2

Sol

ve t

he r

elat

ed e

quat

ion.

Ste

p 3

Use

the

val

ues

from

ste

ps 1

and

2 t

o di

vide

the

num

ber

line

into

reg

ions

.Tes

t a

valu

e in

eac

h re

gion

to

see

whi

ch r

egio

ns s

atis

fy t

he o

rigin

al in

equa

lity.

Sol

ve

�

.

Ste

p 1

Th

e va

lue

of 0

is

excl

ude

d si

nce

th

is v

alu

e w

ould

res

ult

in

a d

enom

inat

or o

f 0.

Ste

p 2

Sol

ve t

he

rela

ted

equ

atio

n.

��

Rel

ated

equ

atio

n

15n�

���

15n�

�M

ultip

ly e

ach

side

by

15n.

10 �

12 �

10n

Sim

plify

.

22 �

10n

Sim

plify

.

2.2

�n

Sim

plify

.

Ste

p 3

Dra

w a

nu

mbe

r w

ith

ver

tica

l li

nes

at

the

ex

clu

ded

valu

e an

d th

e so

luti

on t

o th

e eq

uat

ion

.

Tes

t n

��

1.T

est

n�

1.T

est

n�

3.

��

����

is t

rue.

��

is n

ottr

ue.

��

is t

rue.

Th

e so

luti

on i

s n

�0

or n

�2.

2.

Sol

ve e

ach

in

equ

alit

y.

1.�

32.

�4x

3.�

�1

�a

0

x

�o

r 0

�x

0

�p

�

4.�

5.

��

26.

�1

�2

�x

�0

x�

0 o

r �

x�

1x

��

1 o

r 0

�x

�1

or

x�

5o

r x

�2

1 � 2

2� x

�1

3� x2

�1

5 � x4

� x�

11 � 4

2 � x3 � 2x

39 � 201 � 2

1 � 2

2 � 34 � 5p

1 � 2p1 � x

3� a

�1

2 � 34 � 15

2 � 92 � 3

4 � 52 � 3

2 � 34 � 5

2 � 3

�3

�2

�1

01

22.2 3

2 � 34 � 5n

2 � 3n

2 � 34 � 5n

2 � 3n

2 � 34 � 5n

2 � 3n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nued

)

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

Exer

cise

s

Exam

ple

Chapter 8 A22 Glencoe Algebra 2

Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Wor

d Pr

oble

m P

ract

ice

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Cha

pter

847

Gle

ncoe

Alg

ebra

2

Lesson 8-6

1.H

EIG

HT

Ser

ena

can

be

desc

ribe

d as

bein

g 8

inch

es s

hor

ter

than

her

sis

ter

Mal

ia,o

r as

bei

ng

12.5

% s

hor

ter

than

Mal

ia.I

n o

ther

wor

ds,

�,

wh

ere

His

Ser

ena’

s h

eigh

t in

in

ches

.H

ow t

all

is S

eren

a?

56 in

ches

2.C

RA

NES

For

a w

eddi

ng,

Pau

la w

ants

to

fold

100

0 or

igam

i cr

anes

.

Sh

e do

es n

ot w

ant

to m

ake

anyo

ne

fold

mor

e th

an 1

5 cr

anes

.In

oth

er w

ords

,if

Nis

th

e n

um

ber

of p

eopl

e en

list

ed t

o

fold

cra

nes

,Pau

la w

ants

�

15.

Wh

at i

s th

e m

inim

um

nu

mbe

r of

peo

ple

that

wil

l sa

tisf

y th

is i

neq

ual

ity?

67

3.R

ENTA

LC

arlo

s an

d h

is f

rien

ds r

ent

aca

r.T

hey

spl

it t

he

$200

ren

tal

fee

even

ly.

Car

los,

toge

ther

wit

h ju

st t

wo

of h

isfr

ien

ds,d

ecid

e to

ren

t a

port

able

DV

Dpl

ayer

as

wel

l,an

d sp

lit

the

$30

ren

tal

fee

for

the

DV

D p

laye

r ev

enly

am

ong

them

selv

es.C

arlo

s en

ds u

p sp

endi

ng

$50

for

thes

e re

nta

ls.W

rite

an

equ

atio

nin

volv

ing

N,t

he

nu

mbe

r of

fri

ends

Car

los

has

,usi

ng

this

in

form

atio

n.S

olve

the

equ

atio

n f

or N

.

��

50;

N�

430 � 3

200

� N�

1

1000

�N

1 � 88

� H�

8

4.PR

OJE

CTI

LES

A p

roje

ctil

e ta

rget

is

lau

nch

ed i

nto

th

e ai

r.A

roc

ket

inte

rcep

tor

is fi

red

at t

he

targ

et.T

he

rati

o of

th

e al

titu

de o

f th

e ro

cket

to

the

alti

tude

of

the

proj

ecti

le t

seco

nds

aft

erth

e la

un

ch o

f th

e ro

cket

is

give

n b

y th

e

form

ula

.A

t w

hat

tim

e

are

the

targ

et a

nd

inte

rcep

tor

at t

he

sam

e al

titu

de?

at t

�3

seco

nd

s

FLIG

HT

TIM

EF

or E

xerc

ises

5 a

nd

6,u

seth

e fo

llow

ing

info

rmat

ion

.

Th

e di

stan

ce b

etw

een

New

Yor

k C

ity

and

Los

An

gele

s is

abo

ut

2500

mil

es.L

et S

beth

e ai

rspe

ed o

f a

jet.

Th

e w

ind

spee

d is

100

mil

es p

er h

our.

Bec

ause

of

the

win

d,it

tak

eslo

nge

r to

fly

one

way

th

an t

he

oth

er.

5.W

rite

an

equ

atio

n f

or S

if i

t ta

kes

2 h

ours

an

d 5

min

ute

s lo

nge

r to

fly

betw

een

New

Yor

k an

d L

os A

nge

les

agai

nst

th

e w

ind

vers

us

flyi

ng

wit

h

the

win

d.

��

2

6.S

olve

th

e eq

uat

ion

you

wro

te i

n

Exe

rcis

e 5

for

S.

500

mp

h

7.W

rite

an

equ

atio

n a

nd

fin

d h

ow m

uch

lon

ger

to fl

y be

twee

n N

ew Y

ork

and

Los

An

gele

s if

th

e w

ind

spee

d in

crea

ses

to15

0 m

iles

per

hou

r an

d th

e ai

rspe

ed o

fth

e je

t is

525

mil

es p

er h

our.

��

x;

x�

�2.

96 h

400

� 135

2500

��

525

�15

025

00�

�52

5�

150

1 � 1225

00� S

�10

025

00� S

�10

0

333t

��

��

32t2

�42

0t�

27

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

846

Gle

ncoe

Alg

ebra

2

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

8-6

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

1.�

�16

2.�

1 �

�1,

2

3.�

�,4

4.�

s�

4

5.�

�1

all r

eals

exc

ept

56.

��

0

7.�

t�

�5

or

��

t�

08.

��

9.�

�2

10.5

��

0 �

a�

2

11.

��

0 �

x�

712

.8 �

y

�0

or

y�

2

13.

��

p�

0 o

r p

�14

.�

��

15.g

��

�1

16.b

��

1 �

�2

17.

��

18.

�4

��

�5 3� ,5

19.

��

720

.�

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60%

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e th

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f(x)

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na’s

new

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e

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nsw

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rea

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.

28.O

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�

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lens

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th

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cen

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20 c

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��

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1 � 51 � 20

1 � f1 � q

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9 �

x� 19

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NA

ME

____

____

____

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ATE

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ER

IOD

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_ Copyright ©

Glencoe/M

cGraw

-Hill, a division of T

he McG

raw-H

ill Com

panies, Inc.

Answers (Lesson 8-6)

Chapter 8 A23 Glencoe Algebra 2

An

swer

s

Cop

yrig

ht ©

Gle

ncoe

/McG

raw

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, a

divi

sion

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he M

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c.

Answers (Lesson 8-6)

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Cha

pter

848

Gle

ncoe

Alg

ebra

2

Ob

liqu

e A

sym

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y�

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mes

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ser

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clos

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o th

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ne

as x

→ ∞

or x

→ �

∞.∞

is t

he

mat

hem

atic

al s

ymbo

l fo

r in

fin

ity,

wh

ich

mea

ns

end

less

.

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f(x

) �

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4 �

�2 x� ,y

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�4

is a

n o

bliq

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e be

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se

f(x)

�3x

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��2 x� ,

and

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or �

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d t

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→ 0

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6 is

an

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h f

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NA

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Enri

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__D

ATE

____

____

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8-6

Exam

ple