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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!
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page2
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− + = +
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page3
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page4
7) Completethestepstosolvetheinequality .
(TutorialPage6Example#2)
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 − ≤
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page5
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page6
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page7
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)
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page8
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page9
21) Solve each system by graphing or using a table. Check your answers.
(TutorialPage14Example#1)
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
⎧⎨⎩
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page10
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page11
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page12
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page13
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page14
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.
(TutorialPage24Example#2)
a) 42 3
ii
− −+
b) 31 2ii−
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page15
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page16
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)
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page17
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.
(TutorialPage32Example#1)
a) 138
− b) 4 0.0001−
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page18
45) Simplifyeachradicalexpression.Useabsolutevaluesymbolswhenneeded.(TutorialPage32Example#2)
a) −x3 y63
b) 13313 3x
46) Simplifyeachproduct.
(TutorialPage33Example#1)
a) 2 5 23 336 6x y x y⋅ − b) 5 2 2 53 39 2x y x y− ⋅
47) Rationalizethedenominatorofeachexpression.Assumethatallvariablesarepositive.
(TutorialPage33Example#2)
a)21934
a babc
b)4 94yx
48) Simplify.
(TutorialPage34Example#1)a) 6 3 75− b) 3 3 38 3 24 192x x x− +
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page19
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.
(TutorialPage35Example#3)
a)
12416
825
z
x
−
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
b)
25
10
12
x
y
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page20
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))
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page21
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page22
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.
(TutorialPage40Example#1&2)
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?
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page23
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.
(TutorialPage42Example#1)
a) 3 1464
− = b) 1 188
− =
64) Write each equation in exponential form.
(TutorialPage42Example#2)
a) 491log 72
= b) 51log 4625
= −
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page24
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page25
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page26
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.
(TutorialPage47Example#1&2)
a) 4 = yx
− b) 9 = yx
c) 3 42
yx
= −−
d) 4 8
yx
= −−
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page27
74) Find the vertical asymptotes, holes, and horizontal asymptote for the graph of each rational function.
(TutorialPage48Example#1)
a) 4 + 5 = 3 + 2xyx
b) = 2 9
xyx −
75) Graph each function. Include the asymptotes.
(TutorialPage49Example#2)
a) 4 2 9y
x=
− b)
2 2 2 1
x xyx+ −=−
76) Simplifyeachrationalexpression.Stateanyrestrictionsonthevariable.
(TutorialPage50Example#1)
a)2
2
+ + 2x xx x
b)2
2
3 126
xx x
−− −
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page28
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.
(TutorialPage52Example#2)
a) 2 2
25 6 3 2x
x x x x−
+ + + + b) 2
3 2 +
2 x 4x + −
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page29
80) Solveeachequation.Checkthesolutions.(TutorialPage53Example#1)
a) 2
2 6 6–1x x x x
+ =−
b) 4 5 21 1x x= +
− −
81) Findthesumofeachfiniteseries.
(TutorialPage54Example#1)
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page30
83) Writeeachmeasureinradiansandcheck.
(TutorialPage56Example#1)a)150° b)45°
84) Writeeachmeasureindegreesandcheck.
(TutorialPage56Example#1)
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.
(TutorialPage57Example#1)
a) 1 sin 32
y θ= b) 44sin3
y πθ=
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page31
87) Graph each function. (TutorialPage57Example#2)
a) 12sin2
y θ= − b) 1 sin4
y θ= −
88) Sketchthegraphofeachfunctionintheintervalfrom0to2π.
(TutorialPage58Example#1)
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
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page32
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.
(TutorialPage61Example#1)
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.
(TutorialPage61Example#2)
a) 12sin 12
y x= − − b) y = cos3 x + π
2⎛⎝⎜
⎞⎠⎟+1
MATHACHSPRECALCULUSSUMMERASSIGNMENTJUNE,2017Page33
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