6. Solve by applying the quadratic formula.

Transcription

1 Dae: Chaper 7 Prerequisie Skills BLM 7.. Apply he Eponen Laws. Simplify. Idenify he eponen law ha you used. a) ( c) ( c) ( c) ( y)( y ) c) ( m)( n ). Simplify. Idenify he eponen law ha you used. 8 w a) w ( p)( pq) 8q y c) y. Simplify. Idenify he eponen law ha you used. a) ( ) ( mn) c) ( p q ). Simplify. Idenify he eponen laws ha you used. ( m )( mn ) a) ( mn) ( ) ( y z) y Solve Quadraic Equaions. Solve by facoring. a) =0 6m + m= c) p + p+ 9= 0 d) r + = 9r e) 6y + 9y+ 0= 0 6. Solve by applying he quadraic formula. a) = 0 v = v+ c) d = d d) 6m + m 8= 0 e) a + 7= a Simplify a Radical Epression 7. Simplify. a) c) 6 6+ d) Apply he Power Law of Logarihms 8. Evaluae, using he power law of logarihms. a) log6 6 log c) log 6 d) log Evaluae, correc o hree decimal places. a) log log8 c) log 0 d) log 0. Solve for, correc o wo decimal places. a) = =. c) = 7 d) = 6 Advanced Funcions : Teacher s Resource BLM 7 Prerequisie Skills Copyrigh 008 McGraw-Hill Ryerson Limied

2 Dae: 7. Equivalen Forms of Eponenial Equaions BLM 7... Wrie each epression as a power of he base indicaed. a) 6, base, base c), base. Wrie each epression as a single power of. a) Solve. Check your answers by using graphing echnology. + a) 7 = 9 + = c) 6 = 6. Consider he equaion =. a) Solve his equaion by epressing boh sides as powers of a common base. Solve he same equaion by aking he logarihm, base, of each side.. Solve ( ) =. Check your answer using graphing echnology. 6. a) Solve. Give eac answers. i) = 0 ii) = 0 iii) 7= 0 Use your answers o par a) o sae a formula ha could be used o solve b = 0 for a) Solve 6 = 6 by epressing boh sides of he equaion as powers of. + Solve 6 = 6 by epressing boh sides of he equaion as powers of. + c) Solve 6 = 6 by using graphing echnology. d) Which of he mehods is bes? Eplain. 8. a) Using your knowledge of he base graphs and ransformaions, skech graphs of y = and y = ( ) on he same aes. Use your graphs o esimae he soluion o ( ). c) Use Technology Solve ( ), correc o wo decimal places, by using graphing echnology. 9. Consider he funcions f ( ) =, g( ) =, and h ( ) =. a) Esimae he insananeous rae of change for he hree funcions when =. Repea par a) for = 0. c) Discuss which ype of funcion linear, polynomial, or eponenial has he greaes rae of change. Advanced Funcions : Teacher s Resource BLM 7 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

3 Dae: 7. Techniques for Solving Eponenial Equaions BLM 7... Solve for. Round answers o wo decimal places. a) = (.) 7 = (.06) + c) = 8 d) 6. =.. The ampliude of a vibraion decays according o he equaion ( ) 60 a =. 0.6, where a is he size of he ampliude, in cenimeres, a ime, in seconds. a) Wha will he ampliude be afer 0 s, o he neares enh of a cenimere? How long (o he neares second) will i ake he ampliude o decay o 0% of is original value? c) Would i ake wice as long as your answer for par for he ampliude o decay o % of is original value? Eplain.. Solve each equaion. Leave answers in eac form. a) = 7 =. Consider he equaion () + = 0. a) Wrie he equaion in he form az + bz + c = 0, wh ere z =. Solve he equaion using he quadraic formula. c) Idenify any eraneous roos.. A -mg sample of radioacive francium decays o 0 mg in.6 min. a) Deermine he half-life of francium. Graph he amoun of francium remaining over a -h period. c) On he same se of aes as par, skech he graph of he amoun of an elemen if i) i has he same half-life as francium, bu he original amoun was 0 mg ii) i has he same original mass as he francium sample, bu is half-life is wice as long 6. Solve and check for eraneous roos. Leave answers in eac form. a) 6() 7= 0 = + () c) + = () 7. The populaion of a colony of baceria grows according o he formula P ( ) = (.0), where P is he populaion, in housands, and is he ime, in hours. a) How long does i ake he populaion o reach 0 000, o he neares hour? Calculae he ime i akes for he populaion o double, o he neares hour. 8. The maimum heigh ha a ball reaches afer bounce number n is given by he equaion H =.0(0.90) n, where H is he heigh, in meres. a) Wha is he ball s maimum heigh afer he fifh bounce? Wha is he firs bounce afer which he maimum heigh is less han 0 cm? 9. + Solve + =. 0. Rewrie he equaion P ( ) = (.0) wih base.0 repla ced wih.. The general equaion for populaion R growh is P () = P , w here R is he growh rae, in percen, over ime period 0. Suppose a populaion grew from o 000 in si years. If ime is measured in years, calculae a) he yearly growh rae he growh rae per decade (0 years) 0 Advanced Funcions : Teacher s Resource BLM 7 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

4 Dae: 7. Produc and Quoien Laws of Logarihms BLM 7... Simplify, using he laws of logarihms. Then evaluae, correc o hree decimal places. a) log99 log9 log0 + log. Simplify each algebraic epression. Sae any resricions on he variables. a) log + log ( y) log ( z) l og a logb+ log c. Evaluae, using he laws of logarihms. a) log log6 log6 + log66 + log6 c) log+ loglog8. Wrie as a sum or difference of logarihms. Simplify, if possible. a) log y z log m n. Simplify. Sae any resricions on he variables. a) log( m ) log( m ) + log m log ( ) ( ) ( ) p + log p + log 6 p c) log( 6) log( ) d) log(6 + 6) + log() log( 9) 7. Jorge is offered wo differen jobs. One pays a saring salary of $0 000 per year, wih a guaraneed raise of $000 every year. The oher pays a saring salary of $ 000 wih a guaraneed raise of % per year. a) How long would Jorge have o work before he wo jobs paid he same salary? If salary is he only facor in Jorge s decision, which job offer would be beer? Eplain. c) Would your answer o par change if each company had a pension plan ha paid he average of he bes five years of income as a pension, provided he worker spen 0 years wih he company? Eplain. 8. Use he laws of logarihms o wrie y as a funcion of for each of he following. Then, sae he domain of he funcion. a) log( y) = log( ) log( y) + = log( y+ ) + log( ) c) log = log( + ) y 9. Prove ha + =. log 0 log y0 log y0 log m (Hin: recall ha logn m = ) log n 6. Use Technology a) Graph he funcion f ( ) = log. Graph he funcion g( ) = f( ). c) Graph he funcion h ( ) = f( ). d) How are he funcions g( ) and h ( ) relaed? Wha law of logarihms does his illusrae? Advanced Funcions : Teacher s Resource BLM 7 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

5 Dae: 7. Techniques for Solving Logarihmic Equaions BLM Solve algebraically. a) log( + ) = log( p 8) = 0 c) log ( m + ) =. Use Technology Solve using graphing echnology. a) log( ) = log( v + ) = 0. Solve. Idenify and rejec any eraneous roos. a) log ( ) + log ( ) = log + = log(0 ) c) logm+ log m= 0. Solve. Check for eraneous roos. a) log( + ) = log + 9 =. Solve. a) log + log ( ) = l og( + ) = log( ) 6. Use Technology Find he roos of each equaion, correc o wo decimal places, using graphing echnology. Skech he graphical soluion. a) log = + log( ) log + = log ( ) ( ) 7. The number of years, n, required for an invesmen P o grow o amoun P when ineres is i% per year (epressed as a decimal) is given by he formula log P log P n =. log( + i) a) How many years does i ake for $00 o grow o $80 a a yearly ineres rae of 8%? How much should be invesed, o he neares dollar, in order for i o grow o be $7 in 8 years a a yearly rae of %? c) Wha yearly ineres rae is needed o allow $900 o grow o $689 in 0 years, o he neares enh of a percen? 8. Solve log + log =. 9. a) Calculae ( log)( log8)( log80 ). Compare your answer from par a) o log 0. c) Prove ha log b log c log d = log d. ( )( )( ) a b c a Advanced Funcions : Teacher s Resource BLM 7 6 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

6 Dae: 7. Making Connecions: Mahemaical Modelling Wih BLM Eponenial and Logarihmic Equaions (page ). Refer o he wo eponenial models developed in Eample of he e for he populaion of Decimal Poin since 90. P = 006(.06). P = 000 Decimal Poin will require an upgrade o is waer infrasrucure once he populaion reaches The own mus sar o plan for he cos years before he work mus be done. When should he own sar is financial plan?. Refer o quesion. The populaion of Decimal Poin could be modelled by using an eponenial equaion based on is ripling ime (he ime i akes he populaion o riple). T The equaion would be P = 000, where T is he ripling ime. a) Calculae he value of T: i) Use one of he eising equaions developed in Eample (subsiue 000 for P and solve for ). ii) Use Technology Creae a dynamic model wih a single slider for T and adjus he value of T unil he curve of bes fi is obained. Compare he values of T obained in par a). If here is a difference, eplain why.. Decimal Poin needs $ o begin building a recreaion cenre. Two invesmen opion for a surplus of $0 000 were eplored in Eample of he e. Invesmen Opion Years Before Work Can Begin Lakeland Savings Bond 7.96 Norhern Equiy Muual Fund 7.9 Suppose ha a financial company offered Decimal Poin anoher invesmen opion an ineres rae of.% compounded quarerly, wih he condiion ha should Decimal Poin leave he money invesed for longer han 7 years, Decimal Poin would be paid a bonus of 6% of he original invesmen when he reserve fund is wihdrawn. Should his opion be considered? Jusify your reasoning.. Use Technology A ub of warm waer is lef ouside on a mild winer day. Is emperaure is recorded every min, as shown in he able. Time (min) Temperaure ( C) a) Creae a scaer plo of emperaure versus ime. Does he curve appear linear, quadraic, or eponenial? Jusify your answer. Perform linear, quadraic, and eponenial regression on he daa. Record he equaions, correc o wo decimal places. c) Can any of he models be discouned immediaely? Eplain. Advanced Funcions : Teacher s Resource BLM 7 8 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

7 Dae: BLM (page ) d) Use boh he quadraic model and he eponenial model o calculae he emperaure of he ub of waer afer min. Eplain how his calculaion helps you decide which model is beer. e) Use he model you consider o be he bes o calculae he oudoor emperaure. (Hin: afer a very long ime, he ub of waer will be he same emperaure as he oudoors.) D. The acual equaion ha models heaing or cooling of maerial is T = ( T0 TA)() + TA, where T is he emperaure a ime, T 0 is he iniial emperaure of he maerial, T A is he ambien emperaure (he emperaure of he surroundings), and D is he ime i akes for he difference beween he emperaure of he maerial and he ambien emperaure o be halved. 8 An objec placed in an oven has emperaure modelled by T = 7() + 0. a) How long does i ake he objec s emperaure o reach C? Wha was he original emperaure of he objec? 6. Use Technology Refer o quesion. The able shows he emperaure of maerial as a funcion of ime. Time (min) Temperaure ( C) a) Make a scaer plo of he daa. Esimae he ambien emperaure from your plo. You now know he values of T 0 and T A in D T = ( T0 TA)() + TA. c) Use he esimae for he ambien emperaure o ransform he daa so ha i has an asympoe of 0 insead of he ambien emperaure. Creae a new scaer plo of his ransformed daa. d) Perform an eponenial regression on he ransformed daa. This equaion will be of he form y = k( D. Use his equaion o calculae he value of D in T = ( T0 T )() + T. D D (Hin: remember ha =, so he base in your regression will equal e) Wrie an equaion ha models he given emperaure daa. A D.) A Advanced Funcions : Teacher s Resource BLM 7 8 Secion 7. Pracice Copyrigh 008 McGraw-Hill Ryerson Limied

8 Dae: Chaper 7 Review BLM (page ) 7. Equivalen Forms of Eponenial Equaions. Wrie each as a power of 8. a) 6 c). Wrie each as a power of 8, correc o hree decimal places. a) 0.. Solve. a) = + 7 = 9 7. Techniques for Solving Eponenial Equaions. Solve eacly. = 7. Solve. Round answers o wo decimal places. p+ p = 6. A 0-mg sample of a radioacive isoope decays o 7 mg in. h. a) Calculae is half-life, o wo decimal places. How long will i ake (o he neares hour) unil only mg of he sample remain? 7. Solve. Check for eraneous roos. a) () + =0 = + () 8. The growh of an an hill populaion is D modelled by he equaion P = 00(), where P is he populaion a any ime, in monhs, and D is he amoun of ime needed for he populaion o double, in monhs. If he populaion of he an hill is 00 afer monhs, a) calculae he doubling ime, o wo decimal places deermine how long i will ake for he populaion o reach Produc and Quoien Laws of Logarihms 9. Evaluae, using he laws of logarihms. a) log log log6+ log log 0. Wrie log + log ( ) as a single logarihm.. Wrie log ab c as sums and differences of logarihms. Simplify, if possible.. Simplify and sae resricions necessary on he variable. a) log( ) log(8) log( 7) log( ) 7. Techniques for Solving Logarihmic Equaions. Solve. Check for eraneous roos. a) log ( + 7) = log + = log + ( ) ( ) Advanced Funcions : Teacher s Resource BLM 7 0 Chaper 7 Review Copyrigh 008 McGraw-Hill Ryerson Limied

9 Dae:. Solve log ( ) + log ( + ) =. Round your answers o wo decimal places. Check for eraneous roos.. The ime i akes an oven o prehea is log( 0.T ) given by he formula =, log where is he ime, in minues, and T is he emperaure, in degrees Celsius, a which he oven is se a. a) How long will i ake o prehea an oven o 0 C? If i akes 6 min o prehea he oven, wha was he emperaure seing for he oven? BLM (page ) 7. Making Connecions: Mahemaical Modelling Wih Eponenial and Logarihmic Equaions 6. The populaion of a species of animal in a naure reserve grows by.% each year. Iniially, here are 00 of ha species. a) Wrie an equaion for he populaion of he species as a funcion of ime, in years. Wha will he populaion be afer 0 years? c) How long does i ake he populaion o double? d) Afer 0 years, an epidemic kills all bu 00 of he species. Afer he epidemic, he populaion grows as i did before. Wha will be he equaion modelling he populaion afer he epidemic? e) Skech he graph of he populaion for 0 0. Advanced Funcions : Teacher s Resource BLM 7 0 Chaper 7 Review Copyrigh 008 McGraw-Hill Ryerson Limied

10 Dae: Chaper 7 Tes BLM 7... Wrie as a single logarihm, hen evaluae. log 0 + log 7 log a) log 6 log 6. Solve. a) 9 = 7 k 6 k + = log = log c) ( ) d) log + log = log ( ). Consider he funcions f( ) = log( + ) + and g ( ) = log( ). a) Use your knowledge of he graph of y = log and ransformaions o skech he graphs of y = f( ) and y = g( ). Esimae he poin of inersecion. c) Check your answer by solving log( + ) + = log( ) algebraically.. A radioacive subsance decays from mg o mg in 8.6 min. Calculae he ime required for 90% of he subsance o be decayed. 6. The following daa were gahered during an eperimen. y a) Creae a scaer plo of he daa. Could he daa be fi by drawing an eponenial curve (or a ransformaion of one)? Eplain why or why no. c) Could he daa be fi by drawing a quadraic curve (or a ransformaion of one)? Eplain why or why no. d) Describe an eperimen from which hese daa migh have originaed from ha would be bes fi by i) a quadraic curve ii) an eponenial curve. A car is purchased for $ 000. Three years laer, i is valued a $ 000. a) Calculae he amoun of ime (o wo decimal places) ha i akes for he car s value o be reduced o one-half is original value. How long will i ake for he car s value o be reduced o $8000? Advanced Funcions : Teacher s Resource BLM 7 Chaper 7 Tes Copyrigh 008 McGraw-Hill Ryerson Limied

11 Chaper 7 Pracice Masers Answers BLM 7.. (page ) Prerequisie Skills. a) c y c) 0mn. a) w. a) n. a) m p q c) 8 mn c) z. a) = or = m = or m = c) p = y 6 7 p q. = a) i) = lo g ii) = lo g iii) = log7 = logb 7. a) = = c) = d) Answers may vary. 8. a) Window variables [, 0], y [, 0] d) r = or r = e) y = or y = 6. a) = ± c) d = ± 7 8 6± e) y = v = ± 6 d) m = ± 7. a) 9 c) + d) + 8. a) 8 c) d) 9. a). 0.8 c). d) a) c).0 d) Equivalen Forms of Eponenial Equaions 9. a) 0 c). a) log log. a) = 0 = 8 c) = 7 9 approimaely 0.7 c) Answers may vary. Sample answers: a),,., 000, 600 c) eponenial 7. Techniques for Solving Eponenial Equaions. a).0 0. c). d).7. a).6 cm 68 s c) No, 6 s log log. a) = log log log7 = log7+ log. a) z z+ = 0 log( ± ) = log c) no eraneous roos. a) approimaely. min. a) = = Advanced Funcions : Teacher s Resource BLM 7 Chaper 7 Pracice Masers Answers Copyrigh 008 McGraw-Hill Ryerson Limied

12 Chaper 7 Pracice Masers Answers BLM 7.. (page ) c) i) y = 0. ii) y = 8. c) 6. a) = = log log ± log c) = log 7. a) 6 h 9 h 8. a).8 m 9 9. = log. log 0. P ( ) = (). a) 6.% 60. % 7. Produc and Quoien Laws of Logarihms. a) log,.0 log 60,. y. a) log, 0, 0, z > y > z > log ac b, 0, 0, 0. a) c). a) log + log y log z log + logmlog n. a) log mm>, 0 log pp>, 0 6 c) log, > 6 d) log( ), > 6. a) d) same for > 0; power law 7. a) approimaely years The firs job offer pays beer unil year. c) Yes. If Jorge reires afer 0 years, he second job offer has he beer pension plan. ( ) 8. a) y =, {, > } y = 000, {, > 0, 000} c) y =, {, > 0} + ( ) 7. Techniques for Solving Logarihmic Equaions. a) = 9 p = 8+ 0 c) m =. a) = v = 99. a) = 6 = 0 c) m = 00. a) = = a) = = + 6. a). Advanced Funcions : Teacher s Resource BLM 7 Chaper 7 Pracice Masers Answers Copyrigh 008 McGraw-Hill Ryerson Limied

13 Chaper 7 Pracice Masers Answers BLM 7.. (page ) 0.00 and 8.6 c) 7. a) 6 $800 c) 6.% 8. = 9. a) approimaely. same 7. Making Connecions: Mahemaical Modelling Wih Eponenial and Logarihmic Equaions. 06. a) 68.9 years Answers may vary. Sample answer: There are differences due o judgmen of bes fi.. Yes; reaches $ in 7.90 years.. a) Answers may vary. Sample answer: eponenial T =.76+.8, T = , T = 8.87( 0.88) c) Linear; daa does no follow a sraigh line. d) Quadraic:.7 C; eponenial:.9 C; canno be quadraic, since i predics waer emperaure will increase. e) 0 C. a) approimaely 0 min C 6. a) d) y ; D e) T = () + 0 Chaper 7 Review. a) 8.. a) 8 8 c) log log8 8. a) = 0 = 6 9 log. = log log a) approimaely 8. h h 7. a) = 0 or = = log log 8. a).8 monhs 6.8 monhs 9. a) 0. log ( ). log a+ log b log c +. a) log, > 6 log + + 9, > ( ). a) =..9. a) approimaely min approimaely C 6. a) P = 00(.) 999 c) approimaely 6 years P 00. = d) ( ) 0 = 0 C Advanced Funcions : Teacher s Resource BLM 7 Chaper 7 Pracice Masers Answers Copyrigh 008 McGraw-Hill Ryerson Limied

14 Chaper 7 Pracice Masers Answers BLM 7.. (page ) e) Chaper 7 Tes. a) log6 6 ; log 9 ;. a) = = c) = d) =. a). c) 7+. a). years approimaely 6 years. approimaely 89 min 6. a) Yes; verical reflecion and ranslaion, approaching horizonal asympoe. c) Yes; vere (6, 0). d) i) he heigh of an objec hrown ino he air over ime (heigh reaches a maimum) ii) he warming of an objec over ime (up o 0 C) Advanced Funcions : Teacher s Resource BLM 7 Chaper 7 Pracice Masers Answers Copyrigh 008 McGraw-Hill Ryerson Limied