2 nd Properties of the Exponential Functions

Transcription

1 Dte:/26/25 2 nd Clss Objective: Appl the concept to eplore the properties of functions of the form = bˣ. Appl the concept to grph eponentil functions tht hve bse e Agend: Bell ringer (e: 3,32 pge 44) vocbulr Emples (pge ) Clss work (pge 447) Closing Activit Homework

2 Bell ringer Pge The functions =2(.975)ˣ models the intensit of sunlight beneth the surfce of the ocen. The output represents the percent of surfce sunlight intensit tht reches depth of feet. The models is ccurte from bout 2 feet to bout 6 feet beneth the surfce. ) Find the percent of sunlight 5 feet beneth the surfce of ocen. b) Find the percent of sunlight t depth of 37 feet.

3 st Step : Define the vribles Let the number of feet beneth the surfce of ocen Let The p ercent of surfce sunlight intensit tht reches depthof feet r 5.6 % r.7 % t 5 37 t

4 Pge The popultion of certin niml species decreses t rte of 3.5 % per er. You hve counted 8 of the nimls in the hbitt ou re studing.. Write function tht modes the chnge in the niml popultion. b. Grph the function. Estimte the number of ers until the popultion first drops below 5 nimls. 8, 8 8 r 3.5%

5 t 5 47 ers

6 Pge 444. b h k Horizontl shift = h Verticl shift = k Horizontl smptote is =k

7 Grphing Eponentil Functions Pge 443. A. How does the Grph of =-/3.3ˣ compre to the grph of the prent function? 3 Compress 2 cross..3 shift 3 2 : 9All Reflection Horizontlshift Verticl Horizontl Domin Rnge : of the - is - 3 (NO TRANLATION) (h ) 2 the..3 : Rel 3 (NO TRANLATION) Asmptote (k Number )

8 Grphing Eponentil Functions Pge 443. B. How does the Grph of =-/3.3ˣ compre to the grph of the prent function? 3 Strech Reflection 2..3 shift 3 2 : 9All Horizontlshift Verticl Domin Rnge Horizontl : cross the of is the.66 2 : 6 Rel 8 (NO TRANLATION) (NO TRANLATION) Asmptote (h (k -2 Number ) )

9 Trnslting the prent functions =bˣ pge How does the grph of ech function compre to the grph of the prent function? 2 ( NO NO ( NO NO Verticl Rnge Compress Reflection 2, Horizontl Horizontl Domin 2: : 2 shift All shift (NO of or or 2 4 : cross the - 4 Strech) Asmptote 4 the : Rel TRANLATION, - is) ) (k 4 5 Number 2 )

10 Trnslting the prent functions =bˣ 3. How does the grph of ech function compre to the grph of the prent function? 2 ( NO NO ( NO NO Horizontl Verticl Horizontl Domin Rnge 2-4 Compress Reflection 2, shift shift 2 2 : All4of : 2 or or cross : : Asmptote the 2 Strech), the - is) : 23 5Rel 4 Number

11 Nturl Bse eponentil Continuousl Compounded interest

12 Nturl bse e An irrtionl number defined s follow: As n pproches +, the vlue of ( + /n)ⁿ pproches e e ( ) 2

13 Formul for continuous compounding Where the number of compounding period increses infinitel. Formul is A = P e r t A 8,e (.65)(6)

14 4. Use clcultor to evlute the epression. Round the result to three deciml plces.. e b. e / c. e d. 2e /5. 637

15 Modeling Eponentil Growth For continuous compounded Interest Pge 446 rt A Pe Compound Interest P is initil mount r is the interest rte t is the time period

16 Continuousl Compound interest (pge 447) 5. Suppose ou won contest t the strt of 5 th grde tht deposited $ 3, in n ccount tht ps 5% nnul interest compounded continuousl. How much will ou hve in the ccount when ou enter high school 4 ers lter? Epress the nswer to the nerest dollr.5 r t4 A P3 A 3664 e The mount of the ccount is $3,664

17 Modeling Eponentil Growth 6. You decide to invest $8, for 6 ers ps 6.85 % per er compounded continuousl A P 8 e.685 r t 6 A The blnce fter 6 ers is $2.66.6