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IBMathematicsSL:AnalysisandApproachesSummerHomework
Name______________________________________________________
DearFutureIBMathematicsSLStudent,
IhopeyouareexcitedforyourupcomingyearinIBMathSL!Thepurposebehindthissummerhomeworkpacketistoreacquaintyouwiththenecessaryskillstobesuccessfulinthisyearβsmathcourse.
Atfirstglancethispacketmayseemoverwhelming.However,thereareapproximately9weeksofsummer.Paceyourself.Thereare13Partsofthispacketβcompletetwopartseachweekandyouwilleasilybeabletocompletetheassignmentbeforeyourreturntoschoolinthefall.PleasebepreparedtosubmitthisassignmentduringyoursecondIBMathSLclass.Itwillbegradedforaccuracyaswellascompletion.Workneedstobeshowninaneatandorganizedmanner,anditisperfectlyacceptabletocompletethepacketonseparatesheetsofpaper.Justbesuretostapleanyextrapaperstothepacket.Also,donotrelyonacalculator!
ShowALLworkforeachproblemandtakeyourtime.Remember,thiswillbeyourfirstimpressiontoyournewmathteacher,andyouwanttomakesurethatitisapositiveone!Seebelowfordirectionsandhelpfulwebsites.Wehopeyouhaveawonderfulsummer!
Best,
WarehamHighSchoolMathDepartment
NeedhelpwithyourSummermathpacket???
FeelfreetoemailMrs.Medinaatmmedina@wareham.k12.ma.uswithanyquestionsyoumighthave.Toensurethefastestresponse,pleaseincludeyourname,summerassignmentname,and(if
possible)apictureoftheproblemandyouraccompanyingwork.
Directions:
β’ Beforeansweringanyquestions,readthroughthegivennotesandexamplesforeachtopic.β’ ThispacketistobesubmittedduringyoursecondIBMathSLclassperiod.β’ Allworkmustbeshowninthepacketoronaseparatesheetofpaperstapledtothepacket.β’ Toavoidapenaltyonyourgrade,finalanswersMUSTBEBOXEDorCIRCLED.
2
Part 1 β Functions
Let π(π₯) = π₯!, π(π₯) = 2π₯ + 5, β(π₯) = π₯! β 1, find: 1)
β(β2)
2) π.π(β4)0
3) β(π₯ + 2)
4)
.π(π₯)0! β 2π(π₯)
3
Part 2 β Intercepts of a Graph
Find the π₯ β and π¦ βintercepts for each function below. 5)
π¦ = 2π₯ β 8
6) π¦ = π₯! + 5π₯ + 6
7)
π¦ = βπ₯ + 4
8) π¦ = 3" β 9
4
Part 3 β Points of Intersection
Find the point(s) of intersection of the graphs for the given equations. Write each solution as an ordered pair. 9)
74π₯ β π¦ = 102π₯ + π¦ = 8
10)
7π¦ = 3π₯ β 4π₯! + π¦ = 6
5
Part 4 β Domain and Range
Find the domain and range of each function. Write your answer using interval notation. 11)
π(π₯) = π₯! + 5
12)
π(π₯) = βπ₯ β 3
13)
β(π₯) =2
π₯ β 1
14) π(π₯) = 4"#! + 5
6
Part 5 β Inverses
Find the inverse of each function below. 15)
π(π₯) = 2π₯ + 1
16) (Domain restriction: π₯ β₯ 0) π(π₯) = π₯! β 7
17)
β(π₯) =5
π₯ β 2
18)
π(π₯) = βπ₯ β 4 + 3
19) If the graph of π(π₯) has the point (2, 7), then what is one point that will be on the graph of π!"(π₯)?
20) Explain how the graphs of π(π₯) and π!"(π₯) compare.
7
Part 6 β Transformation of Functions
21) How is the graph of π(π₯) = (π₯ β 3)! + 1 related to the parent function π(π₯) = π₯!? Describe the transformations that occurred from the parent function to the given function.
22) Write an equation for a function, π(π₯), that has the shape of π(π₯) = π₯$, but is translated 6 units to the left and is reflected over the π₯ βaxis.
Part 7 β Vertical Asymptotes
Find the vertical asymptote(s) of each rational function below. 23)
π(π₯) =5
π₯ + 2
24)
π(π₯) =π₯ + 3
π₯! β 6π₯ + 8
25)
β(π₯) =π₯ β 7π₯! β 49
8
Part 8 β Horizontal Asymptotes
Find the horizontal asymptote of each rational function below. 26)
π(π₯) =4π₯ + 102π₯ β 3
27)
π(π₯) =3
π₯ + 9
28)
β(π₯) =π₯! β 1π₯ + 2
29)
π(π₯) =π₯! β 2π₯ + 1π₯" + π₯ β 7
30)
π(π₯) =β5π₯" β 2π₯! + 83π₯" + 4π₯ β 5
31)
β(π₯) =10π₯!
2π₯! β 1
9
Part 9 β Exponential Equations Solve each exponential equation below. 32)
5!"#$ = 5#%"&'(
33)3$"#( = 9'"&)
34)
!16$
!= 216
35)
!14$
"!= 16#$!%&
10
Part 10 β Logarithms The logarithmic equation log% π¦ = π₯ can be written as an exponential equation π" = π¦. They mean the SAME thing. You can use this fact to evaluate logarithms.
Recall the natural logarithmln π₯ = log' π₯. The value of π β 2.718281828β¦ or lim!β#
$1 + $!'!
.
Evaluate each logarithm below. 36)
log& 7
37) log$ 27
38)
log! >116?
39) log!' 5
40) log( 1
41) log!& 9
11
Part 11 β Properties of Logarithms
Use properties of logarithms to evaluate each expression below. 42)
log! 2'
43) log) 9$
44) log! 8$
45)
2*+,2 )-
46)
π*. /
47)
7 ln π
48)
ln π" 49)
ln βπ
50) log34 25 + log34 4
51) log5 40 β log5 5
12
Part 12 β Factoring Trinomials Factor the expression 2π₯! + π₯ β 6 using the box method. Steps to Factor Trinomials in the form ππ₯! + ππ₯ + π by Box Method: 1) Multiply π times π. π β π = (2)(β6) = β12 2) Find two numbers that multiply to π β π = β12 and add up to the coefficient of the middle term π = 1.
(βπ)(π) = β12 βπ + π = 1
3) Rewrite the middle term with these two numbers, put the four terms in a box, and factor the GCF out of each row and column.
2π₯! + ππ β 6 2π₯! β ππ + ππ β 6
2π₯! + π₯ β 6 = (2π₯ β 3)(π₯ + 2)
2π₯! β3π₯
+4π₯ β6
Factor each expression below by using the box method. 52)
5π₯! + 14π₯ + 8
53) 8π! β 10π + 3
54) 2π₯! β 3π₯ β 9
55) 6π₯! β 23π₯ β 4
π₯
+2
2π₯ β 3
13
Part 13 β The Discriminant
Use the discriminant to determine the nature of the roots of each quadratic equation below. 56)
4π₯! β 12π₯ + 9 = 0
57) 7π₯! β π₯ + 2 = 0
58) 3π₯! + 18π₯ = β5
59) 24π₯! + 5 = 14π₯