Explore 1 Graphing and Analyzing f(x) = e x. The following table represents the function ƒ (x) = (1 + 1 x) x for several values of x.

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1 1_ Locker LESSON 13. The Base e Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes of the graphs of ƒ () = b and ƒ () = log b () where b is, 1, and e when ƒ () is replaced b aƒ (), ƒ () + d, and ƒ ( - c ) for specific positive and negative real values of a, c, and d. Also A..A, A.5.B, A.5.D. A.7.I Mathematical Processes A.1.A To appl mathematics to problems arising in everda life, societ, and the workplace. Language Objective.C.3,.D.1,.I., 3.G., 3.G.3 Work with a partner to eplain, in words, how the graph of a transformed eponential function with base e compares to the same transformation on graphs of other eponential functions. ENGAGE Essential Question: How is the graph of g () = a e - h + k related to the graph of f () = e? Possible answer: The graph of g () = ae - h + k involves transformations of the graph of f () = e. In particular, the graph of g () is a vertical stretch or compression of the graph of f () b a factor of a, a reflection of the graph across the -ais if a <, and a translation of the graph h units horizontall and k units verticall. Houghton Mifflin Harcourt Publishing Compan Name Class Date 13. The Base e Essential Question: How is the graph of g () = ae -h + k related to the graph of f () = e? A.5.A Determine the effects on the ke attributes on the graphs of f() = b where b is e when f() is replaced b af(), f() + d, and f( - c) for specific positive and negative real values of a, c, and d. Also A..A, A.5.B, A.5.D, A.7.I/ Eplore 1 Graphing and Analzing f() = e The following table represents the function ƒ () = (1 + 1 ) for several values of f () As the value of increases without bound, the value of ƒ () approaches a number whose decimal value is.718 This number is irrational and is called e. You can write this in smbols as ƒ () e as +. If ou graph ƒ () and the horizontal line = e, ou can see that = e is the horizontal asmptote of ƒ (). 8 6 = e f() 6 Even though e is an irrational number, it can be used as the base of an eponential function. The number e is sometimes called the natural base of an eponential function and is used etensivel in scientific and other applications involving eponential growth and deca. Fill out the table of values below for the function ƒ () = e. Use decimal approimations. 8 Resource Locker f () = e _ e = _ e = PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and wh it would be useful to have a mathematical model for the amount of carbon dioide in the atmosphere. Then preview the Lesson Performance Task. Module Lesson DO NOT EDIT--Changes must be made through File info CorrectionKe=TX-B Name Class Date 13. The Base e Essential Question: How is the graph of g () = ae -h + k related to the graph of f () = e? A.5.A Determine the effects on the ke attributes on the graphs of f() = b where b is e when f() is replaced b af(), f() + d, and f( - c) for specific positive and negative real values of a, c, and d. Also A..A, A.5.B, A.5.D, A.7.I/ The following table represents the function ƒ () = (1 + 1 ) for several values of. Eplore 1 Graphing and Analzing f() = e Houghton Mifflin Harcourt Publishing Compan f () = e f() 6 Resource As the value of increases without bound, the value of ƒ () approaches a number whose decimal value is.718 This number is irrational and is called e. You can write this in smbols as ƒ () e as +. If ou graph ƒ () and the horizontal line = e, ou can see that = e is the horizontal asmptote of ƒ (). Even though e is an irrational number, it can be used as the base of an eponential function. The number e is sometimes called the natural base of an eponential function and is used etensivel in scientific and other applications involving eponential growth and deca. Fill out the table of values below for the function ƒ () = e. Use decimal approimations f () = e e = _ e = Module Lesson A_MTXESE35397_U6M13L.indd 751 1/11/15 :36 PM HARDCOVER PAGES 59 5 Turn to these pages to find this lesson in the hardcover student edition. 751 Lesson 13.

2 B Plot the points on a graph. EXPLORE Graphing and Analzing f () = e C The domain of ƒ () = e is D The range of ƒ () = e is (, -1). 6 8 Is the function increasing or decreasing? For what values of is it increasing/decreasing? The function is increasing throughout its domain for all values of. E ) The function s -intercept is (, 1 because ƒ () = e = 1 and = is in the domain F G of the function. Another point on the graph that can be used as a reference point is (1, e ). Identif the end behavior. ƒ () as ƒ () as - There is a horizontal asmptote at =. Reflect - < < > 1. What is the relationship between the graphs of ƒ () = e, g () =, and h () = 3? (Hint: Sketch the graphs on our own paper.) < e < 3, so < e < 3. Thus g () < f () < h () and the graph of f () is between the graphs of h () and g ().. Houghton Mifflin Harcourt Publishing Compan INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Discuss with students their previous eperience with the irrational number π, which is so important in geometric relationships. Eplain that in this lesson the will learn about another irrational constant, e, which is also applicable to man situations. QUESTIONING STRATEGIES What is f ()? What does this tell ou about the graph of ƒ () = e? 1; the graph passes through the point (,1). Does the graph of ƒ () = e ever intersect the ais? Wh or wh not? No; there is no value for for which e =. As decreases without bound, e approaches but never reaches. INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Have students find the values of 1 for = 1, 1, 1, and 1. Then, have them find the values of for the same values. Ask them to describe what happens as becomes greater. Module Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes In this lesson, students have the opportunit to practice TEKS A.1.A, which asks them to appl mathematics to problems arising in everda life, societ, and the workplace. The natural base, e, is used in man applications of continuous change, from compound interest to applications of probabilit, statistics, and trigonometr. The Base e 75

3 EXPLORE Predicting Transformations of the Graph of f () = e AVOID COMMON ERRORS Students are accustomed to seeing letters representing variables; the ma become confused in working with e. Remind them that, like π, e is a constant representing an irrational number. Eplore Predicting Transformations of the Graph of f () = e The parent function, ƒ () = e, can be transformed into a different eponential function with base e depending on the value and sign of the constant parameters h, k, and a. As in previous transformation of graphs, the effect of h on the graph of g () = ƒ ( - h), the effect of k on the graph of g () = ƒ () + k, and the effect of a on the graph of g () = aƒ () can all be predicted from the value and sign of the parameters. Predict the effect of each transformation on the following graphs and then use a graphing calculator to confirm our prediction. A Transform ƒ () = e into g () = e - 1. The function g () = e - 1 is of the form g () = f ( - h), so g () = e - 1 represents a horizontal translation b 1 units(s) to the right. B Transform ƒ () = e into g () = e + 1. The function g () = e + 1 is of the form g () = f ( - h), so g () = e + 1 represents a horizontal translation b 1 units(s) to the left. QUESTIONING STRATEGIES How do ou know that the graph of g () = e + 3 will be a translation of the graph of ƒ () = e to the left? The epression + 3 equals - (-3), so h is negative. Is the graph of g () = 1 e + 1 a stretch or a shrink of the graph of ƒ () 1 = e? Eplain. It is a vertical shrink b a factor of because f (), not, is being multiplied b the constant. INTEGRATE TECHNOLOGY Use a graphing calculator demonstration or guide students in a calculator investigation to help them see both graphicall and numericall that the domain of all eponential functions is the set of all real numbers, and that the range is directl related to the equation of the horizontal asmptote. Use the TABLE feature to show that the -values get closer and closer to a fied value but never get there, whether decreases or increases, depending on whether ou are looking at an increasing or decreasing function. Houghton Mifflin Harcourt Publishing Compan C Transform ƒ () = e into g () = e +. The function g () = e + is of the form g () = translation b units(s) up. D ƒ () = e into g () = e -. The function g () = e - is of the form g () = translation b units(s) down. E Transform ƒ () = e into g () = e. The function g () = e is of the form g () = b a factor of. F Transform ƒ () = e into g () = 1 e. f () + k, so g () = e + represents a f () + k, so g () = e - represents a, so g () = e represents a vertical The function g () = 1 e is of the form g () = af (), so g () = 1 e represents a vertical compression 1_ b a factor of. G Transform ƒ () = e into g () = - e. The function g () = - e is of the form g () = stretch af () b a factor of and a reflection across the -ais. -af(), so g () = - e represents a vertical vertical vertical H Transform ƒ () = e into g () = - 1 e. The function g () = - 1 e is of the form g () = -af(), so g () = - 1 e represents a vertical compression 1_ b a factor of and a reflection across the -ais. stretch Module Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have pairs of students work together to find real-world data that would fit an eponential growth model of the form ƒ () = ae c where c is a positive constant. Have students write a paragraph about how understanding the eponential growth model can help them predict trends in the data. 753 Lesson 13.

4 Reflect. Discussion Describe the effects of the parameters h, k, and a on the domain, range, and asmptote of g () in regards to the domain, range, and asmptote of the parent function ƒ (). The parameter h shifts the graph horizontall, so the domain and the range do not change. The parameter k shifts the graph verticall, so the domain remains the same and the range changes from { < < } to { k < < }. The asmptote changes from = to = k. The parameter a leaves the domain the same, but if it is negative it changes the range from { < < } to { - < < }. Eplain 1 Graphing Combined Transformations of f () = e When graphing combined transformations of ƒ () = e that result in the function g () = a e - h + k, it helps to focus on two reference points on the graph of ƒ (), (, 1) and (1, e), as well as on the asmptote =. The table shows these reference points and the asmptote = for ƒ () = e and the corresponding points and asmptote for the transformed function, g () = a e - h + k. EXPLAIN 1 Graphing Combined Transformations of f () = e QUESTIONING STRATEGIES What are two reference points on the graph of ƒ () = e that will help when graphing combined transformations? (, 1) and (1, e), as well as the asmptote = f () = e g () = a e - h + k First reference point (, 1) (h, a + k) Second reference point (1, e) (h + 1, ae + k) Asmptote = = k Eample 1 Given a function of the form g () = a e - h + k, identif the reference points and use them to draw the graph. State the transformations that compose the combined transformation, the asmptote, the domain, and range. Write the domain and range using set notation. g () = 3 e Compare g () = 3 e to the general form g () = a e - h + k to find that h = -1, k =, and a = 3. Find the reference points of ƒ () = 3 e (, 1) (h, a + k) = (-1, 3 + ) = (-1, 7) (1, e) (h + 1, ae + k) = (-1 + 1, 3e + ) = (, 3e + ) State the transformations that compose the combined transformation. h = -1, so the graph is translated 1 unit to the left. k =, so the graph is translated units up. Houghton Mifflin Harcourt Publishing Compan a = 3, so the graph is verticall stretched b a factor of 3. a is positive, so the graph is not reflected across the -ais. Module Lesson DIFFERENTIATE INSTRUCTION Curriculum Integration Encourage students to research the histor and significance of the number e, using the Internet and/or a librar. Have students work either individuall or in a small group to write reports on their findings or to present their findings orall to classmates. Both written reports and oral presentations should include graphics. The Base e 75

5 The asmptote is verticall shifted to = k, so =. The domain is - < <. The range is >. Use the information to graph the function g () = 3 e (, 3e + ) 1 8 ( -1, 7) 6 = B g () = -.5 e Compare g () = -.5 e to the general form g () = a e - h + k to find that h =, k = -1, and a = -.5. Houghton Mifflin Harcourt Publishing Compan Find the reference points of g () = -.5 e (, 1) (h, a + k) = (, ) = (, -1.5 ) (1, e) (h + 1, ae + k) = ( + 1, -.5e + -1 ) = ( 3, -.5e -1 ) State the transformations that compose the combined transformation. h =, so the graph is translated units to the right. k = -1, so the graph is translated 1 unit down. a = -.5, so the graph is verticall stretched b a factor of.5. a is negative, so the graph is reflected across the -ais. Module Lesson A_MTXESE35397_U6M13L 755 LANGUAGE SUPPORT Communicate Math Have students work together to complete the table for a specific transformation applied to each of the functions. 1/13/15 :3 PM Tpe of Function Equation Graph Comparison to other graphs in this table Eponential with base Eponential with base 1 Eponential with base e 755 Lesson 13.

6 The asmptote is verticall shifted to = k, so = -1. The domain is - < <. The range is < -1. Use the information to graph the function g () = -.5 e = -1 f() (, -1.5) (3, -.5e-1) - Your Turn Given a function of the form g () = a e - h + k, identif the reference points and use them to draw the graph. State the asmptote, domain, and range. Write the domain and range using set notation. 3. g () = (-1) e a = -1; h = -; k = -3 Reference points: (, 1) (-, -) (1, e) (-1, -e -3) asmptote: = -3 Domain: { - < < } Range: { < -3}. g () = e a = ; h = 1; k = 1 Reference points: (, 1) (1, 3) (1, e) (, e + 1) asmptote: = 1 Domain: { - < < } Range: { > 1} = -3 (-, -) (-1, -3 -e) f() (, e+1) (1, 3) = Houghton Mifflin Harcourt Publishing Compan Module Lesson A_MTXESE35397_U6M13L.indd 756 1/3/15 1:3 AM The Base e 756

7 EXPLAIN Writing Equations for Combined Transformations of f () = e Eplain Writing Equations for Combined Transformations of f () = e If ou are given the transformed graph g () = a e - h + k, it is possible to write the equation of the transformed graph b using the reference points (h, a + k) and (1 + h, ae + k). Eample Write the function whose graph is shown. State the domain and range in set notation. AVOID COMMON ERRORS When students see the equation = ae r, the ma think that e is a variable because the are used to letters representing variables and because there are several other variables in the equation. Make sure that the understand that e is in fact a constant a specific irrational number like π. First, look at the labeled points on the graph. (h, a + k) = (, 6) (1 + h, ae + k) = (5, e + ) Find a, h, and k. (h, a + k) = (, 6), so h =. (1 + h, ae + k) = (5, e + ), so ae + k = e +. Therefore, a = and k =. Write the equation b substituting the values of a, h, and k into the function g () = a e - h + k (5, e+) (, 6) = QUESTIONING STRATEGIES What are two reference points on the graph of ƒ () = e that will help when writing equations for combined transformations? (,1) and (1,e) How is the first reference point transformed from the point (, 1) in the graph of ƒ () = e? The -coordinate is moved to h, and the -coordinate is moved to k and stretched awa from it verticall b a factor of a. Houghton Mifflin Harcourt Publishing Compan g () = e - + State the domain and range. Domain: - < < Range: > First, look at the labeled points on the graph. (h, a + k) = ( -, -8 ) (1 + h, ae + k) = ( -3, -e - 6 ) Find a, h, and k. (h, a + k) = (-, -8), so h = -. (1 + h, ae + k) = (-3, -e - 6), so ae + k = -e - 6. Therefore, a = - and k = = -6 (-, -8) (-3, -e-6) -1 Write the equation b substituting the values of a, h, and k into the function g () = a e -h + k. g () = - e State the domain and range. Domain: - < < Range: < -6 Module Lesson 757 Lesson 13.

8 Your Turn Write the function whose graph is shown. State the domain and range in set notation. 5. Find a, h, and k. (-, ) = 1 (h, a + k) = (-, ), so h = (-1, 1-e) (1 + h, ae + k) = (-1, 1 - e), so ae + k = -e + 1. Therefore, a = -1 and k = 1. - g () = - e Domain: { - < < } Range: { < 1} 6. Find a, h, and k. (h, a + k) = (1, -), so h = 1. (, 3e-5) (1 + h, ae + k) = (, 3e - 5), so - - ae + k = 3e - 5. Therefore, a = 3 and k = (1, -) = -5 g () = 3 e Domain: { - < < } Range: { > -5} EXPLAIN 3 Modeling With Eponential Functions Having Base e INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Have students consider how the would use natural deca functions to solve scientific half-life problems. Eplain 3 Modeling with Eponential Functions Having Base e Although the function ƒ () = e has base e.718, the function g () = e c can have an positive base (other than 1) b choosing an appropriate positive or negative value of the constant c. This is because ou can write g () as ( e c ) b using the Power of a Power Propert of Eponents. Eample 3 Solve each problem using a graphing calculator. Then determine the growth rate or deca rate of the function. The Dow Jones inde is a stock market inde for the New York Stock Echange. The Dow Jones inde for the period 198- can be modeled b V DJ (t) = 878 e.11t, where t is the number of ears after 198. Determine how man ears after 198 the Dow Jones inde will reach 3. Use a graphing calculator to graph the function. The value of the function is about 3 when 1.. So, the Dow Jones inde will reach 3 after 1. ears, or after the ear 199. Houghton Mifflin Harcourt Publishing Compan Module Lesson The Base e 758

9 In an eponential growth model of the form ƒ () = ae c, the growth factor 1 + r is equal to e c. To find r, first rewrite the function in the form ƒ () = a ( e c ). V DJ (t) = 878e.11t = 878 ( e.11 t ) Find r b using 1 + r = e c. 1 + r = e c 1 + r = e.11 r = e So, the growth rate is about 13%. The Nikkei 5 inde is a stock market inde for the Toko Stock Echange. The Nikkei 5 inde for the period can be modeled b V N5 (t) = 3,5 e -.381t, where t is the number of ears after 199. Determine how man ears after 199 the Nikkei 5 inde will reach 15,. Use a graphing calculator to graph the function. The value of the function is about 15, when 1. So, the Nikkei 5 inde will reach 15, after 1 ears, or after the ear. c In an eponential deca model of the form ƒ () = ae, the deca factor 1 - r is equal to e c. To find r, first rewrite the function in the form ƒ () = a ( e c ). V N5 (t) = 3,5 e -.381t t = 3,5 ( e ) Find r b using 1 - r = e c. Houghton Mifflin Harcourt Publishing Compan 1 - r = e c 1 - r = r = e 1-e So, the growth rate is 3.7 %. Module Lesson 759 Lesson 13.

10 Your Turn 7. A paleontologist uncovers a fossil of a saber-toothed cat in California. The paleontologist analzes the fossil and concludes that the specimen contains 15% of its original carbon-1. The percent of original carbon-1 in a specimen after t ears can be modeled b N (t) = 1 e -.1t, where t is the number of ears after the specimen died. Use a graphing calculator to determine the age of the fossil. Then determine the deca rate of the function. Using a graphing calculator gives a value for the function of about 15 when 15,8. So, the fossil is about 15,8 ears old. The deca factor 1 - r is equal to e c in the function f () = a e c. Rewrite the function in the form f () = a ( e c ). N (t) = 1e -.1t = 1 ( e -.1 ) t 1 - r = e So, the growth rate is.1%. ELABORATE QUESTIONING STRATEGIES How do ou find the growth rate of a function c of the form ƒ () = ae where c is a positive c constant? Rewrite the function as f () = a (e ) and subtract 1 from the base e c. If the growth factor 1 + r equals e c, then the growth rate r is e c - 1. How do ou find the deca rate of a function c of the form ƒ () = ae where c is a negative c constant? Rewrite the function as f () = a (e ) and subtract the base e c from 1. If the deca factor 1 - r equals e c, then the deca rate equals 1 - e c. Elaborate 8. Which transformations of ƒ () = e change the function s end behavior? Vertical translations change the function s end behavior because the affect the graph s asmptote. 9. Which transformations change the location of the graph s -intercept? Vertical translations, horizontal translations, vertical stretches/compressions, and reflections across the -ais all change the location of the graph s -intercept. 1. Wh can the function ƒ () = a e c be used as an eponential growth model and as an eponential deca model? How can ou tell if the function represents growth or deca? If ou rewrite the function in the form f () = a ( e c ), e c can be equal to the growth factor, 1 + r, or the deca factor, 1 - r. If the constant c is positive, then the function models eponential growth, and if the constant c is negative, then the function models eponential deca. 11. Essential Question Check-In How are reference points helpful when graphing transformations of ƒ () = e or when writing equations for transformed graphs? Reference points can be found from the equation and then used to graph transformations of f () = e. Reference points can also be used to write the equation for a transformed graph b giving the values of the constants in the equation. Houghton Mifflin Harcourt Publishing Compan Image Credits: Julie Dermansk/Corbis SUMMARIZE THE LESSON How does the graph of ƒ () = e compare to graphs of eponential functions with other bases? Because e > 1, f () is an eponential growth function, so its graph rises from left to right. The graph rises more quickl than the graph of f () = and less quickl than the graph of f () = 3. Module Lesson The Base e 76

11 EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore 1 Graphing and Analzing f () = e Eplore Predicting Transformations of the Graph of f () = e Eample 1 Graphing Combined Transformations of f () = e Eample Writing Equations for Combined Transformations of f () = e Eample 3 Modeling With Eponential Functions Having Base e Practice Eercises 1 Eercises 6 Eercises 7 13 Eercises 1 15 Eercises Evaluate: Homework and Practice 1. What is the greatest value of ƒ () = (1 + 1 ) for an positive value of? The greatest value is e because as, f () e.. Identif the ke attributes of ƒ () = e, including the domain and range in set notation, the end behavior, and all intercepts. Predict the effect of the parameters h, k, or a on the graph of the parent function ƒ () = e. Identif an changes of domain, range, or end behavior. 3. g () = ƒ ( - ) 1_. g () = ƒ () - 5_ 1 The graph is translated unit to the right. There is no effect on the domain, range, or end behavior. Online Homework Hints and Help Etra Practice The domain is { - < < } and the range is { > }. As approaches -, f () approaches and as approaches, f () approaches. The function does not have an -intercepts but has a -intercept of 1. The graph is translated down 5 units. There is no effect on the domain or end behavior but the range changes to - 5 < <. AVOID COMMON ERRORS Students are accustomed to letters representing variables and so ma become confused when working with e. Remind them that, like π, e is a constant representing an irrational number. Houghton Mifflin Harcourt Publishing Compan 5. g () = - 1_ ƒ () 6. g () = 7 _ ƒ () The graph 1 is verticall compressed b a factor of and reflected across the -ais. There is no effect on the domain, but the range changes to - < <. The end behavior as - does not change, but the end behavior as changes from g () to g () -. The graph is verticall stretched b a factor of 7. There is no effect on the domain, range, or end behavior. 7. The graph of ƒ () = ce crosses the -ais at (, c), where c is some constant. Where does the graph of g () = ƒ () - d cross the -ais? (, c - d) Module Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes 1 Skills/Concepts 1.F Analze relationships 6 1 Recall of Information 1.F Analze relationships Recall of Information 1.E Create and use representations Recall of Information 1.F Analze relationships Skills/Concepts 1.A Everda life 3 Strategic Thinking 1.F Analze relationships 761 Lesson 13.

12 Given the function of the form g () = a e - h + k, identif the reference points and use them to draw the graph. State the domain and range in set notation. 8. g () = e g () = - e (, e+) (1, 3) = (-1, -) - = -1 (, -e-1) a = 1; h = 1; k = Reference points: (, 1) (1, 3) (1, e) (, e + ) Domain: { - < < } Range: { > } a = -1; h = -1; k = -1 Reference points: (, 1) (-1, -) (1, e) (, - e - 1) Domain: { - < < } Range: { < -1} 1. g () = 1_ e g () = - 3 _ e + - e (-, + ) 5 (-3, ) = _ a = ; h = -3; k = Reference points: (, 1) ( 5_ -3, ) (1, e) ( e_ -, + ) Domain: { - < < } Range: { > } = (-, - ) -6 3 (-1, - e -) a = - 3_ ; h = -; k = - Reference points: (, 1) ( -, -_ 19 ) (1, e) ( -1, - _ 3 e - ) Domain: { - < < } Range: { < -} Houghton Mifflin Harcourt Publishing Compan Module Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes A_MTXESE35397_U6M13L.indd 76 /3/1 6:6 PM 3 Strategic Thinking 1.G Eplain and justif arguments 1 3 Strategic Thinking 1.A Everda life The Base e 76

13 1. g () = 3 _ e g () = - 5 _ 3 e (, e -3) (1, - ) = = 1 (, ) (5, - e +) 3 3_ a = ; h = 1; k = -3 Reference points: (, 1) ( 1, - 3_ ) (1, e) (, _ 3 e - 3 ) Domain: { - < < } Range: { > -3} a = - 5_ 3 ; h = ; k = Reference points: (, 1) (, 1_ 3 ) (1, e) ( 5, - 5_ 3 e + ) Domain: { - < < } Range: { < } Houghton Mifflin Harcourt Publishing Compan Write the function whose graph is shown. State the domain and range in set notation. 1. (-1, 5e + 3) (-, 8) - Find a, h, and k. 8 = 3 (h, a + k) = (-, 8), so h = -. (1 + h, ae + k) = (-1, 5e + 3), so ae + k = 5e + 3. Therefore, a = 5 and k = 3. g () = 5 e Domain: { - < < } Range: { > 3} = 8 1 (, -) Find a, h, and k. (5, -6e + ) (h, a + k) = (, -), so h =. (1 + h, ae + k) = (5, -6e + ), so ae + k = -6e +. Therefore, a = -6 and k =. g () = -6 e - + Domain: { - < < } Range: { < } Module Lesson A_MTXESE35397_U6M13L 763 1/8/15 9:6 AM 763 Lesson 13.

14 Solve each problem using a graphing calculator. Then determine the growth rate or deca rate of the function. 16. Medicine Technetium-99m, a radioisotope used to image the skeleton and the heart muscle, has a half-life of about 6 hours. Use the deca function N (t) = N e t, where N is the initial amount and t is the time on hours, to determine how man hours it takes for a 5 milligram dose to deca to 16 milligrams. Let N (t) = 16 and N = 5. Using a graphing calculator gives a value for the function of about 16 when. So, it takes approimatel hours. The deca factor 1 - r is equal to e c in the function f () = a e c. Rewrite the function in the form f () = a ( e c ). N (t) = 5 e t = 5 ( e ) t 1 - r = e c 1 - r = e e = r.19 So, the deca rate is 1.9%. 17. Ecolog The George River herd of caribou in Canada was estimated to be about 7 in 195 and grew at an eponential rate to about 7, in 198. Use the eponential growth function P (t) = P e.15t, where P is the initial population, t is the time in ears after 195, and P (t) is the population at time t, to determine after how man ears the herd will be 5 million. Let P (t) = 5,, and P = 7,. Using a graphing calculator gives a value for the function of about 5,, when 6. So, the herd will be 5 million after approimatel 6 ears, or after 1. The growth factor 1 + r is equal to e c. Rewrite the function in the form f () = a ( e c ). P (t) = 7, e.15t = 7, ( e.15 ) t 1 + r = e c 1 + r = e.15 r = e So, the growth rate is 16.6%. Houghton Mifflin Harcourt Publishing Compan Image Credits: Torbjörn Arvidson/Matton Collection/Corbis Module Lesson The Base e 76

15 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Like π, e is an irrational number, so its decimal form never repeats and never terminates. Its value is approimatel The function ƒ () = e is special in mathematics because it is the onl eponential function ƒ () = b whose derivative is equal to itself. For that reason, e is sometimes called the natural base. 18. Chemistr Radioactive plutonium (Pu-39) has a half-life about,11 ears. Use the function N (t) = N e -.9t to find how man ears it will take for grams of Pu-39 to deca to 1 gram. N represents the initial amount of Pu-39 and t is the time in ears. Let N (t) = 1 and N =. Using a graphing calculator gives a value for the function of about 1 when 11,. So, it takes approimatel 11, ears. The deca factor 1 - r is equal to e c. Rewrite the function in the form f () = a ( e c ). N (t) = e -.9t = ( e -.9 ) t 1 - r = e c 1 - r = e e -.9 = r.9 So, the deca rate is.9%. 19. Population The population of a town was estimated to be about 7 in 199 and grew at an eponential rate to about, in 1. Use the eponential growth function P (t) = P e.86t, where P is the initial population, t is the time in ears after 199, and P (t) is the population at time t, to determine after how man ears the population will be 5,. Let P (t) = 5, and P = 7. Using a graphing calculator gives a value for the function of 5, when 3. So, the town population will be 5, after approimatel 3 ears, or after 13. Houghton Mifflin Harcourt Publishing Compan The growth factor 1 + r is equal to e c. Rewrite the function in the form f () = a ( e c ). P (t) = 5, e.86t = 5, ( e.86 ) t 1 + r = e c 1 + r = e.86 r = e So, the growth rate is 9.%. Module Lesson 765 Lesson 13.

16 H.O.T. Focus on Higher Order Thinking. Eplain the Error A classmate claims that the function g () = - e is the parent function ƒ () = e reflected across the -ais, verticall compressed b a factor of, translated to the left 5 units, and translated up 6 units. Eplain what the classmate described incorrectl and describe g () as a series of transformations of ƒ (). The classmate incorrectl described a reflection across the -ais, a vertical compression b a factor of, and a translation to the left 5 units. The function g () = - e is the parent function f () = e reflected across the -ais, verticall stretched b a factor of, translated to the right 5 units, and translated up 6 units. PEER-TO-PEER DISCUSSION Ask students to discuss with a partner the similarities and differences between e and π, as well as their uses in the real world. 1. Multi-Step Newton s law of cooling states that the temperature of an object decreases eponentiall as a function of time, according to T = T s + ( T - T s ) e -kt, where T is the initial temperature of the liquid, T s is the surrounding temperature, and k is a constant. For a time in minutes, the constant for coffee is approimatel.83. The corner coffee shop has an air temperature of 7 F and serves coffee at 6 F. Coffee eperts sa coffee tastes best at 1 F. a. How long does it take for the coffee to reach its best temperature? Let T = 1, T s = 7, T = 6, and k =.83. T = 7 + (6-7) e -.83t = e -.83t Use a graphing calculator gives a value for the function of about 1 when.3. So, it takes approimatel.3 minutes for the coffee to cool to 1 F. b. The air temperature on the patio outside the coffee shop is 86 F. How long does it take for coffee to reach its best temperature there? Let T s = 86. T = 86 + (6-86) e -.83t = e -.83t Using a graphing calculator gives a value of about 1 for the function when.8. So, it takes approimatel.8 minutes for the coffee to cool to 1 F on the patio. Houghton Mifflin Harcourt Publishing Compan Image Credits: Eactostock/Superstock Module Lesson The Base e 766

17 JOURNAL Have students write a journal entr summarizing what the know about e and about the famil of graphs of ƒ () = e. Entries should include information that helps them identif transformations of the graph. c. Find the time it takes for the coffee to cool to 71 F in both the coffee shop and the patio. Eplain how ou found our answer. Coffee Shop: Using a graphing calculator gives a value for the function of about 71 when 17. So, it takes approimatel 17 minutes for the coffee to cool to 71 F inside the coffee shop. Patio: The equation of the horizontal asmptote of the graph of the function for the patio is = 86, the outside air temperature. Thus, the range of the function is{ > 86}. Since 71 < 86, a temperature of 71 F is not possible for this function. So, the coffee will never cool to 71 F on the patio.. Analze Relationships The graphing calculator screen shows the graphs of the functions ƒ () =, ƒ () = 1, and ƒ () = e on the same coordinate grid. Identif the common attributes and common point(s) of the three graphs. Eplain wh the point(s) is(are) common to all three graphs. Common domain: { - < < } Houghton Mifflin Harcourt Publishing Compan Common range: { > } Common -intercept: (, 1) The point (, 1) is common to all three functions because the functions evaluated at are all 1: f () = = 1 f () = 1 = 1 f () = e = 1 Module Lesson 767 Lesson 13.

18 Lesson Performance Task The ever-increasing amount of carbon dioide in Earth s atmosphere is an area of concern for man scientists. In order to more accuratel predict what the future consequences of this could be, scientists make mathematic models to etrapolate past increases into the future. A model developed to predict the annual mean carbon dioide level L in Earth s atmosphere in parts per million t ears after 196 is L (t) = 36.9 e.3t + 8. a. Use the function L (t) to describe the graph of L (t) as a series of transformations of f (t) = e t. b. Find and interpret L (8), the carbon dioide level predicted for the ear. How does it compare to the carbon dioide level in 15? c. Can L (t) be used as a model for all positive values of t? Eplain. Parts per million (ppm) of CO in atmosphere a. The graph of L (t) is a horizontal stretch of f (t) b a factor of.83, a vertical stretch of 36.9, and a translation of.3 8 units verticall. b. L (8) = 36.9 e.3 (8) The annual mean carbon dioide level will be about parts per million in. Net, find t for = 55 L (55) = 36.9 e.3 (55) The annual mean carbon dioide level in will be about parts per million greater than in 15. c. No, because the annual mean carbon dioide level is in parts per million, so L (t) cannot be greater than 1,,. Also, the annual mean carbon dioide level will have to stop growing eponentiall as it approaches 1,, because there are other elements in the atmosphere, such as nitrogen, that cannot disappear. 6 8 Years (t) since 196 Houghton Mifflin Harcourt Publishing Compan LANGUAGE SUPPORT Some students ma not be familiar with the term etrapolate. Eplain that, in this case, it means to use data from the past to predict values in the future. Draw a graph and label the -ais from 196 to 6. Eplain that all values to the left of the current ear have been measured in the past and the can be used to create a model for values to the right of the current ear. QUESTIONING STRATEGIES How is the variable t defined in the Lesson Performance Task? t is the number of ears after 196. What is the domain for the function L (t)? For the mathematical function, the domain is - < t <. However, for this Performance Task, the domain is all integers t such that t and L (t) < 1,,. How can ou write the equation so the variable t is the actual ear itself, and how does this change the graph of L (t)? L (t) = 36.9 e.3 (t - 196) + 8. The graph is shifted to the right b 196 units. Module Lesson EXTENSION ACTIVITY Have students calculate the time t when the carbon dioide level will have doubled from its 196 level. Have them discuss whether this will happen within their lifetimes. Set L (t) = L () and solve for t. The carbon dioide level will have doubled after about 11 ears, or in 61. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. The Base e 768