MAC Final Study Guide
MAC Final Study Guide MAC1105
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This 108 page Study Guide was uploaded by Michaela Maynard on Friday December 4, 2015. The Study Guide belongs to MAC1105 at Florida State University taught by Pennington LeNoir in Summer 2015. Since its upload, it has received 782 views. For similar materials see College Algebra in Calculus and Pre Calculus at Florida State University.
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Date Created: 12/04/15
852quot 5 MAC 1105 Lecture Outline for Dr LeNoir s lecture classes Our course starts with an algebra review in Objectives 15 A lot of this material should be familiar to you 39Please be careful about judging th adj ffiugglmgfd the entire course based on the rst few dams In Preliminary Objectives 14 we will review the basic factoring techniques that will be used throughout our course Preliminary Objective 1 Factoring out the Greatest Common Factor GCF in a polynomial of one variable Note For the variable terms the eb b exponent is the MOQL that s common 48m5 8x3 4132 Laws of Exponents 3 ha m M LIX lax not l am anzom ab Ow b m h a m on a m 2 1 aquot O b b L amnij aim a M Our main emphasis will be factoring out the GOP when there is a negative coef cient on the highest power of 2 As we use this technique in Obj 1 and 3 It will be your choice whether or not to factor out the minus sign in your initial factoring step 52x3 13x4 52x3 1302 Ll gt 13 3 x quot3 0 39 4 Xi H gt 3x3xquot4gt 2235 55a6 22x5 556 MPG 5x my a t Sk H355 5k 2 Preliminary Objective 2 Factoring the Difference of Two Squares mtgEQW quot sum 0 square I 25332 9 25x29 Does n21 fatter 5x 3 5 r 3 Our main emphasis will be factoring the difference of two squares when there is a negative coe icient on the variable term i 25 35 3 g 9 i I 4 amp 3 5L 3 1 432 1 quot39 4132 1 300 Vat quot gvan amt 1 Preliminary Objective 4 The relationship between a b and b a 391 a 1lta bgt 0 rb 1b a 9 boa Q b Conclusion They are n or 2E2 of each other 5 8 33 1 3 22 139 O quot a Slmp lfy 5 8 1 c I 293 3 l Note ab 5 M A Same Preliminary Objective 3 Factoring Quadratic Trinomials Note Many of the problems in our course contain a hint for the trinomial factoring Use It Before we factor recall how to multiply two binomials F O I L Multiply das 52v3 c 5 x ug2xbz33951l 6 mg m We M5 m mm 40 3943 my agx M5 Rx nds We ll use Reverse F O I L in order to factor For some problems one factor will be known you will need to determine the other For some problems you have to be able to factor with no hint Two eGrade examples Complete the factoring 18x2 191 12 21 3ax J b Enter a and b as an ordered pair however if the operator is subtraction then you must enter b as a negative number For example 95123 3 you would enter 2 3 for 21 3Lvoujvould ente rQ3 r 18x2 19x 12ax 3gtQy 4 39Vq Ll th hKc Complete the factoring 1812 331 14 61 7ax l b See eGrade instructions above 18m2 33x14p 7gt3 2 Factor 18x2 51m8 24132 753 5 4A l2c 3 723937 5 4K 5 9x 15 1 qt MOO 54 q x 6l 3quot 5100 It W 1 3x 4 quotfa 8x 5 3x xlYX l39Y 39 quotX39a 3 blab lobx a q JimMg 014 395 These problem are in MultiFormula mode enter factors separated by semicolon lulut iWMh QaUchm I 3quot X 2 39 rst equals a 0 m La t equals a b rm Objective 1 Rational Expressions fractions made up of Ol 3 Note Values that cause division by zero are called r oryn th variable We ll do more with this concept when we study function domains in Obj 100 We will not list restrictions on the vbl as we work in Obj 1 b f O Obj 1a Recognizing Equivalent Rational Expressions In particular dealing with a pesky minus sign when it appears in front of a fraction Consider a numerical fraction 4 3 3 35 We will ask Which are equivalent to the given expression and the problems will be in Multiple Selection mode You will choose all that are equivalent if none are then select that choice Plan of Attack Generate equivalent expressions by distributing the minus through the M then separately through the Q Q Compare these with the answer choices Or you may nd that you can simply compare each choice with the given expression Keep in mind Prelim Obj 4 a b b a Which are equivalent to 33 6 X39U xLp Vb 12 X 2 L1 X U xo U39N c 1 x 1 a6 6 c Answer choices None are equivalent Which are equivalent 0 32 39 jg 4i 3 2 x z X l z X 41 2 b quotz gt Z Z gt m xz Answer choi es x z None are equivalent CL 9 4 b 0 d Objective 1b Multiply Divide Rational Expressions 9 Q 1 lo C Plan of Attack Factor Ever thi Three types of factoring Factor a trinomial ways factor it Ia b Use the hints provided by the other factors Remove the GCF Factor the Difference of Two Perfect Squares x2 y2 When cancelling common factors remember Prelim Obj 1 2 1 2x2 5x 12 M F i x 5 Simplify 29525 m a 39 quot39gt 42 20 3 MC 33fcx 5 9 20939s x W 6x3 lain 3gt an 6 2012 23cc 6 101 Slmpllfy39 9 1632 39 4m2 10x3 I I 5 5 MM W W6 uquot x qua O O 25 Ex4 g2 an 535 QL 9 f559Q S mphfy39 2x2 1313 20 36 A r 5 4 6 2 F Q 5W rat39s Mam on WM can O I DKJ5 01Lgtgt BOILl laba T 3 F 3 21L 0c abs 9115 Objective 1c Add Subtract Rational Expressions Plan of Attack Use the Least Common Multiple to create a common denominator in each fraction Determine the LCM For terms containing the variable use the highest power in any one denomi nator 90 Ob fed Cb GOLUDV nam Era r Create the common denom in each by comparing each fraction s denom with the LCM and them by multiplying each fraction topandbottom by the part missing from the LCM 2 1LCM 29 5 7 CMZ 1h Sunphfy392 6 3 8 582 Slmpllfy 62 6Z5 E a Loa fxa 3L 3amp J5L Zg I 23912 at 9x quot 8k 3 3 5b 4 a W 49x 30x 3 67 w x3 3quot 3 3amp3 if a 3 81 quot 20L 3 Once W 5 H quot 722 Simpl i39 1Z152 5quot Wm one 3 F9 3 L 5 3x 31 qxq11 at 3L3 k j 3 51L 39 3o cqrc lgucg 3 3 9 3mquot 3wquot wabch lyncj e a common Simplify 8 19 X W Y m ma 3 how 5132413 12 2x25x 42 qa 0me 8 Hui q n a mac wgt 57W Samoa lt5 m BCK39LQ a 5 394 BX a 3 439 zaampb7 Cit3 k bax7 nRCCpampu7 3 EWWY mum on o n e 19 7 e wade a wv S mphfy 3x2 11x 20 a23c 10 44604 763w lq d 23 BMW Twila5 W WOW 14x3x 2u 23 5 1 VOL quot W 1quot mittMb Hmt39 inbmml 3 f 0 actor 0 mole an Simplify 4923 3 1 490 5 Simplify 33323 g 1 6 3336 qHSSCK g 3g 5 f 3 Q 3Kq 393 6b 3amp gtL 0 Caliquot at f 39 9 5za was toe 3 3 5 3x123 773 31 Mwbm39f w gy 5 32 004 0 3quot 3x 41 Big La 4 pg 104 33x ll 5 KM 41c 5Kagt 45 5L39gt aw M C a Wan 52 iwr 128 i229 39 41SXLaz gr on ID 33 o quot m Simplify 5x28529116 6352 Simplify 6332131136610 5 3 29 5 KW C gt 134quot g 1 31 lq 3 quotH39 x2lt2x5gt 572x 5mm 9 5 0 quot 5 A 3 N UUatCln neg 35573 763Li3928 by 44516 y 5 37 3uA Art 0255 3mm J r SKI a Lo 46W A Can not 19C f 3L Li 363KAgt T actored7redu39ct 23 Further 3quot Rgt 3V5 drawer 5b 37 3 6 4 39 4 4 U 139 qt 0 Objective 1d Simplify mixed quotients complex fractions There are two common methods used for Simplifying mixed quotients I ll demonstrate the method that I think is best both methods are Shown on the Video clips for this objective 8 n L a 6 Plan of Attack Determine the LCM of all fractions n n U m Olen the h Multiply top and bottom of the blg fractlon by the LCM of all A we t MAI b 3 3 3 d c d 39c 739 d a 9 C Smphfy 3 3c 3 c 3c 3s g ac 3d d c c c c CCQ C Cd c 3 Cd39C2 36 9 3C 0 8 1 LDC 39 36 LOC 36 Jinn d 9c 6 A Mull Chowe 1 fadLCMi 33 1 0 eh L w gt 39439 39 231 I 39 Simplify Elf l V own 3 2 g qw wy H J m3 a i w 5 W quot39 W J39MUH thence wr j s 1w 1a i r 4 4 u lmpl a w a66 TE a m quot quot am a Mb 9 am a Ma a Ma a 9 0 M T 1 FA Simplify 54 2 X rW XL 1 C4 Li L V A XH l m M 111 3 N x H 1 9 W z M YB f3 1 L Sunplify 1 A 39T 39 GTE 5 77 LL31 5 fs 9 4 tj 7 4 o 39gr a q Simplify Beduce completely to a simnle fraction Enter the answer as an ordered pair a b If negative you must enter the minus Sign with a For example 1 3 71 you must enter 13 For integer answers you must enter b as 1 For example 3 55 you must enter 31 3 51 quot5 2 0 Simplify Simplify 4 7 1 3 I 1 2 T I 3 t 5 2 C3 quot39 0 3 73957 31 5 70 0 2 7 7 1 ab a C0 3 39 0 vi 139a 5quot e 30 Y w a JKExMCY C J 5 1l7llcgt39lquotaio 3939 1 5 3 1 1 g 3 p v 2c a Simplify 932 5 Simplify 32 5 3 1 r at 1 lt gt i 5 l 5 3 l r L I quot R 3 5 q aj q 3 75 3 27quot 3 5 5 5 01 KM 5 5 28 9 3 3 539 39 7 3 72 1 I l ak Emen W rEYibfY 356 gal 5 3n 23 Ce Simplify 1 15 4 q 4 Simplify 3 2 a g were 1e if 0 3 a i 13 T 92 3 q 3 q 1 33 69 4 739 3 25 18 L a T L 3 Eheew p w 5 l I I l 7 9r F 3 Ira quotl Uge Mugabe mam a 5 Write in sim lest radical form SRationalize the denominator simplify and reduce completely Enter tmrmn answer negative the minus Sign must be entered with the numerator For example 35 The answer must be entered as 3sqrt25 or as 3sqrt2 5 Vii 035 gig FL wo i 1 5 fr g 43 5 F5quot EE L 4 W rTE rl 4 5quot 51 our r 39 AtaKe w ate4045 4 M c 535 B BsqrtC V Li H g 3 373 qi gz J MWEQ 3 61 3 5 x 5 D 3 3f 3 V 5 clt 5535 5 4 l lm 6 0 quot39quotquotquot I fTg TEESquot 396 quot 3 alumsB 3 WE VE 239 wi Ifquot 4 a quotU qu bc3 E 1 15F5 JE ljlg 6 1 1 5 o a 5 5 SELL 4 I3 614 E H r54 F 5quot 95 543 3 059 22 47 936 F V5 5gt6l5 2 f L 2 sanser 35qlta 3 CiaIS Objective 2 Simplify numbers raised to rational exponents Plan of Attack Rewrite using the de nitions below Remember that cfquot Remember Order of Operations De nition a1 J Provided the root exists ie is a real number I V h M m an JET r oda Simplify 2713 3m 3215 53 51 t Wowe 1614 2512 a5 2512 r i am IR 39 a 5 E n bot ads LANGQRMCI MQHS n63 esz ge 2723 4 j H 345 d H 3 i My q quot 32 43 p 33 lg O IL0 31 271 4 h 3 s 74 ll Tgo 2 W3 27 63W 3 2510 m3 I000 477quot 3 3 256 34 25534 I I via L7 2 quot2 l 4 M23 LI 3956 era593 My 7 Mndtbnbd F 19 32 E 32 5 3 81 3 RT 3 2 1011 72 4f W i w i 3 43 73 Wide vied 9 73 2 32 Objective 3 Special Factoring Techniques Obj 3a Factoring by Grouping Typically used when expression has 4 terms Plan of Attack goup the rst two terms together and factor out the Group the last two terms together and factor out the GCF The binomTaTs must match If they don t check to see if you need to factor out a minus sign Once the binomials match factor out the GCF the common binomial Sometimes you have to rearrange the terms try rearranging the two middle terms Factor 3x 33 my x2 506 l 39K tj H4 mg 3m Of 397 39 so 3y 5 no Which of the following 398 a factor of 3x 33 my 5132 Is a factor of means which would divide evenly into the given expression 96 9 2 w 3 2 Factor 132 by2 ay2 1332 Cuba notquot Haj quot39 aka Qxa 3ab Lisa 09 9 a or K OC gtQ U u l9 13 3 M 0 L CV19 Which of the following is a factor of 1532 by2 ay2 bxz Is a factor of means which would divide evenly into the given expression 11 w K entrangz t m3 gt6 39gaa bgt L 3 6 3 19 Q 61 000 19 Factor xy3ycc3 xv X Jr3lj 3 Xj39gt 33 1 Obj 3b Factor out the GCF with rational exponents 397 on 24 I D Plan of Attack Use the exponent Which is a factor of 2432 3 2223 2 1053 1 9quot quot 3 5136369 m 2 3 b39x4 3 o5c3935k 06 D ax 5latlz5LRgt Rx3935x 1 We l5 Which is a factor of 2403 1 22272 1029 3 3 3 7 39 MY M 4w W ax393lax lx39 5 423 5 4 553 3x1 noneofthese E Factor 2793 2 12x 3 3p 3mP quot 4 3L 3 fz 39LI 3Pu eat SWAG 63f Facto 9354 5x114 6x 34 3 K i c 39 7 3951quot q quot p 9 x g ct x 40 395 X 46 f 39Xquot 396 12 Which is a factor otgxlZ 2x 32 an A 4L 2 1 8x4 2 a 1 7 5b1m o Mum63 9095 E37 Faccox 21 1 4x1 2 2 Factor 43012 17 32 80 72 7 J J quot l 5 q 9 l 4 K 3 ha 4 x quot3amp4393 7 a 5 Obj 3c Factor out the GCF common binomial and rational exponents 397 3939quotquot39 q 6 a he Plan of Attack Simplify rst when needed x Mu W39 g I m Y 6 m 53x14i 13m1 o Simplify 3m D8 15 5 coc to M 5W0 m w r W 3 08 Ln bw H 314 1 Simplify x quot 1X 33 33 32 3 Simplify 293 39 1E 236553 2mm D3 WKRMM 3mm gammy m 2 4 393 13X3 4 C 73 063 K lUL l JrBX pr 3 13 2 3L3 lLIJC Ll K R Factor 1 333 175 Factor 2533 153 f 2 7 4L333I 381076 3 V3 3 30 3x3l95 3 5143 3 g gf 35 3x3gt36 gx3 aw MHXSEQL H 13ch 3C3Hgtas7x 3 0 away3 Elba 0 Factor 923332 3 5 2110 x2 365 l a 7 a Q13gty5 z33gtU5P K23gt5E7gt q x 3 LR3gtV5a x3 3 Objective 4 Review of Eguation Solving Obj 4a Linear Equations KI Plan of Attack Remove groupin symbols Collect like terms Isolate variable solve39 2quot3 3 quot4539 Solve 5x36 x32x 1 4x Rx3x35k45q 5Llxa3KW3qK 54 54 8 443 12 an o 0 Balut non 1 6quot Coy x 3 C 14 mbCr ab 3 Cumquot 5 Obj 4b Quadratic Equations 01132 ba c 0 5t anda Yd Lorna M Obj 4b Method 1 by Factoring Plan of Attack Write in standard form if needed Remember that you can always use the Quadratic Formi aa you don t have to solve by factoring Use the Zero Factor Property If a b 0 then C 0 or b D Solve 8x2 15214x Solve 24x2 83x2520 3K3 amms 0 3x 25 3x i 441 3 ax 5 o P 8L QSQ 01420 23 o 20 8 a5 3 a 25 M a a Set Notablb h 3 N eta hon Obj 4b Method 2 by SquareRoot Property Plan of Attack Used to solve quadratic quations is this special fom Donot Kau 0 319 A t 1 L 7 Enter answers in any order separated by semicolon If there is no solution ent f t39lTe words no U solution with no capital letters no quotation marks and no punctuation marks Use the preview button let it help you Be patient give time to load Solve 32x 12 6 0 I l ola he weng t ewrm 36UCJ39Da 3 LD 39WKe square W 5 rwt W W 139 Dom e variable l 2 m4 and or 3K4 90 ea 0339 alt 17quot lt6 Solve 323 12 12 0 ax 039 39 1F lsqrtaa aleia quotquot 3 392 iW xLra X quotm39xe h X L a wk 07 a if a 22 Solve 2c 32 10 O Solve 2x 32 10 0 4041939240 x5a o W533H Kt32395 m3 H339 No SOLUTioN x2 39315 a l39 x 5 Obj 4b Method 3 by Quadratic Formula akar laquot quotl39c quot O Plan of Attack Be sure the quadratic is in standard form Must correctly identify a b7 and c F 39b I 0quot 40 4 D5Cmn0n c Under 561W 70 root Foracc2bxc0a eGrade Problem Instructions You must simply and reduce completely you must simplify so that d gt O 39 he solution will be entered as an ordered 4tuple abc d where a b c d are given by quot in the nal simpli ed answer You must always enter four numbers you must enter 1 wh e needed x If 2 lt0 then you should multiply both sides of the equation by 1 before using the quadratic formula 0 SOIVQ 4221 Solvex2 63E 0 F 0 39LP 4563 Rx lo Xa tLolc Bto 9535 CIH 052 cl x Ui b qmr X Il quot quot 391 r5quot EC IKHDE39 4 l 5 L0 16 Solve 3326 10 39 quotl Solve x2 3x5 0 No SdU Oh 3L3WC H O a b C X o so am l l Obj 4c Simple Rational Equations M M fractzon fractzon Plan of Attack Just C 035 Values that cause division by zero are called restrictions on the variable We ll do more with this concept when we study function domains in Obj 100 We will not test you on this concept here 31 2 1 3w 2 Solve a 1X5 Solve x 1 3354 9 z ea0 Coll quotlcyH 451644 ave2 5x 5 Xl 2 3 EQRHDEIa ailEVE ll NoT x 3 W W quot Ob J 4d Solvmg higher order degree equations monomzal monomzal Cw 39E rmetQh3 Plan of Attack Our question will be How many olutions are there You answer that question by W and n FACTOR Bring both terms to the left remove the GOP use the ZeroFactor Property 3W the ri h side is a constant then you don t need to factor the equation is ready to solve 17 ho SOlve x7 16563 n3 X 7quotlungs1 X3Lqquot pgt 0 X330 Xqlugo X 39O Xquot L JL W Solve 36 212332 XV lax o X2664 l2 O 4 X2O x 2zo x0 4amp7 3 z B 30utlovx3 X quot quotJ75 Solve x5 25 x5J W3 Solve x5 X5 077quot 0 Xaamp 327o 0 X3R7cgt 3 Solve x8 2033 xaoL3cgt X3 Xs 80 o 5 O Xy iozo Solve x6 z LI if ms 34361 0 x5q O 3 3 qu K W Solve x88 X 1 W No Sdu u 18 Obj 4e Simple Equations with radicals one radical in equation Obj 4e Odd Root Plan of Attack No 11011 lc with Odd Root Raise both sides of the equation to the appropriate integer power to remove the radical Odd Root Don t have to check solutions Solve 7810 1 7 Solve 751 2 Wires W of n1 393 9W 3W x5 38 4 37 x43 E77 Obj 4e Even Root Plan of Attack Even Root Must check solutions quotIIE 39 m must be positive ma Vpositive 2 positive number only gt After you raise both sides to an even power the solutions to the resulting equation may not meet the requirements given above Such solutions are called extraneous Solve W 6 Solve m 7 Jar 6 A 23 7 a 42 34 M 9 gm 4 an 30 M 514quot H7 7quot 7 t bz xquot 3911 39Il x to ii f We 397 3 Chccl igja sko W30 4 55 Km 7 p 39 Lo r 743539 L9 NO Solveltm 533 Solve It 2 17 5 1 3 pg 39 239 2 ALp 39Sk 3 pkr cz x 70 X X H x p oxa x393 7 lt lt lt 3gt 0262gtX393 XO X Rso also X 3quot JECNLKquot r L3 myoJ V SOB IT 39 q 100 CheLK39 HIS43119 39s Vs KW 3 00 41 30V Pogxs 3 ff LHEZW Wa39REW L39 RLHS39 SOD VT 39 8H5 a 2 Olut o39 W91 RHS39 3 a Solve 4 x 5 Solve V211 1 6 1 P akTm squarerooe 0 an 30 Jay s F Cannot W M 25 p X A 397 Om No Soilm0 Salomon Solve m2 1 SolveIO 42 ND 390 KLHxu lgt Eamon Iox Iwu W IO39L x353 m axzhh Ho L4 x49 0 XO Xo0 X X LD Two SOltl t lOnS WIS 5 Objective 5 Linear Inequalities Interval Notatation And and Or Obj 5b Interval Notation Used to represent the solution to inequalities Please Note Inequalities are easier to read if the variable is on the l Interval Notation is always written in ngm DfY Iln C Qrd er 6 9b 37 r ht 37231 533 32No4 43mg CndPOM o BmcKetI usmgu ammo is Interval Notahgn Express in interval notation lt quotI CC 2 64 woo quotaj ltxzxgt7a La a 90 0 a Solve 4 x 5 Solve sz 6 1 2 7M squarerooe c an 30 Apgxs gb W Cannot be M 15 p X A OM NO Son 70 3mmth Solve V211 3 x 39 Edutmhl Solve10 41 O KzaHXL39H 0K X aquotLIL HL La 39039 KL 0 9x27xcp XXO K49 0 X l50 X DO X XLo Two somth CHIS 539 Objective 5 Linear Inequalities Interval Notatation And and Or Obj 5b Interval Notation Used to represent the solution to inequalities Please Note Inequalities are easier to read if the variable is on the l Interval Notation is always written in pang QCY Iln C QfC er ellb 37 rl htgt 37231 353732 Not mama CndPOMb Brawetr usmg empo t Express in interval notation ya I ntcr va N at a b in an S quot 2 a 4 3 00 3 2 gltx 1 X gta M 9 0 qR 39239 gt x gt 0 M 0 O S Llt Z Cy 20 Obj 5c The math terms And and Or Plan of Attack Draw a picture of the two sets on a number line then check if And or Or HIM OR means UYhOh AND means lVl39thS fCt39th Combme cvef thmg over MP Express in interval notation 3 S a and a gt 5 gt gt 5323 02d 5 Hmwcr 5 co HE A 3 5 Express in interval notation x Z 3m lt 5 m HmSw r 3 5 rc tm 5393 lt 0h 5 bc not i 3 5 Ineluoled In both xi Express in interval notation a 2 3am lt 5 My 1 I I A Rh3 J V I l 3 5 Express in interval notation cc 3 3x gt 5 Pom union 0 dudmmt 5913 6 3 w9 4 i i 3 Hn5 w3U5 gt 3 5 Express in interval notation 1 lt 3 gt 5 llv HWO wftttfn quot3 quot 3 over r thW er f i i 339 E 3 3 6 21 Q EVnVUj 36 6quot Obj 5a Solve Linear Inequalities 9 Plan of Attack Like Linear Equations Remove grouping symbols Collect like terms Isolate variable W Unlike Linear Equations If you ever multiply or divide by a negative number must I CMC 5 the inequality sign Inequalities are easier to read if the variable is on the Qt Solve 2 2a 6 Z 4a 5 12 Solve 5 lt 6 2x 1 S 4 T iquahtles 07 9W0 3 lJLROHa 54O azxquotR Ll 2x40 139 II5 8 54 9393 5 4 45 gt 8 aquax om mega H l H5m2awml gtLzo gto x4quota 5 l e Il 4 X org 0 az E0 9 Solve5gt1 zgt15m Solve 26 3mlt2x 1lt9 3a 5Nquot 7316 AND lquotaLgtLS3a 3235 33quot mm 07X 4l 43x 07 15 Zx gt3o3p 854 5544 3 a gt30HUL Sr x cquot 1 337 4 555 ASD X40 34 K Eng Xgt X z 393 a 158 4 P I aquot v d 1 Objective Rectang7ular Coordinate System Distance and MidPoint Formulas Obj 6a Rectangular Coordinate System The notation PI y is used to refer to a point in the coordinate plane It is an BYdfer 039 P101 37 quot 3 O 67 A B C D 22 Obj 6b Distance Formula d K sz r ya 339 a 7 d CX R L 3 33 MQMOHZC For Mu ta Find the distance between the points P1 12 and Q3 4 d 3 ti L Amiga 34 3 gm JEWU 5 N 539 A 7F MuH CIAOICC 2 quot 33 Find the distance between the points P 4 4 and Q8 13 a 39 3 d 5 X W39DK CW 3941 a 4 Jo a 502 2 mam Obj 6c MidPoint Formula 56 Til2 t f ji a gt a 1d a aMule Chmcc Find the mid int of the line segment joining P3 2 and Q 7 1 0 2 A 0 O 4 v 1 w g Find the midpoint of the line segment joining P a 0 and Q0 W igs 13953 so oi 23 Obj 7 General Graphing Principles Obj 7a Point on a graph If a b is on the graph of an W a relationship between x andv then when ll 3 a and j 7quot b the equation is Plan of Attack ll it in glue Vb Y J If 6 c is on the graph of y2 cc 3 nd the value of c Gazar3 A CRQ Mult l t I C 39 quot 3 Chmd If c 27 is on the graph of 93 y 3 nd the value of c Q3 THE 3 La44 sz gm Obj 7 b Intercepts Intercepts Points Where a graph Ly or WUCh 63 a coordinate axis Listntercepts for each graph 5 XX WWD39 e 202 ya We 39 5WWV Com th 52C L 2 2 3 539 IV 5 mom 6 m 39 quotll Find all intercepts algebraically Plan of Attack For xintercepts let quot O solve for cc For yintercepts let LCD solve for y Find all intercepts xy3y20 3 4 2 0 A 3 a 39T and wt let Kcogo r 5 a a 0 IT 591ml 1 ma a 34gt a Jemima 0 y 390 0gtOgt 30 OY a 3M 0 339 m Lia bMCC b quot WV A 0quot L Find all intercepts 512 333 27 MI X Int39 oar sisa7 x Mo m 333a7 a 5eQ Xaz39a l y 3 q A 4 L7IQ A I gate m2 Obj 7c Symmetry Symmetry with respect to raxis To test algebraically replace a ULth neg y le 3 g 33 quotKSee if an equivalent equation results a Symmetry with respect to gzaxis ya To test algebraically ware lb wmh 0133 1401 39 39quot 53 4k See if an equivalent equation results 3 Symmetry with respect to Origin To test algebraically replace so WWH htj zC M EL 1 d b with j quotL1 7 99 Glqu39DE 9 An 39 example of one type of Symmetry problem If 3 6 is a point on the graph of a elation equation with a and y that is symmetric with respect to the xaxg which of these must also be on the graph 3 6 3 3 6 the h over 51 quot ee if an equivalent equation results gL v ge r 0 m 53 l 5 Test for Symm wrt Origin y x2y3cc xax T525 312161 6 both a 33ca 5 3 4963 LU 3gt gtquot 393 2 WW Test for Symm wrt yaxis xzy 32 3 Tat 1619mm x w 56 a 2 y z 3 W C 33 Cf s gt6 5 3 32 7 13 0 av o y k 5 BUWWMCCWC w aspect bD y39all5 Test for Symm wrt Origin 33 gm h 09 if 3 3 SaMMEWm wrt cm x21 3915 vmace both a w y 3 l39 65gt n H gt a T3 33 LCOM39PGVC to 06an L H 00 6H 0quot 831 3 3 W b An online example Which of the following is symm wrt the xaxis 2 ygs 2 y2f5x 3 L a s L3 5 13 gx j SMME hc Ofi ln m2y25 x4y3 2y2y5 6 o395 Lquot 6353a3932 393 5 30 3 46 U 3 02 3 5 3 5 3N5 26 M algax6 Objective 8 Linear Equations Obj 8a Slope The slope of the line through 121 31 and 132312 WW m agl Y f m gigglint ofthelinfB and Chfccam Kquotquotl rmquot 39 1 Find the slope of the line through 4 6 and 2 7 4 6 and 4 2 3 2 and 02 4412 mizLZ az tl wari3 71 u 3 1 0 0395 3 me Wnde hated O S Pe N A 0 N y y 91 Cm 0 3 O l L I I I L quotc x x a axD 2K 0 blwlu me NOT VCRhowl Obj 8e Writing the Equation of a line How do we determine the algebraic rule that de nes a particular line the my equation of a line Each line is uniquely determined by Anl 70 313 2b HE and the 26 The equation of the line througBh 131y1 with slope m is given by C I 5 A B 0 b 5 31 he 90 L Stgndgi d FOYW Slo gemlmcCVCCP n On 0V 39 n B C OFOerm 39Hntto bemula Lquot 3 DYe 3 3i P Obj 8e Write an equation of the line that goes through 1 3 and has slope m 2 Express answer in standard form eGrade Instructions lease remember that standard form is AwBy C where I A BCare integers andAgtO W xECigade 94 5 morn A 2 3 m 2 ozx39 quot5 Dgt33 8L2f 5 17 3 0236 27 fj39aw 34 5 I Obj Se Write an equation of the line that goes through 6 3 and has slope m 2 Express answer ijn sfr iiargl ormgb 3 5 1 5X H 3 p 7 21 5 5quot 30 chqradeC 3 03 7 quotTKquotltquot 5 7 R 3930 W 7 Maeaw amp m SW Obj 8e Write an equation of the line that goes through 3 9 and 12 1 Express answer in slopeintercept form eGrade Instructions Enter only the right hand side of the expressmnm Use W proper or im roper fractions or exact decimals mixed fractions and decimal approximations are t m l4 22 3 3 quotSL LD Wquot J 15 3 3 33 K 2 x 2 e y 306 3quot quot26 w 7 92 Web ry3 5L7 jgn1 9 w3gtq 3 Obj 8e Write an equation of the line that goes through 0 b and has slope m Express answer in slopeintercept form gaunt aawx rb nygayxdavd Formx mg 4 0 1 b Obj 8C Identify the slope of a line Write in slopeintercept 01m Unless special cases of vertical or horizontal line see below Find the slope of the line 2m y 5 Ryn 5 3 M Qx Obj 8b Graph of Linear Equation Plan of Attack Just nd the It isn t necessary to put in slopeintercept form 28 3F 3 WV VCep39bS V X Obj 8b Which of the following most closely resembles the graph of 11 3y 2 36 54 x c O y 7 Ty V 3 a l0 m xv xy 3 OK 1 33 30 392 a3 Obj 8d Summary of facts for Vertical and Horizontal lines 3 3quot quot Xi l2 Horizontal line has equation 5Q has slope m 3 2 except for xaxis has no I nth Vertical line has equation L 5 C has slope Ling C nqlexcept for yaxis has no 5 quot ht Obj 80 Find the slope of the line cum 5ch 9m alrcaouj solvch 13w V Yb lme and med suppCV ho zomtal lint Mco Obj 8d Write an equation of the line that goes through 2 4 that has unde ned slope 3 9 quot W OL V I undclhned N COW E plug vvcr ocm m quotAundc med x c c Obj 8d Write an equation of the line that goes through 2 4 that has no x intercepts hovx conwl a a w Obj 8d Write an equation of the line that goes through 5 1 and 5 4 V fttwl l C 29 Obj 8f Using slope of parallel or perpendicular lines to write an equation of a line Plan of Attack For nonvertical lines M Parallel lines have the a W C 3 M76 Perpendicular lines have slopes whose IUYQjUC b 5 l f CCAVYocal 3 5 3 For example if a line has slope m 3 then a line that s perpendicular has slope W quot 5 For vertical lines we must use geometry Two vertical lines are parallel they never intersect A vertical line and a horizontal are perpendicular they intersect in a 90 angle Sam C OIVH Obj 8f Write an equation of the line that goes through 2 5 that slperpendicular to the line 811 3y 7J The multiple choices are 311 standardform ABC in egers and Agt 0 yawnxh m 83 6345746043 Mfay39 59 is 2 83quot sum 33 gw tour lmC 3 1 940 393 4 t t 83an 73 3 339 V 39 39 V V Rowem 8 lor OH 3 m 3il zwnce m 373 Woof V VH3 lime Obj 8f Write an equation of the line that goes through 4 3 and islparallel to y 2amp1 g39 9 W Mn quot we 30m stoPe 3 3 3L LAB Mun b 9331V gt L 5 4 cho ce39 5gi5x H Obj 8f Write an equation of the line that goes through 1 5 that s parallel to the line 3 3 m L f39 0 quot quot5 L 3 when 3 fo t 30WJG MR x our me 3 vu mcal Obj 8e Linear Applications Writing the equation of a line c My fb Plan of Attack 9 When written in slope intercept form we say 14 F The express sentence tells you how to set up the ordered pairs You cannot lC OY C H b 30 419 As one descel dscgnto the ocean pressure increases linearly The pressure is 15 pounds per square 39 inch on the surface and 30 pounds per square inch 33 feet below the surface Ex re s the ressure p in pounds per square inch in terms of the de th 1 in feet below the surface nter only the right hand side of the expressi Use con rge e Use proper or improper fractions no mixed fractions or exact decimals Do not give decimal approximations k pr s P In terms 04 d d A 5 mgw 35 T I 939 WW b t V n xomered sawi39fdirl 3 5 OMS VRem mlocr to EmmaquotO 173gtquotquotjquotM P use COUCH variable 3330 Blue whales weigh approximately 2 tons when newborn Young whales are nursed for 9 months and by the time of weaning they often weigh 25 tons Let 0 denote the weight of the whale in tons and let 9 be the aggof the whaleln months 39If the welght 1s llnearly related to the age express to 39 in terms of a Same onlme cautlons shown in rst example azz Ehtfir o l K 65 W I q a R l n Ext 3 w m ttrm of a ght 5m w MG H0 51539 23 WV 3971quot OYdQY Cd mixslow 390 Gnvcw o 1 6 lg39mt 6135 The volume v in cubic centimeters of a gas varies linearly with temperature t in degrees Celsius At a temperature of 12 C a gas occupies a volume of 2800m3 When warmed to 66 C the gas occupies a volume of 361cm3 Express 12 in terms of 25 Same eGrade cautions shown in rst C militia v m bcrms at a WIZ 5 OV form V Vi W T quot Mi b 39 V V BXO Jo 2 m 39 go dcrco Vows 1tv euch n 830 0 WA Jr WEBB w kMethod 31mm me lag fm 2quot i v 331 Ho 4 2X0 Vagakb The volume v in cubic centimeters of a gas varies with temperature39t391n degrees Celsius At a temperature of 12 C a gas occupies a volume of 28007123 When warmed to 66 C the gas occupies a volume of 36lcm3 Express t in terms of 22 Same eGrade cautions shown in rst example 1 Lab 439 3 3547633 t39 m terms 05 V Mawass 3 te MVHo t t w V wQ vtlaygfwato aordered WWSZCVJ Q 31 20430 Guvcwaltgoua 39100 6 3 A peanut farmer nds that at an average cost of per bushel 300 bushels can be produced 1200 bushels can be produced at an average of 2 per bushel Assuming that the cost per bushel c and the number of bushels n are linearly related Express c in terms of n Emwess c m terms 0 n m 3quot a quotquotquot 1quot quotquotquotquot C nmrx b aco39mao 700 Ovscme 0539YCgt 39le Cmvcn 3005 6300 Obj 9 CenterRadius Form of the Equation of a Circle centthM 35 Don Fovmma y Mg foamy r thgrby Centtw adm Forge 0 CrC 39 M43 4493 hla a MM3 MfMDYIZc Identify the center and radius x 32y 2225 m22y1r5 x2yl324 3a Ll quot o quot centerz xm 433 3 74 if a T A f K rad lb Jig quot b Cev icK h K 5 Qf39b 3 rodxus J75 Clem O 6 quotl r 3 32 Qattllf Objective 10 Functions A function is an algebraic rule or a correspondence between sets that associates each element x from one set with a M I Mag C element y of another set The set of x s is the 6W0 h The set of y s is the 0 31 gt O Y 9 Obj 10a Identify a Function from graph or set of ogits 3 Genbf F 0 9 Consider x2y24 V L Oa 9 0 3 The graph of a function must pass the VCKU C0 W f t S S an amp Ont ver mm mC v SCCt3 hWC O Ph V at MOST DYVC ixm t t 6 the I 04 80 3 tha h 0 O 39R W mon 3 311 M70 Which represent the graph of a function Read carefully sometimes we ask Which do not represent the graph of a function mks ml y y 5 bvvxe one 1 3 939 NO Q a 8m rms 3 a graph ngh of 0 hm cam 1 w won 4 Which are functions OS 0 n O D RJWCUOY 11 2 3a 2 5v t x 4 308160 all x WM 3 W63 must Show LAP cmmgl 17 2 139 21 13 Lquot I NO a Pumdnom Obj 10b Function Notation Evaluate a function Instead of writing y 3132 5x 8 we now write f 3x2 551 8 faPL Instead of saying let x 2 nd y we now write f2 quotl3 0 2 a 52335a2r ao8 no f ammem 33 awlyt 0 domain vm ocm m quot 392 Punctxons Find each of the following for phe functions given below dfnm Cbn StanL fa 3m2 5x8 927 23 5132 hx 3 5xl mc 3 nc 2 4 quot46 j 5 B a a glt 1gt 3 47 M 3 3a 0 05 20 quot 33 2 3x av5xfa 4 quot 0S 39 3quot z a 3X2 HLHgt 5 733 8wh h sand5 0th x530 3 a0 kindaquot 3 a a Olt6 ICE L I 3 I p d g h 3 3 50 33 quot 1 Bo gs lsr i 2 I h13 b0 539 339 K Hquot 6 5 z z 3 W as 13M m 3 L3 Rwh39h 2151 E mlt3gt 3 M5 5 nlt 3gt a no 2 m2a 2a mxh K39r n n2a a nah 2 END OF Tia 51 1 MATERIAL Obj 100 Domain and Range Give Domain and Range from the graph of a function 339 p s rme v haVHY 33935 3 2 Q 3 m E CD D C J D doe9W3 R i 00gt R3ij 00 D0m0mw vau 3oo4 ch v c 00K bottom4V0 ann vmuteS 34 More Obj 100 Give Domain only from the function rule We re being asked for the 11 set of real numbers that can be substituted in for g What prevents a value from being plugged in for a When might the domain not be all real numbers There are only TWO situations that limit restrict the values that can be used for 13 O NDW SWY 0 wwngatatwc aw twa Q lAnCUDY we dogma vwa exther 0 meme then 0600 Give the Domain fr 3x7 5x4 xg 933 2 3 51 I 17 hc 31 1 06003 DIGOOJND D139 0J Dgt ea53 gt my be pas 36 7 go 31 Z quotI 00gt 7 1 3 y 3 lth S Chd 3th gXQm a Obj 100 Domain and Range lee Domam and Range from the graplil of a functlon D EHJ39J DIM new ij Domamar valu632oo4 c v H m 00 DOWNMAW A anngl vaws 5 34 m Pwnc haMOf j K R 02f 3U C 3 0 R 00gt Exam Two toll 5 More Obj 10c Give Domain only from the function rule We re being asked for the la set of real numbers that can be substituted in for g What prevents a value from being plugged in for 3 When might the domain not be all real numbers 531 There are only TWO situations gat limit restrict the values that can be used for x On 0 w h w V w 3 u c WQUVC For L W0 k quotdcr radical 65 46M Q unCtWY rulf dOCSVH vxvov t 05 thine than 0104290 Give the Domain f9 3x7 5134 395 923 3 52 17 ha 31quot 960005 Dzewlw a Di39 01 0 fx F3quot 9 have be poi 36 7 2 0 3x3 Some on line problems ask For f V33 7 select the values that are in the domain of this function I Wm M valu 63 SC 6 9 awg P Choices 5 6Q 39LUOY quot IAqu POS 905 W a quot dimrfi M 3 CaCuatc lOMOHn choo3e 3K730 322 7 valuf S in dode C 2 quot 73 00 Give the Domain fx c g 75 a7 34 0 3k 239 R Drr39w39 31 kquot 93 anan the vmw fb zor 5 that wound caut c du gOn 7 Give the Domain m E2 10839 039 50 x43 5 n93 g 7 domain 39 O V K 3L 3 TM 3 61 09 a K s ekce t K D 03gtU339 0gt 0R EblLfZE P 5 2113 3 Give the Domain fx cf39b v 3quot 39 quotRWa a p b37 i gt V I c 22 OR Shoo 1c LN M domm D EL x 2 22 Give the Domain fx 258 3 lagM 0 9 OOJQOB 841 No x1 Give the Domain fx 2 7 m quot LQ L Lo 0 2 D ogug3u3oo QC 4 za 0 0R Lgo K1350 31quot 1 Edna 2 U33 A K 4433 Give the Domain fx 2 7 x2x 1 K2A 50 Diik x i E k 006 0 0 D gimi or 6 4 Jig MOOQ Y 52 x UV a 763 36 Some online problems just ask Is the Domain all reals or not all reals When the denominator is Quadratic you can use the Wfrom the quadratic formula W to answer wean bL 1 C 0 31 Solvci 39 b1quot VZbE la I cc 9 2 K C Domain all reals f 3 x2 x 1 f n Ra baa Hac l i 3 ba Mac 1 Q l 394 mob a reals quotNo c quotan ream cc1 m2cc1 badlac lad H00 2 lfq 7 340 quotPm rcols Domain all reals f x 7 Domain all reals my 39539 m 5 W to 39HOC K 2440 7 gquot 3JcquotJlt5 39Not a reals k c a b c 43463903 pr 20 39Not all rtals Domain all reals fac Eff 3T6 fa 2 4 logMac 7 quotla9U 9 WI 9 t l aq 23 40 quot0H fecllS39 Domain all reals fa p xdkl x 2 a1 7 2 3JX2 3 go 039 xxal 0 0 0 6 2 gt 0 1C a V 0 LBC V C Ch quotNot a mob x 752 Dquot Elixiag 37 9quot9 1w Obj 10d Find information from and about the graph of a function There are several different types of problems in this Objective 161 Obj 10d example If fx 2 Q ZC CZ l iSthe point 31 or the graph of y fa 3 l YES or Is QC f a abA 32339 z 5 7 GM 0 Pomt W N W 3L 0h tm raPtx39 H M33 1quot 5 3 30 Obj 10d example For f 2 2 gig if f 51 2 what is x ml 29 6quotc 5 K 29 b 3 y Wl39xKTvSl 35 K 3 K 0 none of these 7 Biff quot7 a 7 x Obj 10d examples h For the function shown g 7 D graF For the function shown 3 4 O 8Y0 h for what 2 is f gt 0 W for what a is f lt 0 l3 above t 33 l5 below 10 Cum Lalm 3 233 Ll LD 9m 0609 x x U 00 Obj 10d examples For the function shown For the function shown for what 13 is fv gt0 3 7O 5 abovC for what a is fxlt0 3 A O a Y 3 m 5 5 b a m below L Cll lb x PMquot 3 3 HSn 2 3 a U 38 xO Obj 10d examples whev 3 3b7 S quot For the function shown 3 7 For the function shown Y a c if m 0 what is fv what N o for what x is fa O 6 eta394 130 3 U Yt PampJL Obj 10d example Answer the questions below for the function shown V f 1 is positive negative when bc l a 3 39VDOES not QHSt 0 55 f 2 is positive negative 64 15 tm asc mpw ce rhhe mevtv twang f 25 i negative unde ned f 3 Q f4 L How many times does the line 3 5 intersect the graph Mg Copyright 2010 present Annette Blackwelder all rights are reserved Reproduction or distri bution of these notes is strictly forbidden 39 Obj 11 Properties of Functions Obj 11a Increasing Decreasing Constant Always look if lll to ght as x values are getting larger The function is ac if the y values are getting larger The function is 66 C if the y values are getting smaller The function is C OY 3t if the g values don t change lllllll 35 45S 00 3 65 Increasing on 3 7 Increasing on w Decreasing on quotgt0 i 3 V 7 00 Decreasing on 39w Constant on Q Constant on kl Obj 11b Local Max Local Min A local man is a point Where the function changes from VC to 6 EC A local WY is a point Where the function changes from 60 to MAC Mt 5 a The maxmin IS the 9 Maude w W w l mun The maxmin IS AT the J Value 0va 5 7 MALm1quot An L V y R78 P2 4 xv Q24 For point P The local Maw For point Q The local Wmn For point R The local gt X is Ll is quot Ll MM is 55 what 5 We qrvalu 397 6 wmrc 3 t39 5 Wm The local mm is at a The local mum is at x j The local W l6 is at x l Obj 11c Average Rate of Change O 9 5 The average rate of change 011 between c and a is the of the secant line between c fE and 51 fz r 3 Cant me Comn66t5 a Average Rate of Change 2 1 two 370mb 0 L C a h MCMofVLC a 9 Formum Ty WRL DW The average rate of change the slope of the graph at x c In calculus the derivative gives the slope of the graph at a c which is the slope of the tangent line at a z c Plan of Attack Set up the two ordered pairs 0 f and x f ll in the values then just nd the slope Always REDUCES A1 Obj 110 Example Find the average rate of change of f between 3 and a for f as 3a2 2x 5 o 1803 3 3333 3 5 R39I W5 33 5 33 W 9 239va Rate Bl Gwaan 003ch 0 0 3239 2 5j 6 C 30quot in 5quot 3933 3x 2a f j 5 an 33 quot x3 X 3 x3 2 33 Bic 0643 31o e 139 7 v 3 2 Obj llc Example Find the average rate of change of f between 4 and 1 for 04 if t 433 HI aggt 5 quot39 quot quot39 202 23 Z39 2 4 ivy Rafe M Chan b 1 T A p K gt 3 b C L quota 2K quotquot2 L K 39l l 3042 f 39 d 2W zq 3n39l M 39 Ubj 11c Example Find the average rate of change of f between 1 and a for f I2 2x R r 2 1 03 26 M 5 54 Jc 2x a a 3 a 3 L32K3 Hv Katt 0Q O M V T 66Q Qquot 47 K 3 F f K H L C KZ R U Q Viv 3 nl m quot Obj 11c Example Find the average rate of chan e of f between 4 and a for f 13 21 mpg Hra we2 no 80 09 Oman 39 H0462 94 21 a 1 at a all 104 1 0 339EI N aaw M l L Ll m w 464 v i quot aw JV 27 L gt Mule A C howe thCC39irmc rationalize numeracowr k WW Obj llc Example Find the average rate of change of f between 5 and 2 for fa 2 2J5 The square root function examples will quiz only not test 5 86 am to W M c Me It to Avg Ratc olquot Chanj 3 4Q 11 a qgg an NS 3 qu 30 40425 L C 71 quot5 EVEng I 6 53 3 539 r Do Noe mutt ow MGM 25 q 2 ML jarfrail 1 Z 17445 Checc Obj 11c Example Find the average rate of change of f etween 3 and 1 for f at xx 1 The square root function examples will quiz only not test 93 WW a 35 mm Irma Rate 0 Cmngu 1900190 27 2 WET r22 K C 39 L g mi9 7 10quot 39Ll M Do 40 malt ouel ZD f 3gt W 39 W Mult Chance Obj 11d Slope of the Secant Line This is the basis of the de nition of the derivative in calculus The Ob 11d problems gill quiz only not test 5 9 LA 7quot Find the slope of the secant line between 1 f x5 and a h f a h for fc x2 21 3 3 m 3 r w h My mm 4 n quot NW0 MD 3 W3 3 x h so h 242 3 KRmhhaah fK w m 26 1 n h snowman s MC3 Rgt a h9L crfs h K x No Cayman Hamilquot Find the slope of the secant line between 13 fx and a h f w h for fa a2 3 New 39uvb za x 11 73 km a 3xh39na393 3 39 war so V 3 a 3 WC jggzmm 33 2 LaRlbh Ma 3 m x I A uhx RW V BW h dbl 7 T T mr h thamp irhh 93 Obj 119 Symmetry Properties for Functions 3 5 3 Qio Er 1 y M 0 VODMCt I n 1 KI L x IL x39 w ODD No rail 15 Graph is symmetric Graph is symmetric 38 mm c Pbr with respect to l39 0 5 with respect to Q 5l Qu nchov flt agt 19L fxgt LEE Plan of Attack To test for Symmetry nd f 50 If f x then function is m If f 513 f then function is Q If f v 2W then function is Mix CVCn of Odd Obj 1 le Even Odd Neither fxx4 3cc25 fxx7 75m53a 5 39 1560 5 Oq3C z7 r5 Pquot quotquot7 73971quot 36quot 39 0quot 3 6qu2 5 2quot 5x5quot3 75H ev n s M 7 Invcoww mhan Qwe 39OM fZ l L74 5b53p gt quotFCQ fxx5 5x2 fc 2x22i3 3 cm 405 5916 w am 31 Ma a t wag 3 026074 2V3 not CVC n 3 we 1w w L rcaraf Mt 1 00 nob We wooing odd 39 9 quot0396 Udd 55k fx35 fa14 3 L nzi 1 t amp zx not 73 we 3 X quot x 21quot amp2q quot even o hO39t QVCY AA a I 39 Nanak ed ltxgt xq NW quot F Z gt 23 Tiquot 9quot 04 odd Obj 12 Collection of 7 basic functions Plan of Attack You DO NOT need to memorize the properties Learn the GRAPH and the FUNCTION rule You should be able to answer any question if you know the GRAPH 1 Special case of a linear function f as 10 y 9160400 2 30 4 0 odd 9on055mm wrt Ofl lh thYEOShij on 39wlw 2 Squaring function f 513 932 wquot D OOIODgt 0 w aCvx 53mm wrt 9 4quot 5 DECICOSn3 IMHO Manama 01 00 3 Cubing function f x3 9 4 91 00 R of Odo ngm Wt 006er intvCaSnrx on 3900 0 AK RV gt39Hr Z quotl o l 4 o x t4 3 395 7 3 0 o J L a 7 M eve v63 4 Square Root function f cc NO 38mm0 am 3 nexmev odd nor even N on 000 020 065 f 5 Cube Root function f x Nab 175 wew odd lune R 4 3 mm m 09V nC On oo 0gt 6 Absolute value function f as a l 39039 quot 5 RI 0 co Even PLAN 3de W f o y aw w ok DEC 0 MIO C on 7 Reciprocal function f 4 K 0 91zw0uo 0gt a wow wow 7 tuft ng decr 000 MO b AR Obj 12 example Enter the equation of the function whose graph is shown Enter both sides of the equation not just the right hand side You must use 3 on the left side of the equation f a is not owed in this 103639 V sMust MCMOI IZC Fume quot Mora for thwi arr 2 W WCs on came Your Answer c 9 l l Dcm It 90 EaventhCMS Ob j 12 example Select all of the following that are symmetric with respect to the yaxis There may be more than one 311 a 3PM x3 PM 3 fx 35 ms 5 lt35 Obj 12 example 7 5 Select all of the following that are true for the function f x ac There may be more than one ltI Domain is 0 00 Domain is 00 O U 000 Range is 00 00 Range is 00 0 U 0 oo raph is symmetric with respect to theyg Graph is symmetric with respect to the origin 763 Obj 13 Functions de ned piecewise A function de ned piecewise has a different function rule for different parts of the domain Obj 13a Evaluate a functions de ned piecewise Plan of Attack Compare the xvalue with the restrictions on the right that tells you which function rule to use DO NOT plug as into each A7 x2 ifxlt3 fc 112 if3ltx5 O ifasgt5 Find each of the following f1 l r R f301 095 90mm f5001 O 1 if x S 1 fx 0 if 1ltalt1 1 if a 21 Find each of the following f 1 x a q H933 1 0 f3 3 5 f35 3 5 R 15 mm O f05 o Obj 13b Graph a functions de ned piecewise A warmup question Graphfxc 2ifx2 1 A g quot74 x T a 1 ng Value hot 0 Pm Pfr AunChon f5 53 Graph faa Qifagt1 II 4y K 2 AR quot l 0 01 Rf Graph f52x ifxlt3 K 7 O 5 1 x43 th 95 Ax 3x 1 ifxlt 1 Graphfx 4 ifxgt 1 Ay Ma a r X H CWCIQ ovt OPB 6 U 4 own male 4391 3 b Ix 0 39 lLl x4quot twn 6 i man 8 3X 5 3 4 4 39 ifxgt0 3 Graph m it i x u i 3Q twat ovc a 2 4 f lt0 v60 oven cwue x X 0 LI 0 R H x 20 I k 40 lme MWEVCW GOUNd I 9 5 th j 4 a 44quot behawm Hum 3 5 AG 333 5 ifxlt2 Graphfm 2 ifxgt2 2 R 393 b H 399 l gt643 M Li 46va 33Lv5 U g Obj 14 Graphing with Re ections Compressions Stretching Translations Obj 14a Graphing with Re ections How do these compare y fx y fx 3 fv We will consider a speci c example to justify the general case y5 czo yE 530 y x LZO 1c f O y across y Krams Aycztxoss i 030 grams g R1 0001 1 j x I x 5 D39 00 O y y R 0 w 011000 X IZ fl 6 o 0 Rquot 0 w 0 4 x l H 3 2 O O quotI l q a m RV another Obj 14a example If 30 is a point on the graph of y f 13 then which of the 39 7 followmg must be on the graph of y f ffja cal X035 3 a 66 03 0 3 3 0 WM another Obj 14a example If 30 is a point on the graph of y f 23 then which of the follo in must be on the ra h of ac w g g p y f aged across zaams 03 0 3gt 30 lk another Obj 14a example If a function f has Domain 2 0 then what must be the Domain Ofyz fW eflcm across lo39asm q sleDomam 02 23 does not 39 Change another Obj 14a example If a function f has Domain 20 then what must be the Domain ofyfcVRHgtec am a ams 201 23 another Obj 14a example If a function f has Range 20 then what must be the Range of y 13 Ken at 004955 Ioasas v A f Do 63 offf Lt Q Q 2 01 27 21 a n5 e another Obj 14a example If a function f has Range 20 then what must be the Range of y Cmy K leu 0WD5 giants I Doc nor 0 2 42 cm v13 e 0H3 t Obj 14c Graphing With Vertical and Horizontal Translations How do these compare yf93 yf3700gt0 yfc ccgt0 We will consider a speci c example to justify the general case mam 43W 2 5 an xL xiii 0390 75 3 5 I l H q 5 How do these compare yfm yfxccgt0 We will consider a speci c example to justify the general case yzx26 ym y sic 3 Smx39b y y gamp 3 Qm0 939 493150 R 01 Obj 14c example Select the function whose graph is the graph of y 35 but is shifted left 3 units down h up 92535 3 ym35 yzx53 63 0U 631 Pl60wh mead V5OQ H t name Ob J 14c example For y f as de ned below graph 3 f as 2 and Slugb V 5hrl39k 35109 2 3 Obj 14b Graphing with Vertical or Horizontal Compression or Stretching How do these compare yf1v ycfmcgt1 ycfx0ltclt1 We will consider a specific example to justify the general case 17211321 R39 013 d39 CHUCK yJ7 Ll L37 H44 10 MR U z rm O PCWUQ How do these compare 921 yf0cgt1 yfcx0ltclt1 We will consider a speci c example to justify the general case 212W PM were RevaW S aV 0 M2 RWMIIAS 3 kW 30 Same 495342wyxae n39value 3 neeoco co mahe 9 so 3 v Gums For 3 f m as de ned below graph 3 3 fx and y f M M W LIW39ECfoPtS A3 VCY b howl E I I M I I 1 x T VVWCC W5 C WNW 3quot Wd Rahgci 0393 4 3 Domam 47 33 OI 1 tRemams tIu mam w I Jame Ea ML RA another Obj 14b example If 2 4 is a point on the graph of y f as then which of the following must be on the graph ofrowe j 44 4 8 lt1 2 lt2 4 28 O 56 value dime a 9 H another Obj 14b example If 40 is a poing on the graph of y f x then which of the following must be on the graph ofw T an ff 8 0 2 0 0 4 08 0 2 W Mult g valwc by 65 another Obj 14b example If a function f has Domain 08 then what must be the Domain of vanrm r y4f quotT39ancr aHeccs 032 02 Grange b9 Ucor o q 1 Mult y value by Ll another Obj 14b example If a function f has Domain 08 then what must be the Domain of y f 456 P FCLBS Narmwcr a 032 08 t W 9 Pach G Ll MMH2 Lrvalufs log 3914 another Obj 14b example If a function f has Ran e 08 then what must be the Range of W Tana OHCL Q quot I 02 08 tw angle f 1 J Mqu q vmws log 4 another Obj 14b example If a function f has Ran e 08 then what must be the Range of yf4 Nawowcr affect3 032 02 km domain KaiMy 6063 Change RR Obj 14 Summary Obj 14a Obj 14b Obj 14b Obj 14c Obj 14c Affects Range y myquot across KrauS y3 m 0quot taller yi w 0464 Showon 3f v2 ohm H yfrc5 SW4 d wn TEEQEEELQX yfw Rm across 301 yf3x 07 warmer yfiv 04C 4 wMEr yfrv2 Sh 39t H45 yfm5 Shr t jv t to51quot1539 Obj 15 Operations on Functions forming new functions by adding subtracting multiplying or dividing existing functions De nition of notation used to denote these functions de ned for all 1 in the domain of 7quot f gm 2 30 3 0 f 9W FCXD 3 0quotquot r fgx PM o 5063 x Z 3 8 Obj 15a examples frr 21 yo N ha 3 mltxgt x Find 9 716 Intege or exact de 39mal only mathematical operators are not allowed mlu Clio mLo 9 U 3 b 20 Z 3 an quotLo 392 3xo Find m h0 Integer or exact decimal only mathematical operators are not allowed Nx0gt hm o 3 3 RR Find 4 If noninteger answer give fraction or exact decimal Don t give decimal approxi h mation 2 4211 11 M43 9 am quot 3 U 3 393 quot 3 Find h f3 If noninteger answer give fraction or exact decimal Don t give decimal approxi mation a a h 3 O 2 39 O a 0 193 3 33 35 3 Obj 15b examples Find the domain of a sum difference product or quotient function Recall that these functions are de ned for all an in the domain of i and 1396 OVCFG t1 0 Y dOCS n 0 ma tth For f3 V211 8 and 93 2 39c 2 nd the domain of f g 320 c2gtCgt 07603 cgt2 D 0 P9 3 l Ll 23 6nttrsech0h 4 f 1 N For f 3 1 x and gc 1 41 nd the domain of f g Ix O Lamas D o 3 696371 I Z 6 l 2 LIX quot L39 2x W 1 la For fxa 4and gxa nd the dom ln offg M HiO Xv20 D 01 Na 3 unde ned z 3L gtlt i 4ND Liz393 4 TM ND vxte ectwj 51A 1179 vmu For fx 2 33 9 and ga Eai S nd the domain of ftlg 5 gt 3xqzo gs 0 D 01 99 5 02 5 3KZ39q 3 AW 5 WV 395 r 3 quot For fc 32 9 and gx nd the domain of f g ag920 airs20 TD 03 j OolBJ 3K3q am 55 Xi39vj gND K5572 W 3 E A 393 53 For fx33c 9and gx x 5 nd the domain of f g 39 Cc 5 D 03 Ric60 D 0 if Eki fag an5 I W 1 1 a 573 For 1653 and 956 nd the domain of fg 973 9 2 5 odd 3Liq f0 025O Domain o t 3 00 32 3Kq 0 655 U23U 3 0gt 3x 5 L55 0R 3 my 7 W MMEMSB T i 3 u Obj 16 Function Composition Given two functions f and 9 then the composite function denoted as Q oq is de ned as follows foga 63gt for allxin the domain ofg such that 91 is in the domain of f cw Q 33 5 than 5 hOS so 06 m cm domain of 3 Obj 16aexamples 3 D h Jcm domawx fx1x 93x2 2 o 9 Find fog 2 Find9 f 2 It mam 309m 9ltT9 5 399 5 3 3355303927 339Jz 8392 quot3 7 M Q Mw39b 3 Iqgt quot73 IB aa TL 6 CWOKLC 39iU Iquot Find f 0 9M N900 Peta a l quotquot PM I d Max3955 Find fof 0600 4amp3 I l re far o 39quotquotquot l x Findltgoggtltx 33003 33 2 739 33 19 2 335L2392 gt4 351 seq 263439Ll 4agt 07313 30 27x 3on3 tax pla Lisa 1 27y 31 136 6 AfMMlb CW9 C Obj 16b Find the domain of a composite function for two rational functions Plan of Attack For 9 O 5 g Method 1 This is the simplest calculation but requires an understanding of the de nition of function composition kigt0 3203 P than g 3 twat nc awhf d sheh gt06 5 rst Puncdpphed a Determine the domain of the rst function applied b Determine if the second function has any restriction any bad value c The rst function a the bad value of the second function So set the rst function bad value of second solve for any additional restriction Method 2 It is easier to remember this process but requires simpli cation of a complex fraction a Determine the domain of the rst function applied b Form the composite function and see if any additional restriction KO For fm i and gx 1 zit 2 Find the domain of fog Find the domain of g o f D M oi raj 2x mag 9 01 30 inlm omggg hr 0 aw 3gtx7 o 3 quot31 b bad value bra 3 a b bad value 0 t S O cvrt gt71 C330 gt0 cho 5 9 I5quot I OxxR x glza l 0 L 2 ya dchbowa ND eesbncbiov 30 x 74 ill For fx 2335 and 923 3366 Find the domain of f o 9 Find the domain of o f D O 33 ELILfRL I03 9 0 r 30 146755 a For 9a mm a ear 1L0 5 5 b bad value 032 4 is 5 17 bad vawc ol 3 b 8 C 9605139 CD C0 R if 7 pg 6 5 gt 06 b ArcA i A gt 07quot 739 3K 5 Rx RxIb 06 EIL I ixa Io O I39O 3 quot 4am Odmhonal rf sicmach This next example illustrates that the domain of the composite function can never be any larger than the domain of the rst function applied Forfxx2andgx 4 D 0 03 X 2443 Find the domain of fog UK SOCgtIK Z L bad value tor I3 home Smce dowwm must be Composed 04 we Pump 1 Exam 5 Matemal 1033 Is Obj 17 Math Models or Constructing Functions Obj 17a Formula problems The surface area S of a right circular cylinder of height h and radius 7 is S 27rr2 27rrh If the height is g of the radius express the surface area S as a function of r or 4no in W 33 lt3arrh wees vacuova b n r 51h m 1 3ar 5 8 ra211r rgt 3 x 21Wquot 3rrrr WSW MuH ehoxce The surface area S of a right circular cylinder of height h and radius r is S 27TT2 27r39rh If the radius is g of the height express the surface area S as a function of h h D V W Sta farQnrh ched r6obon3hp bn rib r 7533 3 nh 31M 7 395 WW7 yamwt CW The volume V of a right circular cone with base radius r and height h is V l7rr2h If the radius 3 Wilt express the volume V as a function of h 4 3k t u Q V 3Tl f WOY V03 VC o h vlg r rsbh V nm h r V 393n Ebhaah R0 Obj 17 b Revenue problems Revenue 2 Price pfr Itch Cquahhda 1 The price p and quantity a sold of a certain product obey the price demand equation p x2050 for 0 S x S 4100 Express the revenue R as a function of 3 Enter only the right hand side of the expression Use correct varia e se proper or improper actions no mixed fractions or exact decimals Do not give decimal approximations C lwc to 496 divas113132 price 905th wNo c paresoolmeh Kw mcea muons bh n P 3 quot39 quot4W 8050 4 quot39 quotin 39f 5105014 1 The price p and quantity 3 sold of a certain product obey the pricedemand equation p 2c2050 for 0 g m lt 4100 Express the revenue R as a function of p Same online problem cautions as 0 Wm given above R5915 xWavvc Rev 03 tune a 3 Non foo r 0l39or5hvp 10 P516 at d EU 80 R PQ i iei Pquot quot 2 5D N For 4 a K HOO 3997P HOO 39 16 WIH taKC OHMfr typo3p 56 fl th 9 f W0 7 The price p and quantity 5 sold of a certain product obey the pricedemand equation 1 p7700 for 0 S p S 3300 Express the revenue R as a function of 3 Same online pro lem cautions as W given above MWOWB R as Pam o 1600 P K 9100 a 5 KPw 16 73 T7700 31 7P 23oo 3523Aoo 3 7P R x3937 3 3300 P quot774 3300 I K 7 330m 71 W m R Obj 170 Area of a Rectangle problems A farmer has 6500 feet of fencing available to enclose a rectangular area Let 1 represent the length if the side of the rectangle as define igfthe gure shown Express the area A of the rectangle a3 Einction of Enter only the righ an 81 e o t e expression Use correct variable Give fractions or exact decimals Do not give decimal approximations wam H as a fume of wCNO x Nfed re IahonShIP bn L 939 39 L VBwlA S CYUCWTf uslng a the mat rlal H Wj 3 92 20500 333 Vt For H 5035250 g APEC iKHDE WSDb L athfr wa A farmer has 8000 feet of fencing available to enclose a rectangular area and divide it into two plots One side will extend beyond a barn that is 100 feet long Fencing will not be needed along the barn Let 13 represent the length of the side of the rectangle as de ned in the gure shown Express the area A of the rectangle as a function of 13 Same online problem cautions as given above Wam H as Fume oi 16No 3 J ona k3 5L1Ild Shaunre uSm 2 material x 3L903 00gt 3000 3 23 8100 50 l f 64 33839I00 3x 454050 4 34 A farmer has 4800 feet of fencing available to enclose a rectangular area and divide it into two plots The rectangular area borders on a river and the side along the river does not require fencing Let 2 represent the length of the side of the rectangle as de ned in the gure shown Express the area A of the rectangle as a function of 11 Noam R 03 Puma of 160w a L Bwld MSm 0 W at en x I 3413 HB OO SOHe 90quot 35 3 jclIB OO39BJc luj Pr 16 01800quot 3 A H H3190 39 363 Obj 17d Distance problems w d my 7 3 Let P 2 be a oint on the a h of 21 3 Ex ress the distance d from P to the oint y p gr p 55 p p 1 4 as a function of an L4 7 W A swam dSt oA w quot 39OLH39 VOYlable I p a vkuuam d 03 645 0 3 4Y 2 n gt 5443 W C 0 55050 swam rs 4mm 254 1 lquot 16 2 NILHt Choice Let Pcy be a point on the graph of y 3x2 2 Express the distance d from P to the point 5 1 as a function of cc C gtP u 2n ant dist bh Kit 339 5 0 d x 5 rjq 7 6 5 4 3L 4 aquot WXM S 3x344I L2 D w x rwwmm Put hHe tfm toct N let lqb 7 11016 gb Mu Chowe Let Pcy be a point on the graph of y 5 22 Express the distance d from P to the point 1 3 asgfunction of 9 d quot39 We 39qu 63x33 79 JC 39quot 93 xC39x3 5 b 33 2 7 j 4 3 01 3 2 Q Rx 39 Vkawdr 4 38 139an Wvg 9 MMK Cmome m 5 5quot 30 pg Obj The Optimal Time problems A car rental agency rents 250 cars per day at a rate of 20 per day For each 1 increase in rate 5 fewer cars are rented Let x represent the number of 1 increases in rate I Express the income from car rentals 239 as a func ion 0 2 n e right a e expression Use correct pvaria 1e e p fractions no mixed fractions or exact decrma s Do no give incomecafs mb gt N wi FORDDY W one 4391 mt z39 450 5gtampOD 350 5J6gtRO gt Two quot39 C F39 1amp50 5395 3014 lime 4 mu 25lgt3933955 JorHHvl 9w 65 CCCPt Y PaLtDY K inc 1 239 11 PDVM or Malt out The owner of an orchard estimates that if 40 fruit trees are planted per acre then each mature tree will yield 60 pounds of fruit per season If 2 additional trees are planted per acre the average yield per tree is reduced by 3 pounds Let 1 represent the number of 2 tree increas planted per acre Express the yield per acre 3 as a function of 51 Same online p blem cautions as given above j Y665gtdgeld yer fe 39 Now 3 7 6390 90gtL3 J an 39 one Jtrce me 3 2 L10 aoo 3 M TWO 3 MC3 10220039 qOQQQaO c 1mm R ce MW 90222ao 33 X R39bvtc 1m oa 3 gt An orchard manager estimates that if the fruit is picked now it will bring 12 per pound with each tree yielding an average of 100 pounds per tree The average yield per tree increases 3 pounds each week he waits but the price drops 1 per pound per week Let 2 represent the number of weeks the manager waits before picking the fruit Express the income from the fruit trees 239 as a function of 51 Same online problem cautions as given above mcovvxc 5 WWC PU va d quot Pound per 66gt M 239 shower mew z39 03quot 06 wait a Wffy st 7 lR l D 39OOTB 0539 Walt M upx oo r3 Wall 0 46st5 Z 673 H30 T3Kgt RR Objective 18 Quadratic Functions 39 os m The simplest quadratic function is f x2 D vac00gt 0 0C 35quot 3 q a a c 43 e quot 39 6 w W 3mm wr j as O o 6 Dec 00 0 Inc 000 Vambom Veal0quot a Objective 18b Quadratic Functions in h k form Applying all of Obj 14 re ections and translations to the function 52le 12 T2055 bl G vertex h r If k Po tbl C S wl k a gt 0 parabola opens g 12 0551o M94 4lequot d rfhccmon quotlb113i a lt 0 parabola opens Own across 6 39am5 Objective 18a Quadratic Functions in Standard form fcacc2bmc A C What s the vertex We could CQm I hm Q 3 wand put it in 1219 form Goodnews OOVVb have no He39s been done For M fcax2brc vertex 390 a gt O parabola opens A B a lt 0 parabola opens My For either quadratic form To nd x intercepts let 9 O solve for c To nd yintercepts let 6 0 solve for 3 Sometimes we ask How many xintercepts are there Standard P39QYM 93 Rat gt0 7 0754 quotWquot or Obj 18a You can use the dlBCfmV6YH b2 0 a aceInc vaLlac 4 0 399 O L lnt For Obj 18b Just Qr Ph b f 0 I43 anm R7 Objective 180 Max Min of Quadratic Function 1 y YalUt A Mow MW 3 Z x4 x oak D RV M m 20 Ob 18a example The information included in this example would be asked in separate on line problems r M a 4 f93 12a92 42 1 mum Vf fx 1 Opens U 96 I M AD LL v eelil A Ircoordinate of vertex 2 756 EEC 9 23924 g o I 3 a 3 How many Isintercepts F c gt quot l 70 41 4 bg39 ac JDZ 40mm 5 WWW gt0 q39gtMiniS pl egt 307qvMgt 2V3gt a 31 3 bMin is atzvz 2 7 W 1 0 ags can we a hem vNot an at once 0 Find intercepts For on line problems Enter them in any order separated by a 2 3 3435 u m ght Hug bgogypaD y SCWHCOth C39W t 4 9O 7 397 ax 9 9 4 a 3K7quot9Kl Drum we 1 o CDZ39OG zK rb 0 0 39 Uquot92Ca 2 z loxquot2 0 2 40 Which of the tol wmg most closely resembles the graph of f x 12m2 411 c y y y n 39 x x x I x an 496an3 down m quot2 2 q s q a 430 W 103415 Ob 18b example The information included in this example would be asked in separate on line problems I H 2 fcax12 8agt0 gt Form 8 L l OpensDown Qgtvertex 391 398 BDHOW many xintercepts Gmh the 3 N l k H Q E l re a Lb mt 00K 56 down 5 HgtMaxis quotquot 39Zalg 5gtMaxis at a 39l 39 V6 4 9 Find all intercepts for f 13 2 222 12 8 For on line problems Enter them in any order separated by a comma wtlet xeo 57 gampo33 752gtX 510 xvlvrt t 370 gt CamelDRE Remn Ob Lug DP NOT mat out RQHYR lI KR f o xli2mz va 1 9 3 1 C 5 k s 3 Find all intercepts for f x 2251 12 10 For on line problems Enter them in any order separated by a comma Manx e39g ago 537 90 3aoa o 5 lo 393 xmt1 eh 7 o a lgta quot39IO 21 4 02 5 2 ZRn250 T QL if 0 quotI x l K quot I 0 Ob 180 example Studies have found that the relationship between advertising dollars a in thou sands and revenue R can be modeled by a quadratic function 11 If Ra 4a2 364a 25695 ow man thousands of advertisin dollars should be spent in order to maximize revenue h Oidfrcd Vans Q Kgt ad a want quotaquot value cl ucreuc Enter number answer integers or exact decimals mathematical operators are not allowed For example 152 must be entered as 75 Don t type any dollar signs commas or units The function given does not represent the results of an actual study 110 50 305 0156 Calculator Ra 74 8 a is 2 39r Arfvf adv W e 8 Ob 18c example A large swimming pool is treated regularly to control the growth of harmful bacteria If the concentration of bacteria 039 per cubic centimeter t days after treatment is given by C39 t 04t2 44t 301 What is the concentration of bacteria Same cautions as in previous example ve zc 6 P wan t C Ct o 4tquot HH r3cgt t c 56 0 60163 Cancenwa OW e c Hu 11 or boobEVIa Ra o X KR8m nb6r fo hfx t SS agtgt also C Ll55339 30 Wmda days Ci F tcxr vatm en b w M 214139 Covwhncratxon o5 baL thO be at 0 mm 397 5 da S Q Objective 19 Power Functions fa xquot where n is an integer n 2 2 The power functions are classi ed into 2 groups f x x where n is an even positive integer n 2 2 For example fr x2 34 fx x6 fr 958 eVCn WSSWHAA Duff 390 Dec 045 0 Ac 000 f as x where n is an odd positive integer n 2 3 For example f9 x3 x5 fa x7 fa 359 L 3 IL Jhgmer Power 3 8 Clef r to Vtgt53 bans near 00 3 59 I bl StefPer a quot 3 D1 00OOgt R6ijgt I WYJL oyj h ViC on P0a ltgtgt 71 buawn out For 0 A 059 500 0b 5 00 Objective 20 Solving Polynomial and Rational Ineggglities that mam that MP Introduction to Obj 20 Ifab 0 then it must be true that 0 0 or b 5 D However if a b gt 0 then it does whave to be true that a 7O or l0 7 Q In fact both number could be negative lt 0 It would require 2 cases to express this solution Case1QquotO and 1070 OrCase2 040 040 bin at are gt 0 bee are me I we consider c gt 0 thffeae numbers multiplied Witlf igbfesi being positi then it would require 4 cases to describe this solution and in only one case is it true that a gt 0 and b gt O and c gt O and We need a process that doesn t require determining and then analyzing cases The process is based on the Factor Theorem and the Intermediate Value Theorem Plan of Attack 1 Factor if needed Watch for Diff of Squares and Factoring y 39 O In the case a 2 Findl ggl 39tionin Points These are values that m che expression quot of a rationa inequa 39ty these are values that make the expression 7 0 from factors in the numerator or make the expression AtId Q n dlfrom factors in the denominator Set each factor equal to O to nd these Partitioning Points 3 Mark these on a number line must be in numberline order 4 Make a SignChart Select a value in each interval that s created by these partitioning points don t use one of endpoints Plug this value into each separate factor and record whether the result is or Consolidate the signs from all the factors The Factor Theorem guarantees that there are no other Partitioning Points beyond those found from the factorization The Intermediate Value Theorem guarantees that there can be no sign change within the intervals created by the Partitioning Points 32K 0 3 2 0 9 4332 SOlve39 1132 5305 Mane 64 076 3 393 Tag 39 3 3 32 40 x 2 L lt3 TESL 3 2K39Z 4 O W 0 C 395 0 9 moat use ioValufS Pd Vougrnowna mequahtv 35 Solve 332142x12lt0 3 55 05 aim62 lactor Pm red lla Rm 3 474 L 33 A O a L go 21x 2LO 6mm 390 X Vquot 2 kquot k r quot no 70ft 39 u 3391 1 1 439 g 3535 quotquot 139 v 39 39 J M aloe 39 39 a 39 9 W ft Cb lftw70 a 1545quot 10 a lL J to Solve 512 362a 14 x 40 Mfr co 39 r r my xx 39 x x eruDa o 9 I C 2 9 O 4 9 K 3w Ll K O 2 p lo I 6 Kai jlp X ct K 0 I L gt I rt 00 V0 F oOgtUL 1 x3x1 x 301 2c SOIVG MOKC Vt MM Solve 0 340 5 I 0 K5 K r 72 A c I Solve x5gt3w4 V X 3Lq gt0 3ampch o nomlal I 7O 4 Jr quot l f d x 33 c9 5 ED x o K73 Ice f lfb 3 Wk 16quot0 163 LVS 52 xncgt kCan t u5 an x 3P5 Par ohonivua PDlh cS town gt0 Solvex523x4 A 653 6920 L9 z 3 10 Q 3 S E Parhhonm POIIA lT5 WWW O l 25 O 3 A 5 Ufaco W Objective 200 How many partitioning points would be needed to solve Plan of Attack Obtain a single fraction on one side with 0 on the other That isz make the problem read to be solved b the si chart method P wt won t souf 5 NOTE You can t C 3905 M u l 39t or multiply by any expression that contains the variable because you won t know if the inequality sign should be reversed NOTE If the denominators are constants then you are allowed to crossmultiply multiply by LCM 3 2 a 4 6x1s2x 3 x2gtx 2 39 52 31 JC i gt O 046 W Wx 3wb Xv 02 xx NaraMm 9 0 Va 70 com ax33 If a blbOle39L grm i a O 7 Par 6 S anall 31933 39 f 3 part quot0l t a 40 Ian H 216 3 7A 35 W Lo 12 o Su 240 3 4 1 20 LC 2 7 gt625 2 3X1 LO one 1762 a l 7 1701M Objective 20d Find the domain when a signchart is needed Recall Obj 10c Give the domain for each fx 2xzo mas3 43 fltxgtm 3 21 20 RKZP3 Dquot0 o Ac 3 Dquot amp 3 Objective 20d Give the domain fx xa3c2 10x3 2 O x 3 3 on330 quot 1 1 quot 1 39 39 9 i Ice 0 Wk lee 5 69 on 165 I x33410X3gtZO mi 5 l X I 0700 5 0 XCan indude so 1 2 30 H55quot 90 EYOhly values 196 025939 5X 3quot 64W b0 0 6 392 5145 my W D I 9 75 U 02 39sz 7R rr f lquot 4gt f V35x 2c2 o 1399 quot 9 Mic MEL 0 3 20 re 39 35X392L2 icy 2 I 70 Numcrator 5 H r39 3 Kl u Domalm 39739 15gt 37450 Apr o kph cannot 0 So can t K 3 Lquota use ParUmOmmzd Pornt f3m fa535x 2a2 kocld mot L all 1 accept D 45 gt0 S xl7x WWW WK den40 D3 in Kf i zquot 15 33 I ND yam CWN G mama These are special cases you may see in your Practice or Hmwk Quiz problems but won t encounter in the Lab Quiz or Test problems We need to be aware that an evenroot radical can have Domain all Reals or even an empty domain fc a225 fz z2 25 at an pas aman 3 neg Doooampgt DUZ Copyright 2010present Annette Blackwelder all rights are reserved Reproduction or distribu tion of these notes is strictly forbidden 7R Objective 21b Example Select the graph of y f 1x 439 390 79 graph IVS over 6 25 fOY unversc I y K y R x39 Squot Lb quot2 L A f 39 unctlon can be 1138 own Inverse ConSIder 9 g i f a 72quot v 4 9539 J p Kl g ITfYQ 13 no LVV V5 Objective 21a Does every function have an inverse NDamast be one tv One 03 have an VEYSC y y k Fa v erb hw ten a g x 4 Noe a 39 RAWme Graph of a function must pass the hOYIZ h e 33k to be be the graph of a one to one function F x 395 to be a Ll f 5 3 3a UHC GOY WM 55 t0 96 one Cb ont 7R Objective 21b Example Select the graph of y f 391x 1 lot 6 29 IVS ovt r zz fair 1mver5c I quot 8 Lb quot x L A f unctlon can be 1138 own Inverse Con31der 9 5 C P z 32quot v 47 75 J v kl K WCYQ l3 no LAVEVSG Objective 21a Does every function have an inverse NDamusb be one w39om a have an mversc y y 39MB verb lune 9349 t 5 x quotquotNO 3 0 L RAWbow Graph of a function must pass the hOYIZ me ctS39b to be be the graph of a one to one function 39 22wa as to be a 6 7H3 5 U 729quot uMUOW mm 55 t0 3906 on b cnt Which are onetoone functions lt12gtlt13gtlt54gt 12gtlt32gtg5 u 5 0 9me at a not 39 e to DV quot lmquot one If a function is not oneto one restrict the domain in order to de ne an inverse function Recall intro to Obj 21 AL y gtltv Objective 21d Given a function nd the function rule for f 1 Plan of AttaclEE Write y for f to simplify the notation Solve for m For applied mathematicians when units are usually associated with the variables ou have the inverse function or our College Algebra course we will interchange a and 1 to write the inverse as a function of x Write f 132 for y to return to function notation Find the function rule for f 1 for each of the following fa3x5 at the quotGYM fx4 a57 w u 5 D 5 3x 5 j c 51 3 l 7 576 313 2x chaqu g i k j 3 5 67 I an 44 J34 via ig afgzm qr5x391 5 5 31 6quot 3 a 705 406 1 51 fltxgteampT 27 fxx1 133 D5 316 7 D g 345qu 7 f3 gay xa a 9397 1 w 93L13993f 2 5 4 C 391 c5 l 466 Kr9 Aquot 3L 4 163 3933 194quot at 2550 3i 7 quot39 l 5 4y553quot J6 j 310 35 x2a 3 DV39OQ quot L l yz 53 2 393 a wtj AKquot 3 x 6 9225 3 6 a 3 quot 7 2 O Jective 22 Exponential Functions f so a a gt 0 a 7E 1 K l 3 30615 l a was by xl 3nj J39b3 39X 73933 Does this de ne a function 4 0V1 M g Ya LS Cd to a PoW fY lVfS J on c a V WC Don t allow base to be negative because could be Yi ed for some 3 ie 539 539 1 X Don t allow base to be 1 because 4 c l quot graph would be linear not exponential What s the domain All reels If so we have to de ne what s meant by irrational exponents For example 4 5 or 4 r We haven t worked with irrational exponents Good News The limiting processes of calculus guarantee that irrational exponents are de ned and C line u as we want That means since 3 lt 7r lt 4 then a3 lt a r lt a4 That means since 1 lt lt 2 then a1 lt a 5 lt a2 52 The exponential functions are classi ed into 2 groups depending on the base x 1 x0lt lt1 fc aagt fc a a We will consider two speci c cases to develop the concept This is not an eGrade example you will not be making tables of values you will not be plotting points Consider 1192 iii Consider f x for an example of a gt 1 for an example of O lt a lt 1 y a y a y a l 3 0 400 000 k l 4 as w 1 s 39 q JD 3 10 f 100 V 6 394 W 100 1i q 0 9 1 IO I g 3 2 q 10 39 3 2 quot7 0 a bl l l 3 l 4 J 39 3 ll 3 4 3 q H 0 v 3 a 1 IO 39IO 10 q 9 2 1q up 10 4 Ll MOHAde q I 100 quotdo 100 000 l quotl 100 11 we nooo 390 w W w j 1 39r v v JPK39OMS 13 an 6131ijwa Objective 22a Properties and Graphs of Exponential Emctions fxa agt1 g0 faaquot 0ltalt1 y l Ly 39 lg exponent J39th39 39 l D 6 01 75 90 CLPOIA W45 K o 0 I 4 neve hlts R x rams x My oi BEL CWCWWO we w 35ij tote 5 j 0 hang R1 lljolB Objective 22b Graphing Exponential Functions with Re ections or Translations Don t nna e 0 laid Don t 4131b b Use Obj 14 W 05 values A 9 4 39e 0 F 6gt 7quot l Cgt Ell 7 rfPl across an ff I across 4615 Select the graph that best represents the graph of each of the following RHFI 05055 44 across fa r0105 MPG 7 mos 3 x I Mi lP mu other remcotton across x or grams un mama 1t Macaw1 ne tyuc w c a me Which function best descrlbes the graph sho n Whlch functlon best descn es the graph shown LQXAS y y 1quot yealt13 em I V f 5 a gt 1 0 4a 4 f25 f25 f x25quot 9325 f 04 21013 04 f0 04 f x 04 More Objective 22b Graphing Exponential Functions with Translations Don t 39 to l Don t 9 06 Point Use Obj 14 6 c quot0 Shvl t Milk 3 3 s Hz 0 2mg MP 3 Egg wot0319Q39C gt5lmh down 93966035h19t mam z39C 3 0 52 Select the graph that best represents the graph of each of the following m 3w swk down 93 3 S A 1 f I 7 a mc olta41 457686 ly y gthgt x I1 x xv 139 x Which function best describes the graph shown Which function best describes the graph shown 040 4 a 7 gt 1 C xv f 2 6 f 6quot 3 me 99543 0 6x3 z 2 x2 522 Objective 22c The exponential function is f1 ex because of so many areas of application 1 n e z 271828 6 umnm 1 Z Grewhface c t DEQWlll not need to Mwoo a 7 l 0 If PYDVC t V 13 7 m C Evaluate eat on a scienti c calculator the Mac calculator in lab class Strontium 90 is a radioactive material that decays over time The amount A in grams of Strontium 90 remaining in a certain sample can be approximated with the functionlA 22560037 where t is the number of years from now How many grams of Strontium 90 i be remaining in this Fr sample after 7 years gpw d F567 a e 037 Cblc 037EEEJ7ET ED g 131 aRSEV 9 17390 I a7525127 Banzai 751312 673 WJ 123 0371317 E10725 El 8000 is invested in a bond trust that earns 59 interest compounded continuously The account balance t years later can be found with the function A 800060 059t How much money will be in the account after 6 years 0 o 54 L03 0 03quot 0006 RA Objective 22d Solving exponential equations when we can obtain the same base Exponential functions are oneto one that means m n 2 I 0 39 a if and nly if m n Rewrite each side if needed in terms of a common base use the smallest base possible Be sure to replace equals 4 x 4 Solve equate BLPDV W Q 07K 1 23K ourl 9 07 gm Equatc F0n9nt5 9kl 0 79 gqgt3 3xgt 4 5 x QK zli 39hwras W W HL8O v IIL8 39 Solve 1 1 573 F5 i quotl X l 7 45 3 I X ca 5 t equate exponmt igazD 72 5 E cei 05 Aax37amp 5 quotCD 3 We 5 iqu 3 RF IllaI5 Objective 23 Logarithmic Functions 0 00 so Consider an exponential function y 095 QKPOH En is 40 rtsule 09 There is no algebraic operation to solve for as What s the inverse function eKPom chlahovw We must de ne a new function y log at p 0 OD a i v I 9 V Ulb 05 Objective 23a Evaluate Logarithmic Functions a 9 1 QKPOW Cntlatwh logz 8 3 i CK PM Twyws 439 R 3 1 quot P 391 lt3 1082557 39Tp w yg JP ab 5 25 3 5 5 1031162 L L meKquot Tl pz 2 46 Pe 2 iPTl I P i r 4 log22 0 LI 392 Ease rtsuie Match logz 1 O r twc I5 1 Which are de ned Be careful sometimes ask Which are unde ned 1081 21 10814 4 logizt 108129 aY HE quotDquot W x eww must be po mv comma we 0 a gamut RR Objective 23b Properties and Graphs of Logarithmic Functions f as log 2 b gt 0 b s 1 The logarithmic functions are classi ed into two groups comparable to the exponential functions Recall Oby 22a y 5 K a 1qu a General was 9 o quotgt ex ylogbbgt1 amt 0 g ylogbx0ltblt1 P K i 00 00 39 xlvwerse gm Moment vert avgo 0Q expOWEyi al xxw t l askum Mncmn 0 b Cons1der fc logy wmc ob Cons1der f x 10 30 m e for an example of b gt 1 5 for an example of 0 lt lt 1 L 9 5 z quot I quot I C a Pick the y values nd the xvalues Pick the y values nd the xvalues a y a y 3 frl w q U 2 1quotle 2 1 L 391 I q 1 20quot 4 1 W gt EVquotquotquot 4 sq 1 yq 1 2 17 Vim 2 27 Objective 23c Graphing Logarithmic Emotions with Re ections or Translations Don t Mal Q table Don t slot VODWtS Use Obj 14 39 W Kerchon grime r aehccuon 319063 across 26 6W5 a 3 acroas y amsi 5 lojapx Select the graph that best represents the graph of each of the following vch acvogtlXacl3 o ab fac loga 0 fwlog x lt gt 4 K lt gt agrm aw aromas Il 0404 Which function best describes the graph shown Which function best describes the graph shown it agt y 39 Graph Mc o y a ogtl no MH l P 5 1 6 0V 0404 s 1 x I re aCr x amo M wglm x QilQEW fx10g52fv f 710g52xquot x 1 g25 x it 10805 13 f a 10825 a ref am quotMzw m0 CWVWY rdll amass sz ablg xoan5 w deC it dec39 aim we VP More Ob39ective 23c Gra hin Lo arithmic Functions with Translations J p g g r9 ma a ma Don t maKe a table Don t aloe wne Use Obj 14 I v 39 W KeHeLuom 39 i lYOhSlamonS W o xNote 3 ny Compansom a 0 C u g 1 oja1cgtkc 3 C dOUJW lj f QOKgt C IhC Re t acre 39 b0g15 Rah across 3 quot 64 0 a 4ch 3g3ec 9 109a OptC 5252 9541 6 she Select the graph that best represents the graph of each of the following fx 10g3acj 2r 5nd fx 10311533 5 Q7 0404 Lav t 131m 4600 034mwa wow 1 Aly I I I 0h mal g I Which function best describes the graph shown Which function best describes the graph shown E Dec 4704014 I lvxc ogt 39 3W k lt5 I m bed nah K dew n I x X f0 10g5293 2 f0 10852 quot 2 an 10g525 3 2 53 10g5293 quot 2 33 z lquotES5 2 W 59 10g25x 2 it 10g255 3 quot 2 chub NO eh m am answer theme 520 Objective 23d Domain of Logarithmic Functions not by graphing Give the domain fa 10ng4 5x 70 q39Squot 70 1314 5 15167 4 k4 4 5 frv15logb 3avg o39nu 00 at UmBquot 70 39 3LgtO DO ogt 16quot0 log3t4 332 3 In 50b 20 439 70 J 042116 3ps o t LO 3X3quot 7 x 5 2 H I la 5 Objective 24 Properties of Logarithmic Functions x As used below a gt 0 a a 1 b gt O b 751 M gt O N gt 011 gt 0 y and 739 represent any real number We c alt 0Q emMC yr 5 went a De nition Obj 24a 5 c logbxi Imeans 2 base Common Logaritth are logarithms base 10 we write 55 instead of 19 aquot O hl g w tbch Hdtca 3 19456 ID a Natural Logarithms are logarithms base 6 we write l h 6 instead of 2 as Objective 24a Example Which of the following is equivalent to In 5 x Agt53x Cgta5e Properties of Logaritth Obj 24b Product Rule logbM39N M l 13 W Must Note logbMN l l0 Ngt Must Note logbMN l M l N I 0 Quotient Rule 10gb III1 quot l N M Must Note 10gb nk 0 Must Note logbM 0 Power Rule logb M quot r W D J lo M h M 4 f ChangeofBase Formula long 2 l b long Y b When Base and Result Match logbb l When Result is 1 logbl O 01 Inverse Emotion Properties Obj 24c Recall Obj 21 f o f 1m a and f 1 o fz a gloggM loga gr E Objective 24c Examples Solve for c if 51 53 15 Solve for a if In 615 3 15x 3 3x 39 5 X 2 1 z 395 5 5 Objective 24b Example Which of the following is equivalent to logba y A logb C both A and B B 10gb cc log y D none is equivalef Objective 24e Examples Evaluating ea or lnx on a scienti c calculator and rounding the result to 3 decimal Places Ar Aux aural m to caxcmator ak When rounding a number to 3 decimal places look at the 4th digit of the decimal If that number is less than 5 then keep the 3rd digit as it is and drop the remaining decimal digits If that number is greater than or equal to 5 then round the 3rd digit up add 1 to the 3rd digit and drop the remaining decimal digits Evaluate round to 3 decimal places e 5 a 1 207 1n 50 z 3 Q a cgtoma7 234130223 f Objective 24be Example Select all that are correct for log3 8 Choice hi8 Choice II E Choice m 1983 Choice m 0528 log3 ln8 Objective 24abc Example Select ALL the correct formulas statements if b gt 0 b a 111 gt 0 y gt 0 eGrade does not warn when a multiple selection problem is left blank loga y logx logy loga y logx logy loga y means 10 c y log 1049 12 then a 3 0 54ch ApplyingLog Properties Objigtive 24d QM 00 em Ru 6 Expand usmg log propertles 10gb P A Lt a M e Ogbba 39 O b 32 0UC glo bx Oo bv l03b2gt R03bx 10303 ogbz Expand using log properties 10gb ma a 1031 umB r 109 053905 6 5bz Haws O b j quot Dab239 O b ltw3 2 Which of the following is eqmvalent to 1032 3 A 2 log 3 1085 3 logb z logbw 3 B 2logba log y log 2 logbw 3 C A and B are the same another Objective 24d Example Write as a single 10 arithm Zlogba 10gby logb z W Rearrangogzt W Ojbz 39 Ogba Powef a Q ond 5 Guohwk 60ij Ojb 03ba La order Mm 1cgbha Ojbjz Oij L 7 IEPe to 97 2 2 Blogbx E A 10 gbwz y 02 another Objective 24d Example Write as a single logarithm 210 w logbw3logbzlogba 1 gbaw Wquot 103 3 i banax Qooyv 05b 0 9 103zw334 bgb 23 105911 Io Ma aLo 603b two 13x v39 03 wk gt Ojb 239902 3953quot W x ougt another Objective 24d Example If log 2 l and log 5 m express 10gb 100 in terms of l and m 595 oaba 1 l03b I39lom a I loo my DabOl39C ogva f loabC W another Objective 24d Example If logb2 l and log 5 m express log 4 log 25 in terms of l and m X omg ieama Objective 25 Solving Exponential Equations when we Mobtain the same base Logarithmic functions are onetoone That means l if and only if M 739 N Objective 25a b N Solve 62 1 143 SS 5 12 UL 7 quot 31 v n a n N r V 3 3 Moms 6 gt Ln 3 9 V N 5 gtn M X V 5xln23lwa quot awlhwhv 3 Vxlqquot g 6 th Mum 3mm hto damp Hm 3mm wa L mman M Wax a lV3U 914 qu MUV H 5ka ITmluo L 03511590 I M nl i Objective 25b Round solution to 3 decimal places Solve 140 e3 1 42000 Solve 4062 120 Q Mmgncal WSC 3 6 7 quot l 3 62 7393 MC 9 3 co xmmu 56 ihB 39 i e3nl3 3 c 5 3 WrinB 039 abz nls 3 HgtV3 2033 XS39Sqq 3K 1 ha 3 Solve 1400 103 42000 Solve 40 39 102 120 n numfnml Dz 73 CoprICIent loan c 30 I cJCij W 9350 W 31641 2amp 1 06 3 36 lo v 3 m A 9 quot39 a Objective 26 Solving Logarithmic Equations Objective 26a All terms in the equation involve log functions SOlve 10 M32 be olve log311 53 log3a7 log3x5 03C xr2gt nayn1 quotgt33 2133 103306 M N M N H i K 5 X C 2gt 539 KL n7 amp w 76 n i39S 7 M N WAN Gnome L2 2 KL i one sdu oy Ka ampa3Lq Q 01705 392 ch tho Nae394 H3 n3 0 K O L lzo gt W C x 320 kt3 v0 ne BV FV 5 smslu s1n Iet CHECKz Uj n vMuag Me I b 9 M N 0t 0 5 03b 03 0quot one soM mon a 691 Solve 1037x8log7x 9 log723 3c Solve log5m4 log53m8 log5x3 OTQK WM 3 05 M quot 55 Lb 32 N pad4 23 32 KgI m 72 23 x L3 qu 0 7 x 7 v7 M4304 r3gt32c 18 L4 th at 415 1513 x 7c b xg39r39bc H2 3 3 X77 F39r39qz 1 0 Ch 394 n N0 Soumom quotH O quot 3 CH thKTPiu m Dbdate the me am use Ob jective 26b Use the De nition of logarithmic notation Solve 5 7log13w5 Solve logc22c111 quot7 03I3K 5390 2 39 39 3X 21 4 H 1 35 IO IO J x 0 00 39gt x K O CHECV1wm g n7 f winter as be gt0 one SOlUmohgt a negv 7 Solve 610g4x5 3 Solve log2x27 4 j 1quot 5gt1jf gv g31 up K 7 Llquota He39s 39 1 a a7 6quot wsfv ass 6 L m on Fi DS mme t wst F0 ctov Objective 27 Solving Equations that involve exponential functions 624x3 x4e2quot2 e3z c3ce34 Solve 843 O Solve 62 O 0 4L3 Cax 027046216 co also ego x o 50320 0 o i aquot lt 9 52quotquot 0 deuuon3 o x2 q 3 0 8quot quotquot x 47x 39 SO 23 zczia L o x 98 4flubon 20123 Solve 2e22 8L4 0 olve 2e23453xe 34x524 0 xaf qcx 0 aem lx533 262K 91645 50 e xo 0 A L LSgtb 39 Ka i 0 e MQLSUJ 1C o W7 q Not allowed m 2 r o L 9 4 0 be 0 X q 7 9L5 erowe 456 Mo olut mh 1 quot 5A 1 L JFK390 so Solve 3x74e4 4 34 4317330 3 78 HMEMV 0 3 0 N 3X l 0 9 3L 7 o 3X 3L 39q M 3x74 7 q 07 Solve Solve m26e3quot 2 5a242e3 0 l2x 35e4 2J Q4Kx 33 0 x28 rt3quot 3 263K gyp IDC KgtH Q qqu 3gts L eqxo fa Q 393 H u o my 0 M a 5 gp z jq w a 9 W va quot1 0 3DW3 L 3quotquot K41qu x2o Can39t We amp 3 W 1 pa 46 aa gt go 3 3gt9 0 6 Xquot gt W84 94 0 w1 Wv wm W b w 2gtltzqgto 2m Rica kW 0 aLtOYe 4 L39Yzo LIL L010 28an 0 N0 Solutio m qu the jug 3IVWPFV 394 Objective 28 Solving Equations that involve natural logarithmic functions Solve 22 70 Solve 44 181n 0 aa xmma7o 44 lvx ltgt g A 339VC 7quot 3L q m hx 39 3 xy mbfr NC 5 29 4 ML 7 ML A f g I E A hb eY e quot 7K C 1 1 EC 61gt G 2421 a 3 31nxgtlt4x3 Solve 2 O Solve 7 O a x8 j gh n 0 nglnx 0 3t32xaln 2C x2 Rhn76 393quot g L LfO YVC 6 Max quotl glquotmmgtb Hunt K JEhtC r I39Hlnz 0 E63 tax 02 3I39Hlvmgt M Seq4 a 1 X 3 4lenx 161nm 311122 y 0 Solve 0 Solve may 3 06 Z lm 9xhz quot 0 IL hm Lolvsx 0 Kw cgzo arm 6an L2 q W 7 7 u 70 quotL LO o vmmho amp 2 X 0 WK 0 9399 0 9 2quot quot NO e0 1C 9 K q 0 quotKg guhon Inc 5 Ian K39s1 3 A32 0 hot de n d 0 CHECK L75 M39 E 33 32 Solomon fag Ob Jective 30 Solve 2 linear equat10ns 1n 2 unknowns a I S I There are 3 possible situations 1476 L 3 Cksgjk g y y y lt 4 x s s 39 f a E tnlutxon5 M33 one SoluhDY No 501mm thhv bc Solwoonb 75 0i M sff fb ow line arc Parallel M63 coma To solve algebraically we will use the a l gn mg g methd Multiply one or both equation if needed by nonzero numbers so that when the equations are added one variable is eliminated 26 33 40 3 Solve Jw5y10 3 3 E23x 2965 11734 2 M 11573 30 4x 4quot 20 Solve 7134y 14 JULl3 quotI 3x 23 4 339 C3 Ht 33 75quot 43 4 ng 932473 Q X 39IDq 5301 6 43 13911 1 431 I 3 935 FI IL iv 9 5 Hr 722 39 awne 3390quotquot C 7 89 1 1 chm V Ml gala gf l atquot 393 gRI ohon QDIVUSWI 3142 11 7 OWM mat ehm mtf x a r 4L1 i Solve x 12y9 X mac 615 a Solve 6x10y 4 l3 mm283 21 39 211 353 14gt 3917 63619 3 Hm 5 Mow339 quot k 2 W 2 43 21 95 3x 53 a 00 b39gtDo Sci O L C3 V7 V4 WIS I a Conwaokcho n 3 W F V Wf and 50L m 0 744 1amplum boluhom bu Ratm A to am hm onUL C aquot 1 quot Objective 31 Solve 2 equations in 2 unknowns First type Quadratic and Linear Algebraic solution 3 I 5 To solve algebraically we will use the 3M U51 MC GVLQ d ft A Substitute the quad b into the i1 90f V w Solve yx22 Solve yx24 12x 4y21 3 45 0 5z 3y15 2 3 a c pizza 5 6 30 A 15 X Lift 2 L534 3ampA2IS Des 4 v 7 a 1quot 0 i336 LIX g l gt O 3C 51C lquot 3 Hot 0ch 0 Kg4216 o C aone P O 0 9 z 5125 i o Rx 3Rz 3 Mysemm a ange Give the m cooiginateSs only of any solutions If multiple solutions then enter the values in any or er separat by a senuco on If there is no solution then enter no solution When there is no solution do not use any capital letters do not use any punctuation marks KE RS I 2123 V x No Solumon 1m FanSlatxogw S 1 we 2325 s 396 C39OXIS y rt t across jr m 590341 3 A 5 90 0 down 339 W 1 Q lek jr39NVCD right Second type 2 Various equations Solve by GRAPHIN G We will ask may IA solutions are there Must know all the basic functions and equations we have graphed and we must know Obj 14 MWCLU DYl and QWSlamo My 4y y 5746 x 2 x i x 3quot Iquot x x 37 35 54 344 x x 2 K 9 a 35 K 3 S l 101 Solve the system of equations by graphing How many real solutions are there 3 125 y lfvlQ l meSwef W one oluhov 00h ma liia 3w d Solve the system of equations by graphing How many real solutions are there yf 3 y 1 3 v onamal E some Eamon Solve the system of equations by graphing How many real solutions are there 3 In x a12y24 wacle I enter I o rco f Copyright 2010present Annette Blackwelder all rights are reserved Reproduction or distribu tion of these notes is strictly forbidden 10
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