STUDENT TEXT AND HOMEWORK HELPER

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1 UNIT 4 EXPONENTIAL FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles Allan E. Bellman Basia Hall William G. Handlin, Sr. Dan Kenned Stuart J. Murph Grant Wiggins Boston, Massachusetts Chandler, Arizona Glenview, Illinois Hoboken, New Jerse

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3 UNIT 4 EXPONENTIAL FUNCTIONS AND EQUATIONS TOPIC 9 Eponential Functions and Equations TOPIC 9 Overview Eponential Functions Eponential Growth and Deca Modeling Eponential Data...40 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS (TEKS) FOCUS (9)(A) (9)(B) (9)(C) Determine the domain and range of eponential functions of the form f () = ab and represent the domain and range using inequalities. Interpret the meaning of the values of a and b in eponential functions of the form f () = ab in real-world problems. Write eponential functions in the form f () = ab (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and deca. (9)(D) Graph eponential functions that model growth and deca and identif ke features, including -intercept and asmptote, in mathematical and real-world problems. (9)(E) Write, using technolog, eponential functions that provide a reasonable fit to data and make predictions for real-world problems. (1)(B) Evaluate functions, epressed in function notation, given one or more elements in their domains. Contents

4 Topic 9 Eponential Functions and Equations TOPIC OVERVIEW VOCABULARY 9-1 Eponential Functions 9- Eponential Growth and Deca 9-3 Modeling Eponential Data English/Spanish Vocabular Audio Online: English Spanish asmptote, p. 386 asíntota compound interest, p. 396 interés compuesto deca factor, p. 396 factor de decremento eponential deca, p. 396 decremento eponencial eponential function, p. 386 función eponencial eponential growth, p. 395 incremento eponencial growth factor, p. 395 factor incremental DIGITAL APPS PRINT and ebook Access Your Homework... ONLINE HOMEWORK You can do all of our homework online with built-in eamples and Show Me How support! When ou log in to our account, ou ll see the homework our teacher has assigned ou. YOUR DIGITAL RESOURCES PearsonTEXAS.com HOMEWORK TUTOR APP Do our homework anwhere! You can access the Practice and Application Eercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on an mobile device. 384 Topic 9 Eponential Functions and Equations STUDENT TEXT AND HOMEWORK HELPER Access the Practice and Application Eercises that ou are assigned for homework in the Student Tet and Homework Helper, which is also available as an electronic book.

5 Big Time Pa Back 3-Act Math 3-Act Math Most people agree that investing our mone is a good idea. Some people might advise ou to put mone into a bank savings account. Other people might sa that ou should invest in the stock market. Still others think that buing bonds is the best investment option. Is a bank savings account a good wa to let our mone grow? Just how much mone can ou make from a savings account? In this video, ou ll see an intriguing situation about an investment option. Scan page to see a video for this 3-Act Math Task. If You Need Help... VOCABULARY ONLINE You ll find definitions of math terms in both English and Spanish. All of the terms have audio support. INTERACTIVE EXPLORATION You ll have access to a robust assortment of interactive eplorations, including interactive concept eplorations, dnamic activitites, and topiclevel eploration activities. LEARNING ANIMATIONS You can also access all of the stepped-out learning animations that ou studied in class. STUDENT COMPANION Refer to our notes and solutions in our Student Companion. Remember that our Student Companion is also available as an ACTIVebook accessible on an digital device. INTERACTIVE MATH TOOLS These interactive math tools give ou opportunities to eplore in greater depth ke concepts to help build understanding. VIRTUAL NERD Not sure how to do some of the practice eercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 385

6 9-1 Eponential Functions TEKS FOCUS TEKS (9)(A) Determine the domain and range of eponential functions of the form f() = ab and represent the domain and range using inequalities. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(A), (1)(D), (9)(C), (9)(D), (1)(B) VOCABULARY Asmptote An asmptote is a line that a graph of a function gets closer to as or gets larger in absolute value. Eponential function An eponential function is a function of the form = ab, where a is a nonzero constant, b 7 0, and b 1. Number sense the understanding of what numbers mean and how the are related ESSENTIAL UNDERSTANDING Some functions model an initial amount that is repeatedl multiplied b the same positive number. In the rules for these functions, the independent variable is an eponent. Definition An eponential function is a function of the form = a # b, where a 0, b 7 0, b 1, and is a real number. Eamples Ke Concept Eponential Function 4 4 O O Lesson 9-1 Eponential Functions

7 Ke Concept Asmptote of an Eponential Function = ab The -ais ( = 0) is the horizontal asmptote for an eponential function of the form = ab. The -values approach zero, but never actuall reach zero. The graphs approach, but do not cross, the -ais. O The -ais is an asmptote. Problem 1 Identifing Linear and Eponential Functions How can ou identif a constant ratio between -values? When ou multipl each -value b the same constant and get the net -value, there is a constant ratio between the values. Does the table or rule represent a linear or an eponential function? Eplain. A The difference between the -values is * * * The ratio between the -values is 3. The table represents an eponential function. There is a common difference between consecutive -values and a common ratio between the corresponding -values. B = 3 The rule represents a linear function because it is in the form = m + b. The independent variable is not an eponent. PearsonTEXAS.com 387

8 Problem TEKS Process Standard (1)(C) Writing an Eponential Function The figures below are stages of the Sierpinski triangle. In Stage 0, there is one black triangle. In each subsequent stage, a white triangle is added to the center of each black triangle, dividing that triangle into 3 black triangles and 1 white triangle. The number of black triangles increases b a factor of 3 in each stage. Stage 0 Stage 1 Stage Stage 3 Stage 4 Is there an stage with eactl 1000 black triangles? No. At each stage the number of black triangles will be a multiple of 3. A Write an eponential function f () = ab to describe this situation. The factor b which the number of black triangles increases at each stage is the value of b. Using the fact that b is 3, ou can show that the value of a is 1, the number of triangles in Stage 0. f () = ab Write the formula for the eponential function. f () = a # 3 Substitute 3 for b. 1 = a # 3 0 From Stage 0, ou know that f(0) = 1. 1 = a # 1 Simplif = a Solve for a. f () = 1 # 3 Substitute 1 for a in the formula f() = a # 3. f () = 3 Simplif. B Determine in what stage there will first be more than 1000 black triangles. You can select techniques such as estimation and mental math to solve this problem. In Stage, ou know there are 9 black triangles. Round 9 to 10 and then multipl b the factor 3 to estimate the number of black triangles in later stages. The estimate of 900 in Stage 6 is high, since ou rounded up in earlier stages. So, ou know the number of black triangles in Stage 6 is less than When ou multipl b 3 again, the number of black triangles will eceed There will first be more than 1000 black triangles in Stage 7. Check You can confirm this result b evaluating f (6) and f (7). f () = 3 6 = 79 f () = 3 7 = 187 Stage Triangle Estimate = 90 Now round to Lesson 9-1 Eponential Functions

9 Problem 3 TEKS Process Standard (1)(A) Evaluating an Eponential Function Wh is the function 30 ~ not ~ 30? In an eponential function, a is the starting value and b is the common ratio. Population Growth Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. The function f () = 30 # gives the population after weeks. How man beetles will there be after 56 das? f () = 30 # = 30 # 8 56 das is equal to 8 weeks. Evaluate the function for = 8. = 30 # 56 Simplif the power. = 7680 Simplif. After 56 das, there will be 7680 beetles. Problem 4 TEKS Process Standard (1)(D) How can ou use the graph to find the domain and range? The domain can be found b using the -coordinates on the graph and the range can be found b using the -coordinates on the graph. Graphing an Eponential Function A What is the graph of = 3 ~? Identif the domain and range. Epress the range using an inequalit. Make a table of - and -values. 1 3 (, ) , , (0, 3) (1, 6) Plot the points (, 1) An value substituted for results in a positive -value. The domain is all real numbers. The range is 7 0. O 4 B What is the -intercept of the eponential function in part (A)? What is the asmptote? The table and the graph show that when = 0, = 3. The -intercept of = 3 # is 3. The table and the graph show that as decreases, the -values get smaller. The value of 3 # will alwas be positive, so the -values will approach zero, but never reach zero. The -ais is the horizontal asmptote for = 3 #. Connect the points with a smooth curve. PearsonTEXAS.com 389

10 Problem 5 Graphing an Eponential Model Maps Computer mapping software allows ou to zoom in on an area to view it in more detail. The function f() = 100 ~ 0.5 models the percent of the original area the map shows after zooming in times. Graph the function. Identif the -intercept, the asmptote, and the domain. ORIGINAL AREA 75 Big Cpress National Preserve 41 Loahatchee NWR 7 Miami Everglades National Park Homestead 1 95 Boca Raton Fort Lauderdale ATLANTIC OCEAN W N E S 0 0 mi f() (, f()) (0, 100) (1, 5) (, 6.5) (3, 1.56) (4, 0.39) ZOOMED IN 1 TIME Everglades National Park Homestead 5 1 Miami Fort Lauderdale 95 Hollwood Biscane Ba ATLANTIC OCEAN W N E S 0 10 mi Should ou connect the points of the graph? No, the data is discrete not continuous. The number of times ou zoom in must be a nonnegative integer O f() 4 41 ZOOMED IN TIMES Pinecrest 81 Hialeah Miami Biscane Ba Aventura ATLANTIC OCEAN W N E S 0 5 mi When = 0, = 100, so the -intercept is 100. The -ais is the horizontal asmptote. The domain is Ú 0, is an integer. Problem 6 TEKS Process Standard (1)(C) Solving One-Variable Equations What is the solution or solutions of = 0.5 +? Step 1 Write each side of the equation as a function equation. f () = and g () = Step Graph the equations using a graphing calculator. Use 1 for f () and for g (). continued on net page 390 Lesson 9-1 Eponential Functions

11 Problem 6 continued How can ou check that the -value is a solution? Substitute for in the original equation. Make sure ou use the same -value for each instance of. Step 3 Use the CALC feature. Chose INTERSECT to find the points where the lines intersect. Intersection The solutions of = are about and Intersection ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Determine whether each table or rule represents a linear or an eponential function. Eplain wh or wh not. For additional support when completing our homework, go to PearsonTEXAS.com = 4 # 5 4. = 1 # 5. =-5 # = Evaluate each function for the given value. 7. f () = 6 for = 8. g(t) = # 0.4 t for t =- 9. = 0 # 0.5 for = h(w) =-0.5 # 4 w for w = Appl Mathematics (1)(A) An investment of $5000 doubles in value ever decade. The function f () = 5000 #, where is the number of decades, models the growth of the value of the investment. How much is the investment worth after 30 r? 1. Appl Mathematics (1)(A) A population of 75 foes in a wildlife preserve quadruples in size ever 15 r. The function = 75 # 4, where is the number of 15-r periods, models the population growth. How man foes will there be after 45 r? Graph each eponential function. Identif the domain, range, -intercept, and asmptote of each function. Epress each range using an inequalit. 13. = = = ( 1 3 ) 16. =-( 1 3 ) 17. = 10 # ( 3 ) 18. = 0.1 # 19. = 1 4 # 0. = 1.5 Select Tools to Solve Problems (1)(C) Use a graph to solve each equation = = 3 PearsonTEXAS.com 391

12 3. Analze Mathematical Relationships (1)(F) A new museum had 7500 visitors this ear. The museum curators epect the number of visitors to grow b 5% each ear. The function = 7500 # 1.05 models the predicted number of visitors each ear after ears. Graph the function. Identif the domain, range, -intercept, and asmptote of the function. Epress the range using an inequalit. 4. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) A solid waste disposal plan proposes to reduce the amount of garbage each person throws out b % each ear. This ear, each person threw out an average of 1500 lb of garbage. The function = 1500 # 0.98 models the average amount of garbage each person will throw out each ear after ears. Graph the function. Identif the domain, range, -intercept, and asmptote of the function. Epress the range using an inequalit. 5. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Compare the rule and the function table below. Which function has the greater value when = 1? Eplain. Function 1 Function = You have just read a journal article about a population of fungi that doubles ever n 3 weeks. The beginning population was 10. The function = 10 # 3 represents the population after n weeks. a. You have a population of 15 of the same fungi. Assuming the journal article gives the correct rate of increase, write the function that represents the population of fungi after n weeks. b. Suppose ou find another article that states that the fungi population triples ever 4 weeks. If there are currentl 15 fungi in our population, write the function that represents the population after n weeks. 7. Eplain Mathematical Ideas (1)(G) Hdra are small freshwater animals. The can double in number ever two das in a laborator tank. Suppose one tank has an initial population of 60 hdra. When will there be more than 5000 hdra? 8. a. Graph =, = 4, and = 0.5 on the same aes. b. What point is on all three graphs? c. Does the graph of an eponential function intersect the -ais? Eplain. d. How does the graph of = b change as the base b increases or decreases? Hdra 39 Lesson 9-1 Eponential Functions

13 Write an eponential function to describe the given sequence of numbers. 9., 8, 3, 18, 51, , 1 9, 1 7, 1 81,... Which function has the greater value for the given value of? 31. = 4 or = 4 for = 3. f () = 10 # or f () = 00 # for = = 3 or = 3 for = f () = or f () = 100 for = Appl Mathematics (1)(A) A computer valued at $1500 loses 0% of its value each ear. a. Write a function rule that models the value of the computer. b. Find the value of the computer after 3 r. c. In how man ears will the value of the computer be less than $500? 36. a. Graph the functions = and = on the same aes. b. What do ou notice about the graphs for the values of between 1 and 3? c. How do ou think the graph of = 8 would compare to the graphs of = and =? 37. Eplain Mathematical Ideas (1)(G) Find the range of the function f () = 500 # 1 using the domain 51,, 3, 4, 56. Eplain wh the definition of eponential function states that b 1. Evaluate each function over the domain {, 1, 0, 1,, 3}. As the values of the domain increase, do the values of the range increase or decrease? 38. f () = = h() = 0.1 Solve each equation. 41. = = # = # - 15 = Suppose (0, 4) and (, 36) are on the graph of an eponential function. a. Use (0, 4) in the general form of an eponential function, = a # b, to find the value of the constant a. b. Use our answer from part (a) and (, 36) to find the value of the constant b. c. Write a rule for the function. d. Evaluate the function for =- and = A population of 150 dandelions is growing in a meadow. A computer model predicts that the population will increase b a factor of 1. ever week. Write an equation to show the population as a function of time, in which is the number of weeks. What will be the approimate population after 4 weeks? PearsonTEXAS.com 393

14 47. A bank offers a special account in which the accept an initial deposit of 3 cents, and then double the value of the account ever month. a. Write an equation to show the value of the account in dollars as a function of months. What is the value in the account after 3 months? b. Select Techniques to Solve Problems (1)(C) Select a technique such as number sense, mental math, or estimation to determine when the value of the account will be more than $15. TEXAS End-of-Course PRACTICE 48. A population of 30 swans doubles ever 10 r. Which graph represents the population growth? A. C Swans Years B. D. 30 Swans 15 Swans Swans Years Years Years 49. Which equation do ou get when ou solve = - 1 for? F. = - 6 H. = G. = + 6 J. = Lesson 9-1 Eponential Functions

15 9- Eponential Growth and Deca TEKS FOCUS TEKS (9)(B) Interpret the meaning of the values of a and b in eponential functions of the form f () = ab in real-world problems. TEKS (1)(A) Appl mathematics to problems arising in everda life, societ, and the workplace. Additional TEKS (1)(G), (9)(C), (9)(D) VOCABULARY Compound interest interest paid on both the principal and the interest that has alread been paid Deca factor 1 minus the percent rate of change epressed as a decimal for an eponential deca situation; the base b in the function = a # b Eponential deca a situation that can be modeled with a function of the form = a # b, where a 7 0 and 0 6 b 6 1 Eponential growth a situation that can be modeled with a function of the form = a # b, where a 7 0 and b 7 1 Growth factor 1 plus the percent rate of change epressed as a decimal for an eponential growth situation; the base b in the function = a # b Appl use knowledge or information for a specific purpose, such as solving a problem ESSENTIAL UNDERSTANDING An eponential function can model growth or deca of an initial amount. Ke Concept Eponential Growth Definitions Eponential growth can be modeled b the function = a # b, where a 7 0 and b 7 1. The base b is the growth factor, which equals 1 plus the percent rate of change epressed as a decimal. Graph Algebra The initial amount (when 0) is also the -intercept. T a b d eponent c The base, which is greater than 1, is the growth factor. a b b 1 O (0, a) PearsonTEXAS.com 395

16 Ke Concept Compound Interest When a bank pas interest on both the principal and the interest an account has alread earned, the bank is paing compound interest. You can use the following formula to find the balance of an account that earns compound interest. A = P(1 + n) r nt A = the balance P = the principal (the initial deposit) r = the annual interest rate (epressed as a decimal) n = the number of times interest is compounded per ear t = the time in ears Ke Concept Eponential Deca Definitions Eponential deca can be modeled b the function = a # b, where a 7 0 and 0 6 b 6 1. The base b is the deca factor, which equals 1 minus the percent rate of change epressed as a decimal. Algebra The initial amount (when 0) is also the -intercept. T a b d eponent c The base is the deca factor. Graph O a b 0 b 1 (0, a) Problem 1 Graphing an Eponential Growth Function You are observing a bacteria population in a laborator culture. The population doubles ever 30 min. The function p() = 500 ~ models the population, where is the number of 30-minute periods since ou began observing. Does the point ( 1, 50) have meaning in this situation? =-1 corresponds to the time 30 min before ou start observing the bacteria population. Thirt minutes before ou start, the number of bacteria is 50. A Graph the function p() = 500 ~, identifing the -intercept and asmptote. The equation p() = 500 # is in the form = a # b, where a = 500 and b =. Since b 7 1, the function represents eponential growth. The graph rises from left to right and has a -intercept of 500. The asmptote is the -ais, with the graph approaching the -ais as decreases. B What is the meaning of the -intercept in this situation? The -intercept 500 occurs when is 0, which corresponds to the time ou start observing the population. There are 500 bacteria present when ou start observing O Lesson 9- Eponential Growth and Deca

17 Problem TEKS Process Standard (1)(A) Modeling Eponential Growth Economics Since 005, the amount of mone spent at restaurants in the United States has increased about 7% each ear. In 005, about $360 billion was spent at restaurants. When can ou use an eponential growth function? You can use an eponential growth function when an initial amount increases b a fied percent each time period. A If the trend continues, about how much will be spent at restaurants in 015? Relate = a # b Use an eponential function. Define Let = the number of ears since 005. Let = the annual amount spent at restaurants (in billions of dollars). Let a = the initial amount spent (in billions of dollars), 360. Let b = the growth factor, which is = Write = 360 # 1.07 Use the equation to predict the annual spending in 015. = 360 # 1.07 = 360 # is 10 r after 005, so substitute 10 for. 708 Round to the nearest billion dollars. About $708 billion will be spent at restaurants in 015 if the trend continues. B What is an epression that represents the equivalent monthl increase of spending at U.S. restaurants in 005? You will need to find an epression of the form r m, where r is approimatel the monthl growth factor and m is the number of months. You know that 1.07 represents the earl increase where is the number of ears = There are 1 months in ears. = Power raised to a power Simplif. = m Let 1 = m, the number of months. The epression m represents the equivalent monthl increase of spending. PearsonTEXAS.com 397

18 Problem 3 Compound Interest Finance Suppose that when our friend was born, our friend s parents deposited $000 in an account paing 4.5% interest compounded quarterl. What will the account balance be after 18 r? Is the formula an eponential growth function? Yes. You can rewrite the formula as A = P r n n 4 t. So, it is an eponential function with initial amount P and growth factor 11 + r n n. $000 principal 4.5% interest interest compounded quarterl A = P(1 + r n) nt Use the compound interest formula. = 000( ) 4 # 18 Substitute the values for P, r, n, and t. = 000(1.0115) 7 Simplif. The balance will be $ after 18 r. Account balance in 18 r Use the compound interest formula. Problem 4 Writing an Eponential Deca Function How do the equations for eponential growth and deca functions compare? The general equation for both functions is f () = ab, but b 7 1 for growth, and 0 6 b 6 1 for deca. The value of a function decreases b a factor of 0.3 with ever unit increase in. When is 0, the value of the function is 1. Write an eponential function in the form f () = ab to describe the situation. f () = ab f () = 1 # b f () = 1 # 0.3 Write the equation for eponential deca. Substitute 1 for a, the initial amount or f(0). Substitute 0.3 for b, the factor b which f() decreases for ever unit increase in. The function that describes the situation is f () = 1 # 0.3. Problem 5 Graphing an Eponential Deca Function Graph the function = 3 ~ 0.5. Identif the -intercept and the asmptote. The equation = 3 # 0.5 is in the form = a # b, where a = 3 and b = 0.5. Since 0 6 b 6 1, the function represents eponential deca. The graph falls from left to right and has a -intercept of 3. The asmptote is the -ais, with the graph approaching the -ais as increases O Lesson 9- Eponential Growth and Deca

19 Problem 6 TEKS Process Standard (1)(G) Modeling Eponential Deca STEM Phsics The kilopascal is a unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For ever 1000-m increase in altitude, the pressure decreases about 11.5%. What is the approimate pressure at an altitude of 3000 m? Will the pressure ever be negative? No. The range of an eponential deca function is all positive real numbers. The graph of an eponential deca function approaches but does not cross the -ais. Relate = a # b Use an eponential function. Define Let = the altitude (in thousands of meters). Let = the atmospheric pressure (in kilopascals). Let a = the initial pressure (in kilopascals), 101. Let b = the deca factor, which is = Write = 101 # Use the equation to estimate the pressure at an altitude of 3000 m. = 101 # = 101 # Substitute 3 for. 70 Round to the nearest kilopascal. The pressure at an altitude of 3000 m is about 70 kilopascals. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXAS.com. 1. Appl Mathematics (1)(A) The value of a stock purchase increases b a factor of 1.5 ever ear. The value of the stock, in dollars, as a function of time, in ears, is given b the equation = 100(1.5). Graph this function and identif the -intercept and the asmptote. What does the -intercept show about the stock?. Appl Mathematics (1)(A) A population of strangler figs, a tpe of vine, was established in the forest man ears ago. The total length of the vines, in feet, over time, in ears, is modeled b the equation = 1() 5. The value = 0 represents the present. Graph this function. What are the -intercept and the asmptote of the graph? What does the -intercept of the graph represent? 3. Eplain Mathematical Ideas (1)(G) An eponential function has the form = a(b) k, where a and b are positive and b 1. Without graphing, how do ou know whether the function will model eponential growth or deca? Eplain our answer. Identif the initial amount a and the growth factor b in each eponential function. 4. g() = 14 # 5. = 150 # = 5,600 # f (t) = 1.4 t PearsonTEXAS.com 399

20 8. The number of students enrolled at a college is 15,000 and grows 4% each ear. a. The initial amount a is.. b. The percent rate of change is 4%, so the growth factor b is 1 + =. c. To find the number of students enrolled after one ear, ou calculate 15,000 #. d. Complete the equation = # to find the number of students enrolled after ears. e. Use our equation to predict the number of students enrolled after 5 r. 9. Appl Mathematics (1)(A) A population of 100 frogs increases at an annual rate of %. How man frogs will there be in 5 ears? Write an epression to represent the equivalent monthl population increase rate. Find the balance in each account after the given period. 10. $4000 principal earning 6% compounded annuall, after 5 r 11. $1,000 principal earning 4.8% compounded annuall, after 7 r 1. $500 principal earning 4% compounded quarterl, after 6 r 13. $5000 deposit earning 1.5% compounded quarterl, after 3 r 14. $775 deposit earning 4.5% compounded annuall, after 1 r 15. $3500 deposit earning 6.75% compounded monthl, after 6 months Write the eponential function that describes each situation. 16. The value of the function decreases b a factor of 1 with each unit increase in. When = 0, the value of the function is The initial amount of the function is 1, and then it decreases b a factor of 4 5 with each unit increase in. 18. The initial value of the function is 3000, and then it loses 0% of its value with each unit increase in. 19. Eplain Mathematical Ideas (1)(G) The graph of an eponential function passes through the points (3, 100) and (5, 5). Write a function f () = ab that describes this situation. Eplain our reasoning. 0. Appl Mathematics (1)(A) A famil bus a car for $0,000. The value of the car decreases about 0% each ear. After 6 r, the famil decides to sell the car. Should the sell it for $4000? Eplain. 1. Appl Mathematics (1)(A) You invest $100 and epect our mone to grow 8% each ear. About how man ears will it take for our investment to double?. Displa Mathematical Ideas (1)(G) Suppose ou start a lawn-mowing business and make a profit of $400 in the first ear. Each ear, our profit increases 5%. a. Write a function that models our annual profit. b. If ou continue our business for 10 r, what will our total profit be? 400 Lesson 9- Eponential Growth and Deca

21 Graph the function. Identif the -intercept and the asmptote. 3. f () = 3( 1 ) 4. f () = 0.9( 3) 5. f () = (0.1) Use Multiple Representations to Communicate Mathematical Ideas (1)(D) The graph of an eponential function is decreasing. The -intercept is 300 and the range of the function is Sketch a graph of the function, and write a possible equation for the function. Identif the initial amount a and the deca factor b in each eponential function. 7. = 5 # f () = 10 # g() = 100( 3) 30. = 0.1 # Appl Mathematics (1)(A) The population of a cit is 45,000 and decreases % each ear. If the trend continues, what will the population be after 15 r? State whether each graph shows an eponential growth function, an eponential deca function, or neither STEM 34. Appl Mathematics (1)(A) Radioactive deca is an eample of eponential deca. The half-life of a radioactive substance is the length of time it takes for half of the atoms in a sample of the substance to deca. Cesium-137 is a radioisotope used in radiolog where levels are measured in millicuries (mci). Use the graph at the right. Write the equation for the graph of Cesium-137 deca. What is a reasonable estimate of the half-life of cesium-137? What is the asmptote of the graph? Eplain what it represents. Level (mci) Cesium-137 Deca Time (r) TEXAS End-of-Course PRACTICE 35. A new fitness center opens with 10 members. Ever month the fitness center increases the number of members b 40 members. How man members will the fitness center have after being open for 8 months? 36. What is the slope of the line that passes through the points (1, 1) and (, -1)? 37. What is the simplified form of 5 -? PearsonTEXAS.com 401

22 9-3 Modeling Eponential Data TEKS FOCUS TEKS (9)(E) Write, using technolog, eponential functions that provide a reasonable fit to data and make predictions for real-world problems. TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. VOCABULARY Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. Additional TEKS (1)(C), (9)(A), (9)(B), (9)(C), (9)(D) ESSENTIAL UNDERSTANDING You can use eponential functions to model some sets of data. When more than three data points suggest an eponential function, ou can use the regression feature of a graphing calculator to find an eponential model. Problem 1 TEKS Process Standard (1)(E) Modeling Real-World Data Transportation The data at the right give the value of a used car over time. Which tpe of function best models the data? Write an equation to model the data. The value of a used car over time The most appropriate model for the data Graph the data and then use differences or ratios to find a model for the situation. Step 1 Step Graph the data. Test for a common ratio. Value of Used Car Years Value ($) 0 1, , Value ($), 1, Years, Years Value ($) 0 1, , ,065 1, , The graph curves and does not look The value of the car is roughl 0.88 quadratic. It ma be eponential. times its value the previous ear. continued on net page 40 Lesson 9-3 Modeling Eponential Data

23 Problem 1 continued Step 3 Step 4 Write an eponential model. Test two points other than (0, 1,575). Relate f () = a # b Test (, 9750): Test (4, 7540): Define Let a = the initial value, 1,575. = 1,575 # 0.88 = 1,575 # Let b = the deca factor, Write f () = 1,575 # 0.88 The point (, 9738) is close to the data point (, 9750). The point (4, 7541) is close to the data point (4, 7540). The equation = 1,575 # 0.88 models the data. Problem TEKS Process Standard (1)(C) Modeling Eponential Data Using Technolog The table shows the numbers of bacteria in a culture after the given numbers of hours. Predict when the number of bacteria will reach 10,000. You can select technolog as a tool to help ou solve this problem. Using a graphing calculator, ou can determine whether an eponential function is a reasonable model. Step 1 Enter the data. L1 L L L3(1)= Hour Bacteria Step Use EpReg. The model is f () = (1.11). Step 3 Graph the data and the function to confirm the model. EpReg = a * b^ a = b = r = r = An eponential model is reasonable. continued on net page PearsonTEXAS.com 403

24 Problem continued How can I see -values that are closer together in the table? Go into the table setup for our calculator. You can change the increment between successive -values. Step 4 Use the model to predict how long it will take for the number of bacteria to reach 10,000. Use tables or the trace feature on the graph. You ma need to adjust the viewing window. X Y1 Since a tenth of an hour is 6 minutes, 15. hours means 15 hours 1 minutes Y1 = Y1 = * 1.1 X = Y = How does the domain for the real-world data differ from the domain for the function? In the function, could be an real number. The bounce number, however, must come from the set {0, 1,, }. So the domain is the set of whole numbers. The number of bacteria will reach 10,000 at about 15 hours 1 minutes. Problem 3 Analzing Features of Eponential Data A ball is dropped from a height of 75 in. The heights of the ball s rebounds after subsequent bounces are shown in the table. A What is an eponential model for this data? Enter the data into lists in a graphing calculator, and perform an eponential regression on the data. An eponential model for the data is f () = 75 # 0.6. B Identif the domain and range of the function in part (A) in the contet of the problem. Epress the domain and range using inequalities. The domain is the bounce number. The bounce number cannot be negative, so the domain is Ú 0, where is an integer. The range is the height of the rebound. The height starts at 75 in. and then decreases, but cannot be negative. It will get closer and closer to 0, but never reach 0 according to the model. The range is 0 6 f () 75, where f () = 75 # 0.6. C What is the -intercept of the function in part (A), and what does it mean in the contet of the problem? The -intercept is the -value when = 0. When = 0, f () = 75 # = 75, so the -intercept is 75. In the contet of the problem, this is the height of the ball after drop 0, or the initial height of the ball, 75 in. D An eponential function is a function in the form f () = a ~ b. What are a and b in the function in part (A)? What do a and b mean in the contet of the problem? In the function f () = 75 # 0.6, a = 75, and b = 0.6. Bounce Height of Rebound (in.) Ball Rebounds The value of a is the initial height of the ball, 75 in. The value of b is the ratio between rebounds. Each rebound is 0.6, or 6%, of the height of the previous rebound Lesson 9-3 Modeling Eponential Data

25 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXAS.com. Identif the domain and range of the function. Epress the range using an inequalit. 1. f () = 15 # f () =-3.5 # 1. f () = 460 # f () =-4 # Find f (0), f (4), and f ( ) for each function. Round to the nearest tenth. 5. f () = 8 #.0 7. f () =-6.4 # f () = 107 # f () =-5 # Select Tools to Solve Problems (1)(C) The table at the right shows the length and weight of a species of fish. a. Select one of the following tools to use to find an eponential model for the data and predict the weight of a fish that is 4 in. long. Eplain our choice. Fish Lengths and Weights Length (in.) Weight (lb) paper and pencil technolog manipulatives b. Find an eponential model for the data. Predict the weight of a fish that is 4 in. long. 10. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) A culture is started with 50,000 bacteria, and then an antibiotic is added to the culture. The table at the right shows the number of bacteria after the antibiotic is added. a. Find an eponential model that is a reasonable fit for the data. b. How long will it take for the number of bacteria to fall below 5000? Bacteria Population Time Since Antibiotic Number of Added (hr) Bacteria 0 50, ,100 35, , ,131 c. What is the -intercept of our function? What does it mean in the contet of the problem? d. What are the values of a and b for our function? What do these values mean in the contet of the problem? e. What are the real-world domain and range of the function? Epress each using inequalities. f. Graph the function. PearsonTEXAS.com 405

26 11. Select Tools to Solve Problems (1)(C) The table below shows the balance in Mr. Harris s retirement savings account for several ears after he retired. Retirement Account Balance Years Since Retirement Balance ($) 0 75, ,375 56, , ,475 a. Use technolog to find an eponential model that is a reasonable fit for the data. b. Predict when the balance will be less than $100,000, assuming that the pattern continues. 1. Use Representations to Communicate Mathematical Ideas (1)(E) Samantha took out a loan that does not require her to make an paments for 1 months, although interest is being charged to her account. The table below shows the total amount she owes each month after taking the loan. Loan Balance Month Total Amount Owed ($) a. Find an eponential model that is a reasonable fit for the data. b. How much will Samantha owe at the end of 1 months, assuming she does not make an paments before then? c. What is the -intercept of our function? What does it mean in the contet of the problem? d. What are the values of a and b for our function? What do these values mean in the contet of the problem? e. What are the real-world domain and range of the function? Epress each using inequalities. f. Graph the function. 13. Appl Mathematics (1)(A) The function f () = # represents the value in millions of dollars of a software compan, where is the number of ears since the compan was started. a. What is the -intercept of the function, and what does it mean in the contet of the problem? b. What are the a and b values of the function, and what do these values mean in the contet of the problem? c. According to the model, predict what the value of the compan will be 10 ears after it was started. 406 Lesson 9-3 Modeling Eponential Data

27 14. The function f () = 346 # 0.95 represents the population of a town, where is the number of ears since a. What is the -intercept of the function, and what does it mean in the contet of the problem? b. What are the a and b values of the function, and what do these values mean in the contet of the problem? c. According to the model, predict what the population will be in 00. For f () = a ~ b, determine whether each statement is alwas, sometimes, or never true. 15. The -intercept is a. 16. The value of f () increases as the value of increases. 17. If a 7 0 and b 7 0, the graph of the function is entirel in Quadrant 1 of the coordinate plane. 18. The range includes negative numbers. TEXAS End-of-Course PRACTICE 19. What is the range of f () = 43 #.4? A. f () 7 0 B. f () 7.4 C. f () D. 0 6 f () The function f () = 4,950 # models the value of a car, where is the age of the car in ears. How old will the car be when the value of the car becomes less than $5000? F. ears G. 3 ears H. 5 ears J. 7 ears 1. The value of Edward s savings account is modeled b the function f () = 975 # 1.08, where is the number of ears since the account was opened. In how man ears will the account value be double the original amount? A. 0 B. 5 C. 30 D. 35. The table shows the number of contestants remaining after each round of a singing competition. What is an eponential model for the data? Use our model to predict how man rounds are required to select a single winner. Round Number of Contestants Remaining Singing Competition PearsonTEXAS.com 407

28 Topic 9 Review TOPIC VOCABULARY asmptote, p. 386 p. 396 p. 396 p. 396, p. 386 p. 395 p. 395 Check Your Understanding Choose the correct term to complete each sentence. 1. For a function = a # b, where a 7 0 and b 7 1, b is the?.. For a function = a # b, where a 7 0 and 0 6 b 6 1, b is the?. 3. The function = a # b models? for a 7 0 and b The function = a # b models? for a 7 0 and 0 6 b Eponential Functions Quick Review An eponential function involves repeated multiplication of an initial amount a b the same positive number b. The general form of an eponential function is = a # b, where a 0, b 7 0, and b 1. Eample What is the graph of = 1 ~ 5? Make a table of values. Graph the ordered pairs O Eercises Evaluate each function for the domain 51,, f () = 4 6. = = 40 ( 1 ) 8. f () = 3 # Graph each function. 9. f () = = 0.5(0.5) 11. f () = 1 # 3 1. = A population of 50 bacteria in a laborator culture doubles ever 30 min. The function p() = 50 # models the population, where is the number of 30-min periods. a. How man bacteria will there be after h? b. How man bacteria will there be after 1 da? 408 Topic 9

29 9- Eponential Growth and Deca Quick Review When a 7 0 and b 7 1, the function = a # b models eponential growth. The base b is called the growth factor. When a 7 0 and 0 6 b 6 1, the function = a # b models eponential deca. In this case the base b is called the deca factor. Eample The population of a cit is 5,000 and decreases 1% each ear. Predict the population after 6 r. = 5,000 # 0.99 Eponential deca function = 5,000 # Substitute 6 for. 3,537 Simplif. The population will be about 3,537 after 6 r. Eercises Tell whether the function represents eponential growth or eponential deca. Identif the growth or deca factor. 14. = 5. # f () = 7 # = 0.15 ( 3 ) 17. g() = 1.3 ( 1 4 ) 18. Suppose $000 is deposited in an account paing.5% interest compounded quarterl. What will the account balance be after 1 r? 19. A band performs a free concert in a local park. There are 00 people in the crowd at the start of the concert. The number of people in the crowd grows 15% ever half hour. How man people are in the crowd after 3 h? Round to the nearest person. 9-3 Modeling Eponential Data Quick Review For some data, an eponential model is a good fit. You can enter the data into our calculator and use the regression feature to find an eponential model. You can then use our model to answer questions about the data b eamining the graphs or tables, or b using the function. Eponential functions are written in the form f () = a # b. Eample The table shows the value of a computer. Find an eponential model, and find the computer s value after 7 ears. Eercises The table shows the population of geese in a park beginning in the ear 003. Use the table to answer Eercises 0 5. Geese Population Year Population Find an eponential model to best fit the data. 1. Predict the population in the ear According to the model, in what ear will the population first be greater than 150 geese? Age Value ($) Computer Value What is the -intercept of our function, and what does it mean in the contet of the problem? 4. What are the a and b values of our function, and what do the mean in the contet of the problem? The data appear to be eponential. The calculator gives a regression model of f () = 00 # Use the table and the eponential model to see that the computer will be worth about $85 after 7 ears. 5. What are the real-world domain and range of the function? PearsonTEXAS.com 409

30 Topic 9 TEKS Cumulative Practice Multiple Choice Read each question. Then write the letter of the correct answer on our paper. 1. If the graph of the function = - 6 were shifted 3 units down, which equation could represent the shifted graph? A. = 3-6 C. = - 9 B. = - 3 D. = 3-3. Which ordered pair is a solution of 3-6 0? F. (7, 1) H. (8, 0) G. (5, -6) J. (-1, -4) 3. Brianna has a clindrical glass that is 15 cm tall. The diameter of the base is 5 cm. About how much water can the glass hold? A. 75 cm 3 C. 95 cm 3 B. 118 cm 3 D cm 3 4. What is the solution of this sstem of equations? -3 + =- + =-6 F. (-, -6) H. (1, 5) G. ( -1, -5) J. ( -3, -) 5. Jeremiah made the graph at the right to show how much mone he saved after working for a few months. Which of the following represents the amount of mone Jeremiah had when he started working? A. -intercept B. -intercept C. slope D. domain Savings, Savings $00 $150 $100 $50 $ Month, 6. Eduardo is drawing the graph of a function. Each time the -value increases b 3, the -value decreases b 4. The function includes the point (1, 3). Which could be Eduardo s graph? F. 4 O G. 4 O H. 4 O J The data shown in the table at the right represent points on a line. What is the -intercept of the line? A. -5 B. -3 C. 0 D..5 O What is the factored form of ? F. ( + ) (3-8) G. ( + 4) (3 - ) H. ( + ) (3-4) J. (3 + ) ( - 4) 9. The formula for the area A of a circle is A = pr, where r is the radius of the circle. Which equation can be used to find the radius? A. r = 1Ap p C. r = A p B. r = A p D. r = 1Ap Topic 9 TEKS Cumulative Practice

31 10. Which function has -values that alwas increase when the corresponding -values increase? F. = G. = + H. = + J. = What is the solution of this sstem of equations? + = =-7 A. (1, 11) C. (-1, -11) B. (-11, 1) D. (11, 1) Gridded Response 1. The formula h =-16t + c can be used to find the height h, in feet, of a falling object t seconds after it is dropped from a height of c feet. Suppose an object falls from a height of 40 ft. How long, in seconds, will the object take to reach the ground? Round our answer to the nearest tenth. 13. A 5-foot ladder rests against a wall. The top of the ladder reaches 0 feet high on the wall. How far awa from the wall is the bottom of the ladder in feet? 0 ft 5 ft Constructed Response 17. The list below shows the heights, in inches, of the students in Core s class. 60, 64, 58, 57, 60, 65, 51, 53, 57, 56 How man students are more than 5 ft tall? 18. Write the sstem of inequalities for the graph below. Show our work. 4 4 O Mr. Wong drove to the grocer store. The graph at the right shows his distance from home during the drive. How man times did Mr. Wong stop the car before reaching the grocer store? 0. The volume of a rectangular prism is 70 in. 3. The height of the prism is 10 in. The width is 4 in. What is the length, in inches? 1. What is the slope of the line below? 6 Distance Time 14. What is the fifth term in the sequence below? 3.5, 4, 4.75, 5.5, c 15. What is the solution of the following proportion? -a - 3(a - ) 4 = Mariah made a model of a square pramid. The height h of the pramid is 6 in. The area of the base B is 36 in.. What is the volume V, in cubic inches, of the pramid? Use the formulav = 1 3 Bh. 4 O 4. Your cellphone plan costs $39.99 per month plus $.10 for ever tet message that ou receive or send. This month, ou receive 7 tet messages and send 10 tet messages. What is our bill, in dollars, for this month? PearsonTEXAS.com 411

32 Additional Practice Topic 9 Lesson 9-1 Evaluate each function over the domain { 1, 0, 1, }. As the values of the domain increase, do the values of the function increase or decrease? 1. = 3. = ( 3 4) 3. = = ( 1 ) # 3 5. =-3 # 7 6. =-(4) 7. = 3 # ( 1 5) 8. = 9. = # = (0.8) 11. =.5 1. =-4 # (0.) Graph each eponential function. Identif the domain, range, -intercept, and asmptote. 13. = 1 # =-4 # 15. = 4 # (0.3) Write and solve an eponential equation to answer each question. 16. Suppose an investment of $5,000 doubles ever 1 ears. How much is the investment worth after 36 ears? After 48 ears? 17. Suppose 15 animals are taken to an island, and then their population triples ever 8 months. How man animals will there be in 4 ears? 18. The population of a cit this ear is 34,500. The population is epected to grow b 3% each ear. What will the population of the cit be in 1 ears? Lesson 9- Identif each function as eponential growth or eponential deca. Then identif the growth factor or deca factor. 19. = 8 0. = = 9( 1 ). = 4 # 9 3. = = 3 # = 5( 1 4) 6. = 0.1 # = 0.7 # 3.3 Graph each eponential function, and identif the -intercept and the asmptote. 8. = 1 ( 1 4) 9. = 4 # (0.) 30. = Additional Practice