You identified, graphed, and described several parent functions. (Lesson 1-5)

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1 You identified, graphed, and described several parent functions. (Lesson 1-5) Evaluate, analyze, and graph exponential functions. Solve problems involving exponential growth and decay.

2 algebraic function transcendental function exponential function natural base continuous compound interest

3

4 Sketch and Analyze Graphs of Exponential Functions A. Sketch and analyze the graph of f (x) = 4x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

5 Sketch and Analyze Graphs of Exponential Functions B. Sketch and analyze the graph of f (x) = 5 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

6 A. Sketch the graph of f (x) = 3 x. A. C. B. D.

7 B. Describe the domain, range, intercepts, asymptotes, end behavior, and when the function is increasing or decreasing for f(x) = 3 x. A. Domain: (, ); Range: (0, ); y-intercept: 1; Asymptote: x-axis; End behavior: ; Decreasing: (, ); B. Domain: (, ); Range: (0, ); y-intercept: 1; Asymptote: x-axis; End behavior: ; Increasing: (, ); C. Domain: (, ); Range: (0, ); y-intercept: 3; Asymptote: y = 2; End behavior: ; Increasing: (, ); D. Domain: (, ); Range: (0, ); y-intercept: 3; Asymptote: y = 2; End behavior: ; Decreasing: (, );

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9 Graph Transformations of Exponential Functions A. Use the graph of to describe the transformation that results in sketch the graph of the function.. Then

10 Graph Transformations of Exponential Functions B. Use the graph of to describe the transformation that results in sketch the graph of the function.. Then

11 Graph Transformations of Exponential Functions C. Use the graph of to describe the transformation that results in sketch the graph of the function.. Then

12 Use the graph of f (x) = 3x to describe the transformation that results in p (x) = 3x 1. Then sketch the graph of the function.

13 A. p (x) is the graph of f (x) = 3x translated 1 unit up. C. p (x) is the graph of f (x) = 3x translated 1 unit to the right. B. p (x) is the graph of f (x) = 3x translated 1 unit down. D. p (x) is the graph of f (x) = 3x translated 1 unit to the left.

14 Graph Natural Base Exponential Functions A. Use the graph of f (x) = e x to describe the transformation that results in g (x) = e3x. Then sketch the graph of the function.

15 Graph Natural Base Exponential Functions B. Use the graph of f (x) = ex to describe the transformation that results in h (x) = e x 1. Then sketch the graph of the function.

16 Graph Natural Base Exponential Functions C. Use the graph of f (x) = ex to describe the transformation that results in j (x) = 2ex. Then sketch the graph of the function.

17 A. Use the graph of f (x) = ex to describe the transformation that results in p (x) = ex + 2. A. p (x) is the graph of f (x) = ex translated 2 units left. B. p (x) is the graph of f (x) = ex translated 2 units right. C. p (x) is the graph of f (x) = ex translated 2 units down. D. p (x) is the graph of f (x) = ex translated 2 units up.

18 B. Sketch the graph of the function p (x) = ex + 2. A. C. B. D.

19

20 Use Compound Interest A. FINANCIAL LITERACY Mrs. Salisman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Salisman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded quarterly? For quarterly compounding, n = 4. A = Compound Interest Formula

21 Use Compound Interest B. FINANCIAL LITERACY Mrs. Salisman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Salisman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded monthly? For monthly compounding, n = 12, since there are 12 months in a year. A = Compound Interest Formula

22 Use Compound Interest C. FINANCIAL LITERACY Mrs. Salisman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Salisman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded daily? For daily compounding, n = 365. A = Compound Interest Formula

23 FINANCIAL LITERACY Mr. Born invested $5000 into a savings account at his local bank. The account has a 3% interest rate. If Mr. Born does not make any other deposits or withdrawals, what will the account balance be after 11 years if the interest is compounded monthly? A. $ B. $ C. $ D. $

24

25 Use Continuous Compound Interest FINANCIAL LITERACY Mrs. Salisman found an account that will pay the 5% interest compounded continuously on her $2000 educational investment. What will be her account balance after 18 years? A = Pert Compound Interest Formula

26 ONLINE BANKING If $1500 is invested in a online savings account earning 4% per year compounded continuously, how much will be in the account at the end of 8 years if there are no other deposits or withdrawals? A. $ B. $ C. $ D. $36,798.80

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28 Model Using Exponential Growth or Decay A. POPULATION A state s population is declining at a rate of 2.6% annually. The state currently has a population of approximately 11 million people. If the population continues to decline at this rate, predict the population of the state in 15 and 30 years. Use the exponential decay formula to write an equation that models this situation. A = N0(1 + r)t Exponential Decay Formula

29 Model Using Exponential Growth or Decay B. POPULATION A state s population is declining at a rate of 2.6% continuously. The state currently has a population of approximately 11 million people. If the population continues to decline at this rate, predict the population of the state in 15 and 30 years. Use the exponential decay formula to write an equation that models this situation. N =N0ekt Continuous Exponential Decay Formula

30 A. POPULATION The population of a town is increasing at a rate of 2% annually. If the current population is 15,260 people, predict the population in 10 and 20 years. A. about 18,602 people in 10 years and 22,676 people in 20 years B. about 18,639 people in 10 years and 22,765 people in 20 years C. about 12,469 people in 10 years and 10,188 people in 20 years D. about 12,494 people in 10 years and 10,229 people in 20 years

31 B. POPULATION The population of a town is increasing at a rate of 2% continuously. If the current population is 15,260 people, predict the population in 10 and 20 years. A. about 18,639 people in 10 years and 22,765 people in 20 years B. about 18,602 people in 10 years and 22,676 people in 20 years C. about 12,494 people in 10 years and 10,229 people in 20 years D. about 12,469 people in 10 years and 10,188 people in 20 years

32 Use the Graph of an Exponential Model A. DEER The table shows the population growth of deer in a forest from 2000 to If the number of deer is increasing at an exponential rate, identify the rate of increase and write an exponential equation to model this situation.

33 Use the Graph of an Exponential Model B. DEER The table shows the population growth of deer in a forest from 2000 to Use your model to predict how many years it will take for the number of deer to reach 500.

34 A. RABBIT POPULATION Use the data in the table and assume that the population is growing exponentially. Identify the rate of growth and write an exponential equation to model this growth. A. 8%; N(t) = 6.6(0.92)t B. 8%; N(t) = 4.5(0.92)t C. 8%; N(t) = 6.6(1.08)t D. 8%; N(t) = 4.5(1.08)t

35 B. RABBIT POPULATION Use the data in the table and assume that the population is growing exponentially. Use your model to predict in which year the rabbit population will surpass 850. A B C D. 2014

36 You graphed and analyzed exponential functions. (Lesson 3-1) Evaluate expressions involving logarithms. Sketch and analyze graphs of logarithmic functions.

37 logarithmic function with base b logarithm common logarithm natural logarithm

38

39 A. Evaluate log Evaluate Logarithms

40 Evaluate Logarithms B. Evaluate.

41 C. Evaluate. Evaluate Logarithms

42 D. Evaluate log Evaluate Logarithms

43 Evaluate. A. 4 B. 4 C. 2 D. 2

44

45 A. Evaluate log Apply Properties of Logarithms

46 B. Evaluate 22 log Apply Properties of Logarithms

47 Evaluate 7 log 7 4. A. 4 B. 7 C. 4 7 D. 7 4

48

49 Common Logarithms A. Evaluate log 10,000.

50 B. Evaluate 10 log 12. Common Logarithms

51 C. Evaluate log 14. Common Logarithms

52 Common Logarithms D. Evaluate log ( 11).

53 Evaluate log A. about 1.04 B. about 1.04 C. no real solution D. about 2.39

54

55 A. Evaluate ln e 4.6. Natural Logarithms

56 Natural Logarithms B. Evaluate ln ( 1.2).

57 C. Evaluate e ln 4. Natural Logarithms

58 D. Evaluate ln 7. Natural Logarithms

59 Evaluate ln e 5.2. A. no real solution B. about C. about 1.65 D. 5.2

60 Graphs of Logarithmic Functions A. Sketch and analyze the graph of f (x) = log 2 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

61 Graphs of Logarithmic Functions B. Sketch and analyze the graph of Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

62 Describe the end behavior of f (x) = log 4 x. A. B. C. D.

63

64 Graph Transformations of Logarithmic Functions A. Use the graph of f (x) = log x to describe the transformation that results in p (x) = log (x + 1). Then sketch the graph of the function.

65 Graph Transformations of Logarithmic Functions B. Use the graph of f (x) = log x to describe the transformation that results in m (x) = log x 2. Then sketch the graph of the function.

66 Graph Transformations of Logarithmic Functions C. Use the graph of f (x) = log x to describe the transformation that results in n (x) = 5 log (x 3). Then sketch the graph of the function.

67 A. Use the graph of f (x) = ln x to describe the transformation that results in p (x) = ln (x 2) + 1. Then sketch the graphs of the functions. A. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit down. C. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit up. B. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit down. D. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit up.

68 Use Logarithmic Functions A. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R, where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. Find the intensity of an earthquake with an amplitude of 250 microns, a period of 2.1 seconds, and B = 5.4.

69 Use Logarithmic Functions

70 Use Logarithmic Functions B. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R, where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. A city is not concerned about earthquakes with an intensity of less than 3.5. An earthquake occurs with an amplitude of 125 microns, a period of 0.33 seconds, and B = 1.2. What is the intensity of the earthquake? Should this earthquake be a concern for the city?

71 Use Logarithmic Functions

72 Use Logarithmic Functions C. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R, where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. Earthquakes with an intensity of 6.1 or greater can cause considerable damage to those living within 100 km of the earthquake s center. Determine the amplitude of an earthquake whose intensity is 6.1 with a period of 3.5 seconds and B = 3.7.

73 Use Logarithmic Functions

74 SOUND The intensity level of a sound, measured in decibels, can also be modeled by the equation d (w) = 10 log (10 12 w) where w is the intensity of the sound in watts per square meter. If the intensity of the sound of a chain saw is 0.1 watts per square meter, what is the intensity level of the sound in decibels? A. 110 decibels B. 100 decibels C. 90 decibels D. 80 decibels

75 You evaluated logarithmic expressions with different bases. (Lesson 3 2) Apply properties of logarithms. Apply the Change of Base Formula.

76

77 Use the Properties of Logarithms A. Express log 96 in terms of log 2 and log 3.

78 Use the Properties of Logarithms B. Express in terms of log 2 and log 3.

79 Express ln in terms of ln 3 and ln 5. A. 3 ln ln 3 B. ln 5 3 ln 3 3 C. 3 ln 5 3 ln 3 D. 3 ln 3 3 ln 5

80 Simplify Logarithms A. Evaluate.

81 Simplify Logarithms B. Evaluate 3 ln e 4 2 ln e 2.

82 Evaluate. A. 4 B. C. D.

83 A. Expand ln 4m 3 n 5. Expand Logarithmic Expressions

84 B. Expand. Expand Logarithmic Expressions

85 Expand. A. 3 ln x ln (x 7) B. 3 ln x + ln (x 7) C. ln (x 7) 3 ln x D. ln x 3 ln (x 7)

86 Condense Logarithmic Expressions A. Condense.

87 Condense Logarithmic Expressions B. Condense 5 ln (x + 1) + 6 ln x.

88 Condense ln x 2 + ln (x + 3) + ln x. A. In x(x + 3) B. C. D.

89

90 A. Evaluate log 6 4. Use the Change of Base Formula

91 B. Evaluate. Use the Change of Base Formula

92 Evaluate. A. 2 B. 0.5 C. 0.5 D. 2

93 Use the Change of Base Formula ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function, where N 1 and N 2 are the numbers of each type of species found in the sample S = ( N 1 + N 2 ). Find the measure of diversity for environments that find 25 and 50 species in the samples.

94 Use the Change of Base Formula

95 Use the Change of Base Formula B. ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function, where N 1 and N 2 are the numbers of each type of species found in the sample S = ( N 1 + N 2 ). Find the measure of diversity for environments that find 10 and 60 species in the samples.

96 Use the Change of Base Formula

97 PHOTOGRAPHY In photography, exposure is the amount of light allowed to strike the film. Exposure can be adjusted by the number of stops used to take a photograph. The change in the number of stops n needed is related to the change in exposure c by n = log 2 c. How many stops would a photographer use to get exposure? A. 2 stops B. 2 stops C. 0.5 D. 0.5

98 You applied the inverse properties of exponents and logarithms to simplify expressions. (Lesson 3-2) Apply the One-to-One Property of Exponential Functions to solve equations. Apply the One-to-One Property of Logarithmic Functions to solve equations.

99

100 A. Solve 4 x + 2 = 16 x 3. Solve Exponential Equations Using One-to- One Property

101 B. Solve. Solve Exponential Equations Using One-to- One Property

102 Solve 25 x + 2 = 5 4x. A. 1 B. C. 2 D. 2

103 Solve Logarithmic Equations Using One-to- One Property A. Solve 2 ln x = 18. Round to the nearest hundredth.

104 Solve Logarithmic Equations Using One-to- One Property B. Solve 7 3 log 10x = 13. Round to the nearest hundredth.

105 Solve Logarithmic Equations Using One-to- One Property C. Solve log 5 x 4 = 20. Round to the nearest hundredth.

106 Solve 2 log 2 x 3 = 18. A. 81 B. 27 C. 9 D. 8

107

108 Solve Exponential Equations Using One-to- One Property A. Solve log 2 5 = log 2 10 log 2 (x 4).

109 Solve Exponential Equations Using One-to- One Property B. Solve log 5 (x 2 + x) = log 5 20.

110 Solve log 3 15 = log 3 x + log 3 (x 2). A. 5 B. 3 C. 3, 5 D. no solution

111 Solve Exponential Equations A. Solve 3 x = 7. Round to the nearest hundredth.

112 Solve Exponential Equations B. Solve e 2x + 1 = 8. Round to the nearest hundredth.

113 Solve 4 x = 9. Round to the nearest hundredth. A B C D. 0.44

114 Solve in Logarithmic Terms Solve 3 6x 3 = 2 4 4x. Round to the nearest hundredth.

115 Solve 4 x + 2 = 3 2 x. Round to the nearest hundredth. A B C D. 0.23

116 Solve e 2x e x 2 = 0. Solve Exponential Equations in Quadratic Form

117 Solve e 2x + e x 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4 D. ln 3, ln ( 4)

118 Solve Logarithmic Equations Solve log x + log (x 3) = log 28.

119 Solve ln x + ln (5 x) = ln 6. A. 2 B. 3 C. 2, 3 D. 2, 3

120 Check for Extraneous Solutions Solve log (3x 4) = 1 + log (2x + 3).

121 Solve log 2 (x 6) = 3 + log 2 (x 1). A. B. no solution C. D. 8

122 Model Exponential Growth A. CELL PHONES This table shows the number of cell phones a new store sold in March and August of the same year. If the number of phones sold per month is increasing at an exponential rate, identify the continuous rate of growth. Then write the exponential equation to model this situation.

123 Model Exponential Growth B. CELL PHONES This table shows the number of cell phones a new store sold in March and August of the same year. Use your model to predict the number of months it will take for the store to sell 500 phones in one month.

124 TECHNOLOGY SERVICE The table below shows the number of service requests received by the TECH SQUAD in the months of May and July in the same year. If the number of service requests is increasing at an continuous exponential rate, identify the rate of growth. A % B % C. 9.45% D %

125 You modeled data using polynomial functions. (Lesson 2-1) Model data using exponential, logarithmic, and logistic functions. Linearize and analyze data.

126 logistic growth function linearize

127 Exponential Regression BACTERIA The growth of a culture of bacteria is shown in the table. Use exponential regression to model the data. Then use your model to predict how many bacteria there will be after 24 hours.

128 Exponential Regression

129 The number of leaves falling per hour from the trees in an arboretum is shown in the table below. Use an exponential regression model to predict how many leaves will fall in the tenth hour. A. 415 B. 500 C. 485 D. 622

130 Logarithmic Regression MEMORY A group of students studied a photograph for 30 seconds. Beginning 1 day later, a test was given each day to test their memory of the photograph. The average score for each day is shown in the table. Use logarithmic regression to model the data. Then use your model to predict the average test score after 2 weeks.

131 Logarithmic Regression

132 MEMORY Students do not remember everything presented to them in a mathematics class. The table below shows the average percentage of information retained t days after the lesson by a group of students. Use a logarithmic regression model to predict the students retention percentage on the tenth day. A. 46.3% B. 45.9% C. 45.1% D. 43.3%

133

134 Logistic Regression ADVERTISING The number of television ads for a certain product affects the percentage of people who purchase the product as shown in the table. Use logistic regression to find a logistic growth function to model the data. Then use your model to predict the limit to the percentage of people who will purchase the product.

135 Logistic Regression

136 MUSIC The probability of a person liking a song increases with the number of friends who say they also like the song. The data shown in the table models this situation. Use a logistic growth function to determine the limit to the probability that a person will like a song based on the number of friends who say they like the song. A. about 31% B. about 27% C. about 56% D. about 29%

137 Choose a Regression INTERNET Use the data in the table to determine a regression equation that best relates the profit of a Web site with the time it has been in business. Then determine the approximate time it will take for the Web site to earn a profit of $100,000 in one year.

138 Choose a Regression

139 BUSINESS Use the data in the table to determine a regression equation that best relates the profit of a business with the time it has been in business. A. LinReg(ax + b): y = 42.05x B. PwrReg: y = x C. ExpREg: y = 2000(1.02) x D. QuadReg: y = 0.42x x

140

141 Linearizing Data Make a scatter plot of the data, and linearize the data assuming a power model. Graph the linearized data, and find the linear regression equation. Then use this linear model to find a model for the original data.

142 Linearizing Data

143 Assuming a power model, linearize the data to find the linear regression equation modeling the linearized data. A. y ˆ= 12x ˆ 8.97 B. y ˆ= 17.07x ˆ+ 6 C. y = 1.92x ˆ ˆ D. y = 1.2x 0.27 ˆ ˆ

144 Use Linearization BACTERIA The table shows the number of bacteria found in a culture. Find an exponential model relating these data by linearizing the data and finding the linear regression equation. Then use your model to predict the number of bacteria after 10 hours.

145 Use Linearization

146 The data in the table is modeled by a quadratic function. Linearize the data. A. C. 46 B. D.

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