3-1: Exponential Functions

Transcription

1 3-1: Exponeial Functions Precalculus Mr. Gallo Exponeial Function Types: Exponeial Growth as x increases, y increases Exponeial Decay as x increases, y decreases approaching zero Asymptote Line the graph approaches but never reaches. Exponeial Decay Exponeial Growth Asymptote 1

2 Exponeial Function For the function, if and 1, the function represes: exponeial growth if and 1, the function represes: exponeial decay In either case, the y-iercept is,, the domain is,, the range is, and the asymptote is Ideify.7 as an example of exponeial growth or decay. What is the y-iercept? a 1 b.7 Since and 1, this is decay. lim f ( x) lim f ( x) x x lim f ( x) lim f ( x) x x y-iercept is,1 Exponeial Growth and Decay In the equation, a is the. initial amou Exponeial Growth: 1 b is the. growth factor Increase written as a decimal is r, rate of increase or. growth rate 1for exponeial growth Exponeial Decay: 1 b is the. decay factor Decay written as a decimal is r,. rate of decay 1for exponeial decay because r is expressed as a negative quaity. 2

3 Families of Exponeial Functions Families of Exponeial Functions Pare Function Stretch 1 Compression (Shrink) 1 Reflection in x-axis Translations (Horizoal by h; Vertical by k) y y b x y ab x h b x k y a b All transformations combined x h k Natural Base Exponeial Functions Have e for a base Most commonly used base (more often than 2 or 1) Simplifies calculations e Irrational number Is an asymptote for graph of Functions have same properties as other exponeial functions. Has the form 3

4 Homework: p. 166 #1-19 odd Compound Ierest Formula If a principal is invested at an annual ierest rate (in decimal form) compounded times a year, then the balance in the accou after years is given by: A P1 r n To allow for quarterly, mohly or even daily compoundings, let be the number of times the ierest is compounded each year Rate per compounding is a fraction of the annual rate Number of compoundings after years is 4

5 Mrs. Salisman invested $2 io an educational accou for her daughter when she was an infa. The accou has a 5% ierest rate. If Mrs. Salisman does not make any other deposits or withdrawals, what will the accou balance be after 18 years if the ierest is compounded: r A P1 n a. Quarterly? 4*18 r.5 A P1 21 $ n 4 b. Mohly? 12*18 r.5 A P1 21 $491.2 n 12 c. Daily? 365*18 r.5 A P1 21 $ n 365 Coinuously Compounded Ierest If a principal is invested at an annual ierest rate (in decimal form) compounded coinuously, then the balance in the accou after years is given by: A t Pe rt Used when there is no waiting period between ierest paymes. 5

6 Mrs. Salisman found an accou that will pay the 5% ierest compounded coinuously on her $2 educational investme. What will be her accou balance after 18 years? P=2 r=.5 t=18 At At At Pe rt 2e The accou will have $ Exponeial Growth or Decay If an initial quaity grows or decays at an exponeial rate or (as a decimal), then the final amou after a time is given by the following formulas: N N r 1 t N N e kt If is a growth rate, then If is a decay rate, then If is a coinuous growth rate, then If is a coinuous decay rate, then Can be used to model investmes, populations of people, animals, bacteria and amous of radioactive material. Models apply to any situation where growth is proportional to the initial size of the quaity being considered. 6

7 A state s population is declining at a rate of 2.6% annually. The state currely has a population of approximately 11 million people. If the population coinues to decline at this rate, predict the population of the state in 15 and 3 years. a. 2.6% annually N N r 1 t t 15 N N 1r 11,, , 49, 298 t 3 N N 1r 11,, ,99,699 b. 2.6% coinuously N N e kt N N e e kt.26*15 11,, 7, 447,626 N N e e kt.26*3 11,, 5,42, 466 Homework: p. 166 #21-26 & WS #6 #1-8 7

8 The table shows the population growth of deer in a forest from 2 to 21. Deer Population Year Deer a. If the number of deer is increasing at an exponeial rate, ideify the rate of increase and write an exponeial equation to model this situation r N t r r 1 b. Use your model to predict how many years it will take for the number of deer to reach years Homework: p. 166 #27, 29, 35-38, 4 8