Function notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property

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1 Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY t give yu the last term * the first term Dn t frget t lk t factr further even after yu finished the first rund f factring Special Trinmials Cubes Difference f squares Perfect square trinmial Frcing the square Sum Difference If a pwer is missing in the dividend, it must be included using a 0 as cefficient Bth expressins must be in the same rder (ascending) Useful: if ne factr f plynmial is knwn, can divide t find the ther factr Remainder Therem When a plynmial f(x) is divided by x-a, the remainder is f(a) Factr Therem Same as remainder therem: if x=a is substituted int plynmial and resulting value is 0, x- a is a factr f the plynmial Factr Prperty If a plynmial has any factr in the frm x-a, then the number a is a factr f the cnstant term f the plynmial 1

2 Slving linear equatins Finding the rts f equatins State restrictins OR n slutin because f cntradictin (dn t frget t d it fr fractins) Slving quadratic equatins by factring using the Zer Prduct Prperty Quadratic frmula Slving plynmial equatins by factring Slving equatin invlving abslute value i squared is -1 Zer Prduct Prperty Set the factrs equal t 0 because that means 1 r bth f the factrs is equal t 0 Discriminant Cnjugates The expressin b 2-4ac tells us the nature f the rts Tw DIFFERENT: Tw EQUAL: NO REAL rts (2 cnjugate cmplex rts: A cmplex number is ne like a+bi Cnjugate f a+bi is a-bi Slving Plynmial Equatin by Factring Lk fr gruping first if it desn t wrk, d lng divisin methd Abslute Value Equatins Remember restrictins & different cases Radical Equatins Islate the radical n ne side Square bth sides Slve Check the rts t identify extraneus rts 2

3 Plynmial inequalities and cnstructing sign charts Y = a ( x 1 ) 2 ( x + 2 ) ( x + 3 ) 2 Include and equal t in final answer if questin has an and equal t Functins with same rts AND same rder are frm same family f functins Dn t frget restrictins n RATIONAL inequalities As sn as yu see n slutin, duble check t make sure there really IS n slutin Example: if yu have a less than r equal t 0 and there is n slutin less than, remember the equal t slutin t! BASICALLY, beware f and equals and duble-check the equal t slutin as a pssiblitiy When finding y-intercept d nt frget t square and multiply by cefficients as well! If there s n cnstant, y-intercept is 0 Always pay attentin t yur slutin t see if it can be cmbined 3

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5 Similar triangles Angle-Angle Side-Side-Side Side-Angle-Side Primary trig ratins in a right triangle Slving a right triangle Trig ratis f special angles (30, 45, 60) Reciprcal trig ratis Trig ratis fr btuse angles Csine law Sine law Ambiguus case Dn t frget apprximately signs Angle f a sun s ray is the same Angle f elevatin vs. angle f depressin Trig ratis cnversin 1 / sin csc 1 / cs sec 1 / tan ct Remember t draw the curve n an angle Ratinalize denminatr! X can be negative! Ambivalent case sin r wrd prblem Used t slve a triangle when given tw sides and the cntained angle OR the measures f 3 sides Used t slve any triangle when given tw angles and any side OR tw sides and an angle ppsite t any f the tw sides Angle in Standard Psitin is ne whse vertex lies at rigin and initial arm lies n psitive x- axis Primary trig ratis sin = y/r, cs = x/r, tan = y/x 5

6 Sin (180 0) = sin 0 Cs (180 0) = - cs 0 Tan (180 0) = - tan 0 When 2 sides and the nn-cntained angle f a triangle are given, it may nt be unique. May have n triangle, ne triangle, r tw triangles with the given measurements 6

7 Cnverting radians t degrees and vice versa Areas f sectrs and lengths f arcs CAST rule & slving fr theta Using the trignmetric ratis f angles (x, y, and r) Trignmetric identities Slving trig equatins using identities 180 = Change yur calculatr between radians and degrees! Watch units (write them in) Rund t what is said Radians Degrees Cycle ne cmplete pattern Perid hrizntal length f ne cycle Amplitude half the difference between maximum value f functin and minimum value 7

8 Vertical Stretch (tells yu amplitude a ) Hrizntal Stretch (tells yu perid 360/k) Vertical Translatin Hrizntal Translatin 8

9 Expnents review Applicatins (expnential grwth/decay) Y = ab x Lgarithms and lg laws Lgarithms with bases ther than 10 Multiplicatin and Divisin Laws f Lgs fr Pwers and Rts Als remember: Translatins 9

10 Finding a lcatin in a matrix Adding matrices Multiplying matrices Applicatin f matrixes Using matrixes t slve linear systems Echeln frm 3 x 4 4 x 5 Rws can be interchanged Rws can be multiplied r divided by any Real number Rws can be added and subtracted frm ne anther CONSISTENT (ne unique slutin) INCONSISTENT (n slutin) DEPENDENT r COINCIDENTAL (infinite slutins) 10

11 Frequency plygn (relative, cumulative, regular) Histgram vs bar graph Range Mean, median, mde Weighted mean Deviatin Percentiles Ppulatin Sample Scre at which a certain percentage f the data lies belw 11