Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... + a x + a 1 1 1 0 P.3 Polyomials ad Special products Expoets must be coutig umbers (1, 2, ) ad the coefficiets ca be ay real umber Stadard form = highest degree first followed by descedig powers 2 7 7 2 4x 5x 2 + 3x = 5x + 4x + 3x 2 Vocab: Stadard form, lead coefficiet, terms, degree II Groupig Whe addig ad subtractig, group like terms Ex 2a (show all steps!) ( 5x 3 7x 3) + ( x 3 + 2x 2 x + 8) Ex 2b ( 7x 4 x 2 4x + 2) ( 3x 4 4x 2 + 3x) III Multiplyig Polyomials A) Distributive property 5(x+7) x(x-7y) -2(x-2) B) FOIL( 2x 4)( x 5) + step up from distributive property C) Multiplyig polyomials bigger tha 2x2 ( x 2 2x + 2)( x 2 + 2x + 2) D) Special Products (p. 30) + = ± = ±2 + ± = ±3 +3 ±
I Equatios ad solutios of equatios Idetity Eq: 1.2 Liear Eqs i Oe Variable 9 = +3 3 How may solutios? Coditioal Eq: 9 = 0 is true for x=3 ad x=-3 oly Cotradictio Eq: +7 = +5 How may solutios? II Liear Eqs i oe variable p. 87 Stadard form: + = 0 Liear equatios have exactly solutio(s) Equivalet Equatios: have same solutio(s) 2 = 4 is equivalet to = 4 2 = 6 is equivalet to x=3 III Equatios that lead to liear equatios EX 3 Fractio equatios + = 2 +3 4 = 2 +4 1 Ex 4 Extraeous solutios = IV Itercepts algebraically: to fid the -itercept, you plug i for ad to fid the -itercpt you plug i for. Ex: Fid the itercepts for 3+2+5 = 15
I Itro to problem solvig page Key words for additio: Key words for subtractio: Key words for multiplicatio: Key words for divisio: 1.3 Modelig Liear Equatios Draw the picture, list what you have, what you are lookig for, choose the formula, double check your aswers, state aswer clearly. II Usig math modelig Moey ad Percet #49 Dimesios #57 Distace formula d=rt #63 Similar triagles (height problems) #69
III Mixture Problems Ivetory #73 IV Commo formulas Box o page 101, has formulas for what? Ex 9 Cat food ti Which formula do we eed? Why? P.4 Factorig Polyomials I Commo Factors read p. 37 A) Defiitios a. Completely factored: b. Prime polyomial: B) Commo factors distributive property i reserve directio (Fid GCF) 3 a. Ex 1a 6x 4x b. Ex 1c ( x + )( x) + ( x + ) 2 2 2 3 C) Special Factorig techiques a. P. 38 for table b. New special factorig forms i. ( )( ) ( )( ) u + v = u + v u uv + v 3 3 2 2 u v = u v u + uv + v 3 3 2 2 sig ote: 2 16x + 24x + 9 x + 2 y ex 4 2 x 10x + 25 Ex 3 ( ) 2 2 #50 3 8 y + 125
II No special patter, triomials with biomial factors 2 Ex 7 x 7x + 12 2 2x + x 15 III Factorig by groupig read p. 41 paragraph 3 2 Ex #72 x + 5x 5x 25 #80 2 2x + 9x + 9 IV Applicatio: Geometric modelig p.44 #131 1.4 Quadratic Equatios ad Applicatios I Factorig Stadard form for a Quadratic equatio is Highest expoet is, so the degree of this polyomial is Quadratic equatios are. Solvig a quadratic equatio by factorig oly works if Ex 12 +9+7 = 3 II Extractig Square Roots If the quadratic equatio is =, the its equivalet equatio is This is a special product: = So aswers are +3 = 25 4 +7 = 44 5 2 +100 = 2
II Completig the Square + is completed by addig to both sides of the equatio So + +! " = Whe the leadig coefficiet is oe Ex 3 +2 6 = 0 Whe the leadig coefficiet is ot 1: Ex 5 3 4 5 = 0 Iside a algebraic expressio: Ex IV Quadratic Formula: Give the stadard form quadratic equatio + +# = 0, we ca complete the square ad derive the quadratic equatio: =!±! %& % Proof o page 111 Solutios to the quadratic formula are based o the discrimiat, 4# 1) It is positive: 2) It is zero: 3) It is egative: Usig the quadratic formula: Ex 6 +3 9 = 0 V Applicatios Dimesios #114 fallig time ad positio #119
3 5 4 36 36 125 64 32 81 ( a ) = ( a) m m a * b = ab a a b m a = = a b m = a page 20, geeralizatios about th root of umbers a P.2: Radicals if ' is eve, a = a If ' is odd, a = a IV Simplifyig Radicals 3 75x 3 6 40x 3 16x 54x 3 4 V Ratioalizig Deomiators ad Numerators ( 4 5 2 3 2 3+ 7 *Always try to keep radicals out of the deomiator! The same techique works for umerators! V Ratioal Expoets a a 1 m = a 1/ is the ratioal expoet ( a ) m = m/ is the ratioal expoet Ex 16c (radicals to expoets) 4 2x x 3 2 2 Ex17a (expoets to radicals) ( ) 3/ 2 x + y
I What are they? Imagiary uit is = 1.5 Imagiary Numbers Complex umbers: add real umbers to imagiary uits + **Formal defiitio p. 122 Equality i Imagiary umbers +) = # +) meas II Operatios with complex umbers A) Additio/subtractio Add or subtract the, the add or subtract the Ex 1 a) 4+7)+1 6) c)3) 2+3) 2+5) B) Multiplyig Ex 2b) 2 )4+3) d) 3+2) III Complex Cojugates Complex cojugates ad whe multiplied together produce, ad elimiate Writig a quotiet i stadard form: Ex 4 * * IV Complex Equatios of Quadratic Equatios **Always pull out ) first! Simplifyig Complex Radicals: Ex 5b 48 27 Complex umbers ad quadratic equatios Ex 6b 3 2 +5 = 0
1.6 Other types of Equatios I Polyomial equatios of degree 3 or higher Solve by factorig (eed =0) Ex 1 3 = 48 Like Ex 2, #14 +2 +3 +6 = 0 Some polyomials are Quadratic i ature (or type) i.e. A example similar to Example 3, #20: 36, +29, 7 = 0 *Note: plug the origial solvig for the origial variable II Equatios ivolvig radicals Ex 4 2 +7 = 2 4 3 = 2 5+2 #52 +3 / = 8
III Equatios with fractios or absolute value: key here is to multiply by the LCD of all fractios + = 0 / #66 = 1 I absolute value problems, we have to accout for both the positive ad egative outcomes. Beware of extraeous aswers +1 = 5 +10 #74 +6 = 3 +18 IV Applicatios you are resposible for readig the examples o your ow. There are may differet examples with all the types of equatios we have discussed today. 1.7 Liear Iequalities i oe variable Read pages 140 ad 141 o your ow, will ot cover i class, but use them a lot I Ve diagram Bouded Ubouded II Solvig Liear equalities Whe solvig iequalities, you eed to solve it as a. But, it takes to satisfy it make sure you sketch all parts!
Just like solvig liear equatios Similar to Ex 2: #40 4 +1 < 2 +3 Sig flips? Similar to Ex 3: #37 2 1 1 5 Double iequalities! Whatever you do, do it to ALL sides! Similar to Ex 4: #48 8 3 +5 < 13 III Iequalities ivolvig absolute value - Blue box page 144, it s importat. Like whe we solved absolute value eqs i 1.6, we eed to solve them cosiderig 2 outcomes ad. Similar to Ex 5a 5 < 2 ca be read as #62 7 < 5 #64 8 0 IV Applicatios #127 p. 149 1.8 Other types of iequalities I Polyomial iequalities Key umbers (p.150):. Test itervals (p.150): Ex: 2 3 = +1 3, so its zeros are ad The zeros divide it ito test itervals, which are O a test iterval, the fuctio is either all positive or all egative.
Solvig a polyomial iequality Similar to Ex 1&3, #22 > 2 +8 Use table to solve ad look at all the pieces! Similar to Ex2, #28 2 +13 8 46 6 We ca check out solutios algebraically ad graphically: Algebraically: plug i a x-value ito the origial iequality Graphically: sketch the graph of the polyomial ad see what happes with the graph ad the x-axis. Uusual solutio sets Ex4 +2 +4 > 0 +2 +1 0 +3 +5 < 0 4 +4 > 0 Try #30 +3 +8 > 0
II Ratioal Iequalities Key umbers i a ratioal expressio are Ex 5 / 6 3 III Fidig the domai of a expressio All we are doig is buildig o the stuff that we leared i P.5 whe we covered ratioal expressios Similar to Ex7: #64 7 0 I Is it a solutio? Ex1 y=10x-7 poits (2,13) (-1,-3) Sectio 1.1 Graphs of Equatios.
II Key top graphig: plug i x-values ad produce solutios Ex 2 y=7-3x x Y=7-3x (x,y) -1 0 1 2 Ex 2 = 2 x Y=x^2-2 (x,y) -1 0 1 2 III Fidig Itercepts: always put aswer i To fid x itercept, set y=0 = 8 4 ^3 8 +32 = 64 To fid y itercept, set x=0 = 8 4 ^3 8 +32 = 64 IV Symmetry p. 80-81 replace somethig, if get same as started with the it is symmetric Respect to x-axis Respect to y-axis Respect to origi plug i for y, simplify plug i for x, simplify plug i for x, for y, simplify
Ex. 5 Fid symmetry algebraically = 2 Symmetric to x-axis? Symmetric to y-axis? Symmetric to origi? Sketch the graph of the eq: EX 7 = 1 Check for symmetry first ad itercepts x y (x,y) -2-1 0 1 2 V Circles Stadard form is. h + : =; with radius ; ad its ceter at h,: If 3,4 is a poit o a circle ad its ceter is 1,2, what is the equatio of the circle? To fid radius use distace formula == + Give +8+ 2=32, write it i stadard form of a circle by completig the square twice
I Usig Slope Slope-itercept form: If we plug i x=0 Graphig: slope is rise over ru, so if you kow m= >, 2.1 Liear equatios i 2 variables the you go to the right ad up. Horizotal lies Graph y=-1 Vertical lies Graph x=4 Graphig a liear equatio: x+y=2 Cotext is importat, do ot cofuse the poit x=0 with the lie x=0 (or y=4 with the poit y=4) 3x+1=y II Fidig slope p.172 Slope (for o-vertical lies) is over, or m= Order is importat! Like Ex 2: fid slope betwee (4,5) ad (-2,5) Ca zero be o the bottom? if you get zero o the bottom, the the graph is a. Picture of positive, egative, ad horizotal slopes bottom page 172 III Writig Liear Equatios i two variables Poit slope form is this is a tool to help us get to #66: If lie goes through the poits (4,3) ad (-4,-4), the what is its equatio i slope itercept form?
IV Parallel ad Perpedicular lies Parallel Lies have the Perpedicular lies have slopes as i.e. if slope was 6 the the perpedicular lie would have a slope of / Note, the perpedicular lie of a verical lie (=5) would be a horizotal lie (ad vice versa)! Like Ex 4: #88 Fid parallel lie ad perpedicular lie to +=7, 3,2 V Applicatios Slope ca represet i the real world p. 182 #130: The Uiversity of Florida had erollmets of 46,107 studets i 2000 ad 51,413 studets i 2008. (a) What was the average aual chage i erollmet from 2000 to 2008? (b) Use the average aual chage i erollmet to estimate the erollmets i 2002 (cut back o how much for sake of time) (c) Write a equatio of a lie that represets the data i terms of year T what t=0 is 2000. What s the slope? Iterpret the slope i cotext of the problem
2.2 Fuctios I Itro A fuctio is (p. 185) Characteristics of a fuctio are (blue box p.185) p. 186 blue box Four ways to represet a fuctio Determiig a fuctio visually Draw a example of a fuctio Draw a example that is ot a fuctio Fidig fuctios algebraically rearrage to solve for Y Ex 2a +=1 Ex2b + =1 II Fuctio Notatio Iput x Output f(x) f of x Equatio @=1 Evaluatig a fuctio Ex 3 A= +4+1 A) G(2) B) G(-3) C ) G(x+2) Piecewise fuctios Like ex 4: @=B +1, 1 C Evaluate at x=0, 1, 3 3+2,>1 Fidig values whe @=0 set eq equal to zero, you are fidig the X values Ex 5a 2+10 Ex 5b @= 5+6 Fidig values whe @=A Set eqs equal to each other Ex 6a @= +1 ad A=3
III Domais of fuctios Have studet read the paragraph p.190 startig at the title ad edig at Ex 7 We are just buildig o what we already kow Ex 7: a) @:{ 3,0, 1,4,0,2,2,2,4, 1} b) A= 6 c) Volume of a sphere: = G; d) h= 4 3 IV Applicatios #93 Path of a Ball: The height y (i feet) of a baseball throw by a child is = H +3+6 where x is the horizotal distace (i feet) from where the ball was throw. Will the ball fly over the head of aother child 30 feet away tryig to catch the ball? (he holds a baseball glove at 5 feet tall) V Differece Quotiets Ex11: For @= 4+7, fid IJI J A summary of terms ca be foud o page 193, this is expected that you kow ad I will be usig them without goig over them agai
Review Outlie Exam 1 (Sectios 1.1-1.8, 2.1-2.4) Sectio 1.1 Determie if a poit is a solutio to a graph. Graph a equatio by plottig poits. (*Must show your x/y chart!) Fid the x ad y itercepts of a graph. (*Itercepts must be writte as coordiate poits!) Determie both algebraically ad graphically if a graph is symmetrical about the x-axis, the y- axis, or the origi. Sectio 1.2 Solve a liear equatio Solve a equatio with a fractioal expressio. Solve a equatio with extraeous solutios. Fid the x ad y itercepts of a equatio. Sectio 1.3 Write ad solve a mathematical model. Sectio 1.4 Solve a quadratic equatio by factorig, usig square roots, completig the square, ad usig the quadratic formula. Determie the umber of solutios to a quadratic equatio by usig the discrimiat. Sectio 1.5 Add, subtract, ad multiply complex umbers. Use the complex cojugate to write a complex quotiet i stadard form. Solve a quadratic equatio with complex solutios. Sectio 1.6 Solve a polyomial equatio with a degree of three or higher. Solve a equatio ivolvig radicals. (*You must check your solutios!) Solve a equatio ivolvig fractios. (*You must check your solutios!) Solve a equatio with absolute value. (*You must check your solutios!) Sectio 1.7 Represet a iequality o a umber lie. Represet a iequality i iterval otatio. Solve liear iequalities i oe variable. Solve double iequalities. Solve liear iequalities ivolvig absolute value. Sectio 1.8 Fid key umbers. Idetify ad test the test itervals for solutios sets. (*You must show your work for your tests!) Solve polyomial iequalities ad write your aswers i iterval otatio. Solve ratioal iequalities ad write your aswers i iterval otatio. Fid domai for a square root fuctio. (*Write a iequality to do this.) Sectio 2.1 Graph a lie i slope-itercept form (y = mx+b). Fid the slope of a lie. (*Slope formula must y2 y1 be memorized! m = ) x 2 x 1 Write the equatio of a lie i poit-slope form y y = m( x x )). ( 1 1 Write the equatio of a lie i slope-itercept form. Describe what it meas for two lies to be parallel. Describe what it meas for two lies to be perpedicular. Write the equatio of a lie parallel or perpedicular to a give lie. Sectio 2.2 Determie if a relatio is a fuctio. Use fuctio otatio ad evaluate a fuctio. Evaluate a differece quotiet. Evaluate a piecewise fuctio. Fid values for which f(x)=0. Fid values for which f(x)=g(x) Fid the domai of a fuctio.