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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.

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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

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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 .

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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− + =

+ − =

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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

−−

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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− + =

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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= +

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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= −

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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

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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+ − − − =

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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

−

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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

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39. Grapheachequationwithoutacalculator:

a) b)

40. Grapheachequationwithoutacalculator:

a) b)

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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)

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k) l)\

m) n)

43. Rewriteasasinglelog: 1log2 log4 log42

x x+ −

44. Expand:log!!"!!!!

!! !

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45.Graph,nameandstatethedomainandrangeofeachofthesefunctions.

Eachgraphshouldhaveaminimumoffourpoints.

( )( )( )( )( )

( )x

xf

xxfxxf

xxfxxfxxf

1

3

2

=

=

=

=

=

=