13.3 Exponential Decay Functions

Transcription

1 Locker LESSON. Eponential Deca Functions Teas Math Standards The student is epected to: A.5.B Formulate eponential and logarithmic equations that model real-world situations, including eponential relationships written in recursive notation. A..A, A.5.A, A.5.D, A.7.I Mathematical Processes A..A To appl mathematics to problems arising in everda life, societ, and the workplace Language Objective.B.,.D.,.I.,.I.,.D,.E Work with a partner to compare and contrast eponential deca and eponential growth functions. ENGAGE Essential Question: How is the graph of g () = ab - h + k where < b < related to the graph of ƒ () = b? Possible answer: The graph of g () = ab - h + k involves transformations of the graph of ƒ () = b. In particular, the graph of g () is a vertical stretch or compression of the graph of ƒ () 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. Eponential Deca Functions Essential Question: How is the graph of g () = a b h + k where < b < related to the graph of f () = b? A.5.B Formulate eponential equations that model real-world situations Also A..A, A.5.A, A.5.D, A.7.I Eplore Graphing and Analzing f () = ( and f () = ( ) Resource Locker ) Eponential deca functions are eponential functions with bases between and assuming a positive leading coefficient. These functions can be transformed in a manner similar to eponential growth functions. Begin b plotting the parent functions of two of the more commonl used bases: and. To begin, fill in the table in order to find points along the function ƒ () = ( ). You ma need to review the rules of the properties of eponents, including negative eponents. What does the end behavior of this function appear to be as increases? f () approaches. Plot the points on the graph and draw a smooth curve through them. Complete the table for ƒ () = (. ) Plot the points on the graph and draw a smooth curve through them. f () = ( ) f () = ( ) PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and wh it might be important to know the amount of a radioactive isotope remaining in the environment. Then preview the Lesson Performance Task. Module 7 Lesson DO NOT EDIT--Changes must be made through File info CorrectionKe=TX-B Name Class Date. Eponential Deca Functions Essential Question: How is the graph of g () = a b h + k where < b < related to the graph of f () = b? Houghton Mifflin Harcourt Publishing Compan A.5.B Formulate eponential equations that model real-world situations Also A..A, A.5.A, A.5.D, A.7.I Eplore Graphing and Analzing f () = ( and f () = ( ) - - and. ) Resource Eponential deca functions are eponential functions with bases between and assuming a positive leading coefficient. These functions can be transformed in a manner similar to eponential growth functions. Begin b plotting the parent functions of two of the more commonl used bases: To begin, fill in the table in order to find points along the function ƒ () = ( eponents, including negative eponents. ). You ma need to review the rules of the properties of What does the end behavior of this function appear to be as increases? f () approaches. Plot the points on the graph and draw a smooth curve through them. Complete the table for ƒ () = ( ). Plot the points on the graph and draw a smooth curve through them. f () = ( ) f () = ( ) - Module 7 Lesson A_MTXESE597_U6ML.indd 7 //5 :6 PM - - HARDCOVER PAGES 57 5 Turn to these pages to find this lesson in the hardcover student edition. 7 Lesson.

2 F G Fill in the following table of properties: f () = ( ) f () = ( Domain - < < Range ) > > End behavior as f () f () End behavior as - f () f () -intercept (, ) (, ) Both of these functions [decrease/increase] throughout the domain. ( Of the two functions, ƒ () = ) decreases faster. - < < EXPLORE Graphing and Analzing ƒ () = ( and ƒ () = ( ) INTEGRATE TECHNOLOGY ) Make sure that students are comfortable using their calculators to graph eponential functions. The ma need to practice putting in the appropriate domains and ranges. QUESTIONING STRATEGIES What is the deca factor in an eponential deca function? the base b of an eponential deca function = ab where a > Reflect. Make a Conjecture Look at the table of properties for the functions. What do ou notice? Make a conjecture about these properties for eponential deca functions of the form ƒ () = ( n ), where n is a constant. The domain, range, end behavior, and -intercept are the same for both functions. These same properties appl to all eponential deca functions of the form f () = ( n ). Eplore Predicting Transformations of the Graphs of f () = ( ) and f () = ( ) Based on our eperience with transforming the parent function ƒ () in previous lessons, make predictions about the effect of varing the parameters in g () = aƒ (-c) + d. Confirm our predictions using a graphing calculator. A The graph of g () = ( factor of. ) ( The graph of g () = ) factor of. will be the graph of ƒ () = ( verticall ) will be the graph of ƒ () = ( verticall ) stretched b a compressed b a Module 7 Lesson PROFESSIONAL DEVELOPMENT Math Background Students will graph most of the eponential functions in this lesson b hand. The will see that the graphs of eponential deca functions approach the positive -ais as increases without bound, so the -ais is an asmptote for the graph of an function of the form ƒ () = b where b > and b. Students will also transform the graphs of eponential functions and discover how the transformations affect the asmptote, -intercept, and rate of increase or decrease, and write transformed functions for graphs based upon the asmptote and two points on the graph, the reference points. Houghton Mifflin Harcourt Publishing Compan What is the parent function for eponential deca functions? ƒ () = b where < b < is the parent function for the famil of eponential deca functions with base b. EXPLORE Predicting Transformations of the Graphs of ƒ () = ( AVOID COMMON ERRORS ) and ƒ () = ( ) Some students ma think that a horizontal shift in the graph of an eponential function affects the domain. Demonstrate that the domain of all eponential functions and their translations is the set of all real numbers, just as with quadratic functions. Go back to the definition of domain and point out that the value of can be an real number in an eponential growth or deca function, or an translation of these functions. You might use a graphing calculator demonstration to reinforce this idea visuall. Eponential Deca Functions 7

3 QUESTIONING STRATEGIES Can ou automaticall conclude that an eponential function model decas if the base of the power is a fraction or decimal? Eplain. No, some fractions and decimals 7 have a value greater than one, such as.5 and, and these bases produce eponential growth functions. B The graph of g () = - ( ) will be the graph of ƒ () = ( and verticall compressed b a factor of. The graph of g () = -5 ( ) will be the graph of ƒ () = ( and verticall stretched 5 b a factor of. ) reflected across the - ais ) reflected across the - ais The graph of q () = - 5 ( ) will be the graph of ƒ () = ( ) reflected across 5 -ais stretched the and verticall b a factor of. The graph of q () = - ( ) will be the graph of ƒ () = ( ) reflected across -ais compressed the and verticall b a factor of. _ C The graph of g () = ( ) + will be the graph of ƒ () = ( The graph of g () = ( ) - will be the graph of ƒ () = ( right the. ) translated unit to the left. ) translated units to + The graph of q () = ( ) will be the graph of ƒ () = ( ) translated units to left the. - The graph of q () = ( ) will be the graph of ƒ () = ( ) translated units to right the. D The graph of g () = ( ) + will be the graph of ƒ () = ( The graph of g () = ( ) - 5 will be the graph of ƒ () = ( The graph of q () = ( The graph of q () = ( ) + 5 will be the graph of ƒ () = ( ) - will be the graph of ƒ () = ( ) translated units up. ) translated.5 units down. ) translated 5 units up. ) translated units down. Houghton Mifflin Harcourt Publishing Compan Reflect. Which parameters make the domain and range of g () differ from those of the parent function? Write the transformed domain and range for g () in set notation. None of the parameters alter the domain, which is all real numbers for both the parent and transformed functions. The parameter a alters the range if it is less than, and the parameter k alters the finite end of the range. - < < ; Range (a > ) : > k ; Range (a < ) : < k Module 75 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have pairs of students work together to create a graphic organizer to compare and contrast eponential growth functions and eponential deca functions. 75 Lesson.

4 Eplain Graphing Combined Transformations of f () = b Where < b < When graphing transformations of ƒ () = b where < b <, it is helpful to consider the effect of the transformation on two reference points, (, ) and ( -, b), vas well as the effect on the asmptote, =. The table shows these reference points and the asmptote = for ƒ () = b and the corresponding points and asmptote for the transformed function, g () = ab -h + k. EXPLAIN Graphing Combined Transformations of ƒ () = b Where < b < Eample f () = b g () = a b -h + k First reference point (, ) (h, a + k) Second reference point ( -, a_ b) ( h -, b + k ) Asmptote = = k For each of the transformed functions, use the reference points and the asmptote to draw the transformed function on the grid with the parent function. Then describe the domain and range of the transformed function using set notation. INTEGRATE MATHEMATICAL PROPERTIES Focus on Modeling As a starting point for graphing combined transformations, students should be aware that the graph of ƒ () = ab alwas passes through the points (, a) and (, ab). g () = ( ) - - Identif parameters: a = b = Find reference points: (h, a + k) = (, - ) = (, ) ( h -, _ a b + k ) ( = -, _ - = (, ) ) Find the asmptote: = - Plot the points and draw the asmptote. Then connect the points with a smooth curve that approaches the asmptote without crossing it. - < < > - h = k = - = - - = - Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES What is the horizontal asmptote of the graph ƒ () = a b - h + k where b =? The horizontal asmptote of the graph ƒ () is = k. For the graph of ƒ () = a b - h + k where b = and (h, k) is at the origin, what are the reference points? (, a) and (-, a) Module 76 Lesson DIFFERENTIATE INSTRUCTION Communicating Math Have students make up their own functions in the form = ab - h + k for different a, h, and k values, and then discuss the transformations with each other. Eponential Deca Functions 76

5 CONNECT VOCABULARY Connect the terms deca and growth to life. When something alive grows, it tends to become taller and larger. When something alive decas, it tends to get smaller; it takes up less space. B g () = - ( ) + + Identif parameters: a = - b = h = - k = Find reference points: (h, a + k ) = (-, - + ) (-, 7) ) (- -, _ - + = - ( h -, a _ b + k ) = (, - ) Find the asmptote: = = Plot the points and draw the asmptote. Then connect the points with a smooth curve that approaches the asmptote without crossing it. - < < < - - Your Turn For the transformed function, use the reference points and the asmptote to draw the transformed function on the grid with the parent function. Then describe the domain and range of the transformed function using set notation. Houghton Mifflin Harcourt Publishing Compan. g () = ( ) + - Identif parameters: a = ; b = ; h = -; k = - Find reference points: ( (h, a + k) = (-, - ) = (-, - ) h -, a_ b + k ) = ( Find the asmptote: = - - < < _ - -,, - = (-, 5) ) = - > - Module 77 Lesson 77 Lesson.

6 Eplain Writing Equations for Combined Transformations of f () = b where < b < Given a graph of an eponential deca function, g () = ab - h + k, the reference points and the asmptote can be used to identif the transformation parameters in order to write the function rule. Eample Write the function represented b this graph and state the domain and range using set notation. - = (-, ) - - (-, -) - EXPLAIN Writing Equations for Combined Transformations of ƒ () = b where < b < INTEGRATE TECHNOLOGY Students can check the equations the write b graphing the functions on their graphing calculators. Have them use the TRACE or TABLE feature to identif coordinates of points in the resulting graph. Find k from the asmptote: k =. The first reference point is at (-,). Equate point value with parameters-based epression. (-, ) = (h, a + k) Use the -coordinate to solve for h. h = - Use the -coordinate to solve for a. a = - k = - The second reference point is at (-, -). Equate point value with parameters-based epression. (-, -) = ( h -, Equate -coordinate with parameters. Solve for b. g () = -( + ) + - < < < _- b + = - _- b = -6 b = _- -6 = a_ b + k ) Houghton Mifflin Harcourt Publishing Compan Module 7 Lesson Eponential Deca Functions 7

7 QUESTIONING STRATEGIES For a given value of k and a first reference point of (6, ), how do ou find the values of h and a? The value of h is 6 and the value of a is - k. For a given value of k and a and a second reference point of (, ), how do ou find the values of b? a Set the value of + k to, substitute b the values of a and k, and solve for b. Houghton Mifflin Harcourt Publishing Compan B - - = - (, ) (, - ) Find k from the asmptote: k = -. The first reference point is at (, - ), so (, - ) = ( h, a + k ) h = a = - k = a_ The second reference point is at (, ), so (, ) = (h -, b ) + k _ - = b _ b = 5 b = _ 5 = g () = ( ) < < > Module 79 Lesson 79 Lesson.

8 Reflect. Compare the -intercept and the asmptote of the function shown in this table to the function plotted in Eample A g () _ 9 6 The -intercept appears to have moved down b units from ( 5_ 5_ to ), and the asmptote appears to have moved down b units as well (from to ). 5. Compare the -intercept and the asmptote of the function shown in this table to the function plotted in Eample B g () The -intercept is not apparent in the graph but must be larger than, while the table shows that the - intercept is at The asmptote of both functions appears to be -. Your Turn Write the function represented b this graph and state the domain and range using set notation. 6. Asmptote: = -, so k = - First reference point: (, -) (, -) = (h, a + k) = - h = a = - - (-) = Second reference point: (, ) (, ) = ( a_ h -, b + k ) _ b - _ = b = b = g () = ( - ) - Houghton Mifflin Harcourt Publishing Compan - < < > - Module 7 Lesson Eponential Deca Functions 7

9 EXPLAIN Modeling With Eponential Deca Functions QUESTIONING STRATEGIES How is the deca factor related to the percent of decrease in value? The sum of the deca factor and the percent of decrease is. For eample, for a percent of decrease of %, the deca factor is.77. Eplain Modeling with Eponential Deca Functions Eponential deca functions can be applied to situations in which a quantit decreases b a constant percentage for each unit increase in time. ƒ (t) = a ( - r) t In this form of the deca function, r (which must be epressed as a decimal or a fraction rather than a percentage) is called the deca rate. The term ( - r) is known as the deca factor. The vertical stretch parameter, a, is also the value of the deca function at the start (when t = ). Eample Given the description of the deca terms, write the eponential deca function in the form f (t) = a ( - r) t and graph it with a graphing calculator. The value of a truck purchased new for $, decreases b 9.5% each ear. Write an eponential function for this situation and graph it using a calculator. Use the graph to predict after how man ears the value of the truck will be $5. Purchased new for $,... a =,...decreases b 9.5% each ear. r =.95 Substitute parameter values. Simplif. V T (t) =, ( -.95 ) t V T (t) =, (.95 ) t Graph the function with a graphing calculator. Use WINDOW to adjust the graph settings so that ou can see the function and the function values that are important. Houghton Mifflin Harcourt Publishing Compan Image Credits: Transtock Inc./Superstock Find when the value reaches $5 b finding the intersection between V T ( t ) =, (.95) t and V T ( t ) = 5 on the calculator. The intersection is at the point (7.6, 5), which means after 7.6 ears, the truck will have a value of $5. Module 7 Lesson LANGUAGE SUPPORT Graphic Organizers Have each pair of students complete a compare and contrast Venn diagram to show the similarities and differences between eponential deca and eponential growth functions. Encourage students to discuss and show the similarities and differences between their graphs, their equations, and so on. 7 Lesson.

10 B The value of a sports car purchased new for $5, decreases b 5% each ear. Write an eponential function for the depreciation of the sports car, and plot it along with the previous eample. After how man ears will the two vehicles have the same value if the are purchased at the same time? Purchased new for $5,... a = 5,...decreases b 5% each ear. r =.5 t Substitute parameter values. V c ( t ) = 5, ( -.5 ) t Simplif. V c ( t ) = 5, (.5 ) Add this plot to the graph for the truck value from Eample A and find the intersection of the two functions to determine when the values are the same. The intersection point is (, ). After ,55 ears, the values of both vehicles will be INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Eplain the concept of depreciation and how it is used in the business world. Discuss the difference between an item losing the same dollar amount of value each ear, known as straight-line depreciation, and the item losing the same percent of its value each ear, a tpe of depreciation which is based on an eponential deca model. $,55. Reflect 7. What reference points could ou use if ou plotted the value function for the sports car on graph paper? Confirm that the graph passes through them using the calculate feature on a graphing calculator. The transformation parameters are a = 5,, h =, and k =. The parent function is b =.5. The reference points are (h, a + k) = (, 5,) and ( h -, a_ b + k ) = (-, 5,9). Using the calculate feature confirms the graph passes through (, 5,) and (, 5,9).. Using the sports car from eample B, calculate the average rate of change over the course of the first ear and the second ear of ownership. What happens to the absolute value of the rate of change from the first interval to the second? What does this mean in this situation? Average rate of change during the interval from t to t = f ( t ) - f ( t ) t - t 5,.5-5, First ear: rate of change = = -$675 per ear - Second ear: rate of change = 5, (.5) - 5,.5 = -$577.5 per ear - The absolute value of the rate of change decreased during the second interval. This means that the car depreciates less each ear than the ear before. Houghton Mifflin Harcourt Publishing Compan Module 7 Lesson Eponential Deca Functions 7

11 ELABORATE QUESTIONING STRATEGIES How do ou rewrite eponential deca functions to answer questions about the functions? Properties of eponents can be used to rewrite eponential functions to show specific growth or deca factors. SUMMARIZE THE LESSON What does the graph of an eponential deca function look like? An eponential deca function is a function of the form = ab, with a > and < b <. Eponential deca models describe situations in which a quantit decreases b a fied percent each time period. The graph of an eponential deca function is a curve that falls from left to right and gets less and less steep as increases. The -ais, or a line parallel to it, is a horizontal asmptote of the graph. Your Turn 9. On federal income ta returns, self-emploed people can depreciate the value of business equipment. Suppose a computer valued at $765 depreciates at a rate of % per ear. Use a graphing calculator to determine the number of ears it will take for the computer s value to be $5. v (t) = 765 ( -.) t = 765 (.7) t Intersect with v (t) = 5 Using a graphing calculator, the intersection point is at (5.79, 5). It will take about 5.79 ears for the value of the computer to drop to $5. Elaborate ) or ƒ () = (. Which transformations of ƒ () = ( ) change the function s end behavior? Vertical translations change the horizontal asmptote and thus the end behavior as increases without bound. Reflections across the -ais change the end behavior as decreases without bound, from approaching positive infinit to approaching negative infinit.. 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 -intercept.. Discussion How are reference points and asmptotes helpful when graphing transformations of ƒ () = ( ) or ƒ () = ( ) or when writing equations for transformed graphs? Reference points and asmptotes are eas to transform and have a simple relationship to the function parameters (a, h, and k) associated with the transformation. The point (, ) becomes (h, a + k), the point ( -, becomes b) ( h -, a_ b + k ), and the asmptote = becomes = k. Houghton Mifflin Harcourt Publishing Compan. Give the general form of an eponential deca function based on a known deca rate and describe its parameters. f (t) = a ( - r) t a is the starting value, or the value at t =. r is the deca rate, or what fraction of the value is lost per unit of time. ( - r) is the deca factor, or what fraction of the previous value remains after the passage of a unit of time.. Essential Question Check-In How is the graph of ƒ () = b used to help graph the function g () = a b - h + k? The graph of g () = a b -h + k can be derived from the basic shape of the parent function, f () = b, using transformations based on the parameters a, h, and k. Module 7 Lesson 7 Lesson.

12 Evaluate: Homework and Practice. Graph the function ƒ () = ( ) b plotting points with integer -values from to. Online Homework Hints and Help Etra Practice EVALUATE f () Describe the transformation(s) from each parent function and give the domain and range of each function.. g () = ( +. g () = ) ( ) + Vertical translation up b. Domain: - < < >. g () = - ( ) g () = ( Reflection across the -ais, translation right b and up b. Domain: - < < < Horizontal translation left b. - < < > ) Vertical stretch b a factor of, translation left b and down b 6. - < < > 6 Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Graphing and Analzing ƒ () = ( and ƒ () = ( ) ) Eplore Predicting Transformations of the Graphs of ƒ () = ( ) and ƒ () = ( ) Eample Graphing Combined Transformations of ƒ () = b where < b <. Eample Writing Equations for Combined Transformations of ƒ () = b where < b < Eample Modeling With Eponential Deca Functions Practice Eercise Eercises 5 Eercises 6 Eercises Eercises 6 Module 7 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information.E Create and use representations 5 Skills/Concepts.F Analze relationships 6 Skills/Concepts.E Create and use representations Skills/Concepts.F Analze relationships 6 Skills/Concepts.A Everda life 7 Strategic Thinking.D Multiple representations Strategic Thinking.F Analze relationships Eponential Deca Functions 7

13 AVOID COMMON ERRORS Quickl check for students who identif a domain other than - to for an eponential function or its transformation (other than when restricted b a real-world constraint). Note that eponential functions do not have a vertical asmptote, so the continue unbounded in each direction. For each of the transformed functions, use the reference points and the asmptote to draw the transformed function on the grid. Then describe the domain and range of the transformed function using set notation. 6. g () = - ( - ) Identif parameters: a = -; b = ; h = ; k = Find reference points: (h, a a + k) = (, - + ) = (, ) (h -, b + k ) = ( -, - + = (, -) ) Find the asmptote: = - < < < 7. g () = ( ) + + Identif parameters: a = ; b = ; h = -; k = Find reference points: Houghton Mifflin Harcourt Publishing Compan g () = ( = - - ) = = = (h, a + k) = (-, + ) = (-, ) a_ h -, b + k ) = - -, ( + = (-, 7) ) ( Find the asmptote: = - < < > Identif parameters: a = ; b = ; h = ; k = Find reference points: (h, a + k) = (, + ) = ( 5_, ) -, _ a_ ( h -, b + k ) ( = Find the asmptote: = - < < > + ) = ( -, 7_ ) Module 75 Lesson A_MTXESE597_U6ML.indd 75 //5 : AM 75 Lesson.

14 9. g () = ( ) = - Identif parameters: a = ; b = ; h = ; k = - Find reference points: (h, a + k) = (, - ) = (, -) a ( h -, -, - = (, 5) ) b + k ) = ( Find the asmptote: = - Domain: - < < > -. g () = - ( ) = 7 = Identif parameters: a = -; b = ; h = - ; k = 7 Find reference points: ( h -, a _ (h, a + k) = (-, - + 7) = (-, ) b + k ) = ( - -, _ - Find the asmptote: = 7 - < < + 7 = (-, ) ) < 7. g () = - ( ) _ = = Identif parameters: a = -; b = _ ; h = -; k = Find reference points: (h, a + k) = ( -, - + -, - b + k ) = ( ) = ( ) ( h -, _ a - -,_ + Find the asmptote: = ǀ- < < Range: ǀ < ) = (-, -) Houghton Mifflin Harcourt Publishing Compan Module 76 Lesson A_MTXESE597_U6ML.indd 76 //5 : AM Eponential Deca Functions 76

15 Write the function represented b each graph and state the domain and range using set notation. Asmptote:. = - (, 7) k = - 6 First reference point: (, ) (, ) = (h, a + k) (, ) h = a = - (-) a = - 6 = - Second reference point: (, 7) - (, 7) = ( h -, _ a b + k ) _ b - = 7 _ b = b = _ b = g () = ( ) - - ǀ- < < Range: ǀ > - Houghton Mifflin Harcourt Publishing Compan. -6 = (-, ) (-5, -) - Asmptote: = k = First reference point: (-, ) (-, ) = (h, a + k) h = - a = - a = - Second reference point: (-5, -) (-5, -) = ( h -, _ a b + k ) _- b + = - _- b = -6 b = _- -6 b = g () = -( + ) + ǀ- < < Range: ǀ < Module 77 Lesson 77 Lesson.

16 Write the eponential deca function described in the situation and use a graphing calculator to answer each question asked.. Medicine A quantit of insulin used to regulate sugar in the bloodstream breaks down b about 5% each minute after the injection. A bodweight-adjusted dose is generall units. How long does it take for the remaining insulin to be half of the original injection? l (t) = ( -.5) t = (.95) t Half of the original injection () is 5. Use a graph to find the intersection with l (t) = 5. Intersection point is at (.5, 5). It takes about.5 minutes. PEER-TO-PEER DISCUSSION Ask students to brainstorm and discuss two specific, real-world situations, one of eponential growth and one of eponential deca. For each situation, ask students to discuss how the graphs should look and tell how the would go about creating equations to help them model real-world data. 5. Paleontolog Carbon- is a radioactive isotope of carbon that is used to date fossils. There are about.5 atoms of carbon- for ever trillion atoms of carbon in the atmosphere, which known as.5 ppt (parts per trillion). Carbon in a living organism has the same concentration as carbon-. When an organism dies, the carbon- content decas at a rate of.% per millennium ( ears). Write the equation for carbon- concentration (in ppt) as a function of time (in millennia) and determine how old a fossil must be that has a measured concentration of. ppt. c (t) =.5 ( -.) t =.5 (.6) t Intersection point is at (6.65,.). The fossil is about 6.65 millennia, or 6,65 ears old. 6. Music Stringed instruments like guitars and pianos create a note when a string vibrates back and forth. The distance that the middle of the string moves from the center is called the amplitude (a), and for a guitar, it starts at.75 mm when a note is first struck. Amplitude decas at a rate that depends on the individual instrument and the note, but a deca rate of about 5% per second is tpical. Calculate the time it takes for an amplitude of.75 mm to reach. mm. a (t) =.75 ( -.5) t =.75 (.75) t Intersection point is at (7.,.). The amplitude will reach. mm in about 7 seconds. Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Eliana Aponte/Reuters/Corbis; (b) Joshua David Treisner/Shutterstock Module 7 Lesson Eponential Deca Functions 7

17 JOURNAL Have students write about the two tpes of eponential models, and describe how the differ from polnomial models such as quadratic and cubic. H.O.T. Focus on Higher Order Thinking 7. Analze Relationships Compare the graphs of ƒ () = ( Which of the following properties are the same? Eplain. a. Domain ǀ- < < ; ǀ b. Range c. End behavior as increases ǀ > ; ǀ d. End behavior as decreases ) and g () =. f () ; g () f () ; g () is not defined for values less than. None are the same.. Communicate Mathematical Ideas A quantit is reduced to half of its original amount during each given time period. Another quantit is reduced to one quarter of its original amount during the same given time period. Determine each deca rate, state which is greater, and eplain our results. The deca rate of the first quantit is 5% because the deca factor is - r =, so the deca rate equals, or 5%. The deca rate of the second quantit is 75% because the deca factor is - r =, so the deca rate is, or 75%. The deca rate of the second quantit is greater. 9. Multiple Representations Eponential deca functions are written as transformations of the function ƒ () = b, where < b <. However, it also possible to use negative eponents as the basis of an eponential deca function. Use the properties of eponents to show wh the function ƒ () = - is an eponential deca function. Houghton Mifflin Harcourt Publishing Compan Given f () = - Power of a power propert = ( - ) Propert of negative eponents = ( ) The last result is in the form f () = b where < b < and is therefore an eponential deca function.. Represent Real-World Problems You bu a video game console for $5 and sell it 5 ears later for $. The resale value decas eponentiall over time. Write a function that represents the resale value, R, in dollars, over the time, t, in ears. Eplain how ou determined our function. R (t) = 5 (.75) t ; Sample answer: I used the general eponential deca function f () = ab and substituted for f (), 5 for a, and 5 for, resulting in = 5 ( b 5 ). I then solved for b b dividing b 5 to get. and then took the fifth root of., resulting in.7779, which I rounded to.75. Module 79 Lesson 79 Lesson.

18 Lesson Performance Task Sodium- is a radioactive isotope of sodium used as a diagnostic aid in medicine. It undergoes radioactive deca to form the stable isotope magnesium- and has a half-life of about 5 hours. This means that, in this time, half the amount of a sample mass of sodium- decas to magnesium-. Suppose we start with an initial mass of of grams sodium-. a. Use the half-life of sodium- to write an eponential deca function of the form m Na (t) = m ( - r) t, where m is the initial mass of sodium-, r is the deca rate, t is the time in hours, and m Na (t) is the mass of sodium- at time t. What is the meaning of r? a. Substitute 5 for m Na (t), for a, and 5 for t in the function. 5 = ( - r) 5.5 = ( - r) = 5 ( - r) r.5 r m Na (t) = (.955) t The value of r means that the mass of sodium- is reduced b.5% each hour. b. The combined amounts of sodium- and magnesium- must equal m, or, for all possible values of t. Show how to write a function for m Mg (t), the mass of magnesium- as a function of t. b. The sum of the mass of magnesium- and sodium- is equal to m, which is. m Mg (t) + m Na (t) = m m Mg (t) + m Na (t) = Solve for m Mg (t). m Mg (t) = - m Na (t) Substitute (.955) t for m Na (t) m Mg (t) = - (.955) t c. Use a graphing calculator to graph m Na (t) and m Mg (t). Describe the graph of m Mg (t) as a series of transformations of m Na (t). What does the intersection of the graphs represent? c. The graph of m Mg (t) is a reflection of the graph of m Na (t) across the t-ais and a translation of units verticall. The intersection of the graphs represents the point where the mass of sodium- is equal to the mass of magnesium-, which occurs at the first half-life of sodium-. Mass of Na/Mg (g) 6 Mg Na 5 Time in Minutes (t) 5 Module 75 Lesson Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS Students ma set r equal to -.5 because this is a deca situation. However, this would make the term ( - r) t greater than one, and the function m Na (t) would become a growth function. Eplain to students that the term ( - r) t alread contains the minus sign that turns m Na (t) into a deca function. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Have students consider the graphs for m Na (t) and m Mg (t), and have them eplain which is eponential deca and which is eponential growth, based on the properties of the graphs. Have students discuss whether the can determine from the graphs the final values of the functions as t gets ver large. EXTENSION ACTIVITY Have students research the half-life of technetium-99m, another rad ioactive isotope widel used in medicine. Have students write an eponential deca function for an initial mass of grams. Then have students graph this function and compare it to the one for sodium-. Have students discuss the difference in deca rates and how that might affect a real-world situation. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 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. Eponential Deca Functions 75