Algebra Study Guide

8/21/2019 Algebra Study Guide

1/54

End of Course Algebra I Study Guide:November, 2011

End-of-Course Exam Items Estimated Number of Questions (37

uestions!

Numbers & Operations 6-8

Function Characteristics 11-13

Linear Functions & Ineua!ities 11-13

"ata #na!$sis 6-8

Format o% eam uestions' 2( mu!tip!e choice, ) comp!etion items an* 3 short

ans+er uestions # raphin ca!cu!ator can be use* %or a!! uestions an* +i!! be

c!eare* at the start o% the eam # %ormu!a sheet an* raph paper +i!! be provi*e*

in the test boo.!et

Study Guide "rgani#ation:

On the net pae is the $e%ie& 'oi) Index /hese items are .e$ reen

stan*ar*s that +i!! be assesse* %or ra*uation /he number o% test uestions %or

each topic hea*in is sho+n to he!p ui*e $our stu*$ priorities Fo!!o+in the In*e

are Samle Questions %or each topic #%ter each set o% uestions, there are

STUDY NOTES that ep!ain the ans+ers

1


2/54

$e%ie& 'oi) Index:

(A*+ A7! Numbers , "erations -. uestions

Compare & Or*er ea! Numbers

va!uate epressions +ith variab!es

ponents, roots, use properties to eva!uate

va!uate ponentia! Functions#rithmetic & 4eometric 5euences

5o!vin euations +ith severa! variab!es' e # 7 prt

(A3! /un)tion C0ara)teristi)s 11-13 uestions

e!ations an* Functions

"omain, rane, 9n*in roots

Functions *e9ne* piece+ise

In*epen*ent, *epen*ent variab!es

:u!tip!e representations' s$mbo!ic, raph, tab!e, +or*s

Connections bet+een representations

va!uate % at a, so!vin %

(A2! inear /un)tions , Ineualities 11-13 uestions

;rite & 5o!ve Linear uations

4raph Linear uations


3/54

Samle Question A*A1: Or*er the %o!!o+in %rom reatest to !east

, 3 , 8(, 8, , (3

o # 8, 8(, , (3 , 3

o @ , 3 , 8, 8(, , (3

o C (3 , 3 , , , 8(, 8

o " 3 , (3 , , , 8(, 8

Ans&er: 4

Samle Question A*A*: # starAs co!or ives an in*ication o% temperature an*

ae /he chart sho+s %our t$pes o% stars an* the !o+est temperature o% each t$pe

List the temperatures in or*er %rom !o+est to hihest

'ye o&est

'emerature

Color

# 13) @!ue-

;hite@ 208 @!ue

4 (0 Be!!o+

< ) @!ue

o # 13) , 208 , ) , (0

o @ 13) , ) 208 , (0

o C (0 , 13) , 208 , )

o " (0 , ) 13) , 208 ,

Ans&er: C

STUDY NOTES:

;hen or*erin numbers, a!+a$s be sure +hich or*er, reatest to !east or !east to

reatest

3


4/54

ea! numbers inc!u*e scienti9c notation, %ractions, *ecima!s, eponents an*

ra*ica!s 5ubsets o% rea! numbers are natura!, +ho!e, countin, inteers, rationa!

an* irrationa! numbers Not inc!u*e*' imainar$ numbers i

;hen comparin or or*erin numbers, the %astest +a$ is to use $our ca!cu!ator an*

convert a!! numbers to *ecima! approimations, then or*er 5cienti9c notation is a

+a$ to +rite ver$ !are or ver$ sma!! numbers /he base number is a!+a$s +ritten

as one p!ace %o!!o+e* b$ a *ecima!' 1233 , not as 1233

to convert %rom to stan*ar*, move the *ecima! 3 p!aces to the riht, From

)2 to )200

to convert %rom to stan*ar*, move the *ecima! 3 p!aces to the !e%t, From

)2 to 000)2

a*ica!s' ' to approimate, use $our ca!cu!ator It is the opposite o% eponents,

it un*oes an eponent

/o convert %rom a %raction to a *ecima!, *ivi*e the top b$ the bottom

5amp!e item %or


5/54

Ans&er: A

)


6/54

(A*5! Numbers , "erationsva!uate epressions +ith variab!es

Samle Question A*51: For +hat va!ues o% a is an inteerE

o # a 7 1, 0o @ a 1o C a 7 1, a G 0o " a H 1

Ans&er: C

Samle Question A*5*: va!uate 2+ 6$2 %or + 7 an* $ 7 3

o # 330o

@ 62o C 60o "

Ans&er: 5

STUDY NOTES:

va!uate means to 9n* the va!ue o% an a!ebraic epression b$ substitutin a

number %or each variab!e an* simp!i%$in b$ usin or*er o% operations


7/54

7 3, 7 3, #bso!ute va!ue is the *istance %rom 0 to the epression insi*e

the brac.ets

(A*C! Numbers , "erations

ponents, roots, use properties to eva!uate

Samle Question A*C1: 5imp!i%$' 2-2 32)

223-3)2

o # 3)

)2

o @ 232)o C 3)

2)

o " 32

2)

Ans&er: C

Samle Question A*C*: 5imp!i%$'

o #

o @

o C correct

o "

Samle Question A*C3: 5imp!i%$ the epression usin positive eponents

o # 3

o @ )

o C 20

o "

Ans&er: C

STUDY NOTES:


8/54

27 2M2M2M2 7 16

;hen simp!i%$in eponents, chane neative eponents to %raction %orm as sho+n

be!o+

1

23

/o simp!i%$ suare or cube roots, !oo. %or %actors that are per%ect suares' 7

7 2

Or per%ect cube roots'

(A7A5! Numbers , "erationsva!uate ponentia! Functions, approimate so!utions usin raphs or tab!es

Samle Question A7A1: Bou +on a *oor prie an* are iven a choice bet+een

t+o options #' 1)0 investe* %or 10 $ears at P compoun*e* annua!!$, or @' 200

investe* %or 10 $ears at 3P compoun*e* annua!!$ ;hich p!an is best an* +hat is

the 9na! amount o% the investmentE Investment 7


9/54

/$pica! eamp!es are compoun* interest an* ro+th rates /he raph o% an

eponentia! %unctions is not !inear, it increases at a proressive!$ %aster rate

ponentia!!$

For eamp!e' popu!ation ro+th o% cric.ets' % 72 mo*e!s the ro+th in +ee.s

o% initia! cric.ets, +ith a base ro+th rate o% 2 In 8 +ee.s, +e +i!! have %8

728 7 1,02 cric.ets

#n eamp!e o% an eponentia! raph o% these


10/54

o "

Ans&er: A

(A3! /un)tion C0ara)teristi)s

$elations and /un)tions

Samle Question A3A1: ;hich o% the %o!!o+in euations !iste* be!o+ *etermine$ as a %unction o% EA6x7 3Qy7 2x 156y 7x2 1Qx7 0C6 x2y27 1Q x7y46 3xJ 2y7 Q y7 2x

Ans&er: 4

5amp!e item %or


11/54

Ans&er: C

Samle Question A3A3: ;hich o% the %o!!o+in tab!es, situations, or raphsrepresent a %unctionE

A6 /he ae in $ears o% each stu*ent in $our math c!ass an* each stu*entAs shoesie56 /he number o% *erees a person rotates a spiot an* the vo!ume o% +ater thatcomes out o% the spiotC6Roursstu*ie*

2 2 3 1

5core ontest

8)P

(2P

(0P

0P

46

Ans&er: 5

STUDY NOTES:

# %unction is a re!ation +here ever$ input has eact!$ one output Functions can beepresse* in severa! +a$s'

A story roblem6 ver$ stu*ent in the c!ass an* their e$e co!or +ou!* be a%unction since ever$one on!$ has one e$e co!or It is OS that t+o stu*ents have thesame co!or Ro+ever, e$e co!or an* ever$ stu*ent +ou!* not be a %unction since theco!or b!ue miht have more than one stu*ent associate* +ith it

A gra06 /o 9ure out i% a raph is a %unction, +e *o +hat is ca!!e* the vertica! !inetest I% $ou ta.e a vertica! !ine an* s+eep the entire raph %rom riht to !e%t an* iton!$ hits the raph once at a!! times, then it is a %unction 5o the 9rst raph is a%unction, but the secon* raph isnAt'

11


12/54

A table6 /he !e%t han* si*e o% the tab!e i% in co!umns or the top o% the tab!e i% inro+s is usua!!$ the input, or va!ues o% the %unction /he riht han* si*e orbottom is usua!!$ the output I% there are an$ repeats o% the input or va!ues+here the output is *i?erent, then it is not a %unction /he 9rst tab!e is a %unction,the secon* tab!e is not

1 2 3

$ 3 3 )

$1 31 2 )3 6

An euation: f(x! /his notation, rea* % o% , sa$s that % is a %unction +here isthe input #n$ time $ou see an euation +ritten as % 7 it is sa$in it is a%unction

y I% the euation has on!$ $As an* As in it, then this is a bit har*er #s!on as $ou can manipu!ate the euation so it is a pure $ 7 then it is a %unction $2

7 *oesnAt count/he 9rst 3 are %unctions, the !ast 2 are not'

yJ 3x7 2y 1 1)( 2 = xxf 93 += xy x2y27 1 3=y


4omain and $ange

Samle Question A3A2: # %unction f(n)7 60nis use* to mo*e! the *istance inmi!es trave!e* b$ a car trave!in 60 mi!es per hour in nhours I*enti%$ the *omainan* rane o% this %unction ;hat restrictions on the *omain o% this %unction shou!*be consi*ere* %or the mo*e! to correct!$ reTect the situationEA6 "omain' hours car trave!e* ane' *istance car trave!e* estriction' hours must be reater than or eua! to ero

56 "omain' *istance car trave!e* ane' hours car trave!e* estriction' *istance must be reater than or eua! to eroC6 "omain' hours car trave!e* ane' mi!es per hour car trave!e* estriction' hours must be reater than or eua! to ero46 "omain' *istance car trave!e* ane' mi!es per hour car trave!e* estriction' *istance must be reater than or eua! to ero

12


13/54

Ans&er: A

Samle Question A3A: ;hat is the *omain an* rane o% 5)( = xxf E

A6 "omain' xcan be a!! rea! numbers ane' f(x)can be a!! rea! numbers

56 "omain' x7 ) ane' f(x) 7 0C6 "omain' 5x ane' f(x)is !ess than or eua! to ero46 "omain' 5x ane' f(x)is reater than or eua! to ero

Ans&er: 4

Samle Question A3A: @e!o+ is the raph o% 512)( += xxf "etermine the

*omain an* rane o% this %unction

A6 "omain' 0x ane' 0)( xf

56 "omain' 5x

ane' 5)( xfC6 "omain' xcan be a!! rea! numbers

ane' 5)( xf46 "omain' xcan be a!! rea! numbers

13


14/54

ane' f(x)can be a!! rea! numbers

Ans&er: C

STUDY NOTES:

4omainis the input, norma!!$x, o% the %unction 5ometimes it +ou!* be easier tothin. o% +hat canAt be in the *omain ane is the output, norma!!$y or f(x). /hereare %our +a$s to thin. o% *omain an* rane''able: /hexco!umn or ro+ *epen*in on ho+ the tab!e is oriente* sho+s theva!ues o% the *omain an* theyco!umn or ro+ sho+s the va!ues o% the rane@e!o+ the *omain +ou!* be 1, 2, 3, an* the rane +ou!* be ), 6, , 8 1 2 3 $ ) 6 8Gra0: Loo. a!on thex-ais I% there is a raph *ra+n above, throuh, or be!o+thex-ais at that spot, then thatx-va!ue is in the *omain Li.e+ise, !oo. a!on the

y-ais I% there is a raph *ra+n to the !e%t, throuh, or to the riht o% that spot,

then thaty-va!ue is in the rane @e!o+ the *omain +ou!* be %rom 56 x estimation an* the rane +ou!* be %rom 5)(2 xf the rane cou!* a!so be

+ritten as 52 y

Euation: Loo.in at *omain 9rst, there are on!$ t+o thins $ou canAt *o' *ivi*eb$ ero or ta.e the suare root o% a neative number Other+ise the *omain can be

a!! rea! numbers 5o %or eamp!e, 24)( = xxf has no *ivision sin an* no suare

root sin so the *omain is a!! rea! numbers Ro+ever, 9)( = xxf has a suare

root sin over thexso the *omain can on!$ be numbers reater than or eua! to (,

since i% I +ere to put an 8 into the euation that +ou!* be the suare root o%

neative 1 Li.e+ise,x

xf 9)( = has a *ivision sin +herexis in the bottom so the

*omain can be ever$thin ecept +hat ma.es the bottom ero In this case the*omain is ever$thin but 0

No+ consi*er rane /his is much har*er in an euation First *etermine the*omain an* then thin. o% a!! the va!ues that %unction can be %or those va!ues in the

*omain /a.in our previous eamp!es 9)( = xxf has a rane that is reater

than or eua! to ero because puttin in a (, 10, 11, etc +ou!* ive $ou ero or a

1


15/54

positive number Li.e+ise,x

xf 9)( = can be a!! numbers but ero since +e can have

both positive an* neativex-va!ues

Story 8roblem: ;hatever is the input, or the in*epen*ent va!ue, o% the stor$prob!em is the *omain I% there is a stor$ prob!em about peop!e, then $our *omaincanAt be neative or %ractions since part o% a person *oesnAt ma.e since Li.e+ise, i%the stor$ prob!em is about mi!es, then $our *omain must be reater than or eua! toero since it *oesnAt ma.e sense to have neative mi!es ;ith rane, it is the sameconsi*eration as *omain ecept rane is the output, or the *epen*ent va!ue 5o i%+e have a %unction +here the input is hours an* the output is mi!es, both *omainan* rane +ou!* be reater than or eua! to ero

1)


16/54


/un)tions 4e9ned 8ie)e&ise

Samle Question A3A7: #ccor*in to the >5


17/54

STUDY NOTES:/un)tions 4e9ned 8ie)e&ise


18/54

Samle Question A3A11: :ari!$n .ept trac. o% ho+ ta!! her son +as ever$ $earsince he +as born I% :ari!$n +ere to +rite a %unction re!atin heiht, h, an* $ears,

y, +hat +ou!* be the in*epen*ent variab!eEA6 heiht, h56 $ears,yC6 heiht per $ear, h/y46 $ear per heiht,y/h

Ans&er: 5

Samle Question A3A1*: /amm$ +ante* to be a pro%essiona! bas.etba!! p!a$erver since she +as a chi!* she +ou!* .eep trac. o% +hat +as the averae sa!ar$ %ora %ema!e bas.etba!! p!a$er since the $ear 1(() I% /amm$ +ere to +rite a %unctionre!atin sa!ar$, s, an* $ears,y, +hat +ou!* be the *epen*ent variab!eEA6 sa!ar$, s56 $ears,yC6 sa!ar$ per $ear, s/y46 $ear per sa!ar$,y/s

Ans&er: A

STUDY NOTES:Indeendent , 4eendent ;ariables

/o ana!$e in*epen*ent an* *epen*ent variab!es, it is o%ten easier to 9ureout the *epen*ent variab!e 9rst /he deendentvariab!e *epen*s on the

indeendentvariab!e For eamp!e, i% a %unction is re!atin mi!es an* hours *rivenone cou!* see that the number o% mi!es +ou!* *epen* on ho+ !on one is on theroa*, hours /here%ore mi!es +ou!* be the *epen*ent variab!e an* hours +ou!* bethe in*epen*ent variab!e Li.e+ise, a heiht o% a tree *epen*s on the number o%$ears it has ro+n, not the other +a$ aroun*, so heiht +ou!* be the *epen*entvariab!e an* $ears +ou!* be the in*epen*ent variab!e

I% a re!ationship is +ritten in %unction notation %or our tree eamp!e so thath(t)=3t+here tis time in $ears an* h(t)is heiht o% the tree in inches, then the*epen*ent variab!e +ou!* be the variab!e insi*e the h(t), name!$ t, an* thein*epen*ent variab!e +ou!* be outsi*e o% the notation, name!$ h

18


19/54


=ultile $eresentations

Samle Question A351: /he chart sho+s the amount o% tota! sa!ar$ commissionp!us base sa!ar$ pai* to emp!o$ees o% a car *ea!ership in one +ee.

1000(0080000600)0000300200

1 2 3 ) 6 8 (

Number of

cars sold

1 2 3 ) 8

Total Salary 200

300

00

600

800

(00

;hich euation best represents tota! sa!ar$ T that an emp!o$ee ma.es %or se!!inan$ number o% cars, c, in a +ee.EA6 T(c)7 100c 10056 T(c)7 100c 200C6 T(c) 7 100c 20046 T(c)7 c 200

Ans&er: A

1(


20/54

Samle Question A35*: # popu!ation o% bacteria *oub!e* ever$ hour, startin+ith bacteria /he chart be!o+ represents the tota! popu!ation,p, at a iven hour,t

0

10

20

30

40

50

60

70

1 2 3 4

Number of

bacteria per

hour

;hich euation best represents the tota! popu!ation P o% bacteria at a iven hour,tA6 P(t)7 2t56 P(t)7 2t C6 P(t)7 t

46 P(t) = 2t

Ans&er: 4

Samle Question A353: /he raph be!o+ sho+s a mo*e! %or the re!ationshipbet+een the ban.in an!e an* the turnin ra*ius %or a bic$c!e trave!in at aparticu!ar spee* For the va!ues sho+n, the ban.in ane! Ban* the turnin ra*iusrcan be approimate* b$ an inverse variation

Radius 2 8Banking

Angle)60

280

160

10

;hich euation best represents the ban.in an!e B%or the turnin ra*ius rEA6 B(r) = 28r56 B(r) 7 2rJ 28

20

our 0 1 2 3Number ofbacteria

8 16

32


21/54

C6 3134)( += rrB

46r

rB 112)( =

Ans&er: 4

STUDY NOTES:=ultile $eresentations

ver$ chart, tab!e, euation, an* stor$ prob!em correspon*s to a set o%or*ere* pairs /o match a raph to a tab!e, or a tab!e to an euation, or an euationto a raph, the or*ere* pairs in one %orm must match the other %orm For eamp!e,i% a raph oes throuh the points 2, ), 3, 6 an* , it +ou!* correspon* to theeuation f(x)7x 3 because f(2)7 ), f(3) 7 6 an* f(4)7 Li.e+ise i% a raphoes throuh the same points, it +ou!* correspon* to the tab!e'

! 2 3

f"!#

) 6


Conne)tions 5et&een $eresentations

Samle Question A352: ;hich euation !iste* be!o+ best represents the set o%*ata in the tab!eE

! -2

-1

1 3 6

y -

-

1

) 1

1

2

0

A6y7 3x 256y7 J 3x 2C6y7xJ 246y7 JxJ 6

Ans&er: A

Samle Question A35: ;hich %unction inc!u*es a!! o% the or*ere* pairs in thetab!eE

! -2

0 1 2 3

f"!#

) -3

-1

) 1)

A6 f(x)7xJ 356 f(x)7x C6 f(x)7 2x2J 3

46 27)( += xxf

21


22/54

Ans&er: C

Samle Question A35: ;hich euation !iste* be!o+ best represents the set o%*ata in the tab!eE

! -3

-1

2 3 )

y -

8

-

2 8

A6xJy7 256 2xJy7 2C6xy7 J 246x 2y7 J 2

Ans&er: 5

STUDY NOTES:Conne)tions 5et&een $eresentations

For euations to match *ata, the euation must be true %or a!! *ata points Ina tab!e, the top ro+ or !e%t co!umn i% vertica! is +hat $ou input into the euation

/he bottom ro+ or riht co!umn i% vertica! is +hat $ou shou!* et out o% theeuation I% ever$ *ata point matches %or an euation, that euation bestrepresents the *ata For eamp!e, consi*er the tab!e'

! -2

0 3 )

y -2

-6

3 10

1(

/he euation %or this tab!e isy7x2J 6 because it +or.s %or a!! va!ues y = x+or.s%or -2, -2 but not %or 0, -6 y = xJ 6 +or.s %or 0, -6 but not %or 3, 3

For euations to match raphs, the same thin app!ies ach point a!on theraph can be represente* as x,y 5o an euation must enerate a!! the points onthe raph Note that i% an euation is represente* in %unction notation, f(x), thenthe euation +or.s %or *ata i% +hen substitutinxinto the euation it ives $ou f(x).


E%aluate f"!#at a+ sol%ing f"!#

Samle Question A3C1: For the %unction 13)( += xxf , 9n* )5(f

A6 J 1256 J 2C6 1246 18

22


23/54

Ans&er: C

Samle Question A3C*: For the %unction 64)( = xxf , 9n* )2(f A6 J 156 J C6 J 246 2

Ans&er: A

Samle Question A3C3: For the %unction 3)( += xxf , 9n* )6(f

A62

1

56 6

C6 346 (

Ans&er: C

Samle Question A3C2: For the %unction 63)( += xxf , 9n*x+hen 0)( =xf A6 J 256 0C6 646 (

Ans&er: A

Samle Question A3C: For the %unction 179)( = xxf , 9n*x+hen 1)( =xf A6 J 156 J 8C6 146 2

Ans&er: 4

Samle Question A3C: For the %unction 92)( = xxf , 9n*x+hen 3)( =xf A6 J (56 J 3C6 3

23


24/54

46 6

Ans&er: 4

STUDY NOTES:

E%aluate f"!#at a+ sol%ing f"!#

f(x) means the %unction +ith respect tox It *oes NO/ mean f timesx ;hen $ousee f(x)$ou automatica!!$ .no+ that +hat %o!!o+s the eua! sin is a %unction /hein*epen*ent variab!e, or the input, isxan* the *epen*ent variab!e, or the output isf(x) /a.e the %unction f(x) = 3x 5ometimes it is easier Kust to thin. o% it asy73x since that is ho+ +e +rite euations more o%ten

E%aluating f"!# $ 3!> 7 at *6 In ot0er &ords+ 9nd f "%#&>sin the %ami!iar euationy7 3x , I cou!* as. +hat is the va!ue o%yi%xis 2E

/o ans+er this I +ou!* substitute a 2 into thexan* ety7 13 #nother +a$ to as.the uestion +ou!* be to use the 9rst notation an* sa$ 9n* f 2 I +ou!*substitute a 2 into thexan* et f 2)7 13 5ee the %o!!o+in eamp!es %oreva!uatin a %unction at a particu!ar point'

Fin* f 3 i% f(x)7 xJ #ns+er'ep!ace thex+ith a 3f 3 7 3 J 3 J 7 J 15o f 3 7 J 1

Fin* f i% 9)( = xxf

#ns+er'ep!ace thex+ith a

f 7 94

5594 ==

5o f 7 )

2


25/54

Sol%ing f"!#? 3!> 7 for!&0en f(!! ? 16

>sin the %ami!iar euationy7 3x , I cou!* as. +hat is the va!ue o%x+henyis1E /o ans+er this I +ou!* substitute a 1 %or theyan* so!ve %orx 1 7 3x +ou!* ive me an ans+er o%x7 J 2 #nother +a$ to as. the uestion +ou!* be touse the 9rst notation an* sa$ 9n*x+hen f(x)7 1 I +ou!* substitute a 1 %or thef(x)an* et 1 7 3x 5ee the %o!!o+in eamp!es %or so!vin a %unction %orx+hen f(x)is iven'

For the %unction f(x) 7 xJ 1, 9n*xi% fx 7 3#ns+er'ep!ace f(x)+ith a 33 7 xJ 13 1 7 xJ 1 1

7 xD 7 xD1 7x5ox7 1 +hen f(x)7 3

For the %unction f(x)7 2xJ 20, 9n*xi% f(x)7 8#ns+er'ep!ace f(x)+ith an 88 7 2xJ 208 20 7 2xJ 20 2028 7 2x28D2 7 2xD2

1 7x5ox7 1 +hen f(x)7 8

2)


26/54

(A2! Core Content: inear fun)tions+ euations+ and ineualities

(A2A! @rite and sol%e linear euations and ineualities in one %ariable

Samle Question A2A1'

5o!ve the euation an* chose the correct ans+er'

y 12 = 4

#y7 -16@y7 -8Cy7 8"y7 16

Ans&er: C

STUDY NOTES:#** 12 to both si*es o% the euation

Samle Question A2A*'

5o!ve the ineua!it$ an* chose the correct ans+er'

x + 5 8

# xH -3@ xH 3

C xH 11" xV 3

#ns+er' @

STUDY NOTES:#** -) to both si*es o% the ineua!it$


26


27/54

4eore has eiht more than t+ice the number o% marb!es $ou have ;rite anineua!it$ %or the number o% marb!es 4eore has

Chose the correct ans+er'

# 8 J 2n@ 2n- 8

C 8 -2n" 2n 8

Ans&er: 4

STUDY NOTES:Let n be the un.no+n so t+ice the un.no+n +ou!* be 2n iht more than t+icethe un.no+n +ou!* be 2n 8


;rite an abso!ute va!ue euation %or a!! the numbers 2 units %rom

Chose the correct ans+er'

# x +7 = 2

@ x + 2 =7

C x 7 = 2

" x 2 =7

Ans&er: C

STUDY NOTES:

x b =a

is an abso!ute va!ue representin a!! the numbers that are aunits %rom b

Samle Question A2A'

/he euation 2x 1 10 = 4 has t+o rea! so!utions

2


28/54

"etermine the neative so!ution o% the euation an* chose the correct ans+er

# -1@ -2C -3" -

Ans&er: 5

STUDY NOTES:Iso!ate the abso!ute va!ue 9rst usin the properties o% eua!it$ #** 10 to each si*ean* *ivi*e b$ 2 Bou +i!! then have

-1 7 3 or -17-3 /he neative so!ution is -2

Samle Question A2A'

5o!ve %or ' 2 J 3 7 1 2

Choose the correct ans+er'

# 2@ )C " 6

Ans&er: 5

STUDY NOTES:2 - 3 7 1 2

*istribute 2 J 6 7 1 2combine !i.e terms 6 - 6 7 1 2a** -2 -2 -2

J 6 7 1a** 6 7 20*ivi*e b$ 7)


5o!ve x 6 4

Choose the correct ans+er' an* be ab!e to raph the so!ution on a number !ine

# 2 VxV 10

28

x =1 3


29/54

@ 2 HxH 10CxV 2 an*xV 10"xH 2 an*xH 10

Ans&er: 5

STUDY NOTES:

means J b V c an* J b H -c

STUDY NOTES:

;hen so!vin !inear euations $ou must use the properties o% eua!it$I% a7b then ac 7 bc, a** the same number to each si*eor a-c7b-c , subtract the same number %rom each si*eor a c 7 b c , mu!tip!$ the same number b$ each si*e

an* i% c is not 0 then

;hat $ou *o to one si*e o% the euation $ou *o to the other

#!so remember to simp!i%$ each si*e o% the euation b$ usin the *istributivepropert$ ab c 7 ab ac an* combinin !i.e terms %or eamp!e 2x + 5x7 1shou!* be simp!i9e* to 7x

5o!vin !inear ineua!ities +or. the same +a$ +ith one eception' +hen $oumu!tip!$ or *ivi*e each si*e b$ a neative

2(

x b c

a

c

b

c

=


30/54

(A25! @rite and gra0 an euation for a line gi%en t0e sloe and t0e y-inter)et+ t0e sloe and a oint on t0e line+ or t&o oints on t0e line+ andtranslate bet&een forms of linear euations6

5amp!e item %or


31/54

STUDY NOTES:se the euation y 7 = 2(x 2) +ithout s.etchin the raph to *escribe theraph

# the raph has a neative s!ope an*y-intercept o% -2@ the raph has a neative s!ope an*y-intercept o% 3C the raph has a positive s!ope an*y-intercept o% -2" the raph has a positive s!ope an*y-intercept o% 3

Ans&er: 4

STUDY NOTES:>sin the point s!ope %orm o% the !ine, the raph has a positive s!ope o% 2 an* oesthrouh the point 2, 5o!vin the euation %or $ $ou +ou!* et $72 3 5o, theraph has a positive s!ope an* a $-intercept o% 3

Samle Question A25'

;rite the euation 3x + 2y = 5 in s!ope-intercept %orm

31


32/54

5!ope Intercept %orm is' WWWWWWWWWWWWWWWWWWWWWWW

Ans&er:2

5

2

3+= xy

STUDY NOTES:5o!ve the euation %or $ First subtract 3 %rom both si*es then *ivi*e both si*es b$2

Samle Question A25'

;rite the euation y 1= 2(x 2) in stan*ar* %orm

# 2 $ 7-3@ -2 $ 7 3C 2x y =3

" $ 7 2 - 3

Ans&er: C

STUDY NOTES:5tan*ar* %orm is +ritten # @$ 7 C, +here #,@, an* C are inteers an* # ispositive

Samle Question A257'

;rite the euation y =1

2x + 5 in stan*ar* %orm

# x + 2y =10

@ x 2y = 10

C y 1

2x = 5

" y 1

2x =10

32


33/54

Ans&er: 5

STUDY NOTES:5tan*ar* %orm is +ritten # @$ 7 C, +here #,@, an* C are inteers an* # ispositive :u!tip!$ both si*es o% the euation b$ 2, ettin 2$ 7 10, thensubtract both si*es o% the euation b$ 2$ an* 10 ettin J 2$ 7 -10

33


34/54

(A2C! Identify and interret t0e sloe and inter)ets of a linear fun)tion+in)luding euations for arallel and erendi)ular lines6

e%er to the raph be!o+ to ans+er the %o!!o+in uestions about three *i?erent ce!!phone p!ans' @!ue, 4reen, an* e* /he raphs re!ate tota! minutes per month use*%or each ce!! phone p!an an* month!$ cost in *o!!ars

5amp!e item %or


35/54

Ans&er: ASamle Question A2C1'

;hat can be sai* about the initia! cost o% each ce!! phone p!anE

# #!! the p!ans bein +ith the same cost

@ /he e*

Ans&er: A

STUDY NOTES:

#t 300 minutes' @!ue p!an- 0 e* p!an- )0 4reen p!an-60

Samle Question A2C3'

;hat is the !east number o% minutes $ou can use an* have the 4reen an* e*

# 100 minutes@ 200 minutesC 2)0 minutes" 300 minutes 00 minutes

Ans&er: A

STUDY NOTES:/he e* an* 4reen !ines intersect t+ice /he !east number o% minute +ou!* be 100minutes an* $ou +ou!* pa$ 30 %or each p!an


3)


36/54

I% someone nee*e* to use their ce!! %or more than 600 minutes ever$ month, +hichp!an +ou!* $ou recommen* so the$ pai* the sma!!est month!$ %eeE

# @!ue

# unre!ate*@ para!!e!C perpen*icu!ar" opposite

36


37/54

Ans&er: 5

STUDY NOTES:ach s!ope riseDrun is X /he$ are para!!e!

Samle Question A2C'

/he s!ope o% si*es #" an* "C are

# unre!ate*@ para!!e!C perpen*icu!ar" opposite

Ans&er: C

STUDY NOTES:

/he s!ope o% #" is -2 an* the s!ope o% "C is X /he$ are opposite reciproca!s sothe$ are perpen*icu!ar


/he actua! s!ope o% si*e #@ is'

# Y@ J YC X

" J X 2F -2

Ans&er: C

STUDY NOTES:/o o %rom point # to point @ $ou o up positive 2 units an* over a positive /hes!ope is X

3


38/54

38


39/54

(A24! @rite and sol%e systems of t&o linear euations and ineualities int&o %ariables6

5amp!e item %or


40/54

;hat is the so!ution to the simu!taneous !inear s$stem picture* aboveE

# 0, @ -2, 0C -3, -1" -2, 0

Ans&er: C

STUDY NOTES:/o so!ve a s$stem o% euations raphica!!$, $ou +ou!* !oo. %or the point o%

intersection an* then p!u the point into both o% the euations to see i% the point isa so!ution to both euations

Samle Question A24*'

5o!ve the %o!!o+in simu!taneous !inear euations a!ebraica!!$'

2x + y = 2

0


41/54

x +y = 1

# -1,-2@ -1D3,-D3C 0,-1" -1,0

Ans&er: 4

STUDY NOTES:;hen so!vin a s$stem o% euations $ou have t+o choices'5o!vin b$ substitution- so!vin one o% the euations %or either or $ an* thensubstitute into the other euation5o!vin b$ !inear combinations- mu!tip!$in one or both o% the euations b$ anumber so that +hen $ou a** the euations toether one o% the variab!es ise!iminate*

Samle Question A243'

#n aca*emic team is oin to a state mathematics competition /here are 30peop!e oin on the trip /here are ) peop!e +ho can *rive an* 2 t$pes o%=ehic!esQ vans an* cars # van seats 8 peop!e, an* a car seats peop!e, inc!u*in*rivers Let v7 number o% vans an* c7 number o% cars

v c )8v cH 30

Ro+ man$ vans an* cars *oes the team nee* %or the tripE

# 3 cars an* 3 vans

@ 3 vans an* 2 carsC 3 cars an* 2 vans" 3 vans an* 1 car

Ans&er: 5

STUDY NOTES:/he on!$ combination that +ou!* satis%$ both ineua!ities +ou!* be 3 vans2peop!e an* 2 cars 8 peop!e Bou +ou!* a!so have the 9ve *rivers, 3 in vansan* 2 in cars

1


42/54

(A2E! 4es)ribe 0o& )0anges in t0e arameters of linear fun)tions andfun)tions )ontaining an absolute %alue of a linear exression ae)t t0eirgra0s and t0e relations0is t0ey reresent6

>se the %o!!o+in situation %or the net three uestions'

# as stationAs 10,000-a!!on un*erroun* storae tan. contains 1,000 a!!ons o%aso!ine /an.er truc.s pump aso!ine into the tan. at a rate o% 00 a!!ons perminute Ro+ !on +i!! it ta.e to 9!! the tan.E

Samle Question A2E1'

Fin* a %unction that represents this situation

# y =400x +10,000

@ y =400x +1,000

C y = 400x +10,000

" y = 400x + 9,000

Ans&er: 5

STUDY NOTES:#t 0 minutes, there is 1,000 a!!ons o% aso!ine an* the rate o% chane is 00a!!onsper minute /his in%ormation 9ts into $7m b %ormat

Samle Question A2E*'

I% the To+ rate increases %rom 00 to )00 a!!ons per minute, ho+ +i!! the raph o%the %unction chaneE

# /he $-intercept +i!! increase@ /he $-intercept +i!! *ecreaseC /he s!ope +i!! *ecrease" /he s!ope +i!! increase

Ans&er: 4

STUDY NOTES:/he rate o% chane increases %rom 00 to )00 a!!ons per minute, this is the s!ope/he startin point o% 1,000 a!!ons remains unchane* /he euation +ou!* be$7)001000

2


43/54

Samle Question A2E3'

I% the initia! amount o% aso!ine in the tan. chanes %rom 1,000 to 2,000 a!!ons,ho+ +i!! the raph o% the %unction chaneE

/he $-intercept +i!! increaseF /he $-intercept +i!! *ecrease

4 /he s!ope +i!! *ecreaseR /he s!ope +i!! increase

Ans&er: A

STUDY NOTES:#t 0 minutes the number o% a!!ons is 2,000 /here%ore, the $-intercept +ou!* no+be 0,2000

e%er to the %o!!o+in t+o %unctions to ans+er the net 3 uestions

Function #' y=3 x Function @' y =

1

3 x

5amp!e item %or


44/54

Function # opens up+ar*s in a v shape an* the s!ope o% the !ines is D-3 Function@ opens *o+n+ar* an* has a s!ope o% D- 1D3 /he$ *o not reTect because theirs!opes are *i?erent

Samle Question A2E'

;hich %unctionAs en* behavior is neativeE

# Function #@ Function @C Neither" @oth

Ans&er: 5

STUDY NOTES:@ecause Function @ opens *o+n+ar*, $ *ecreases as increases an* *ecreases

Samle Question A2E'

;hich %unction has a reater rate o% chaneE

# Function #@ Function @C Neither" @oth

Ans&er: A

STUDY NOTES:/he s!ope o% Function # is D- 3 an* the s!ope o% Function @ is D- 1D3 Function #has a reater rate o% chane


45/54

5amp!e item %or


46/54

Ans&er: 4

6


47/54

(AA! 4es)ribing and Assessing 4ata Sets

>se an* eva!uate the accurac$ o% summar$ statistics to *escribe an* compare *ata

sets

Samle Question AA1:

# !oca! compan$ has a sa!ar$ *ispute /he +or.ers c!aim that the$ are un*erpai* ascompare* to the nationa! averae :anaement c!aims that the$ are actua!!$

ovrpa!"as compare* to the nationa! averae

/hins to consi*er +hen ans+erin the %o!!o+in uestions'

In this compan$ there are on!$ a %e+ o% the +or.ers earnin sa!aries +hich

are uite hihQ in %act these sa!aries are +a$ above the nationa! averae

@oth o% the c!aims ma*e can be accurate!$ supporte* b$ appropriate

summar$ statistics

;ou!* it be in the manaementsA best interest to use the mean or the me*ian +henuotin the [averaeA sa!ar$ o% the +or.ers at this !oca! compan$E p!ain

;hat +ou!* be in the +or.ersA best interest +hen uotin the [averaeA o% !oca!

sa!ariesE p!ain

Ans&er:

/he manaement +ou!* +ant to use the mean /he %e+ hih sa!aries +ou!* pu!! the

mean up an* not a?ect the me*ian /his +ou!* cause the mean to be reater than

the me*ian an* support their case that the +or.ers +ere bein overpai* as

compare* to the nationa! averae

/he +or.ers +ou!* +ant to use the me*ian %or the same ep!anation iven above/he me*ian +ou!* be !o+er than the mean an* use* as support %or the c!aim that

the +or.ers are bein un*erpai*


48/54

Compan$ #

Compan$ @

Compan$ C

Samle Question AA*:

ach bo-an*-+his.er p!ot sho+s the prices o% use* cars in thousan*s o% *o!!ars

a*vertise* %or sa!e at three *i?erent car *ea!ers I% $ou +ant to o to the *ea!er

+hose prices seem !east epensive, +hich *ea!er +ou!* $ou o toE >se statistics%rom the *isp!a$s to Kusti%$ $our ans+er

p!ain $our ans+er be!o+'

Note:/he sca!e oes %rom 0 to 10

Ans&er:

Compan$ C shou!* be chosen /he raph nee*s to be use* +hen provi*in support

o% the choice 5peci9ca!!$, the uarti!es shou!* be compare* For eamp!e,

Compan$ C shou!* be pic.e* /his is because since it has the !o+est thir* uarti!e,

)P o% its cars are cheaper than )P o% the cars %rom the other t+o companies

8


49/54

Samle Question AA3:

Ca!cu!ate the mean an* stan*ar* *eviation o% the %o!!o+ *ata set'

1), 18, 21, 1, 23, 12, 1), 18, 20, 20, 1(, 18

o # mean 7 1) stan*ar* *eviation 7 31(o @ mean 7 1) stan*ar* *eviation 7 306

o C mean 7 18 stan*ar* *eviation 7 1018

o " mean 7 18 stan*ar* *eviation 7 31(

Ans&er: A

Samle Question AA2:

Ca!cu!ate the me*ian o% the %o!!o+ *ata set'

1), 18, 21, 1, 23, 12, 1), 18, 20, 20, 1(, 18

o # me*ian 7 1)

o @ me*ian 7 13)

o C me*ian 7 1)

o " me*ian 7 18

Ans&er: 4

(A5! Inferen)es and Con)lusions

:a.e va!i* in%erences an* *ra+ conc!usions base* on *ata

Samle Question A51:

(


50/54

:r 5hapiro %oun* that the amount o% time his stu*ents spent *oin mathematics

home+or. is positive!$ corre!ate* +ith test ra*es in his c!ass Re is conc!u*in that

*oin home+or. ma.es his stu*entsA test scores hiher

p!ain an$ Ta+s in :r 5hapiroAs reasonin'

Ans&er:

Corre!ation *oes not imp!$ causation For eamp!e, stu*ents +ho are *oin +e!! in

mathematics an*Dor enKo$ it ma$ there%ore *o the home+or. more %reuent!$ than

those +ho are stru!in an*Dor *o not enKo$ the subKect /hus, it miht not be the

act o% *oin the home+or. that is causin the hih scores, but another %actor +hich

is ma.in both o% the variab!es test scores an* home+or. comp!etion o up

Samle Question A5*:

"r ;i!son notice* that as her patients ate more brocco!i, their cho!estero! !eve!s

*roppe* 5he conc!u*e* that eatin brocco!i causes !o+er cho!estero! !eve!s in the

b!oo*

p!ain an$ Ta+s in "r ;i!sonAs conc!usion'

Ans&er:

Corre!ation *oes not imp!$ causation For eamp!e, patients +ho are eatin brocco!i

ma$ be *oin other thins that contribute to !o+erin cho!estero! !eve!s, !i.e

eercise /hus, it miht not be the act o% eatin brocco!i that is causin the !o+er

cho!estero!, but another %actor +hich is ma.in the variab!e move !i.e eercise

Samle Question A53:

)0


51/54

/he %o!!o+in scatter p!ot sho+s the tota! %ue! cost %or a train run %rom 5an Francisco

to 5eatt!e %or trains +ith *i?erent numbers o% cars attache* to the !ea* enine

/he associate* !east-

suares reression

euation is'

Fue!

Cost 7 08Number

o% Cars 213

>se the above euation to pre*ict the tota! %ue! cost %or a train +ith )0 cars

Comment on an$ concerns $ou ma$ have +ith usin this euation to ma.e such a

pre*iction'

Ans&er:

Fue! Cost 7 08)0 213

Fue! Cost 7 13 or ,130

STUDY NOTES:

5ince this euation +as create* %rom *ata that ha* a rane o% 10 to 32 train cars, it

is not a oo* i*ea to use this euation to ma.e estimates be$on* this rane /hetren* ma$ chane, the *ata ma$ not continue to have the same pattern "aner o%

trapo!ation

(AC! inear 'ransformations

"escribe ho+ !inear trans%ormations a?ect the center an* sprea* o% univariate *ata

)1

5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

Fuel Cost vs Train Length

Fuel Cost (in

thousands)

Number of Train Cars


52/54

Samle Question AC1:

"ue to bu*et constraints at a particu!ar compan$, ever$ emp!o$ee receives a 2P

*ecrease to their sa!ar$

;hat impact *oes this sa!ar$ *ecrease have on the mean an* on the rane o%

emp!o$ee sa!aries at the compan$E

o # /he mean an* rane both *ecrease

o @ /he mean an* rane *o not chane

o C /he mean *oes not chane but the rane *ecreases

o " /he mean *ecreases but the rane *oes not chane

Ans&er: A

Samle Question AC*:

#t a particu!ar compan$, ever$ emp!o$ee receives a P cost-o%-!ivin increase to

their sa!ar$

;hat impact *oes this cost-o%-!ivin increase have on the mean an* on the rane o%

emp!o$ee sa!aries at this compan$E

o # /he mean increases but the rane *oes not chane

o @ /he mean *oes not chane but the rane increases

o C /he mean an* rane both increase

o " /he mean an* rane *o not chane

Ans&er: C

Samle Question AC3:

)2


53/54

5uppose that a *istribution o% a set o% numbers has a mean 20 an* a stan*ar*

*eviation o% ) I% 2 is a**e* to each score, +hat +i!! the ne+ mean an* stan*ar*

*eviation o% this set o% numbersE

mean standard de%iation

o # 20

o @ 20 )o C 22 )

o " 22

Ans&er: C

STUDY NOTES:

I% $ou use the *ata set \1), 20, 2)] an* a** 2 to each number $ou et a ne+ *ata

set o% \1, 22, 2] It is eas$ to see the ne+ mean is 22 +ith a stan*ar* *eviation

o% )

Samle Question AC2:

5uppose that a *istribution o% a set o% numbers has a mean 20 an* a stan*ar*

*eviation o% ) I% 2 is mu!tip!ie* to each score, +hat +i!! the ne+ mean an* stan*ar*

*eviation o% this set o% numbersE

mean standard de%iation

o # 20 10

o @ 20 )

o C 0 )

o " 0 10

Ans&er: 4

STUDY NOTES:

I% $ou use the *ata set \1), 20, 2)] an* mu!tip!$ each number b$ 2 $ou et a ne+

*ata set o% \30, 0, )0] It is c!ear that the ne+ mean is 0 +ith a stan*ar*

*eviation o% 10

)3


54/54

5amp!e item %or

Algebra Study Guide

Documents