precalculus summer assignment - home - abraham...

MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page1

Name:______________________________________________________________

Pre-CalculusSummerAssignmentDueDate:ThebeginningofclassonSeptember8,2017.

ThepurposeofthisassignmentistohaveyoupracticethemathematicalskillsnecessarytobesuccessfulinPre-Calculus.AlloftheskillscoveredinthispacketarefromAlgebra2andAlgebra1.Thematerialcoveredisfromourdistrictapproved,PearsonAlgebra2CommonCore,textbook.Ifyouneedto,youmayusereferencematerialstorefreshyourmemory(oldnotes,textbooks,onlineresources,etc.).Whilegraphingcalculatorswillbeusedduringafewtestsandquizzes,themajorityofinclassassessmentsarenon-calculator.Youareencouragedtolearnhowtobecalculator-independent.Attheendofthispage,therearelinkstosomesuggestedonlinecalculators.

Pre-CalculusisafastpacedcoursethatistaughtatthecollegeleveltoprepareyouforAPCalculus.Thereisalotofmaterialinthecurriculumthatmustbecoveredbeforetheendoftheyear.Therefore,wecannotspendalotofclasstimere-teachingprerequisiteskills.Thisiswhyyouhavethispacket.SpendsometimewithitandmakesureyouareclearoneverythingcoveredinthispacketsothatyouwillbesuccessfulinCalculus.Ofcourse,youarealwayswelcomedtoseekhelpfromyourteacherifnecessary.

Thisassignmentwillbecollectedandgradedasyourfirsttest,thelastclassdayofthefirstweekofschool.Besuretoshowallappropriateworktosupportyouranswers.Inaddition,theremaybeaquizonthismaterialduringthefirstquarter.Allquestionsmustbecompletewiththecorrectwork.YoumustreturninSeptemberknowinghowtodoallthematerialinthispacket.

Forassistancewiththepacketyoumaycontactmeatgnaem@roselleschools.org.Emailsmaytakeafewdaysduringsummerforaresponse.Pleasebespecificinyouremailforwhatyouneedassistancewith,includethesectionandthequestionnumberaswell.Foreachquestioninthepacket,thereisanexampleintheseparatetutorialpacket.Thetutorialpacketispostedonlineonourschool’swebsite.Referencetopageandexamplenumbersislistedundereachquestion. CalculatorsLinksOnline Calculator

https://www.desmos.com/calculator

https://mathway.com/graph

http://www.emathhelp.net/calculators/calculus-1/online-

graphing-calculator/

Emulator for Download

https://wabbit.codeplex.com/

http://lpg.ticalc.org/prj_tilem/download.html

GoodLuck!


1) Findthemissingnumberintheequation.

(TutorialPage2Example#1&2)

a)+(–4)=–4 b)8(–3)+8• x=8• ()c)• 3

2=1 d)–2(+1)=• 5–2• 1

2) Writeanalgebraicexpressionthatmodelseachwordphrase.

(TutorialPage3Example#1)a)Theproductof2dividedbythenumberhand8morethanthenumberk.

b)Twodecreasedbythequotientofthenumberaand7andincreasedbyamultipliedby3.

3) Simplifythealgebraicexpression.Thenevaluatethesimplifiedexpressionforthegivenvaluesofthe

variable.(TutorialPage3Example#2)

a)(4x+1)+2x;x=3

b)6p2–(3p2+2q2);p=1,q=5

c) 1 ; 1, 02 3 4 5r s r r s+ − + = − =

d) 3 1( ) ( ); 6, 24 4m n m n m n+ − − = =

4) Solveeachequationfortheindicatedvariable.

(TutorialPage4Example#1)

a) 2 5 1 , for3 12f g fg f+ = −

b) 3 4 , fora y a y yb− + = +


c) 3 ,4

x kj+ = forx

d) 12r+3s=1,forr

5) Solveeachinequality.Graphthesolution.

(TutorialPage5Example#1&2)a)4–(2x–4)≥5–(4x+3)

b)7–7(x–7)>–4+5x

6) Solveeachabsolutevalueequation.Checkyourwork.

(TutorialPage6Example#1)a) 2 3x − –4=3 b) 3 6x − +1=13


7) Completethestepstosolvetheinequality .


a) ≤ – 42x ≤ Rewriteasacompoundinequality.

b) ≤2x ≤ Addtoeachpart.

c) ≤ x ≤ Multiplyeachpartby .d)Whatisthesolution?

8) Determinewhethereachrelationisafunction.Explainyouranswer.Findthedomainandrangeofeach

relation.(TutorialPage6Example#1)

a){(1,2),(1,3),(1,4),(1,5),(1,6)} b){(0,−1),(1,2),(−1,−1),(−2,5),(2,9)}

9) Evaluateeachfunctionforthegivenvalueofx,andwritetheinputandtheoutputasanorderedpair.

(TutorialPage7Example#3)a)h(x)=12xforx=4 b)t(x)=8x−5forx=7

10) Foreachfunction,determinewhetheryvariesdirectlywithx.Ifso,findtheconstantofvariation.

(TutorialPage8Example#1&2)a)

x y4 16 28 3

b)34y−17x=0

4 32x − ≤


11) Findthemissingvalueforeachdirectvariation.(TutorialPage8Example#3)

a)Ify=8whenx=4,findywhenx=6 b)Ify=9whenx=3,findxwheny=7

12) Write an equation for each line.

(TutorialPage9Example#1) a)m = 4; contains (3, 2) b)m = −1; contains (0, 7)

c) d)

13) Graph each equation.

(TutorialPage9Example#2) a)−3x + 2y = 6 b)3y + x = 3


14) Usingpoint-slopeform,writeanequationofthelinethrougheachpairofpoints.

(TutorialPage10Example#1)a)(−2,−5)and(8,−3) b)(3,5)and(0,7)

15) Writeanequationofeachlineinslope-interceptform.

(TutorialPage10Example#2&3)a)Through(−2,−2)andparalleltoy=−5x−4 b)Through(−4,1)andperpendiculartoy=−3x+7

c)2y − 6 = 0 d)−2x + 4y − 3 = 0


16) Identify the type of translation of f (x) = x .

(TutorialPage11Example#1&2) a) 2g(x)= x – b) 3g(x)= x –

17) Graph each translation of f(x) = .x

(TutorialPage11Example#1)a) 1 5g(x)= x – – b) 4 2g(x)= x + +

18) Describe the transformations of f(x) that produce g(x).

(TutorialPage11Example#2) a)f(x) = −5x to g(x) = x

b)f(x) = x2 to g(x) = 2 (x2 − 3)


19) Forthefunctionf(x)=|x|,answerthefollowingquestions.(TutorialPage12Example#1&2)

a) Makeatableofvaluesforeachequation.Thengraphtheequation.

2 + 1 – 5y x=

b) Withoutgraphing,identifythevertex,axisofsymmetry,andtransformationsfromtheparentfunctionf(x)=|x|for 4 – 5 + 3y x=

20) Graph each inequality.

(TutorialPage13Example#1) a)3x − 2 ≤ 5x + y b) < + 2 – 4y x


21) Solve each system by graphing or using a table. Check your answers.


a)2 + + 3= 0 – 1 = x y

x y−⎧

⎨⎩

b) + = –2

–2 + 3 = –3x yx y

−⎧⎨⎩

22) Solveeachsystem.

(TutorialPage15Example#1&2)a)Solveeachsystembysubstitution.

–2 + = 6–7 + 6 = 1m nm n

⎧⎨⎩

b)Solveeachsystembyelimination.

5 + 4 = 6–2 – 3 = –1f mf m

⎧⎨⎩


23) Solve each system of inequalities by graphing. (TutorialPage16Example#1)

a)4 + 1

+ 2 –1x yx y

≤⎧⎨ ≤⎩

b) 2 + > 3

– < 2x yx y

⎧⎨⎩

24) Grapheachfunction.Identifythevertexandaxisofsymmetry.

(TutorialPage17Example#1)a)y=y=(x+4)2−2 b)y=2(x−1)2+3


25) Grapheachparabola.Labelthevertexandtheaxisofsymmetry.

(TutorialPage18Example#1)a)y=−3x2+6x−9 b)y=2x2−8x+1

26) Writeeachfunctioninvertexform.Checkyouranswers.

(TutorialPage19Example#2)a)y=−x2+4x+6 b)y=x2−2x−3


27) Writeeachfunctioninstandardform.(TutorialPage19Example#2)

a)y=2(x−1)2−3 b)y=−3(x+4)2+1

28) Factoreachexpression.

(TutorialPage20Example#1&2)a)x2+6x+8

b)2x2−6x+4

c)9x2−6x+1

d)3x2+2x−8

e)27x2−12

f)16x2−32x+16

g)x2−64

h)125x2−100x+20


29) Solveeachequationbyfactoring.Checkyouranswers.(TutorialPage21Example#1)

a)x2−10x+16=0 b)2x2=−5x+12

c)2x2+10x=0 d)3x2−5x+2=0

30) Whatvaluecompletesthesquareforeachexpression?

(TutorialPage22Example#1)a)3x2+12x b)−7x2+14x

31) Rewriteeachequationinvertexform.

(TutorialPage20Example#2)a)y=x2+8x+13 c)y=2x2+4x−3


32) Whatarethesolutionsforeachequation?UsetheQuadraticFormula.

(TutorialPage23Example#1)a)−x2+7x−3=0 b)2x2+1=5−7x

33) Whatisthevalueofthediscriminantandwhatisthenumberofrealsolutionsforeachequation?

(TutorialPage23Example#2)a)x2+x−42=0 b)2x2+7=5x

34) Simplify each expression.

(TutorialPage24Example#1) a)(5−2i)(−3+4i) b)(5 + 6i) + (−2 + 4i)

35) Write each quotient as a complex number.


a) 42 3

ii

− −+

b) 31 2ii−


36) Solveeachsystem.(TutorialPage25Example#1&2)

a)2

2

62

y xy x

⎧ < − +⎪⎨

> −⎪⎩ b)

22 53 1

y x xy x

⎧ = − −⎨

= +⎩

37) What is the classification of each polynomial by its degree? By its number of terms? What is its end behavior?

(TutorialPage26Example#1) a)8 − 6x3 + 3x + x3 − 2 b)15x7 − 7

38) Writeapolynomialfunctioninstandardformwiththegivenzeros.

(TutorialPage27Example#1)a)2,1,3 b)2,3,−3,−1


39) Writeathird-degreepolynomialfunctiony=P(x)withrationalcoefficientssothatP(x)=0hasthegivenroots.(TutorialPage27Example#2)

a)1,2−i b)1,5i

40) Findtherealorimaginarysolutionsofeachpolynomialequation.

(TutorialPage28Example#1)a)4x3+4=0 c)8x3+27=0

41) Divideusingpolynomiallongorsyntheticdivision.

(TutorialPage29Example#1&2)a)(x2+3x−8)÷(x−5) b)(x3+2x2−20x+4)÷(x+7)


c)(3x4+x3−6x2−9x+12)÷(x+1) d)(x4−12x3−18x2+10)÷(x+4)

42) Writetheexpansionofeachbinomial.

(TutorialPage30Example#1)a) 4( )x y− b) 5( 1)r +

43) Determinetheequationofthegraphofy=x3undereachsetoftransformations.

(TutorialPage31Example#1)a)Areflectionacrossthex-axis,averticaltranslation5unitsup,andahorizontaltranslation8unitsright

b)Averticalstretchbyafactorof6,ahorizontaltranslation3unitsleft,andaverticaltranslation1unitup

44) Findthereal-numberrootsofeachradicalexpression.


a) 138

− b) 4 0.0001−


45) Simplifyeachradicalexpression.Useabsolutevaluesymbolswhenneeded.(TutorialPage32Example#2)

a) −x3 y63

b) 13313 3x

46) Simplifyeachproduct.


a) 2 5 23 336 6x y x y⋅ − b) 5 2 2 53 39 2x y x y− ⋅

47) Rationalizethedenominatorofeachexpression.Assumethatallvariablesarepositive.


a)21934

a babc

b)4 94yx

48) Simplify.

(TutorialPage34Example#1)a) 6 3 75− b) 3 3 38 3 24 192x x x− +


49) Simplify.Rationalizealldenominators.(TutorialPage34Example#2)

a) (4 6 1)( 6 4)− + b) 2 72 7−+

50) Simplify each expression. Assume that all variables are positive.

(TutorialPage35Example#2) a)

1134(2 )(3 )y y b)

1 26 6( 3 )(7 )x x−

51) Write each expression in simplest form. Assume that all variables are positive.


a)

12416

825

z

x

−

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

b)

25

10

12

x

y

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠


52) Solve. Check your solutions. (TutorialPage36Example#1)

a) 3 2 2 3x− − = b) 3 5 2 3 0x + − =

53) Solve. Check for extraneous solutions.

(TutorialPage36Example#2) a) 2 10 5x x− = − b) 1 7x x− + =

54) Let f (x) = 4x − 3 and g(x) = x2 + 2. Perform each function operation and then find the domain of the result.

(TutorialPage37Example#1&2) a) (f · g)(x) b) ( f − g)(x)

c) f(g(2)) d) f(g( −5))


55) Find the inverse of each function. (TutorialPage38Example#1)

a) ( ) 2f x x= + b)f(x) = x + 3

56) Name the domain and range of the inverse of the function.

(TutorialPage38Example#2) a) 5y x= + b) 3 2y x= +

57) Graph each function.

(TutorialPage39Example#1) a) y = 2 x + 3 + 4 b) y = 3 x + 2


58) Solve the equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why. (TutorialPage39Example#2)

a) 3 1 5x+ = b) 2 5 4x x− = −

59) Determine whether the function represents exponential growth or exponential decay. Then find the y-intercept.


a) 1152

x

y ⎛ ⎞= ⎜ ⎟⎝ ⎠ b) 5( ) 6

2

x

f x ⎛ ⎞= ⎜ ⎟⎝ ⎠

60) Write an exponential function to model each situation. Find each amount after the specified time.

(TutorialPage40Example#1&2) a)A tree 3 ft tall grows 8% each year. How tall will the tree be at the end of 14 yr? Round the answer to the nearest hundredth.

b)The price of a new home is $126,000. The value of the home appreciates 2% each year. How much will the home be worth in 10 yr?


61) Graph each exponential function. (TutorialPage41Example#1)

a) y = 3 1

2⎛⎝⎜

⎞⎠⎟

x+1

+ 2 b) y = 1

22( )x−1

− 3

62) Suppose you invest $7500 at an annual interest of 7% compounded continuously.

(TutorialPage41Example#2) a) How much will you have in the account in 10 years? b)How long will it take for the account to reach

$20,000?

63) Write each equation in logarithmic form.


a) 3 1464

− = b) 1 188

− =

64) Write each equation in exponential form.


a) 491log 72

= b) 51log 4625

= −


65) Evaluate the logarithm. (TutorialPage42Example#3)

a) 9log 3 b) 2log 64

66) Write each logarithmic expression as a single logarithm.

(TutorialPage43Example#1) a) 2 2log 16 log 8− b) 5 5log 3logx y+

67) Write each logarithm as a quotient of two common logarithms.

(TutorialPage43Example#2) a) 5log 16 b) 9log 32

68) Solve each equation. Round the answer to the nearest hundredth.

(TutorialPage44Example#1) a)7 − 2x+7 = 5 b)43x + 2=3


69) Solve each equation. Round the answer to the nearest thousandth. (TutorialPage44Example#2)

a)2 log 250x − 6 = 4 b)5 + log (2x + 1) = 6

70) Use natural logarithms to solve each equation. Round your answer to the nearest thousandth. Check your

answers. (TutorialPage45Example#1)

a)2ex = 4 b)12e3x−2 = 8

71) Solve each equation. Round your answer to the nearest thousandth. Check your answers.

(TutorialPage45Example#2) a)1 + ln x2 = 2 b)ln (t − 4)2 + 2 = 5


72) Answerthefollowing:(TutorialPage46Example#1&2)

a)Dothedatainthetablerepresentadirectvariation,inversevariation,orneither?

x 5 10 15 20y 10 20 30 40

b)Thetimetneededtocompleteataskvariesinverselyasthenumberofpeoplep.Ittakes5hforsevenmentoinstallanewroof.Howlongdoesittaketenmentocompletethejob?

73) Graph each function. Include the asymptotes.


a) 4 = yx

− b) 9 = yx

c) 3 42

yx

= −−

d) 4 8

yx

= −−


74) Find the vertical asymptotes, holes, and horizontal asymptote for the graph of each rational function.


a) 4 + 5 = 3 + 2xyx

b) = 2 9

xyx −

75) Graph each function. Include the asymptotes.


a) 4 2 9y

x=

− b)

2 2 2 1

x xyx+ −=−

76) Simplifyeachrationalexpression.Stateanyrestrictionsonthevariable.


a)2

2

+ + 2x xx x

b)2

2

3 126

xx x

−− −


77) Divide.Stateanyrestrictionsonthevariables.(TutorialPage51Example#2)

a)2

2

3 12 8 16 2 8 8 16x x xx x x+ + +÷− − +

b)24 16 2 8

4 3 6x x xx x− − −÷

+

78) Assumethatthepolynomialsgivenarethedenominatorsofrationalexpressions.FindtheLCDofeach

set.(TutorialPage52Example#1)

a)x2+7x+12andx+4 b)x2–9andx2+2x–3

79) Simplifyeachsumordifference.Stateanyrestrictionsonthevariable.


a) 2 2

25 6 3 2x

x x x x−

+ + + + b) 2

3 2 +

2 x 4x + −


80) Solveeachequation.Checkthesolutions.(TutorialPage53Example#1)

a) 2

2 6 6–1x x x x

+ =−

b) 4 5 21 1x x= +

− −

81) Findthesumofeachfiniteseries.


a) ( )34

1n

n−∑

= b) ( )9

4 23

nn

−∑=

82) Findthecenterandradiusofeachcircle.

(TutorialPage55Example#1)a)x2+y2+2x–6y=15 b)x2+y2–10x–4y=–20


83) Writeeachmeasureinradiansandcheck.

(TutorialPage56Example#1)a)150° b)45°

84) Writeeachmeasureindegreesandcheck.


a) 76π− b) 5

3π

85) Themeasureθofanangleinstandardpositionisgiven.Findtheexactvaluesofcosθandsinθfor

eachanglemeasure.(TutorialPage56Example#2)

a) 3 radians4π b) 2 radians

3π−

86) Find the amplitude and period of each sine function.


a) 1 sin 32

y θ= b) 44sin3

y πθ=


87) Graph each function. (TutorialPage57Example#2)

a) 12sin2

y θ= − b) 1 sin4

y θ= −

88) Sketchthegraphofeachfunctionintheintervalfrom0to2π.


a) 1cos4

y πθ= b) 1cos22

y θ=

89) Find the period and two asymptotes of the graph of each tangent function. Then find two points on each graph

that are not on the x-axis. (TutorialPage59Example#1)

a) y = 4 tan θ b) y = − tan 2θ


90) Graph at least three cycles of each tangent function. (TutorialPage60Example#2)

a)y = 3 tan θ b)y = −2 tan 4θ

91) Determine the amplitude, period, and any phase shift or vertical shift in the graphs of the functions.


a) 2 sin( 3 )3

y x π π= + − b) 3cos 124

y x π⎛ ⎞= − + +⎜ ⎟⎝ ⎠

92) Sketch each graph in the interval from 0 to 2π.


a) 12sin 12

y x= − − b) y = cos3 x + π

2⎛⎝⎜

⎞⎠⎟+1


93) Find the exact value of each expression. Do not use a calculator. (TutorialPage62Example#1)

a) sec − 3π

4⎛⎝⎜

⎞⎠⎟

b) csc −

π2

⎛⎝⎜

⎞⎠⎟

94) Sketch each graph in the interval from 0 to 2π.

(TutorialPage62Example#2) a) y = − sec 2θ b) y = cot 3θ

precalculus summer assignment - home - abraham...

Documents