7-1, and 7-2 exponential growth exponential decay algebra-2

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7-1, and 7-2 7-1, and 7-2 Exponential Growth Exponential Growth Exponential Decay Exponential Decay Algebra-2 Algebra-2

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Page 1: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

7-1, and 7-27-1, and 7-2

Exponential GrowthExponential Growth

Exponential DecayExponential Decay

Algebra-2Algebra-2

Page 2: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Lesson ObjectivesLesson Objectives1. Be able to explain how a 1. Be able to explain how a powerpower is similar to is similar to

and and exponentialexponential function and how it is different. function and how it is different.

2. Be able to explain why the graph of an 2. Be able to explain why the graph of an exponential function has the shape it does.exponential function has the shape it does.

3. Know what 3. Know what input valueinput value results in the results in the initial initial valuevalue of the function. of the function.

4. Explain how the “4. Explain how the “initial valueinitial value” got its name.” got its name.

Page 3: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Lesson ObjectivesLesson Objectives4. Describe how a given function is a 4. Describe how a given function is a

transformationtransformation of the of the parent exponential parent exponential functionfunction..

5. Describe the difference between exponential 5. Describe the difference between exponential “growth” and exponential “decay.”“growth” and exponential “decay.”

6. Know how the base determines whether the 6. Know how the base determines whether the function exhibits growth or decay.function exhibits growth or decay.

7. Solve simple problems involving exponential 7. Solve simple problems involving exponential growth or decay.growth or decay.

Page 4: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Arrange the following Arrange the following equations into equations into 3 different 3 different groups groups

)2(

3

xy

xy 2

4)2(3 xy

3)2(3 xy

2xy

xy

1

33xy

45 xy

4)1(3

2

xy

13*4 xy

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

10.10.

Page 5: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turnYour turn

PowerPower: An : An expressionexpression formed by repeated formed by repeatedMultiplication of the same Multiplication of the same factorfactor..

43xcoefficientcoefficient BaseBase

ExponentExponent

1. 1. Define what a Define what a powerpower is. is.

2. 2. Give an example of a Give an example of a powerpower. Label the parts.. Label the parts.

Page 6: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

ExponentialExponential Function Function

xabxf )(

PowerPower: “x” (input variable) is the base and a : “x” (input variable) is the base and a number is the exponent. number is the exponent.

baxxf )(xxf )2(3)(

Power Power FunctionFunction

23)( xxf 3. 3. Describe how you can tell the Describe how you can tell the power functionpower function

from the from the exponential functionexponential function..

ExponentialExponential: “x” is the exponent and a number is : “x” is the exponent and a number is the base.the base.

Page 7: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:4. 4. What happens to population as time goes by?What happens to population as time goes by?

5. 5. The population of the USA is now about 300 The population of the USA is now about 300 (million). Make a graph that shows how the (million). Make a graph that shows how the population will change as time goes by.population will change as time goes by.

Page 8: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:6. 6. “Plug in” the following input values in order to “fill in” “Plug in” the following input values in order to “fill in”

the table below. Round to the 0.1 decimal position.the table below. Round to the 0.1 decimal position.

xxf )2(3)(

x -5 -3 -0.5 0 1 2

y

7. 7. Graph the points Graph the points on an x-y plot.on an x-y plot.

0.10.1 2.12.1 33 66 1212

0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x)

(o

utp

ut

valu

es)

0.40.4

Page 9: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

VocabularyVocabularyAsymptoteAsymptote: a line that the graph of a function : a line that the graph of a function

approaches but never reaches.approaches but never reaches.

Page 10: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Negative Input valuesNegative Input values xxf )2()( x -5 -3 -1 0 1 Will f(x) ever Will f(x) ever

equal equal zerozero??

0.50.5 11 22

52x2

0.0310.031 0.1250.125

32 12 02 1252

132

112

1 1 2321

81

21 1 2

Horizontal asymptote: Horizontal asymptote: y = 0 y = 0

Page 11: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:

8. 8. Graph the exponential function on your calculator Graph the exponential function on your calculator then copy the graph to your answer sheet. then copy the graph to your answer sheet.

xxf )3(4)(

9. 9. Adjust your window to “ZOOM in”. Copy this to your Adjust your window to “ZOOM in”. Copy this to your answer sheetanswer sheet

10. 10. What is the horizontal asymptote?What is the horizontal asymptote?

Page 12: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:xxf )2(3)(

12. 12. f(0) = ? f(0) = ?

11. 11. What is the “base” of the exponential function? What is the “base” of the exponential function?

Page 13: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

The “Initial” ValueThe “Initial” Value

ttf )2(3)( Since “negative time” doesn’t make sense, Since “negative time” doesn’t make sense, what is the what is the “domain“domain” of this function? ” of this function?

(( what input values are allowed?) what input values are allowed?)

If the input variable was If the input variable was timetime, the previous function, the previous function would look like: would look like:

The The initial valueinitial value occurs when t = 0. occurs when t = 0.

What is the What is the initial valueinitial value of f(t) ?? of f(t) ??

f(0) = ?f(0) = ? ?)2(3)0( 0 f

3)0( f

Page 14: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Vocabulary: The “Initial” Vocabulary: The “Initial” ValueValue

xxf )3(7)(

The The initial valueinitial value of the function of the function is the coefficientis the coefficient of the power. of the power.

xxg )5(3)(

What is he What is he initial valueinitial value of the following functions ? of the following functions ?

xxg 7)(

Page 15: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:ttf )3(5.0)(

xabxf )(

14. 14. What is the initial value of: What is the initial value of:

15. 15. What is the initial value of: What is the initial value of:

16. 16. The ‘y’ intercept is a point on the y-axis. What The ‘y’ intercept is a point on the y-axis. What input value (for x) causes a y-intercept ?input value (for x) causes a y-intercept ?

0.50.5

aa

00

17. 17. Find Find f(0) for the following function: for the following function: xxf )10(2)(

f(0) = 2

Page 16: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

TransformationsTransformations of the “exponential of the “exponential function”function”

The The parent functionparent function: : xxf 2)(

AddingAdding 2 (to the parent function): 2 (to the parent function):

22)( xxf

Replacing xReplacing x with (x – 2) (in the with (x – 2) (in the parent function):parent function): )2(2)( xxf

MultiplyingMultiplying the parent function by 3: the parent function by 3:

)2(3)( xxf

Translates right 2Translates right 2

Translates up 2Translates up 2

Vertically stretches by a Vertically stretches by a factor of 3factor of 3

Page 17: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Identifying the Parts of the Identifying the Parts of the function:function: dabxf x )(

‘‘aa’ is the ’ is the initial valueinitial value f(0) = ‘a’ f(0) = ‘a’

‘‘bb’ is called the ’ is called the growth factorgrowth factor

2)4(10)( xxf

Initial value: Initial value: 1010 f(0) = 10 + 2 = 12 f(0) = 10 + 2 = 12

Growth factor: Growth factor: 44

‘‘dd’ shifts graph up/down ’ shifts graph up/down andand is the horizontal asymptote is the horizontal asymptote

Horizontal asymptote: Horizontal asymptote: y = 2y = 2

Page 18: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

TransformationsTransformations of the “exponential of the “exponential function”function”

The The parent functionparent function: : xxf 2)(

MultiplyingMultiplying the parent function by -1: the parent function by -1:

)2()( xxf Reflects across x-axisReflects across x-axis

Combinations of transformations:Combinations of transformations:

5)2(4)( )3( xxfReflects across x-axisReflects across x-axis

Vertically stretched Vertically stretched by a factor of 4by a factor of 4

Translated Translated leftleft 3 3

Translated Translated downdown 5 5

5)2(4)( )3( xxf

Page 19: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn: Your turn: What is the transformation of the What is the transformation of the parent function: parent function: xxf 3)(

xxf )3(2)( 18. 18.

19. 19.

20. 20.

Reflected across x-axis Reflected across x-axis and and vertically stretched vertically stretched by a factor of 2by a factor of 2

TranslatedTranslated rightright 7 and 7 and upup 5 5

vertically stretched vertically stretched by a by a factor of ½ and factor of ½ and translated translated downdown 4 4

53)( )7( xxf

4)3)(21()( xxf

Page 20: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn: Your turn: What is the horizontal What is the horizontal asymptote? asymptote?

xxf )3(2)( 21. 21.

22. 22.

23. 23.

y = 0y = 0

y = 5y = 5

y = -4y = -4

53)( )7( xxf

4)3)(21()( xxf

Page 21: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Exponential GrowthExponential Growth dabxf x )(

‘‘a’ is the a’ is the initial valueinitial value f(0) = ‘a’ f(0) = ‘a’ ‘‘b’ is called the b’ is called the growth factorgrowth factor

‘‘b’ > 1b’ > 1

xxf )2(3)( Table of valuesTable of values

xx f(x)f(x)x)2(3

000)2(3 33

111)2(3 66

0

5

10

15

20

25

30

35

40

45

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

222)2(3 1212

333)2(3 2424

44 4)2(3 4848

22

22

22

22-1-1 1)2(3

1.51.5-2-2

2)2(3 0.750.75

‘‘d’ is the d’ is the horizontal asymptotehorizontal asymptote

Page 22: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Exponential GrowthExponential Growthxabxf )(

Does the output value ever reach ‘0’ ?Does the output value ever reach ‘0’ ?

What do we call the line: y = 0 ?What do we call the line: y = 0 ?

“ “ Growth” occursGrowth” occurs when the growthwhen the growth factor ‘b’ > 1factor ‘b’ > 1

xxf )2(3)(

0

5

10

15

20

25

30

35

40

45

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

Horizontal asymptoteHorizontal asymptote

Page 23: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: xxf )2(5)( Graph the function:Graph the function:

24. 24. Where does it cross the y-axis?Where does it cross the y-axis?

25. 25. What is the “intial value of f(x) ? What is the “intial value of f(x) ?

26. 26. What is the horizontal asymptote?What is the horizontal asymptote?

y = 5y = 5

55

y = 0y = 0

27. 27. What is the growth factor?What is the growth factor? 22

Page 24: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Exponential DecayExponential Decay dabxf x )(

‘‘a’ is the a’ is the initial valueinitial value f(0) = ‘a’ f(0) = ‘a’ ‘‘b’ is called the b’ is called the decay factordecay factor

0 < ‘b’ < 10 < ‘b’ < 1

xxf )5.0(4)( Table of valuesTable of values

xx f(x)f(x)x)5.0(4

000)5.0(4 44

111)5.0(4 22

0

2

4

6

8

10

12

14

16

18

20

-3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

222)5.0(4 11

333)5.0(4 0.50.5

44 4)5.0(4 0.250.25

½½

½½

½½

½½-1-1 1)5.0(4

88-2-2

2)5.0(4 1616

‘‘d’ shifts everything up or downd’ shifts everything up or down

Page 25: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:

xxf )2(5)( 28. 28. Is the following function Is the following function growthgrowth or or decaydecay? ?

29. 29. Is the following function Is the following function growthgrowth or or decaydecay??

257)( x)( -xf

30. 30. Is he following function Is he following function growthgrowth or or decaydecay??

xxf )5.0(2)(

Page 26: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Population GrowthPopulation GrowthtrPtP )()( 0

PopulationPopulation (as a (as a function of time)function of time)

InitialInitial populationpopulation

GrowthGrowth raterate

time time

It’s just a formula!!!It’s just a formula!!!

The initial population of a colony of bacteriaThe initial population of a colony of bacteria is 1000. The population is 1000. The population doublesdoubles every hour. What every hour. What is the population after 5 hours?is the population after 5 hours?

5)2(1000)5( P 000,32)5( P5)2(1000)5( P

Page 27: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: trPtP )1()( 0 Huntsville has a population of 600 people. The Huntsville has a population of 600 people. The population increases by 3% every year. What will population increases by 3% every year. What will the population be in 50 years?the population be in 50 years?

50)03.01(600)50( P

2630)50( P

31. 31.

50)03.1(600)50( P

Page 28: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: trPtP )1()( 0 The population of Detroit, Michigan The population of Detroit, Michigan decreasesdecreases by by 2% every year. If the population is 750,000 right 2% every year. If the population is 750,000 right now, what will the population be in 12 yearsnow, what will the population be in 12 years

12)02.01(000,750)12( P

538,588)12( P

32. 32.

12)98.0(000,750)12( P

Page 29: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turnYour turnAmountAmount (as a (as a function of time)function of time) Initial amountInitial amount

(“principle”)(“principle”)GrowthGrowth rate rate

time time

You spend 20% of your savings every month (80% You spend 20% of your savings every month (80% remains at the end of each month). How much remains at the end of each month). How much money will you have left in 10 months if you started money will you have left in 10 months if you started with $500?with $500?

10)8.0(500$)10( A

69.53$)5( A

33. 33.

trAtA )1()( 0

Page 30: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: trPtA )1()( Initial amountInitial amount (“principle”)(“principle”)

A bank account pays 3.5% interest per year.A bank account pays 3.5% interest per year. If you initially invest $200, how much moneyIf you initially invest $200, how much money will you have after 5 years? will you have after 5 years?

5)035.01(200$)5( A 54.237$)5( A

34. 34.

5)035.1(200$)5( A

Page 31: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: trPtA )1()( A bank account pays 14% interest per year.A bank account pays 14% interest per year. If you initially invest $2500, how much moneyIf you initially invest $2500, how much money will you have after 7 years? will you have after 7 years?

35.35.

Page 32: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn:

36. 36. f(0) = ?f(0) = ?

37. 37. f(1) = ?f(1) = ?

38. 38. Horizontal asymptote = ?Horizontal asymptote = ?

5)4.0(3)( xxf

dabxf x )(

38. 38. Domain = ?Domain = ?

39. 39. range = ?range = ?

Page 33: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: trPtP )1()( 0

The population of a small town was 1500 in the The population of a small town was 1500 in the population increases by 3% every year. population increases by 3% every year.

37. 37. What is the population in 2009? What is the population in 2009?

Page 34: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Transforming Exponential Transforming Exponential FunctionsFunctions

xxf 2)(

xxf 2)(xxf 2)(

The graph of can be obtainedThe graph of can be obtained from by from by reflectingreflecting it across it across the y-axis. the y-axis.

xxf 2)(

xxxf2

12)(

Page 35: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Putting it all together:Putting it all together:

dabxf cx ))(1()1()(If negative:If negative:

Reflect across x-axisReflect across x-axis

2)4.0(10)( xxf

Initial value:Initial value:Crosses y-axis hereCrosses y-axis here

Growth factor:Growth factor:

If negative:If negative:Reflect across y-axisReflect across y-axis

Horizontal shiftHorizontal shift

vertical shiftvertical shift

2)5(3)( xxf

xxf 2)7.0(4)(

5)1.1(6)( 2 xxf

Page 36: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: dabxf x )(

For each of the following what is the:For each of the following what is the: a. “initial value”?a. “initial value”? b. “decay factor”?b. “decay factor”? c. “horizontal asymptote”c. “horizontal asymptote” d. Any reflections (across x-axis or y-axis)d. Any reflections (across x-axis or y-axis)

3)3.0(2)( xxf38.38.

39.39. xxf )5(10)(

40.40. 45.0)( xxf

Page 37: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Identifying the Parts of the Identifying the Parts of the function:function: dabxf x )(

‘‘a’ is the a’ is the initial valueinitial value f(0) = ‘a’ (plus ‘d’) f(0) = ‘a’ (plus ‘d’)

‘‘b’ is called the b’ is called the decay factordecay factor

2)4.0(10)( xxf

Initial value: Initial value: 1010 f(0) = 10 + 2 = 12 f(0) = 10 + 2 = 12

Decay factor: Decay factor: 0.40.4

‘‘d’ shifts graph up/down d’ shifts graph up/down andand is the horizontal asymptote is the horizontal asymptote

Horizontal asymptote: Horizontal asymptote: 22

Page 38: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Graphing Exponential DecayGraphing Exponential DecayUse the “power of the calculator” or: Use the “power of the calculator” or:

0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x)

(o

utp

ut

valu

es)

xxf )5.0(6)(

f(1) = ?f(1) = ?

3. Horizontal 3. Horizontal asymptoteasymptote

1. f(0) = ?1. f(0) = ?

2. Some other point2. Some other point

f(0) = 6f(0) = 6

f(1) = 3f(1) = 3

y = 0y = 0

Domain = ?Domain = ? Range = ?Range = ?All real #’sAll real #’s y > 0 y > 0

Page 39: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Exponential Growth and Exponential Growth and DecayDecay

xabxf )(exponential growthexponential growth: growth factor > 1: growth factor > 1

exponential decayexponential decay: growth factor 0 < b < 1 : growth factor 0 < b < 1

Page 40: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

What 3 things do you need to What 3 things do you need to graph exponential growth?graph exponential growth? dabxf x )(

f(1) = ? f(1) = ? f(1) = 6 + 5 = 11f(1) = 6 + 5 = 11

5)2(3)( xxf

0

5

10

15

20

25

30

35

40

45

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

3. Horizontal asymptote3. Horizontal asymptote

1. f(0) = ?1. f(0) = ? f(0) = 3 + 5 f(0) = 3 + 5

2. Some other point2. Some other point

y = 5y = 5

Domain = ?Domain = ? Range = ?Range = ?All real #’sAll real #’s y > 5 y > 5

Page 41: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

What 3 things do you need to What 3 things do you need to graph exponential decay?graph exponential decay? dabxf x )(

10)4.0(3)( xxf

0

5

10

15

20

25

30

35

40

45

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

x (input value)

f(x) (

outp

ut v

alue

s)

3. Horizontal asymptote3. Horizontal asymptote

1. f(0) = ?1. f(0) = ? f(0) = 3 + 10 f(0) = 3 + 10

2. Some other point2. Some other point

y = 10y = 10

Domain = ?Domain = ? Range = ?Range = ?All real #’sAll real #’s y > 10 y > 10

f(1) = ? f(-1) = 7.5 + 10 = 17.5f(-1) = 7.5 + 10 = 17.5

Page 42: 7-1, and 7-2 Exponential Growth Exponential Decay Algebra-2

Your turn:Your turn: dabxf x )(

For each of the following what is the:For each of the following what is the: a. “initial value”?a. “initial value”? b. “growth factor”?b. “growth factor”? c. “horizontal asymptote”c. “horizontal asymptote”

xxf )3(4)( 41.41.

42.42.xxf )06.1(000,10)(

43.43. 42)( xxf