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LESSON 15.3 Name Constructing Exponential Functions Class Date 15.3 Constructing Exponential Functions Essential Question: What are discrete exponential functions and how do you represent them? Common Core Math Standards Resource Locker The student is expected to: Explore F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Also F-IF.2, F-IF.7e Understanding Discrete Exponential Functions Recall that a discrete function has a graph consisting of isolated points. Mathematical Practices The table represents the cost of tickets to an annual event as a function of the number t of tickets purchased. Complete the table by adding 10 to each successive cost. Plot each ordered pair from the table. MP.6 Precision Tickets t Cost ($) (t, f(t)) 1 10 (1, 10) 2 20 (2, 20) 3 30 4 40 5 50 Language Objective Explain to a partner what the graph of a discrete exponential function looks like. Possible answer: Discrete exponential functions are exponential functions whose domains are limited, such as the set of integers, making the graph appear as isolated points instead of a continuous curve. 4, 40 5, 50 40 30 20 10 x 0 1 2 3 4 5 Tickets Module 15 be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 3 693 gh "File info" made throu Date Class tructi 15.3 Cons tions Func ion: What F-LE.2 For F-IF.7e A1_MNLESE368187_U6M15L3.indd 693 ntial ng Expone Name Quest Essential ons and ential functi te expon are discre the full text rds, of these standa see the table ing Discrete Explore Understand how do you starting on represent page CA2. them? Resource Locker Also F-IF.2, l Exponentia HARDCOVER PAGES 545554 Functions . isolated points of consisting er t of has a graph of the numb d function a function each ordere a discrete l event as cost. Plot an annua successive tickets to 10 to each the cost of by adding represents lete the table The table ased. Comp tickets purch (t, f(t)) the table. pair from Cost ($) (1, 10) Tickets t 10 1 (2, 20) 20 2 3, 30 30 3 4, 40 Turn to these pages to find this lesson in the hardcover student edition. Recall that ( 40 4 5 y g Compan ( ) ( ) 5, 50 50 ) y 60 n Mifflin Harcour t Cost ($) Publishin 50 40 30 © Houghto 20 x 10 0 1 3 2 4 5 Tickets Lesson 3 693 Module 15 Lesson 15.3 L3.indd 7_U6M15 SE36818 A1_MNLE 693 ) ) ) 50 PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and why the population of invasive species might grow exponentially. Then preview the Lesson Performance Task. 3, 30 y 60 Cost ($) Essential Question: What are discrete exponential functions and how do you represent them? © Houghton Mifflin Harcourt Publishing Company ENGAGE ( ( ( 693 3/24/14 11:20 AM 3/24/14 11:15 AM B The number of people attending an event doubles each year. The table represents the total attendance at each annual event as a function of the event number n. Complete the table by multiplying each successive attendance by 2. Plot each ordered pair from the table. Event Number n Attendance (n, g(n)) 1 20 (1, 20) 2 40 (2, 40) 3 80 4 160 5 320 360 ( ( ( 3, 80 4, 160 5, 320 EXPLORE Understanding Discrete Exponential Functions ) ) ) INTEGRATE TECHNOLOGY Have students complete the Explore activity in either the book or online lesson. QUESTIONING STRATEGIES y How are the tables for the functions different? They are different because successive output values of the first function have a common difference, while successive output values of the second function have a common ratio. 320 Attendance 280 240 200 160 120 80 © Houghton Mifflin Harcourt Publishing Company 40 x 0 1 2 3 4 5 Event Number C Complete the table. Function Linear? Discrete? f(t) Yes Yes g(n) No Yes Reflect 1. Communicate Mathematical Ideas What are the limitations on the domains of these functions? Why? The domains include only whole numbers. You cannot buy a fraction of a ticket to an event, nor stage a fraction of an event. Module 15 694 Lesson 3 PROFESSIONAL DEVELOPMENT A1_MNLESE368187_U6M15L3.indd 694 Learning Progressions 3/24/14 11:15 AM In this lesson, students learn how to identify and represent exponential functions. Some key understandings for students are as follows: • An exponential function can be represented by an equation of the form f (x)= abx, where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1. • For an exponential function ƒ(x), if x is in the domain of the function, then the ratio of ƒ(x+1) to ƒ(x) is a constant ratio. In the next module, students will learn more about exponential functions, including exponential growth and decay functions. Constructing Exponential Functions 694 Representing Discrete Exponential Functions Explain 1 EXPLAIN 1 An exponential function is a function whose successive output values are related by a constant ratio. An exponential function can be represented by an equation of the form ƒ(x) = ab x, where a, b, and x are real numbers, a ≠ 0, b > 0, and b ≠ 1. The constant ratio is the base b. Representing Discrete Exponential Functions When evaluating exponential functions, you will need to use the properties of exponents, including zero and negative exponents. Recall that, for any nonzero number c: c 0 = 1, c ≠ 0 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Point out that the constant b in a function of 1 , c ≠ 0. c -n = _ cn Example 1 the form f (x) = abx, where a and b are real numbers, a ≠ 0, b > 0, and b ≠ 1, corresponds to the common ratio, r, in a geometric sequence. Explain to students that if the domain of the function is the set of positive integers, the points of the graph of a discrete exponential function represent a geometric sequence. The x-values are the term numbers of the sequence and the y-values are the corresponding terms. 2 = __ 2 = 2. reciprocal of the denominator: 1∙ __ 1 1 2 © Houghton Mifflin Harcourt Publishing Company __1 ( ) 1 x with a domain of ⎧⎨-1, 0, 1, 2, 3, 4⎫⎬ ƒ(x) = 3 ⋅ _ ⎭ ⎩ 2 -1 1 = 3⋅ _ 1 = 3⋅2= 6 ƒ(-1) = 3 ⋅ _ 2 1 _ 2 QUESTIONING STRATEGIES How do you evaluate a complex fraction, such 1 ? Multiply the numerator by the as __ Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table. ( ) () (x, f(x)) x f(x) -1 6 (-1, 6) 0 3 (0, 3) 1 1 1_ 2 2 3 _ 4 3 3 _ 8 4 3 _ 16 (1, 1_12 ) (2, _34 ) (3, _38 ) 3 (4, _ 16 ) y 6 5 4 3 2 1 x -1 0 1 Module 15 2 3 695 4 5 Lesson 3 COLLABORATIVE LEARNING A1_MNLESE368187_U6M15L3.indd 695 Peer-to-Peer Activity Have students work in pairs. Instruct one student in each pair to construct a table of values and graph for the function defined by h(x) = 3x with the domain ⎧ ⎫ ⎨-2, -1, 0, 2⎬. Instruct the other student to do the same for the function defined ⎩ ⎭ by g(x) = 3x with the same domain. Have the students compare and contrast the tables of values and the graphs. 695 Lesson 15.3 8/21/14 4:18 PM B ( ) 4 x; domain = ⎧⎨-2, -1, 0, 1, 2, 3⎫⎬ ƒ(x) = 3 _ ⎩ ⎭ 3 () 4 ƒ(-2) = 3 _ 3 -2 ( ) 3 x 2 9 3 27 11 1 = 3⋅ _ = 3 ⋅ _2 = 3 ⋅ _ = _ = 1_ 2 16 4 16 16 4 _ 2 f(x) -2 11 1__ 16 -1 1 2_ 4 0 3 1 4 2 1 5_ 3 3 1 7_ 9 ( ( (x, f(x)) ) _ ) ( ) ( ) ( _) __ -2, 1 16 11 -1, 214 0, 3 1, 4 2, 513 (3, 7_91 ) y 8 6 4 © Houghton Mifflin Harcourt Publishing Company 2 x -4 -2 0 2 4 Reflect 2. What If What would happen to the function ƒ(x) = ab x if a were 0? What if b were 1? If a = 0 then ƒ(x) = 0 for all x, and if b = 1, then ƒ(x) = a for all x. Neither of these constant functions are exponential functions. 3. Discussion Why is a geometric sequence a discrete exponential function? A sequence is a function. When the input values increase by 1, the successive output values are related by a constant ratio, the common ratio. The general recursive rule for a geometric sequence is the form of an exponential function. Module 15 696 Lesson 3 DIFFERENTIATE INSTRUCTION A1_MNLESE368187_U6M15L3.indd 696 Cognitive Strategies 8/21/14 4:18 PM Encourage students to isolate the necessary information from any problem scenario they are given. It may help to circle or underline values or to write the important information on a separate sheet of paper. Constructing Exponential Functions 696 Your Turn EXPLAIN 2 Make a table for the function using the given domain. Then graph the function using the ordered pairs from the table. Constructing Exponential Functions from Verbal Descriptions 4. () x 3 ; domain = ⎧⎨-3, -2, -1, 0, 1, 2⎫⎬ (x) = 4 _ ⎩ ⎭ 2 y x 8 MODELING -3 6 Have students fold a piece of paper in half repeatedly. Have them make a table representing the number of layers of paper after 0 folds, 1 fold, 2 folds, and so on. -2 4 -1 2 x -4 -2 0 2 4 QUESTIONING STRATEGIES How do you find the value of b in order to write an exponential function in the form ƒ(x) = abx ? Choose a number x in the domain of the function. Then divide f(x + 1) by f(x) to get b. Explain 2 f(x) 32 5 = 1_ (_ 27 ) 27 16 7 = 1_ (_ 9 9) 8 2 _ _ (3) = 23 (x, f(x)) 5 (-3, 1_ 27 ) (-2, 1_79 ) (-1, 2_23 ) 0 4 (0, 4) 1 6 (1, 6) 2 9 (2, 9) Constructing Exponential Functions from Verbal Descriptions © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital Vision/Getty Images You can write an equation for an exponential function f(x) = ab x by finding or calculating the values of a and b. The value of a is the value of the function when x = 0. The value of b is the common ratio of successive function ƒ(x + 1) values, b = _______ . For discrete functions with integer or whole number domains, these will be successive values ƒ(x) of the function. Example 2 Write an equation for the function. When a piece of paper is folded in half, the total thickness doubles. Suppose an unfolded piece of paper is 0.1 millimeter thick. The total thickness t(n) of the paper is an exponential function of the number of folds n. The value of a is the original thickness of the paper before any folds are made, or 0.1 millimeter. Because the thickness doubles with each fold, the value of b (the constant ratio) is 2. The equation for the function is t(n) = 0.1(2) . n Module 15 697 Lesson 3 LANGUAGE SUPPORT A1_MNLESE368187_U6M15L3.indd 697 Connect Vocabulary In this lesson there are several key words that begin with the prefix in-. Learning the meaning of prefixes helps all students of math and science, since many key terms in both disciplines originated in Latin or have Latin prefixes. In this lesson, the words increase and input both start with the prefix in-, which means in, into, on, near, or towards. The prefix in- can also mean not, as in the word inverse. Another key prefix in this lesson is non-, which means not. The term nonzero, for example, means not zero. 697 Lesson 15.3 8/21/14 4:18 PM A savings account with an initial balance of $1000 earns 1% interest per month. That means that the account balance grows by a factor of 1.01 each month if no deposits or withdrawals are made. The account balance in dollars B(t) is an exponential function of the time t in months after the initial deposit. EXPLAIN 3 Constructing Exponential Functions from Input-Output Pairs Let B represent the balance in dollars as a function of time t in months. 1000 . The value of a is the original balance, The value of b is the factor by which the balance changes every month, ) ( The equation for the function is B(t) = 1000 1.01 . 1.01 . t AVOID COMMON ERRORS When calculating the value of b, students may divide in the wrong order. Remind them that they should always divide an output value by the preceding output value. For example, if they are using (4, 27) and (5, 9), they should divide the function value of the second pair by the function value of the first pair: Reflect 5. Why is the exponential function in the paper-folding example discrete? The paper can only be folded a whole number of times. Your Turn 6. A piece of paper that is 0.2 millimeters thick is folded. Write an equation for the thickness t of the paper in millimeters as a function of the number n of folds. Initial thickness (n = 0) a = 0.2 Change per fold: double b=2 t(n) = 0.2 ⋅ 2 n Equation: Explain 3 9 = __ 1. b = ___ 27 3 INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 After students have found an equation for an Constructing Exponential Functions from Input-Output Pairs You can use given two successive values of a discrete exponential function to write an equation for the function. Example 3 (3, 12) and (4, 24) 24 = 2. Find b by dividing the function value of the second pair by the function value of the first: b = _ 12 Evaluate the function for x = 3 and solve for a. Write the general form. ƒ(x) = ab x Substitute the value for b. ƒ(x) = a ⋅ 2 x Substitute a pair of input-output values. 12 = a ⋅ 2 3 Simplify. 12 = a ⋅ 8 Solve for a. Use a and b to write an equation for the function. Module 15 A1_MNLESE368187_U6M15L3.indd 698 © Houghton Mifflin Harcourt Publishing Company exponential function, have them check both pairs in the equation to verify that the equation is correct. Write an equation for the function that includes the points. 3 a= _ 2 3 ⋅ 2x ƒ(x) = _ 2 698 Lesson 3 8/21/14 4:18 PM Constructing Exponential Functions 698 B QUESTIONING STRATEGIES 9 ÷3= Find b by dividing the function value of the second pair by the first: b = _ 4 Write the general form. ƒ(x) = ab x If an exponential function includes the points (4, 64) and (5, 32), what is the common ratio? Explain how you found it. The common ratio is 0.5. 64 = __ 1 = 0.5 . I found it by dividing 32 by 64: ___ 32 2 Substitute the value for b. Substitute a pair of input-output values. ELABORATE CRITICAL THINKING . 4 x (_34) 1 ( ) 3 3 = a⋅ _ 4 3= a⋅ Solve for a. a= ƒ(x) = _3 4 4 () 3 4 _ 4 x Your Turn SUMMARIZE THE LESSON (-2, _52 ) and (-1, 2) 2 =5 b=2÷_ 5 a 2 = a ⋅ 5 -1; 2 = _; a = 10 5 f(x) = 10 ⋅ 5 x Elaborate © Houghton Mifflin Harcourt Publishing Company 8. Explain why the following statement is true: For 0 < b < 1 and a > 0, the function ƒ(x) = ab decreases as x increases. When you multiply a number a by a fraction b between 0 and 1, the product is less than x the original number a. When you multiply by this fraction b repeatedly, the results decrease further. 9. Explain why the following statement is true: For b > 1 and a > 0, the function ƒ(x) = ab increases as x increases. When you multiply a number a by a number b greater than 1, the product is greater x than the original number a. When you multiply by this number b repeatedly, the results increase further. 10. Essential Question Check-In What property do all pairs of adjacent points of a discrete exponential function share? The ratio of function values is the same for any two pairs. Module 15 A1_MNLESE368187_U6M15L3.indd 699 Lesson 15.3 _3 Write an equation for the function that includes the points. 7. 699 ƒ(x) = a ⋅ Simplify. Use a and b to write an equation for the function. Have students give an example of an increasing exponential function and of a decreasing exponential function. What are discrete exponential functions and how can you represent them? A discrete exponential function is a function of the form f (x) = abx , with a ≠ 0, b > 1, and b ≠ 1, such that the graph of the function is a set of isolated points. For example, the domain of the function could be the set of whole numbers. Such a function can be represented by a set of input and output values in a table or by a graph of ordered pairs. ( ) 9 (1, 3) and 2, _ 4 699 Lesson 3 8/21/14 4:18 PM EVALUATE Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice Complete the table for each function using the given domain. Then graph the function using the ordered pairs from the table. 1. 1 ⋅ 4 x; domain = ⎧⎨-2, -1, 0, 1, 2⎫⎬. ƒ(x) = _ ⎩ ⎭ 2 x y 8 -2 6 -4 2. -2 2 0 _1 1 2 2 8 0 2 4 () 32 ASSIGNMENT GUIDE (-1, _18) (0, _12) _1 4 8 2 (1, 2) (2, 8) x 1 ; domain = ⎧⎨0, 1, 2, 3, 4, 5⎫⎬. ƒ(x) = 9 _ ⎩ ⎭ 3 f(x) 8 0 9 6 1 3 4 2 1 2 3 _1 4 _1 5 1 __ x f(x) -1 9 x 0 2 4 6 8 () (x, f(x)) (0, 9) (1, 3) (2, 1) (3, _13) (4, _19) 1 (5, __ 27 ) 3 9 27 x 2 ; domain = ⎧⎨-1, 0, 1, 2, 3, 4⎫⎬. ƒ(x) = 6 _ ⎩ ⎭ 3 y 8 6 0 6 4 1 4 2 2 2 2_ 3 3 7 1_ 9 4 5 1__ 27 x 0 2 4 6 8 Module 15 Exercise A1_MNLESE368187_U6M15L3.indd 700 (x, f(x)) (-1, 9) (0, 6) (1, 4) (2, 2_23) (3, 1_79) 5 (4, 1__ 27 ) Depth of Knowledge (D.O.K.) Mathematical Practices 1 Recall of Information MP.6 Precision 5–8 2 Skills/Concepts MP.4 Modeling 9–21 2 Skills/Concepts MP.2 Reasoning 1 Recall of Information MP.2 Reasoning 3 Strategic Thinking MP.3 Logic 23–24 Practice Explore Understanding Discrete Exponential Functions Exercise 19 Example 1 Representing Discrete Exponential Functions Exercises 1–4, 22–23 Example 2 Constructing Exponential Functions from Verbal Descriptions Exercises 5–8, 20–21 Example 3 Constructing Exponential Functions from Input-Output Pairs Exercises 9–18, 24 Lesson 3 700 1–4 22 Concepts and Skills © Houghton Mifflin Harcourt Publishing Company x y 3. 1 __ 32 -1 x (x, f(x) ) (-2, __1 ) f(x) 3/24/14 11:15 AM Constructing Exponential Functions 700 4. QUESTIONING STRATEGIES () x 4 ; domain = ⎧⎨-3, -2, -1, 0, 1⎫⎬ ƒ(x) = 6 _ ⎩ ⎭ 3 Why is the restriction b ≠ 1 in the definition of an exponential function necessary? What would happen if b = 1? 1 raised to any power is 1, so the function would always have the same value. It would not increase or decrease. 8 y 6 4 2 x -8 -6 -4 -2 0 (x, f(x)) 17 ) (-3, 2__ x f(x) -3 17 2__ 32 -2 3 3_ -1 1 4_ 2 0 6 (0, 6) 1 8 (1, 8) 8 32 (-2, 3_38) (-1, 4_12) Write an equation for each function. 5. Business A recent trend in advertising is viral marketing. The goal is to convince viewers to share an amusing advertisement by e-mail or social networking. Imagine that the video is sent to 100 people on day 1. Each person agrees to send the video to 5 people the next day, and to request that each of those people send it to 5 people the day after they receive it. The number of viewers v(n) is an exponential function of the number n of days since the video was first shown. a = 100 b=5 v(n) = 100 ⋅ 5 n © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David J. Green - technology/Alamy 6. A pharmaceutical company is testing a new antibiotic. The number of bacteria present in a sample when the antibiotic is applied is 100,000. Each hour, the number of bacteria present decreases by half. The number of bacteria remaining r(n) is an exponential function of the number n of hours since the antibiotic was applied. a = 100,000 1 b=_ 2 7. a=5 b = 0.99, (power remaining after 1% lost) p(n) = 5(0.99) A1_MNLESE368187_U6M15L3 701 Lesson 15.3 n Optics A laser beam with an output of 5 milliwatts is directed into a series of mirrors. The laser beam loses 1% of its power every time it reflects off of a mirror. The power p(n) is a function of the number n of reflections. Module 15 701 () 1 r(n) = 100,000 _ 2 n 701 Lesson 3 6/8/15 12:35 PM 8. The NCAA basketball tournament begins with 64 teams, and after each round, half the teams are eliminated. The number of remaining teams t(n) is an exponential function of the number n of rounds already played. a = 64 1 b=_ 2 AVOID COMMON ERRORS Students may forget to find the value of both a and b when writing a function rule for an exponential function of the form ƒ(x) = ab x. Remind students that they need to determine the common ratio and the initial value of an exponential function in order to write the function rule. () 1 t(n) = 64 _ 2 n Write an equation for the function that includes the points. 9. (2, 100) and (3, 1000) 10. (-2, 4) and (-1, 8) 8 b= 4 =2 _ 1000 b = ____ 100 = 10 8 = a ⋅ 2 -1 a 8= 2 a = 16 100 = a ⋅ 10 2 _ 100 = 100a a=1 f(x) = 16 ⋅ 2 x f(x) = 10 x 11. (1, _54 ) and (2, _32 ) 12. (_23) (_45) (-3, _161 ) and (-2, _83 ) (_38) 1 (__ 16 ) b = ___ b =___ 2 _ ⋅5 =_ 3 4 3 __ ⋅ 16 =_ 8 1 _4 = a ⋅ (_5)1 5 6 _4 = a ⋅ _5 5 6 4 _ a=_ ⋅6 5 5 24 = __ 25 x 24 _ f(x) = __ ⋅ (5) 1 __ = a ⋅ 6 -3 16 1 1 __ = a ⋅ ___ 16 216 1 a = __ ⋅ 216 5 =_ 6 16 = 13.5 © Houghton Mifflin Harcourt Publishing Company 25 =6 f(x) = 13.5 ⋅ 6 x 6 Use two points to write an equation for the function. 13. x f(x) 1 2 2 2 _ 7 3 2 _ 49 4 2 _ 343 Module 15 A1_MNLESE368187_U6M15L3.indd 702 (_27) b = ___ 2 1 =_ 7 () 1 2=a⋅ _ 7 1 1 2=a⋅_ 7 a = 14 1 f(x) = 14 ⋅ _ 7 () 702 x Lesson 3 3/24/14 11:15 AM Constructing Exponential Functions 702 Use points (-2, 53) and (-1, 530) 14. x f(x) -4 0.53 -3 5.3 -2 53 -1 530 530 b = ___ 53 = 10 530 = a ⋅ (10) -1 1 530 = a ⋅ __ 10 a = 5300 f(x) = 5300 ⋅ (10) x 15. 8 2 6 b=_ 3 (1, 6) 6 4 Use points (0, 3) and (1, 6) y =2 a=3 (0, 3) f(x) = 3 ⋅ 2 x x -6 -4 0 -2 2 16. 8 y Use points (1, 8) and (2, 4) (1, 8) 4 b=_ 8 1 =_ 2 6 4 © Houghton Mifflin Harcourt Publishing Company 2 x 0 2 4 () 1 8=a _ 2 (2, 4) 6 8 a 8=_ 2 a = 16 1 () 1 f(x) = 16 ⋅ _ 2 x 17. The height h(n) of a bouncing ball is an exponential function of the number n of bounces. One ball is dropped and on the first bounce reaches a height of 6 feet. On the second bounce it reaches a height of 4 feet. 4 b=_ 6 2 =_ 3 () 2 6=a _ 3 1 2 6=a⋅_ 3 3 a=6⋅_ 2 =9 () 2 h(n) = 9 ⋅ _ 3 Module 15 A1_MNLESE368187_U6M15L3.indd 703 703 Lesson 15.3 n 703 Lesson 3 8/21/14 4:18 PM 18. A child starts a playground swing from standing and doesn’t use her legs to keep swinging. On the first swing she swings forward by 18 degrees, and on the second swing she only comes 13.5 degrees forward. The measure in degrees of the angle m(n) is an exponential function of the number n of swings. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Point out to students that, unlike linear 13.5 b = ___ 18 3 =_ 4 functions, the letter b in an exponential function of the form ƒ(x) = ab x does not represent the y-intercept. () 3 18 = a _ 4 1 3 18 = a ⋅ _ 4 4 a = 18 ⋅ _ 3 = 24 3 m(n) = 24 _ 4 () n 19. Make a Prediction A town’s population has been declining in recent years. The table shows the population since 1980. Is this data consistent with an exponential function? Explain. If so, predict the population for 2010 assuming the trend holds. Year Population 1980 5000 1990 4000 2000 3200 2010 2560 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Steve Hix/ Somos Images/Corbis To check for exponential behavior, find the ratios between successive values of the function. 4000 4 ____ 4 ____ =_ , 3200 = _ 5000 5 4000 5 4 The ratio between successive function values is _ , so this is consistent with 5 an exponential function. 4 3200 ⋅ _ = 2560 5 20. A piece of paper has a thickness of 0.15 millimeters. Write an equation to describe the thickness t(n) of the paper when it is repeatedly folded in thirds, where n is the number of foldings. b=3 a = 0.15 t(n) = 0.15 ⋅ 3 n Module 15 A1_MNLESE368187_U6M15L3 704 704 Lesson 3 6/12/15 9:53 AM Constructing Exponential Functions 704 21. Probability The probability of getting heads on a single coin flip is __12 . The probability of getting nothing but heads on a series of coin flips decreases by __12 for each additional coin flip. Write an exponential function for the probability p(n) of getting all heads in a series of n coin flips. JOURNAL Have students describe a specific discrete exponential function in at least two different ways. They can describe it with words, a function rule, a table, and/or a graph. 1 b=_ 2 _1 = a(_1) 2 1 2 _1 = a ⋅ _1 2 2 a=1 n p(n) = 1 2 (_) 22. Multipart Classification Determine whether each of the functions is exponential or not. a. ƒ(x) = x 2 b. ƒ(x) = 3 ⋅ 2 x 1 c. ƒ(x) = 3 ⋅ _ x 2 d. ƒ(x) = 1.001 x e. ƒ(x) = 2 ⋅ x 3 1 f. ƒ(x) = _ ⋅ 5 x 10 Exponential Not exponential Exponential Not exponential Exponential Not exponential Exponential Not exponential Exponential Not exponential Exponential Not exponential © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©mj007/ Shutterstock H.O.T. Focus on Higher Order Thinking 23. Explain the Error Biff observes that in every math test he has taken this year, he has scored 2 points higher than the previous test. His score on the first test was 56. He models his test scores with the exponential function s(n) = 28 · 2 n where s(n) is the score on his nth test. Is this a reasonable model? Explain. No; Biff’s test scores do not increase by a constant ratio. Instead they go up by 2 points each test. His scores are not consistent with an exponential function. 24. Find the Error Kaylee needed to write the equation of an exponential function from points on the graph of the function. To determine the value of b, Kaylee chose the ordered pairs (1, 6) and (3, 54) and divided 54 by 6. She determined that the value of b was 9. What error did Kaylee make? She should use ordered pairs with x-values that differ by 1. The value Kaylee found is not the constant ratio of successive values. Module 15 A1_MNLESE368187_U6M15L3.indd 705 705 Lesson 15.3 705 Lesson 3 6/10/15 12:49 AM Lesson Performance Task INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students explain why they would choose In ecology, an invasive species is a plant or animal species newly introduced to an ecosystem, often by human activity. Because invasive species often lack predators in their new habitat, their populations typically experience exponential growth. A small initial population grows to a large population that drastically alters an ecosystem. Feral rabbits that populate Australia and zebra mussels in the Great Lakes are two examples of problematic invasive species that grew exponentially from a small initial population. 3 or its decimal equivalent, 1.5, to use the fraction __ 2 when solving Part C of the Lesson Performance Task. An ecologist monitoring a local stream has been collecting samples of an unfamiliar fish species over the past four years and has summarized the data in the table. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 After students have completed the graph of Here are the results so far: Year Average Population Per Mile 2009 32 2010 48 2011 72 2012 108 the populations versus time, discuss how the value of b, when a is positive, causes the graph to increase over time. Then discuss what happens to the graph t when students let b = __23 in f (t) = 32 __23 . 2013 2014 () a. Look at the data in the table and confirm that the growth pattern is exponential. 48 _ = 1.5 32 72 _ = 1.5 48 108 _ = 1.5 72 Since there is a common ratio, r = 1.5, the growth pattern is exponential. b. Write the equation that represents the average population per square meter as a function of years since 2009. () t 3 The function is f(t) = 32 _ . 2 © Houghton Mifflin Harcourt Publishing Company c. Predict the average populations expected for 2013 and 2014. 2013: 162 individuals per mile d. Graph the population versus time since 2009 and include the predicted values. Average population per mile 2014: 243 individuals per mile 250 P 200 150 100 50 t 0 1 2 3 4 5 Time (years) Module 15 706 Lesson 3 EXTENSION ACTIVITY A1_MNLESE368187_U6M15L3.indd 706 Have students research the feral rabbit population in Australia or the zebra mussel population in the Great Lakes. Have students find out how the population was introduced into the new habitat and describe efforts to control the population. 3/24/14 11:15 AM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Constructing Exponential Functions 706