3.1 Exponential Functions and Their Graphs

Transcription

1 . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions. Eponentil Function: function of the form f b ;, b, b : initil vlue (-intercept); f b b: bse Evluting n Eponentil Function E: Determine the vlue for f 4. Note: A common mistke is multipling the nd 4. Remember the order of opertions (eponents before multipliction).. f b. f Grphs of Eponentil Functions E: Plot points nd grph the functions. Eponentil Growth Domin: Rnge: Intercepts: Asmptotes: End Behvior: Incresing/decresing Continuous? Eponentil Dec Domin: Rnge: Intercepts: Asmptotes: End Behvior: Incresing/decresing Continuous? Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

2 Grphs of Eponentil Functions: Grph is INCREASING when b & ; grph is DECREASING when b &. Trnsformtions of Eponentil Functions: h b k Verticl Stretch: Verticl Shrink: Reflection over -is: Reflection over -is: Horizontl Trnsltion: h Verticl Trnsltion: k Horizontl Asmptote: k Investigte the following trnsformtions: f( ) vs. f( ) 4 f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) E: Grph the eponentil function Trnsformtion of.. Rewrite s. Horizontl Trnsltion: Reflect: Verticl Trnsltion: -intercept: f Horizontl Asmptote: E4: Use clcultor to evlute the function f ( ) e when =- E5: Sketch the grph of the nturl eponentil function. f ( ) e.4 Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

3 Properties of Eponents: E. 6.) b.) c.) d.) e.) ( b ) f.) ( ) b g.) Eponentil Model: b Eponentil Growth: & b ; b is the growth fctor Eponentil Dec: & b ; b is the dec fctor Applictions of the Eponentil Model E7: CCSD s student popultion went from,4 in 956 to 9,5 students in 5. Write n eponentil function tht represents the student popultion. Predict the popultion in. Let represent the er 95 nd represent the number of students. Substitute the given vlues into the eponentil model b nd solve for nd b. r Formul for Compound Interest: A P n nt A = blnce r = nnul interest rte P = principl t = time in ers n = number of times interest is compounded ech er Note: Annull = time per er Seminnull = times per er Qurterl = 4 times per er Monthl = times per er Weekl = 5 times per er Interest Compounded Continuousl: A Pe rt A P E8: Clculte the blnce if $ is invested for ers t 6% compounded weekl. r n nt A? P t r n Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

4 The vlues of re pproching.78. This is the pproimte vlue of the trnscendentl number e. Nturl Bse e: e lim.7888 Interest Compounded Continuousl: A Pe rt Nturl Eponentil Function: k f e Grph of e : E. 9 A totl of $, is invested t n nnul interest rte of %. Find the blnce fter 4 ers if the interest is compounded continuousl. You Tr: Describe the trnsformtions needed to drw the grph of grph. f 4. Sketch the QOD: Using tble of vlues, how cn ou determine whether the hve n eponentil reltionship? Pge 4 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

5 . Logrithmic Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of n eponentil, logistic or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions. 9.5 The student will grph the inverse of n eponentil or logrithmic function. Review: Solve the following for Eplortion: Use our clcultor to write 997 s power of. Tr Note: 4, , 584.9, 995.(close!) Now use our clcultor to find LOG 997. log (Compre to bove) A LOGARITHM IS AN EXPONENT. Logrithm (bse b): b. log b log b (Red s log bse b of. ) for, b nd b if nd onl if b Note: You cnnot tke the log of negtive number! Common Logrithm: Given positive number p, the solution to p is clled the bse- logrithm of p, epressed s log p, or simpl s log p. (When no bse is specified, it is understood to be bse.) Nturl Logrithmic Function: logrithm bse e, log e, written ln Nottions:. log is used to represent log (common logrithm). ln is used to represent log e (nturl logrithm) Pge 5 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

6 E: Rewrite ech eqution (eponentil form) to logrithmic form.. b. Bse (b) = Eponent = 4 6 Bse (b) = Eponent = E: Rewrite ech eqution (logrithmic form) to eponentil form.. log5 5 Bse (b) = Eponent = b. log, 5 Bse (b) = Eponent = Evluting Logrithms: logrithm is n eponent E: Evlute the logrithms.. log 8 b. f, f log 4 6 c. log5 d. f., f log f log. Clcultor: Grphing Logrithmic Functions: logrithmic function is the inverse of n eponentil function E4: Use tble of vlues to sketch the grph of log grph nd compre the grph to the grph of.. Discuss the chrcteristics of the Note: To crete the tble, it is helpful to rewrite the function s nd choose vlues for. Domin: Rnge: Intercepts: Asmptote: Incresing: End Behvior: lim f lim f The function log is the inverse of, so its grph is the reflection of over the line. Pge 6 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

7 Trnsformtions of log b : k log b h Verticl Stretch:, Horizontl Stretch:, Reflection over -is: Reflection over -is: Horizontl Trnsltion: h Verticl Trnsltion: k E5: Describe the trnsformtions used to grph the function log 5. Then sketch the grph. Properties of Logrithms: A.) log becuse B.) log becuse C.) log D.) If log Note: Grph in red is the grph of log. becuse log, then Evluting Logrithmic Epressions: E6: Evlute.5 log4. log nd..5 log Let log.5. Rewrite in eponentil form: log4. Let log4.. Rewrite in logrithmic form: Note: B the properties of inverses, we could hve evluted the bove emples without rewriting using the following: log nd log Solving Logrithmic Equtions: E7: Solve the equtions for.. log log 6 The bses re equl, so 7 7 Note: Both solutions work in the originl eqution.. log9 Rewrite eponentill:. 4 ln e Rewrite eponentill: Note: Remember tht the bse of ln is e. You Tr: Sketch the grph of function on the sme grid with its inverse. f ln Pge 7 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

8 Pge 8 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

9 . Properties of Logrithms Sllbus Objective: 9. The student will ppl the properties of logrithms to evlute epressions, chnge bses, nd re-epress dt. Eplortion: Use our clcultor to find log nd log. Evlute the following logrithms on our clcultor, then speculte how ou could clculte them using onl the vlues of log nd/or log. log. nd log.477. log6 log log8 log8.9. log log.76 Recll: Properties of Eponents b b b Becuse logrithm is n eponent, the rules re the sme! Properties of Logrithms: Condensed Epnded logb RS logb R logb S R logb logb R logb S S log b c R c log R b E: Use the properties of logrithms to epnd the following epressions. b b b b b.) log 5 b.) ln c.) log d.) z log E: Use the properties of logrithms to condense the following epressions..) log log4 b.) log log c.) ln ln ln Pge 9 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

10 Evluting Logrithmic Epressions with Bse b Let log b. Rewrite in eponentil form: b. Tke the log of both sides. log b log Use the properties of logs to solve for. log b log Note: This will work for logrithm of n bse, including the nturl log. log log b Chnge of Bse Formul: log b log log b ln ln b E: Evlute the epression log5 8. B the chnge of bse formul,. Using the clcultor, E4: Evlute the epression log64. B the chnge of bse formul,. Using the clcultor, Note This did NOT require the use of clcultor! We know tht So log64 6 Grphing Logrithmic Functions on the Clcultor E5: Grph log nd log 5 on the sme grid on the grphing clcultor. Some clcultors cnnot tpe in log bse into the clcultor, so we must rewrite the functions using the chnge of bse formul. Our Clcultor: Tpe green lph f, no need to use chnge of bse. log log log log 5 5 log log Cution: The grph creted b the clcultor is misleding t the smptote! You Tr: Epnd the epression using the properties of logrithms. ln z. QOD: When is it pproprite to use the chnge of bse formul? Eplin how to evlute logrithm of bse b without the chnge of bse formul. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

11 .4 Solving Eponentil nd Logrithmic Equtions Sllbus Objectives: 9.4 The student will solve eponentil, logrithmic nd logistic equtions nd inequlities. 9.6 The student will compre equivlent logrithmic nd eponentil equtions. Strtegies for Solving n Eponentil Eqution: Rewrite both sides with the sme bse Tke the log of both sides fter isolting the eponentil E: Solve the following eponentil equtions..) 4 Rewrite with bse : Both sides hve the sme bse, so the eponents must be equl: b.) 8 6 We cnnot rewrite both sides with the sme bse, so tke the log of both sides. c.) 7 e 5 Isolte the eponentil: Tke the nturl log of both sides: Note: We could hve determined tht immeditel using the eqution e. Strtegies for Solving Logrithmic Eqution: Condense n logrithms with the sme bse using the properties of logs Rewrite the eqution in eponentil form Check for etrneous solutions E: Solve the following logrithmic equtions..) ln ln5 Condense: Rewrite in eponentil form: Solve for : Check: b.) log 65 4 Rewrite in ep. form: Tke the log of both sides: Rewrite in eponentil form: Note: We could hve determined tht 5 immeditel using the eqution root of ech side. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter 4 65, b tking the 4 th

12 c.) log log 8 Condense: Rewrite in eponentil form: Solve for : Check: (remember cn t tke the log of negtive!) Note: You must check ever possible solution for etrneous solutions. All negtive nswers re not necessril etrneous! E: Solve the eqution log5 4. Rewrite in eponentil form: Solve for : Check: E4: Solve the eqution Rewrite using properties of logs: log 6. Rewrite in eponentil form: Check: Chllenge Problems: Use our rsenl of eponentil nd logrithmic properties! E5: Solve the eqution 6 7. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

13 E6: Solve the eqution Tke the log of both sides: 4 7. Rewrite: Solve for : Check You Tr: Solve the eqution e 5. QOD: Compre nd contrst the methods for solving eponentil nd logrithmic equtions. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

14 .5 Eponentil nd Logrithmic Models Sllbus Objective: 9.7 The student will solve ppliction problems involving eponentil nd logrithmic functions. kt Newton s Lw of Cooling: The temperture T of n object t time t is T t Ts T TS e, where T s is the surrounding temperture nd T is the initil temperture of the object. E: A 5 F potto is left out in 7 F room for minutes, nd its temperture dropped to 5 F. How mn more minutes will it tke to rech F? Solve for k using the given informtion: T t Ts T TS e kt Use k to solve for t: It tkes bout minutes for the potto to cool to F. This is minutes longer. Compound Interest: A P r n nt A = blnce mount P = principl (beginning) mount r = nnul interest rte (deciml) n = # of times compounded in er t = time in ers E: How long will it tke for n investment of $, t 6% compounded semi-nnull to rech $5? A P r n nt It will tke bout ers. Pge 4 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

15 A Pe E: How long will it tke for n investment of $, t 6% compounded continuousl to rech $5? rt It will tke bout ers. Annul Percentge Yield (APY): the rte, compounded nnull ( t ), tht would ield the sme return For A P r n nt n r, APY n E4: An mount of $4 is invested for 8 ers t 5% compounded qurterl. Wht is the equivlent APY? You Tr: Determine the mount of mone tht should be invested t 9% interest compounded monthl to produce blnce of $, in 5 ers..6 Nonliner Models QOD: Wh is using the nnul percentge ield more fir w to compre investments? Pge 5 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

16 Sllbus Objectives: 9.7 The student will solve ppliction problems involving eponentil nd logrithmic functions. 9.4 The student will solve eponentil, logrithmic nd logistic equtions nd inequlities. t Eponentil Model: P t Pb P t : popultion t time t P : initil popultion Growth Model: b r; b is clled the Growth Fctor Dec Model: b r; b is clled the Dec Fctor E: Write n eponentil function tht models the popultion of Smllville if the initil popultion ws,85, nd it is decresing b.% ech er. Predict how long it will tke for the popultion to fll to. P r, so b Pt Solve for t when Pt t It will tke bout ers. E: The popultion of nts is incresing eponentill such tht on d there re nts, nd on d 4 there re nts. How mn nts re there on d 5? t Use P t Pb nd write sstem of equtions with the given informtion: Pb Pb Solve the sstem b dividing the equtions: P b P P P t 5 P t P 5 nts Note: Students could hve clled d t to come up with the sme solution. 4 Eponentil Regression Pge 6 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

17 E: Find n eponentil regression for the popultion of Ls Vegs using the tble below. Then predict the popultion in ,4 4,64 5,787 58,95 54,847 9: Rdioctive Dec: the process in which the number of toms of specific element chnge from rdioctive stte to nonrdioctive stte Hlf-Life: the time it tkes for hlf of smple of rdioctive substnce to chnge its stte E4: The hlf-life of rdioctive Strontium is 8 ds. Write n eqution nd predict the mount of 5 grm smple tht remins fter ds. t Use P t Pb : Solve for b when P, P.5, nd t 8. c Logistic Function: f c (constnt): limit to growth (mimum) b The logistic function is used for popultions tht will be limited in their bilit to grow due to limited resources or spce. Think About It: Wht would limit popultion growth? Grph of Logistic Function: Domin: f e Rnge: Alws Incresing Horizontl Asmptotes: Pge 7 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter

18 E5: Estimte the mimum popultion for Dlls nd find the popultion for the er 8,,64 given the function Pt () tht models the popultion from t.6e Mimum popultion: c,,64 Popultion in 8: t P (8) E6: Sketch the grph of the function. Stte the -intercept nd horizontl smptotes..4 -intercept: H.A.: Writing Logistic Function f E5: Find logistic function tht stisfies the given conditions: Initil vlue = 4; limit to growth = ; psses through the point,. c b c, 4,, b? You Tr: Complete the tble. Isotope Hlf-Life (ers) Initil Quntit Amount After Yers 4 C 57 5 grms 4 C 57.7 grm 6 R 6 grms Reflection: Eplin how to determine if n eponentil function is growth or dec model. Pge 8 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter