Exponential Growth and Decay S E C T I O N 6. 3

Transcription

1 Exponenial Growh and Decay S E C T I O N 6. 3

2 The Grea Divide 10 minues o complee Follow up Quesions (Wih your parner be prepared o answer he following quesions abou his aciviy) Do he graphs represen a consan rae of change? Do he graphs appear quadraic? How do he ables verify your observaions? Wha is he y-inercep for each graph? Where do you find he y-inercep recorded on he able? (in each problem) Wha is he raio beween any wo successive given y- values? (in each problem) Is he successive quoien for each pair of y-values consan?(in each problem)

3 The Sky Is Falling The daa given in he able below is a resul of placing a ma on a fla surface and dropping beans ono he ma. For each bean ha landed in a shaded area of he ma, you added one addiional bean o your oal number of beans. Shaded Ma Bean Drop Daa Number of Trials, x Number of Beans, y

4 The Sky is Falling Creae a scaerplo of he daa and record your window. Use he Sky Is Falling ables you have been given o wrie a linear and an exponenial model for your daa. Round answers o he neares housandhs. Bean Drop Daa Number of Trials, x Number of Beans, y Window for his graph: x-min: 0 x-max: 10 x-scale: 1 y-min: 0 y-max: 125 y-scale: 10

5 The Sky Is Falling - Tables Aach he Sky is Falling Table ino your noes

6 The Sky Is Falling On your graphing calculaor, graph boh he linear and he exponenial models you creaed. Was here a consan rae of change for your models? Was here a consan successive quoien for your models? How did you deermine he iniial values from your ables? Which one of he above models is a beer fi for he daa? Why?

7 The Sky Is Falling - Tables Aach he Sky is Falling Table o your noes y = x y = 10(1.433) x

8 Wha s Going On? Before we begin Wha is occurring mahemaically in order for a siuaion o be modeled wih a linear funcion? Consan Rae Of Change Wha values are necessary o wrie a linear model? Slope and y-inercep Wha is occurring mahemaically in order for a siuaion o be modeled wih a exponenial funcion? Consan successive quoiens Wha values are necessary o wrie an exponenial model? The a Iniial Value and successive quoiens o find he base b Wha is occurring mahemaically in order for a siuaion o be modeled wih neiher a linear nor exponenial funcion? Will no have a consan rae of change or consan successive quoiens

9 Wha s Going On? 15 minues o complee How do we handle problems like 2 and 3 when given a rae, r%, of he growh or decay? Are he dependen values in a siuaion increasing or decreasing each ime? How migh his effec he base value in our exponenial model? Increasing values would show growh Decreasing values would show decay (1±r)

10 Exponenial Growh Model When a real-life quaniy increases by a fixed percen each year (or oher ime period), he amoun y of he quaniy afer years can be modeled by he equaion: y = a b y = a (1 + r ) b = (1 + r) where a is he iniial amoun and r is he percen increase expressed as a decimal. NOTE: b is he growh facor. Someimes he equaion is wrien, A=P(1+r), where A sands for he balance amoun and P sands for he principal, or iniial amoun.

11 Exponenial Decay Model When a real-life quaniy decreases by a fixed percen each year (or oher ime period), he amoun y of he quaniy afer years can be modeled by he equaion: y = a b y = a (1 r ) b = (1 r) where a is he iniial amoun and r is he percen decrease expressed as a decimal. NOTE: b is he decay facor. Someimes he equaion is wrien, A=P(1 - r), where A sands for he balance amoun and P sands for he principal, or iniial amoun.

12 Example 1: In 1996, here were 2573 compuer viruses and oher compuer securiy incidens. During he nex 7 years, he number of incidens increased by abou 92% each year. a. Idenify he consans a, r, and b: a = 2573 r =.92 Growh, so b = (1+r) = (1.92) b. Wrie an exponenial model o represen he siuaion given. y y = a 1 + r or y = a b ( ) = 2573(1 +.92) or y = 2573(1.92) c. Abou how many incidens were here in 2003? = = 7 years y = (1.92) y 247,485 viruses

13 Example 1: (coninued) In 1996, here were 2573 compuer viruses and oher compuer securiy incidens. During he nex 7 years, he number of incidens increased by abou 92% each year. y a b y = = 2573(1.92) d. Esimae he year when here were abou 125,000 compuer securiy incidens. Using your calculaor: is approximaely 6, so in he year 2002, here were abou 125,000 securiy incidens.

14 Example 2: A new snowmobile coss $4200. The value of he snowmobile decreases by 10% each year. a. Idenify he consans a, r, and b: a = 4200 r = 0.1 Decay, so b = (1 r) = (0.9) b. Wrie an exponenial model o represen he siuaion given. y y = a 1 r or y = a b ( ) = 4200(1.1) or y = 4200(.9) c. Esimae he value afer 3 years. y = 4200(.9) 3 = $

15 Example 2: (coninued) A new snowmobile coss $4200. The value of he snowmobile decreases by 10% each year. y y = = a b 4200(.9) d. Esimae when he value of he snowmobile will be $2500. y y 1 2 = 4200(.9) = 2500 x x x MIN MAX SCL = 1 y = 1 MIN = 10 y = 4000 MAX = 1 y = 500 SCL Find he poin of inersecion. Afer abou 5 years

16 Aach his noe page ino your noebook Complee all examples.

17 Sky Is Falling Tables: Complee and Aach o Noes