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# COLLEGE ALGEBRA MAC 1105

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This 20 page Class Notes was uploaded by Nathen Fadel on Thursday September 17, 2015. The Class Notes belongs to MAC 1105 at Florida State University taught by Staff in Fall. Since its upload, it has received 49 views. For similar materials see /class/205606/mac-1105-florida-state-university in Calculus and Pre Calculus at Florida State University.

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Date Created: 09/17/15

Review 39l39est 4 41 42 45 44 4546 Name Find the indicated composite for the pair 39of functions 1 Given fx 7 f f and gx 1 Xfind g o fx x ANS g o fx X39 From the answer what is the relationship between gx and x ANS gx and fx are inverses of one another 39For the function f use composition of functions to show that fquot1 is as given 1 xl 2 Let fx 1 xx Show that f391x Find the inverse relation 4 quot4 3 61 quot2 81 A 4 quot3 quot4 quot 61 quot2 11 8 C 4 quot4 6 6 quot4 2 B quot41 6 31 quot3943 quot21 6 1 D quot31 45 11 6 61 quot6 quot2 Determine whether the function is one to one 4 fx x2 1 A Yes B No 5 fx x 4 A No amp B Yes Using the horizontal line test determine whether the function is one to one Q O l s N d d s s d i 39 o ul d u 10 Circle Answer YES NO Circle answer YES N 0 Find the domain and range of the inverse of the given function 8 fx x 2 A Domain 0 co range 2 oo B Domain and range all real numbers C Domain 2 00 range 2 co D Domain 2 co range 0 co Determine whether the given function is one to one If so find a formula for the inverse 9 fx 5x5 19x3 2x2 5x 1 5 A Not one to one B fquot1x 353 1 5 D f 1x 2 5x5 19x3 2x2 5x 1 10 fx 2 2x C f 1x B f 1X A Not one to one 9 2 D fquot1x 2x42 X 9R 1 00 D OO 00 Moved right 2 units moved down 1 units 16 fx t l I I I I a I I I I I I l I I I I t I r 14 e497 Find each of the following to four decimal places using a grapher ANS I Etc A 545981 2 A 0 1353 1 x s s s s s s s s s s s s s s s s s s u s s s s s s s s s s s s s s gtI u s s s s s I 5 s s s u s s s u I I x 5 300 where x is the number of people in the group and Cx is in dollars Determine Cquot1x 3 3X42 1 to fx Cx B 73891 Graph the function Describe its position relative to the graph of the given equation 13 An organization determines that the Cost per person of chartering a bus is given by the formula 6 u u I I I I I I I I I I BX 300 6x x IO I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I B 2980958 X I I a o I I a I I I I r v a t o a 0 I r I o I a I I I I I a a Find a formula for the inverse of the function described below a o a l l I l I I I a v o a v a a I I I a C 73891 C 0 0183 D 0 0003 D 545981 11 fx 12 00 A f 1x 5X x9 fquot 1gtlt Graph the function and its inverse using the same set of axes Use any method X2 9Xgt0 log5 X E f 1 X 2 x39 9 C Not one to one D f 1x Q9 Graph the function 17 f x eX 6 4 y Iy s 1 s d s s s s l s s x x s s a x s s s x a s s s s x y s s 1 x s s x a s 5d s s 5 s 1 s n 4 1 s s x s s e 4 s x s s s I s I s s x l J l l l l l l J J I J J 39 I I I I I I I I I I quot I I I quot 5 L4 2 quot 1 2 3 5X 3 5X x 1 4 4 x s l s s s s l u x s s s s 5 l s I s s u a 5 1 5 I x x 1 1 x y s 4 u s s s s 1 s x x 4 x s I s 1 s 1 u s I s 4 s s a x u s ANS v 39 V D 00 00 Shlfted down 6 umts R 6 oo 18 4 quotX y e s u y s x i d V u s s s 1 s s s 5 d 5 s x 4 x s s x s x s 104 x x s s s s 1 x s s x x 1 s s s x 1 x x s 5 5 Lb u n A s u s I x s s I 1 l l J l J l l 1 l l l J l J l l J l l l l l l J J J J l J l 39 I I I I I I I I I I I I I I I I 139 39 I I I I I I I I I I I I I I I I l39 d I 2 4 6 8 X 4 2 2 4 6 8 X s s 1 s s s s u x u s s x a s s s s x s s d x s s s s s x s IO 4 s x s s u IO x s x x 1 s s s s x s s 1 s s s s s s s s s s s s I s s x s s all i S i s 20 1 s x 20 s s s s v D 00 00 Re ected about the y ax1s I R 4 00 Shlfted up 4 umts X 19 fx 2 e s s s s y s s s s s 104 s x s x a y s I s s s x u s s 1 5 s I s 1 s s s 51s s s x s u l s s x I s s s s s 4 x s s x s s l s x s s ANS Re ected about the y axis Re ected about the x axis Shifted up 2 units Graph the function Describe its position relative to the graph of the given equation 20 f x 2gtlt2 to fx 2X ANS Moved left 2 unit 5 I re ected across x axis horizontal asymptote y 0 ANS Moved left 1 unit 5 R 00 4 Re ect about xaxis moved down 4 units 22 fx e X5tofx eX E A Re ected over the y axis moved right 5 units 1 r u r I go I I l I I I I I I I H O I I I I I I I I I I I I I Match the function with one of the graphs 23 fx 3gtlt 31 A X U1 h gtlt Use a grapher to find the points of intersection of the graphs of the following pairs of equations 24 y 5X y 51X A 15 4 25 y3X eX2ye2X X A 0776 2172 C 0 1 0325 1592 Use a grapher Solve graphically 26 S X lt l A 00 0 Find the following 1 27 log5 3 55 A5 28 log32 2 2 A5 B oo 0 B 5 B5 B 5ty I I I I J I I I I 5 quot 5 T T E 5I v C None B 1955 1975 0466 2075 D None C 0 co D 0 oo C 2 D2 1 C D 5 5 29 log 1000 A 3 B 10 C 3 D 1 Convert to a logarithmic equation 30 63 216 Alog216 6 3 39 B log6 3 216 C log3 216 6 D 10556 216 3 1 31 43 64 A log3 1 4 B log4 3 1 C log4 1 3 D log164 4 3 64 64 64 2 32 e 01353 A 01353 log2 e B e log 2 01353 C 01353 2 loge 2 D 2 loge 01353 33 yZ 8 Azlog9y Bzlogy8 Cylog92 Dy logZS 34 et 3000 a Ans ln 3000 t b Solve for t t In 3000 35 100e2t 2000 hint divide by 100 rst a Ans ln 20 2 2t b Solve for t t lnzzo Convert to an exponential equation 36 logg 1 0 A081 B810 C801 D108 37 log5 TE 3 A 13 Z 5 B 35 z 1 C 5 3 1 D 5125 3 125 125 125 38 In 39 36636 A e36636 1 B e36636 39 C e39 36636 D e36636 2 In 39 39 6 logt180 A t6 180 B 1180 6 C 6t 130 D 180t 6 40 ln 45 12 A 12e 45 B ea 125 C el2 45 D e4 125 Find the following using a grapher Round to four decimal places 41 In 56 A 17482 B 02477 C 206642 D 40254 42 In 0 A 04874 B 05231 C 04985 D Does not exist 43 log 062 A 0478 B 0022 C 07076 D 02076 44 ln 1156 A Does not exist B 1056 C 24476 D 1256 Find the logarithm using the change of base formula 45 logzg 7836 A 18941 B 27986 C 07640 D 13088 46 log6 1 53 A 10843 B 07243 C 08689 D 09223 Solve the problem 47 The number of bacteria growing in an incubation culture increases With time according to B 5800 5XWhere X is time in days a Find the number of bacteria at intial time hint X 0 ANS 5800 b Find the number of bacteria when x 3 ANS 725000 48 The value of a stock is given by the function Where V is the value of the stock after time t in months Vt 651 e14t 35 a What is the value of the stock after 2 months ANS 9600 1mg In 32 In 65 14 14 b When will value of the stock be 68 ANS z 506 months leave answer in terms of natural logs Graph the function and its inverse using the same set of axes Use any method 49 fx log3 X f 1X 3X f of J 5 L J 5 V Note notice the inverses have symmetry about dashed line yx Above graphs DO NOT include asymptotes Graph the function Describe its position relative to the graph of the given equation 50 f x log2x 2 to f x logzx 10 ANS D 2 00 Moved left 2 units R 00 00 Vertical Asymptote x 2 51 fx lnx 2tofx lnx a ANS D 2 00 Moved right 2 units R 00 00 Vertical Asymptote x 2 ll t l ltI t I 52 fxlnx 2tofxlnx Moved right 2 units Vertical Asymptote X 0 Illtlrlllt D 00 0 Re ected about the y axis R39 00 00 Vertical Asymptote x 0 Re ected across x axis moved up 6 units Vertical Asymptote x 0 Solve 55 The population growth of an animal species is described by Ft 400 log 2t 3 where t is measured in months a Find the population of this species 6 months ANS 470 b When will the population reach 800 ANS 485 months 56 The population growth of an animal species is described by Ft 400 90 log3 2t 1 where t is measured in months Find the population of this species in an area 13 months after the species is introduced A 345 B 2830 C 670 D 1435 57 The height in meters of girls of a certain tribe is approximated by h 052 2 log t 3 where t is the girl39s age in years and 1 s t s 20 Estimate the height to the nearest hundredth of a meter of a girl of the tribe 4 years of age A 096 m B 052 m C 077 m D 112 m ExPress as a sum of logarithms 58 log4 6439 256 Alog644log256 4 Clog43log44 59 logd n A logdd logd6 logdn C loge6n 1ogd6h ExPress as a product 60 logj t5 i A logj 5t 2 Blogj5logjt 61 log 3 K 4 Alogc KlogC 4 B 416ch Express as a difference of logarithms M 62 log 39 g 30 A log M log 30 B log M log 30 2g 2s g g 63 loga A loga C logp B loga Cloga P EXpress in terms of sums and differences of logarithms 64 loga3x4yz5 A 201oga3xyz C loga3 4logax 1ogay Slogaz gig S Anogm 9 10g10 s 65 log10 C log10 9 3 log10 r log10 s 6 9 m 66 16 L gb n8134 A logbm6 logbp9 logbn8 logbb4 C m6P9 n8b4 X5 3 67 10gb y729 A 3710ng 10gby Blogbz C Elglogbx5 logby7 logbz9 C loga C loga P Blog464log42561og44 D log4 64 log4 256 B log6d lognd D loge6 logdn Clog 5 D510gjt ClogCKlogCK DKlogC 4 c1 301 M D1 30 1 M 080 Ugo Osg egg D logp 10ga C B loga3 4logax logay Slogaz B 10810 s 10610 9 51510610 I D 1og10 9 log10 r log10 s B 610gbm 910ng 8logbn 4 D 6logbm 910ng Slogbn 4 B 310ng glogby Blogbz D Slogbx 7logby 9logbz 4 x5b6 68 lo gm y7zl6 Alogbx 6 710ng 16logbz 39 5 3 7 o QZlogbx T2 Zlogby 4logbz 6b7 a 69 1 0g a2b4 A 4logba 3 B logba4c logbB Express as a single logarithm and simplify if possible 70 ln 42 1n 6 Aln B ln36 J 71 logax 4 loga y 2 loga x Agtlogax4y4 B10ga Y4 72 In 17x 2ln x ln y A In quot34xy B 34 In xy 73 ln x2 81 ln x 9 Alnx9 Blnx 9 74 In W 32ln W 5 In W 5 2 A 1n M 3 1n W 52 w 53 Simplify 75 ln e7 A 1 B 7 ln e i 76 410554 3x 39 A 3x B 43X 77 1010st A lot B t Answer the question 78 Given fx 2X evaluate the following a flog2 6 ANS a 6 b flog 2 7 ln 2 ANS b 7 ln 2 ANS 2 ln 2 c fllogz In 2 B i logbx5 logb b6 logby7 logb216 5 3 7 D 74 10ng E Zlogby 4logbz C Zlogba 3 D Blogba 3 C ln D ln7 C loga x2 Y4 D logaX3 2 17x3 17x2 C 1n yz D In yz Cln x81 D1nx2 9 39 2 1 WW53 Dil WW5 C n W 56 W 5 C 7 p D e7 C 1 D 4 C 10 D 1 79 Given fx In x evaluate the following a fez ANS a 2 b fe1n 2 ANS b ln 2 39 c fe3 1n 2 ANS C 31n 2 or ln 8 DOIVE we pro mem 124 Answer A 114 Answer D 56 Answer 80 Gwen that logb A 4 and 10gb B 3 nd 10gb AB 57 AnsWer C Jan 12 B 7 C D 12 58 Answer D 39 59 Answer D 81 leen that 10gb A 3470 and logblB 0385 find 10gb 60 Answer D A3855 B3470 13085 D1r337 61 Answer B 62 Answer B 82 Given that log 10 2 03010 and log 10 3 04771 nd 10510 24 63 Answer C A 05044 B 13301 C 04303 1317323 64 Answer B 1 65 Answer D 83 Given that log b 5 12301 and log b 7 14873 find log39b 5b 66 Answer D A 27174 132301 011757 b 1312301 67 Answer B 4 68 Answer C Q 4 Solve the exPonential eguation algebraically Then check using a grapher 69 Answer C 23 84 16X163 70 Answer D E a A 3 EMS 23 D 16 1 71 Answer D lt lt 72 Answer C all Cc9 35 410 2X 4096 39 1 quot A 2 73 Answer B C m5 39 C 1024 m2 74 Answer C lt3 lt3 86 7x5 I E 75 Answer C 1H5 39 1117 76 Answer A g 5 A In 5 1n 7 B a ln 7 C 1n 2001 5 Dquot 1n5 77 Answer 13 i i a 39 Answer 243915439951395 78 Answer a Solve the equation If necessary round to thousandths 106 Answer A 79 Answer 37 53X1 15 107 Answer C 30 Answer39 C A 0700 B 0228 C 0394 D 1333 81 Answer C a I 82 Answer B Solve the exponential equation algebraically Then check using a grapher 83 Answer B 88 52x 572 A 36 B 18 33 6 D5 84 Answer C 39 39 39 85 Answer D 89 27 35X 9x2 i 86 Answer 39B 11 1 3 87 Answer C A3 B 3 C3339 D3 3 2 2 1 Answer x 125 A s 2 Answer Answers may vary One possible solution is 4 4 3 Answer D I nswer B 1 1 44 Answer A 5 126 Answer c A I 127 Answer 1 f l o fx f 1fx f11 XjX 1 x 1 1 x x 1 X 45 Answer D 1 3 x X 46 Answer D 28 Answer C it 39 1 1 52 Answer Moved down 2 units 1 X quot 1 X 1 1 2f0f 1xff 1x A lt ltgt ELM 11 1 1 x 88 Answerzc 12 Answer X X 1 30 Answer D I 89 Answer B 3 AnsWer B 14 Answer B 22AnSWer1 A 31 Answerva 90 Answer C 4 Answer B 4 15 Answer C 23 Answer D 32 Answer D 91 Answer D 5 Answer A 24 Answer A H 33 Answer B 92 Answer D 6 Answer No I 25 Answer B 36 Answer C 93 Answer 7 Answer39Yes J 26 Answer D 37 Answer C 94 Answer A 8 Answer A A 5 27 Answer C 38 Answer B 95 Answer 9 Answer A c a 28 AHSWEI I C 39 Answer A 96 Answer 10 Answer D we M 29 AnSWerl A I 40 Answer C 97 Answer C 41 AWw A J 1 IL I AnSWerl D 98 Answer A Solve the equation it necessary round to thousandths 90 e9X e3X e6 A 0595 B 1640 C 0500 D 0640 Solve the exPonential equation algebraically Then check using a grapher 2 1 x 5x 91 3 81 A 14 B1 4 C14 D 1 4 92 27X 105X A 27 B 5 C 135 D 0 X ln116 ln1t ln16 ln16 x2 x 934 3 ln43 ln4 ln3 ln4ln3 Solve the equation If necessary round to thousandths 94 5X 8 92x 034 A 4891 13 4891 C 9783 D 2020 Solve the exponential equation algebraically Then check using a grapher 004t M 95 e 02 ANS t 004 96 1000e4t 90 39 ANS t Em Solve the equation If necessary round to thousandths 97 3e9gtlt5 6 A 1191 B 2303 C 0479 39 D 2149 Solve the eXponential equation algebraically Then check using a grapher 98 350 175X 0 A 104678 B 117018 C 114801 D 92338 Solve the logarithm equation algebraically Then check using a grapher 99 log x 8 ANS 108 100000000 A 6 39 102 10g53 7x 4 ANS 100 In x 17 ANS e17 5 39 241549515 531430 103 108 x 3 1 416g x ANS 5 101 log911 3x 6 ANS 3 104 log9 x 2 logg x 2 1 ANS 5 105 ln6x 1 ln 3 In x 3 ANS 196 Use only a grapher Find the approximate solutions of the equation 106 e33X 406535 A 042500 B 014994 C 123192 D No solution 107 log42x 5 log4x 2 1 A 2408 B 3125 C 65 D No solution Use only a grapher Find the approximate points of intersection of the pair of equations 108 23x 38y 114 y 31 lnx 205 ANS 34041 9396 109 y 1n 3x y 3x 8 ANS 3445 2336 Solve for the indicated variable 110 P P P P 10 kt fort ANS39 t 1 lo P PO 0 1 O 39 quot k g P1PO 111 vevoekt 1000 fort ANS t 1000 16 3 7 k vO Solve 112 In 1999 the population of a city was 1885162 The exponential growth rate was 15 per year39Find the exponential growth function A Pt 1885162 620051 B 131 z 1885162 605 r C Pt 1885162 e5t D Pt 1885162 e6t 113 How long will it take for the population of a certain country to double if its annual growth rate is 2 Round to the nearest year A 35 yr W 13 100 yr C 15 yr D 1 yr Solve the problem 114 Suppose the consumption of electricity grows at 53 per year compounded continuously Find the number of years before the use of electricity has tripled Round the answer to the nearest hundreth A 207 years B 021 years C 5660 years D 2073 years Solve 115 The number of books in a small library increases according to the function 130 600 4000 004t where t is measxi red in years 39 I E a How many books did the library start off with ANS 4000 books b How many books will the library have after 9 years ANS 6033 books c Estimate how many years it will take to reach 6300 years ANS 113 years The next four questions deal with the interest compounded continuously At I ert Where P gt principal r gt interest rate tgttime and A t gt Amount in account after t years 4 variables 116 Suppose that 4000 is invested at an interest rate of 53 per year compounded continuously What is the balance after 3 years Hint Solving for At ANS468935 117 Randy invested his inheritance in an account that paid 67 interest Assuming compounding continuously after 6 years he found that he now had 5527364 What was the original amount of his inheritance Hint Solving for P ANS 36977 118 Kimberly invested 4000 in her savings account for 7 years When she withdrew it she now had 476498 What was the interest rate on the account Hint Solve for r ANS 25 119 How long will it take for 2800 to grow to 30000 at an interest rate of 15 if the interest is compounded continuously Round the number of years to the nearest hundredth Hint Solve for t ANS 15811 yr 120 The number of farms in a certain state has declined continually since 1950 In 1950 t0 there were 92120 farms and in 1995 t 45 that number had decreased to 27339 Assuming the number of farms decreased according to the exponential model find the value of k and write an exponential function that describes the number of farms after time t in years where t is the number of years since 1950 a Assuming the number of farms decreased according to the exponential model find the value of k Hint Solve for k ANS k 0027 b write an exponential function that describes the number of farms after time t in years where t is the number of years since 1950 ANS Pt 92120 e0027 t c Using from part b predict the number of farms in 2005 t55 ANS P55 79407 d Using the model from part b about what year will the number of farms reach 80 486 leave answer in terms of natural logs 1 5749 6580 In 5749 ln 6580 v I 121 In 1985 t0 the number of female athletes participating in Summer Olympic Type Games was 550 In 1996 t11 about 3650 participated in the Summer Olympics in Atlanta a Assuming that exponential model applies find the value of k rounded to the hundredths place ANS k 2 0172 39 b Write the function ANS Pt 550e0172t c Using the function from part b nd the number female athletes that will particpate in the 2006 t21 ANS P21 20372 39 1 Using the function from part b about what year will the number of females particpating be 61111 people I leave answer in terms of natural logs 6111 In 550 1 6111 1 550 II I1 ANS 172 172 14 gt in year 1999 A 122 x4 7 8x2 hint let u x2 A 11 x7171 Bii x7 7 C ii x71 71 D 11 WW 123 3m 22 63m 2 5 0 hint let u 3m 2 7 3 1 7 1 c 1 13 1 124 x24 12x1411390 hintux14 A 1 14641 E 114641 C 1 14641 D 1 1114 125 6623 26613 24 hint Let u 6613 A 64 B 64 216 C 64 216 D 4096 46656 126 IQ 12 x hint u A 16 9 B quotbx1239 39 C 9 D 7 127 eX 6e lt 1 hint 1 multiply everything ex ANS x ln3 2 let u eX 128 log3 x2 10g szi Hint 1 Simplify using log properities 2 Let u 2 log3 x A 1 3 B 9 e 1 9 D 0 3 EXTRA QUESTIONS TO REVIEW FOR FINAL UPDATED MARCH 5 2006 1 Use the graph to answer the question State the solution of the inequality in interval notationi a lt 0 b g 0 2 Starbucks found that if they can predict the pro ts of the certain store by the using number of X cups of co eei Given the following function and the graph of the function belowi Px 39 52 4 3 Find the interval when Starbucks has PROFIT GAIN Hint Px gt 0 b Find the interval when Starbucks has BREAK EVEN Hint Px 0 x mtercepts 30 70 3 Starbucks discovered that it can estimate the store s daily costs based on the number of cups of expresso sold that day7 where x is the number of cups and C35 is the pro t The store derived the following equation7 Cx 153422 24x 3 What is the daily costs for if one store if it sold 12 cups of expresso b What is the number of cups of expresso sold if costs were 3915395000 Assuming a lmeal xelatlonshlp between the numbel ol ploducts sold and dam pxo t ol opezation Each day you must pay 1 on to opelam the oompany At the end ol each day you nds that ox each pxoduct sold pzo ts muease an addltonal 2 on a pzo t ol opezation b an a certain day the dalbl pro t olopexatlon was 7 on Fmd the Killme ofpxoducts sold that day c llyou sold 2 ploducts on a celtam day what was the dalbl pzo t7 Use the gaph to answer each quesn39on belcrw a Staue the muemal m which the gaph c State what interval satis es 2 g o Usmg the gaphlng calculatox GRAPH z e m Az e 2 Round to A decimals plaos a Determine the re 39ve maximum and minimum b Detemnne whale the function is decreasing 4221524n u a M Y Y 2 v hint nd 2 gt o 922 4 z a hint nd 2 lt 0 John owns a oo ee shop He has lound that his revenue is xepzesenued by the equation Rc 722 Ac a with Rh being the zevenue lzomsales and a the numbez ol cups ol oo ee sold a How many cups ol oo ee must he sell to earn the maxlmwe xevel39lue7 b What is the maximumzevsmue7 10 For the following function7 1 given by x 72m2 7 z 7 4 nd the difference quotient Answer each of the following to help get answer SIMPLIFY after STEP 3 Step 1 fz h Step 2 fz h 7 x fltzh 7m H H There is linear relationship between the speed at which your car is traveling and the braking distance7 distance from when your put foot on the brake to when you come to a complete stop A car traveling at 35mph had a braking distance of 6ft Traveling at 75mph had a braking distance of 12ft a Write a linear equation which represents the problem above b ls the speed of the car was 50mph7 would would be the breaking distance c If the braking distance of a certain car was 845 ft How fast was the car traveling H to Solve the following equations a lx75l2 b lgz74l8 c l6zl724 13 Solve the following inequalities State your answer in interval notation a 7397 b l z2llt1 Cl71l25 14 Solve each inequality State you answer in interval notation a 72mx73m24gt0 b m427x 0 15 Given the following graph of the function what is the solution of the inequality below the graph in interval notation given x efxesliz e n rmlelcepts 10 I 16 When work is necessary show it below with all steps It is not required but if needed you may use the graph provided to graph the function If the information requested does not exist please write none in the answer Given fz 72y e in 710 nd the following a vertex e domain h Minimum value b eeinteroepts f range l Increasing o yrlntercep s g Maximum value Decreasing d Axis of symmetry

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