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SummerPacket2020
HonorsPreCalculus
HonorsPre-Calculusisademandingcoursethatreliesheavilyuponastudent’salgebra,geometry,andtrigonometryskills.YouareexpectedtoknowthesetopicsbeforeenteringHonorsPre-Calc.Youshouldbeabletodo95%ofthispacketwithoutacalculator!
DUE:1stdayofschool
Thefirstmajorassessmentoftheyearwillcoverthefollowingpre-requisitetopics:
• Linearfunctions(graphing,findingintercepts,writingequations,slope,parallelvs.perpendicular,etc.)• Solvingsystemsoflinearandnon-linearequations• Quadraticfunctions(graphing,vertexvs.generalform,solvingequations,etc.)• Functions(notation,domainandrange,operationsonfunctions,compositionoffunctions,testingfor
functions,evaluatingfunctions,findingintercepts,graphing,parentfunctionsetc.)• Dividingpolynomials• Factoringpolynomials• Rulesofexponents• Rulesofradicals• Complexnumbers• Righttriangletrigonometry• Pythagoreantheorem• Specialrighttriangles• Circles(graphing,findingthecenterandradius,standardform,etc.)• Unitcircletrigonometry• LawofSinesandLawofCosines• Evaluatinglogarithmsandpropertiesoflogarithms• Solvingexponentialandlogarithmicequations• Transformationsofexponentialandlogarithmicfunctions
Eachofthesetopicshasbeencoveredinyourpreviousmathclassesandisconsideredpre-requisitematerialforthiscourse.Exampleproblemsforeachtopicareincludedinthesummerpacket.Solutionstothepacketwillbeprovideduponreturningtoschool.Youwillbeassessedonthismaterialinthefirstfewdaysofschool.
2
1. Findthex-andy-intercepts.Graphthefunction
4 8x y− =
X:___________ Y:___________
2. Arethepointssolutionsto 4 8x y− = ?Justify.
a) (3,4) b)(-2,6)
3. Findtheslopeofeachlineandgraph.
a) 4y = − m=__________ b) 2x = m=__________
4. Ifthepoint −3, !!liesonthegraphoftheequation2𝑥 + 𝑘𝑦 = −11,findthevalueofk.
5. Alinehasequation 2 5x y+ = .Findtheslopeofalinethatis
a)parallel b)perpendicular
3
6. Put7 9 18x y+ = intoslopeinterceptformandgraph.
7. FindanequationinStandardFormofthelinewithx–intercept5andy-intercept3.
8. FindanequationinSlope-InterceptFormofthelinethatgoesthrough(8,3)and(2,-1).
9. FindanequationinStandardFormofthelinethrough(1,-4)thatisperpendiculartotheline 2 4x y− = .
10. FindanequationinStandardFormfortheperpendicularbisectorofthesegmentjoining(0,3)and(-4,5).
11. Usesubstitutiontofindallpointsofintersectionofthegraphsof 2=− yx and 1656 =− yx .
4
12. Useeliminationtofindallpointsofintersectionofthegraphsof 1023 =+ yx and 352 =+ yx .
13. Findallpointsofintersectionofthesystemof 2 2x y− = − and 4 6x y− = − bygraphing.
14. Solveeachsystemofequations.
a) b)
15. Withoutputtingtheparabolainvertexform,identifythevertex.Isthevertexamaximumorminimum?
a) 𝑓 𝑥 = 2𝑥! + 8𝑥 + 7 b)𝑦 = −𝑥! + 6𝑥 − 8
2
2
04 0
x yx x y+ =
− − = 2 2
2 3 04 0
x yx y x− + =
+ − =
5
16. Findtheintercepts,zeros,axisofsymmetryandvertexofthegraphof 1032 −+= xxy .
Graphtheparabola.17. Changeintovertexform:
a) b)
18. Simplify.
a) b)
c) d)
2( ) 6 8f x x x= − + − 21( ) 2 124
f x x x= − −
23
1i
2(6 7 )i−
5 41 2
ii
+
−
164 2i
−−
6
19. Solvethequadraticequation.Solutionsmayberealorcomplex.
a) b)
20. Factoreachexpression.
a) 2 2 2x xy x y− + − b) 4 81b −
c) 3 23 54x x x− − d) 38 125m −
e) 3 2 2 32 5x y x y xy xy− + + f) 26 11 2n n− −
g) 2 8 8x x− − h) 2 215 10a b ab−
i) 22 98c − j) 28 2 6a a+ −
2 2 2 0x x− + = 22 6 9 03x x− + =
7
21. Divide
a) Usesyntheticdivision. b)Uselongdivision
2865 3
+
++
xxx
!"!!!!"!!!!"!!!"
!!!!
22. Findthex-andy-interceptsofthegraphoftheequation.
a) b)
c) d)
23. Findthedomainandrangeofeach:
a)𝑦 = 3𝑥 − 9 b) 2+−= xy c) 762 2 −+= xxy
4y x= + 10y x= − +
4 25y x= − 2 1y x= +
8
24. Let𝑓 𝑥 = 𝑥! + 2𝑥 + 1,𝑔 𝑥 = 𝑥 + 2andℎ 𝑥 = 4− 𝑥!.Find:
a) 𝑔 ∘ ℎ 𝑥 b) ℎ 𝑔 𝑥
𝑐) 𝑓(𝑥 − 2) d) ))(( xgf −
e) ( )f x b+ f) ))(( xgf ⋅
g) ( )xgh⎟⎟⎠
⎞⎜⎜⎝
⎛
(includedomainrestrictions)
25. Determinewhethertherelationdeterminesafunction.
a) Apersonandtheirbirthday f)
b)
c)
d)
e)
2 5 10x y+ =
216y x= −
75y = −
4y x= −
9
26. Evaluateeach.
a) Givenxxf 32)( ⋅= ,find )5(f . b)Given
xxf 221)( ⋅= ,find )3(−f .
27. Simplifyeach
a)!!!!!!!!!!
!"!!!!!!!!!
!! b)
2 2
1 1
x yx y
− −
− −
−
−
Changethefollowingequationsintotheform .Graphthefunctions.
28.
Equation_________________
Center___________
Radius__________
2 2 2( ) ( )x h y k r− + − =
040214 22 =++++ yyxx
10
29.
Equation___________________
Center___________
Radius__________
30. Findtheequationofacircle.
a)withcenter(2,-3)andpoint(5,1)
b)withendpointsofthediameterat(-4,0)and(-4,-6)
31. InrighttriangleABC,withangleBbeingtherightangle,
a) IfAB=16andm∠A=53°,findBC b)IfAB=36andAC=47,findm∠A
2 24 4 4 4 7 0y x y x+ − − − =
11
32. Solvefortheunknownsides.Leaveexact!
a) b)
45⁰8in.
15cm. 60⁰
33. Findtheunknowns.
a) FindBC. b)Findm C∠
c)FindAB. d)Given ABCΔ with 034=∠B , b = 15cm,
and .20cmc = , solve the triangle.
34. Converteachanglefromradianstodegrees.
a)37π
b)1712π
−
12
35. Converteachanglefromdegreestoradiansintermsofpi.
a)370° b)-775°
36. Findapositiveandnegativeco-terminalangleforeach.
a)-500° b)37π
37. Findthereferenceangleforeach.
a)700° b)611π
−
38. Findtheexactvalueforeachtrigfunctionwithoutacalculator.
a) sin 225° b) cos( 30 )− °
c) tan120° d)csc35π
e)sec(45π
− ) f)cot613π
g) sin 270° h) tan( )2π
13
39. Grapheachequationwithoutacalculator:
a) b)
40. Grapheachequationwithoutacalculator:
a) b)
14
42.Solveeachequation
a)22(3) 5x = b)5log( 2) 11x − =
c)512 10 7x+= − d) 2log 5x =
e) 43(5 ) 15x−
= f) 6 6log log ( 1) 2x x− + =
g) 5 5 5log log ( 1) log 20x x+ + = h)1 6416
x⎛ ⎞ =⎜ ⎟⎝ ⎠
i) j)
15
k) l)\
m) n)
43. Rewriteasasinglelog: 1log2 log4 log42
x x+ −
44. Expand:log!!"!!!!
!! !
16
45.Graph,nameandstatethedomainandrangeofeachofthesefunctions.
Eachgraphshouldhaveaminimumoffourpoints.
( )( )( )( )( )
( )x
xf
xxfxxf
xxfxxfxxf
1
3
2
=
=
=
=
=
=