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PrerequisiteSkills– Page1

PartI.LinearEquationsinTwoVariables:NotesandReview

SlopeFormula: 𝑚 = $%&$'(%&('

Equationsoflines:VerticalLine:𝑥 = # Slopeofaverticallineisundefinedornoslope.HorizontalLine:𝑦 = # Slopeofahorizontallineis0.Slope-InterceptForm:𝑦 = 𝑚𝑥 + 𝑏where𝑚istheslope(riseoverrun)and𝑏 isthey-intercept(0, 𝑏).Point-SlopeForm:𝑦–𝑦4 = 𝑚(𝑥–𝑥4)StandardForm(GeneralForm):𝐴𝑥 + 𝐵𝑦 = 𝐶 where𝐴, 𝐵,and𝐶 areintegers(nofractionsordecimals)and

𝐴 ispositive.

ParallelandPerpendicularLines:Parallellineshavethesameslope.Perpendicularlineshaveslopesthatarenegativereciprocalsofeachother,thatis,𝑚4 =

&48%.Theproductof

theslopesoftwoperpendicularlinesis−1.

Directions: Foreachlinearequation,findtheslope,y-intercept,andgraphtheline.

1.𝑦 = 4;𝑥 − 4 2.2𝑥 − 6 = 0 3.𝑦 + 3 = 2

4.4𝑥 − 2𝑦 = 10 5.2𝑥 + 3𝑦 = 3 6.−2𝑦 = 5𝑥 − 4

7.4𝑦 + 3𝑥 = 0 8.6𝑥 − 3 = −15 9.−4𝑥 + 4𝑦 = 16

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

𝑚 =

𝑏 =

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PrerequisiteSkills– Page2

Directions:Readeachquestion.Showallyourwork.Writeyouranswerneatly.

1.Findtheslopeofthelinepassingthroughthepoints(−5, 4)and(7, 8)

2.Findtheslope ofthelinepassingthrough C

D, 4Eand &I

E, JE

3.Write theequationofalinepassingthroughthepoint(−3, 4)withaslopeof½ inpoint-slopeform.

4. Takeyourequationfromquestion#3andwriteitinslope-interceptform.

5. Now takeyourequationfromquestion#4andwriteitinstandardform.

6.Determinewhetherthelinespassingthroughthefollowingpairsofpointsareparallel,perpendicular,orneither:Line1:(−4, 6), (5, 9)Line2:(0, −½), (3,½)

7.Writetheequationofalinewithaslopeofzeropassingthroughthepoint(−6, 7).

8.Writetheequationofalinewithundefined slopepassingthroughthepoint(−6, 7).

9.Writetheequationof alineperpendiculartotheline𝑦 = ¾𝑥– 2passingthroughthepoint(0, −5) instandardform.

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PrerequisiteSkills– Page3

PartII.Factoring:NotesandReview

1.GCF(GreatestCommonFactor)FirststartbyfactoringouttheGCF(greatestcommonfactor).Askyourself,areallofthetermsdivisiblebythesamenumber?Dotheyallhavethesamevariable(s)incommon?

Example:24𝑥E − 12𝑥J + 48𝑥IWhatnumberareallofthetermsdivisibleby?12Whatvariablesdotheyhaveincommon?𝑥JGCF=12𝑥JFactoredform:24𝑥E − 12𝑥J + 48𝑥E = 12𝑥J(3𝑥 − 1 + 4𝑥;)

2.DifferenceofTwoSquares:𝑎; − 𝑏; = 𝑎 + 𝑏 𝑎 − 𝑏 or𝑎; − 𝑏; = (𝑎 − 𝑏)(𝑎 + 𝑏)Perfectsquaresare1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,…basicallyanypositiveintegerthatwhensquarerootedequalsaninteger(nodecimals).AFTERlookingfortheGCF(alwaysthefirststepinfactoring),askyourself:isitabinomial(twoterms)?Arethetwotermsseparatedbyaminussign?Arebothtermsperfectsquares?

Example:4𝑥; − 36𝑦;LookoutforanyGCFs.Inthiscasebothtermsaredivisibleby4.Whenyoufactoroutthe4,you’releftwith:4(𝑥; − 9𝑦;).Noticethat𝑥; − 9𝑦; isadifferenceoftwosquares!Thisisthefactoredformof𝑥; − 9𝑦; = (𝑥 + 3𝑦)(𝑥 − 3𝑦).Therefore,4𝑥; − 36𝑦; = 4 𝑥 + 3𝑦 𝑥 − 3𝑦 .

3.Factoringbygrouping:𝑎𝑥 + 𝑎𝑦 + 𝑏𝑥 + 𝑏𝑦Thistechniqueisusedwhenthereareanevennumberoftermssuchas4or6.AfterfindingtheGCFoftheentirepolynomial,groupthefirsttwotermstogetherandthelasttwotermstogether.FindtheGCFofeachgroup.

Example#1:𝑎𝑥 + 𝑎𝑦 + 𝑏𝑥 + 𝑏𝑦 = 𝑎𝑥 + 𝑎𝑦 + (𝑏𝑥 + 𝑏𝑦).Afterseparatingthe4termsintotwogroupsof2,lookatthefirstgroupandfactoroutitsGCF:𝑎𝑥 + 𝑎𝑦 = 𝑎(𝑥 + 𝑦) andlookatthesecondgroupandfactoroutitsGCF:𝑏𝑥 + 𝑏𝑦 = 𝑏(𝑥 + 𝑦).Sofar:𝑎𝑥 + 𝑎𝑦 + 𝑏𝑥 + 𝑏𝑦 = 𝑎 𝑥 + 𝑦 + 𝑏 𝑥 + 𝑦 .Whatdotheynowhaveincommon?𝑥 + 𝑦Factoritout: 𝑥 + 𝑦 𝑎 + 𝑏

Example#2:𝑥J + 2𝑥; − 3𝑥 − 6TheGCFofthefirsttwotermsis𝑥; andtheGCFofthelasttwotermsis−3.So𝑥J + 2𝑥; − 3𝑥 − 6 = 𝑥; 𝑥 + 2 − 3 𝑥 + 2 = (𝑥; − 3)(𝑥 + 2)Notice:ifthefirstfactorwould’vebeen𝑥; − 4,wecould’vecontinuedfactoring.

4.Trinomialswithaleadingcoefficientof1:𝑥; + 𝑏𝑥 + 𝑐AfterfactoringoutanyGCFs,doesithavethreetermsintheform𝑥; + 𝑏𝑥 + 𝑐?Lookat𝑐 andaskyourself,whatnumbersmultiplytogethertogiveme𝑐 (inotherwords,whatarefactorsof𝑐)?ThenlooktoseewhatsetoffactorswhenADDEDtogetherequal𝑏 (remembernegativefactors).

Example#1: 2𝑥; + 28𝑥 + 80TheGCFis2.Factoritouttoget2(𝑥; + 14𝑥 + 40).Sonowlet’slookat40andlistitsfactors:1x40,2x20,4x10,5x8.Whichsetoffactorsadduptoequal14?So2𝑥; + 28𝑥 + 80 = 2 𝑥 + 4 𝑥 + 10 .

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PrerequisiteSkills– Page4

Example#2:𝑥; − 5𝑥 − 14TheGCFis1sononeedtofactoranythingout.Factorsof-14are1x-14,2x-7,-1x14,-2x7.Whichofthosefactorsaddtoequal-5?Thefactors2x-7.Solution:𝑥; − 5𝑥 − 14 = (𝑥 + 2)(𝑥 − 7)

5.Trinomialswithaleadingcoefficientnotequalto1:𝑎𝑥; + 𝑏𝑥 + 𝑐, 𝑎 ≠ 1Thereareseveraltechniquestofactorthesetypesoftrinomials,includingtrialanderror.Thetechniqueexplainedhereisgoingtousefactoringbygrouping.Asalways,beginbylookingforaGCFforthetrinomial.Thenmultiply𝑎and𝑐.Nowyouhavetofindthefactorsofthatnewnumber𝑎𝑐 thatwhenaddedwillgiveyoub.

Soforexample#1:6𝑥; + 11𝑥 − 10.Startbymultiplying6and-10.Thisequals-60.Whatarethefactorsof-60:1x-60,2x-30,3x-20,4x-15,5x-12,6x-10,10x-6,12x-5,15x-4,20x-3,30x-2,60x-1.Ofthesepossibilities,whichfactorsadduptoequalpositive11?Positive15times-4.Sothenextstepistorewritethetrinomialasapolynomialof4terms.

6𝑥; + 11𝑥 − 10 = 6𝑥; + 15𝑥 − 4𝑥 − 10Proceedtofactorbygrouping. = 3𝑥 2𝑥 + 5 − 2 2𝑥 + 5

= (3𝑥 − 2)(2𝑥 + 5)

Example#2:2𝑥; + 7𝑥 − 4Multiply2times-4=-8.Listthefactorsof-8:1x-8,2x-4,4x-2,8x-1.Whichfactorsadduptoequalpositive7?8andnegative1.Rewritethetrinomialsoyoucanfactorbygrouping.

2𝑥; + 7𝑥 − 4 = 2𝑥; + 8𝑥 − 𝑥 − 4= 2𝑥 𝑥 + 4 − 1 𝑥 + 4= 2𝑥 − 1 𝑥 + 4

6.SumorDifferenceofCubes:𝑎J + 𝑏J = (𝑎 + 𝑏)(𝑎; − 𝑎𝑏 + 𝑏;) and𝑎J − 𝑏J = 𝑎 − 𝑏 𝑎; + 𝑎𝑏 + 𝑏;Youfirstneedtorecognizeperfectcubes:1,8,27,64,125,216,343,512,729,1000,etc.

Example#1:𝑦J − 64Startbyrewritingitsothatyoucanseewhatnumbercubedis64:𝑦J − 4J.Now,ignoringthecubes,writedownexactlywhatyousee:𝑦 − 4.Thisisthefirstfactor.Thenyou’releftwiththefirstterm,𝑦,squared.Thefirstsignisalwayspositive.Sinceherethefactoris𝑦 − 4,thesignthatfollows𝑦; willbeplus(oppositesigns).Multiply4 and𝑦togethertogetthemiddleterm,andfinally,squarethelastterm4 toget16.Thelastsignisalwaysplus.𝑦J − 64 = (𝑦 − 4)(𝑦; + 4𝑦 + 16)whereas𝑦J + 64 = 𝑦 + 4 𝑦; − 4𝑦 + 16 .

Example#2:125 + 8𝑥J = 5J + 2J𝑥J = (5 + 2𝑥)(25 − 10𝑥 + 4𝑥;)

Ignorethecubesandyou’releftwith5 + 2𝑥Where5 isthe

firsttermand2𝑥isthesecond

term Squarethefirstterm:5; = 25

Oppositesigns Alwaysplus

Multiplythetwotermstogether.Ignorethesigns.So

here,5 times2𝑥 is10𝑥

Squarethesecondterm:(2𝑥); = 4𝑥;

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PrerequisiteSkills– Page5

Directions:Factorcompletely.Showallworkwherenecessary.

1.𝑥; + 5𝑥 + 6 2.𝑦; + 15𝑦 + 36 3.𝑝; − 12𝑝 + 27

4.𝑎; + 10𝑎 − 75 5.−𝑥; − 𝑥 + 20 6.𝑥; − 16

7.2𝑥; − 8 8.25𝑥; − 1 9.81𝑥; − 1

10.2𝑥; − 27𝑥 + 36 11. 3𝑥; + 7𝑥 − 20 12.16𝑥; − 48𝑥

13.2𝑦; − 16𝑦 + 32 14.−6𝑎; − 600 15.𝑥; − 𝑦;

16.2𝑝; − 11𝑝 + 5 17.4𝑦; + 15𝑦 − 4 18. 𝑥E − 𝑦E

19.𝑥J + 2𝑥; + 5𝑥 + 10 20.𝑥J + 2𝑥; − 3𝑥 − 6 21.𝑥J − 125

22. 8𝑥J + 1 23.125𝑥J − 27 24.216𝑥J𝑦J − 343𝑧J

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PrerequisiteSkills– Page6

PartIII.LibraryofFunctions:NotesandReview

D: −∞,∞R: −∞,∞x-int:0y-int:0slope=1Symmetricw/respecttotheorigin(Odd)inc:(−∞,∞)

D: −∞,∞R:𝑦 > 0vertex:(0,0)x-int:0y-int:0Symmetricw/respecttothey-axis(Even)dec:(−∞, 0)inc:(0,∞)abs.minat

𝑥 = 0

Linear:𝒇(𝒙) = 𝒙 Quadratic:𝒇(𝒙) = 𝒙𝟐

D: −∞,∞R:𝑦 > 0x-int:0y-int:0Symmetricw/respecttothey-axis(Even)dec:(−∞, 0)inc:(0,∞)abs.minat

𝑥 = 0

D: −∞,∞R: −∞,∞x-int:0y-int:0Symmetricw/respecttotheorigin(Odd)inc:(−∞,∞)

AbsoluteValue:𝒇(𝒙) = 𝒙 Cubic:𝒇(𝒙) = 𝒙𝟑

D:𝑥 ≥ 0R:𝑦 ≥ 0x-int:0y-int:0Notevenoroddinc:(0,∞)abs.minof0

at𝑥 = 0

D:𝑥 ≠ 0R:𝑦 ≠ 0Nox- ory-intSymmetricw/respecttotheorigin(Odd)dec:(−∞, 0) ∪(0,∞)

SquareRoot:𝒇(𝒙) = 𝒙� Reciprocal:𝒇(𝒙) = 𝟏𝒙

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PrerequisiteSkills– Page7

PartIII.LibraryofFunctions:NotesandReview(cont’d)

D: −∞,∞R: −∞,∞x-int:0y-int:0Symmetricw/respecttotheorigin(Odd)inc:(−∞,∞)

D: −∞,∞R:𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠x-int:[0,1)y-int:0NotevenoroddConstant:[𝑘, 𝑘 +1)𝑓𝑜𝑟𝑘𝑎𝑛𝑖𝑛𝑡𝑒𝑔𝑒𝑟Discontinuous

CubeRoot:𝒇(𝒙) = 𝒙𝟑 GreatestInteger(Step):𝒇(𝒙) = 𝒙

PartIV.GraphingFunctionsusingTransformations:NotesandReviewTypesofTransformations

𝒚 = 𝒂𝒇 𝒙 , 𝟎 < 𝒂 < 𝟏𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛𝑠(𝑔𝑟𝑎𝑝ℎ𝑔𝑒𝑡𝑠𝑤𝑖𝑑𝑒𝑟)

𝒚 = 𝒂𝒇 𝒙 , 𝐚 > 𝟏𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑠𝑡𝑟𝑒𝑡𝑐ℎ

𝑔𝑟𝑎𝑝ℎ𝑔𝑒𝑡𝑠𝑛𝑎𝑟𝑟𝑜𝑤𝑒𝑟

𝒚 = −𝒇 𝒙𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑏𝑜𝑢𝑡𝑡ℎ𝑒𝑥 − 𝑎𝑥𝑖𝑠

𝒚 = 𝒇 𝒙 ± 𝒌𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛

+ up,− down

𝒚 = 𝒇(𝒙 ± 𝒉)ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛

+ left,− right

𝒚 = 𝒇 −𝒙𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑏𝑜𝑢𝑡𝑡ℎ𝑒𝑦 − 𝑎𝑥𝑖𝑠

𝑓 𝑥 = 𝑥; + 3 𝑓 𝑥 = (−𝑥 + 3);𝑓 𝑥 = (𝑥 + 3);

𝑓 𝑥 = −𝑥; 𝑓 𝑥 =13𝑥

;𝑓 𝑥 = 3𝑥;

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PrerequisiteSkills– Page8

Directions: Foreachproblem,describeinwordsthetransformationshown (up4units,left3units,verticalcompression,reflectionaboutthex-axis,etc.).Thengraphthefunctionalongwithitsparentfunction.Useddashedlinesordifferentcolorstodistinguishbetweentheparentfunctiongraphandthegraphofthegivenfunction.

1. 2. 3.

4. 5. 6.

7. 8. 9.

𝑓 𝑥 = 𝑥 − 2 𝑓 𝑥 =12 𝑥 − 4𝑓 𝑥 = 𝑥 − 1 + 2

𝑓 𝑥 = 𝑥 + 4� − 1 𝑓 𝑥 = 2 𝑥 − 1�𝑓 𝑥 = −𝑥� − 2

𝑓 𝑥 = −𝑥J + 2 𝑓 𝑥 = −𝑥x𝑓 𝑥 =

1𝑥 + 2

1.Graph thecirclecenteredat(−2,−1)withadiameterof4.Writeitsequation.

2.Write theequationofthecirclecenteredattheoriginwithadiameterof9.Thengraphthecircle.

3.Writetheequationofthecirclecentered at(3,2)withadiameterof2.Thengraphthecircle.

4.Findtheequationofthecirclewithendpoints(0,0)and(2,4).Thengraphthecircle.

5.Look atthegraphofthecircle.Writeitscenter,radiusandthenitsequation.

Center:Radius:

6.Lookatthegraphofthecircle.Writeitscenter,radiusandthenitsequation.

Center:Radius:

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PrerequisiteSkills– Page9

PartV.Circles:NotesandReview

Directions: Reacheachquestion.Showallworkwhereneeded.

Standardformoftheequationofacircle:(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐,withitscenterat ℎ, 𝑘 andaradiusof𝑟.Circlewithcenterattheorigin:𝐱𝟐 + 𝐲𝟐 = 𝐫𝟐, 𝑤𝑖𝑡ℎ𝑖𝑡𝑠𝑐𝑒𝑛𝑡𝑒𝑟𝑎𝑡𝑡ℎ𝑒𝑜𝑟𝑖𝑔𝑖𝑛 0, 0 𝑎𝑛𝑑𝑎𝑟𝑎𝑑𝑖𝑢𝑠𝑜𝑓𝑟.Giventwoendpoints,findthecenterofthecirclebyusingthemidpointformula:𝑀 = ((%~('

;, $%~$'

;)

Giventhecenterandoneendpoint,findtheradiusofthecirclebyusingthedistanceformula:𝑑 = (𝑥; − 𝑥4); + (𝑦; − 𝑦4);

�

Giventwoendpoints,findtheradiusofthecirclebyusingthedistanceformulatofindthediameterandthendividingby2tofindtheradius(rememberradius=½diameter)

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PartVI.ExponentialFunctions:NotesandReview𝒚 = 𝒂𝒃𝒙

1. 2. 3.

4. 5. 6.

𝑓 𝑥 = 3( + 1 𝑓 𝑥 = 2(~;𝑓 𝑥 = (0.25)(−2

𝑓 𝑥 = −3( + 1 𝑓 𝑥 = 3(0.5)(𝑓 𝑥 = 2 + 4(&4

If𝑎 > 0 and𝑏 > 1,thefunctionrepresentsexponentialgrowth.

𝒚 = 𝟐𝒙

If𝑎 > 0 and0 < 𝑏 < 1,thefunctionrepresentsexponentialdecay.

Domain:(−∞,∞)Range:𝑦 > 0

y-intercept:(0,a)x-intercept:noneasymptote:𝑦 = 0one-to-onefunction

SmoothContinuous

𝒚 = (𝟎. 𝟓)𝒙

Directions: Grapheachexponentialfunction.Makesuretoincludetheasymptote.

Asymptote:𝑦 = 0

Asymptote:𝑦 = 0

Asymptote:𝑦 = Asymptote:𝑦 =Asymptote:𝑦 =

Asymptote:𝑦 = Asymptote:𝑦 =Asymptote:𝑦 =