7-1, and 7-2 exponential growth exponential decay algebra-2
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7-1, and 7-27-1, and 7-2
Exponential GrowthExponential Growth
Exponential DecayExponential Decay
Algebra-2Algebra-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.
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.
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.
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.
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.
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.
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
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.
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
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?
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?
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
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)(
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
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
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
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
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
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
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
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
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
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
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)(
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
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
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
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
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
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.
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 = ?
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?
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)(
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
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
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
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
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
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
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
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