15.2 Graphing Logarithmic

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1 _ Locker LESSON 5. Graphing Logarithmic Functions Teas Math Standards The student is epected to: A.5.A Determine the effects on the ke attributes on the graphs of f () = b and f () = log b () where b is,, and 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.7.I Mathematical Processes A..E Create and use representations to organize, record, and communicate mathematical ideas. Language Objective.D.,.I., 3.B.3, 3.H.3,.D,.F Work with a partner to compare and contrast the graphs of eponential and logarithmic functions. ENGAGE Essential Question: How is the graph of g () = a log b ( - h) + k where b > and b related to the graph of f ( ) = log b? The graph of g () = alog b ( - h) + k involves transformations of the graph of f () = log b. 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. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and have students suggest the first step the would take to find models for this data set. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Name Class Date 5. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b related to the graph of f () = log b? A.5.A Determine the effects on the ke attributes on the graphs of f () = log b () where b is,, and 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.7.I Eplore Graphing and Analzing Parent Logarithmic Functions The graph of the logarithmic function ƒ () = log, which ou analzed in the previous lesson, is shown. In this Eplore, ou ll graph and analze other basic logarithmic functions. Complete the table for the function ƒ () = log. (Remember that when the base of a logarithmic function is not specified, it is understood to be.) Then plot and label the ordered pairs from the table and draw a smooth curve through the points to obtain the graph of the function.. f () = log - Complete the table for the function ƒ () = ln. (Remember that the base of this function is e). Then plot and label the ordered pairs from the table and draw a smooth curve through the points to obtain the graph of the function. _ e.368 e.7 e 7.39 f () = ln (, ) (, ) (, ) (8, 3) Resource Locker 6 8 (.5, -) (, ) 6 8 (., -) (e, ) (, ) ( e, ) (e, ) 6 8 (, ) Module Lesson DO NOT EDIT--Changes must be made through File info CorrectionKe=TX-B Name Class Date 5. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b related to the graph of f () = log b? A.5.A Determine the effects on the ke attributes on the graphs of f () = log b () where b is,, and 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.7.I Houghton Mifflin Harcourt Publishing Compan Eplore Graphing and Analzing Parent Logarithmic Functions The graph of the logarithmic function ƒ () = log, which ou analzed in the previous lesson, is shown. In this Eplore, ou ll graph and analze other basic logarithmic functions. Complete the table for the function ƒ () = log. (Remember that when the base of a logarithmic function is not specified, it is understood to be.) Then plot and label the ordered pairs from the table and draw a smooth curve through the points to obtain the graph of the function. f () = log -. Complete the table for the function ƒ () = ln. (Remember that the base of this function is e). Then plot and label the ordered pairs from the table and draw a smooth curve through the points to obtain the graph of the function. f () = ln e.368 e.7 e Resource (8, 3) (, ) (, ) (, ) 6 8 (.5, -) (, ) (, ) 6 8 (., -) (e, ) (, ) (e, ) 6 8 ( e, ) Module Lesson A_MTXESE35397_U6M5L 839 //5 :7 PM HARDCOVER PAGES 59 6 Turn to these pages to find this lesson in the hardcover student edition. 839 Lesson 5.

2 C Analze the two graphs from Steps A and B, and then complete the table. Function f () = lo g () f () = log f () = ln Domain > Range < < > - < < + > - < < + EXPLORE Graphing and Analzing Parent Logarithmic Functions End behavior Vertical and horizontal asmptotes As +, f () +. As +, f () -. Vertical asmptote at = ; no horizontal asmptote As +, f () +. As +, f () -. Vertical asmptote at = ; no horizontal asmptote As +, f () +. As +, f () -. Vertical asmptote at = ; no horizontal asmptote INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activit either in the book or online. Intervals where increasing or decreasing Intercepts Intervals where positive or negative Increasing throughout its domain -intercept at (, ); no -intercepts Positive on (, + ) ; negative on (, ) Increasing throughout its domain -intercept at (, ); no -intercepts Positive on (, + ) ; negative on (, ) Increasing throughout its domain -intercept at (, ); no -intercepts Positive on (, + ) ; negative on (, ) QUESTIONING STRATEGIES What point or points do all of the graphs have in common? (, ) Wh does each of the graphs contain this point? Because log b = for all nonzero values of b. Reflect. What similarities do ou notice about all logarithmic functions of the form ƒ () = lo g b where b >? What differences do ou notice? The all have a vertical asmptote at =, an -intercept at, are alwas increasing, and the have the same end behavior. The rates of change are different. Houghton Mifflin Harcourt Publishing Compan Module 5 8 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions In previous lessons, students learned how different parameters affect the graphs of various tpes of functions, including quadratic, radical, and eponential functions. In this lesson, the complete their stud of transformations in Algebra b appling what the ve learned to the graphs of logarithmic functions. The predict how changing the parameters of a logarithmic function will transform the graph of the function, and how those changes affect the attributes of the function, including its domain and range, its asmptote, and its end behavior. Graphing Logarithmic Functions 8

3 EXPLORE Predicting Transformations of the Graphs of Parent Logarithmic Functions Eplore Predicting Transformations of the Graphs of Parent Logarithmic Functions You can graph the logarithmic function ƒ () = log b where b > and b on a graphing calculator b specifing the base when ou enter the function s rule using the LOG ke after pressing the Y= ke. For instance, the first calculator screen shows how to enter the function ƒ () = log, and the second screen shows the function s graph. Notice that the graph passes through the point (, ) as ou would epect. AVOID COMMON ERRORS Students ma make errors identifing the direction of the horizontal translation. Use their earlier eperiences with transformations of other tpes of functions to remind them that the horizontal shift is represented b the number subtracted from, and is to the right when that number is positive, and to the left when that number is negative. Thus, an argument of ( ) translates the graph to the right, and an argument of ( + ) translates it to the left. In this Eplore, ou will predict the effects of parameters on the graphs of logarithmic functions with bases,, and e. You will then confirm our predictions b graphing the transformed functions on a graphing calculator. Predict the effect of the parameter h on the graph of g () = log b ( - h) for each function. horizontal translation a. The graph of g () = log ( - ) is a of the graph of ƒ () = log [right/left/up/down] units. horizontal translation b. The graph of g () = log ( + ) is a of the graph of ƒ () = log [right/left/up/down] units. horizontal translation c. The graph of g () = log ( - ) is a of the graph of Houghton Mifflin Harcourt Publishing Compan ƒ () = log [right/left/up/down] unit. horizontal translation d. The graph of g () = log ( + ) is a of the graph of ƒ () = log [right/left/up/down] unit. horizontal translation e. The graph of g () = ln ( - 3) is a of the graph of ƒ () = ln [right/left/up/down] 3 units. horizontal translation f. The graph of g () = ln ( + 3) is a of the graph of ƒ () = ln [right/left/up/down] 3 units. Check our predictions using a graphing calculator. Module 5 8 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Provide each pair with an eponential function. Have them graph their function, and then use the points on the graph to graph the inverse of the function. Have them write an equation of the inverse function using what the ve learned about inverse functions and transformations. Then have each pair present their work to the class. 8 Lesson 5.

4 B Predict the effect of the parameter k on the graph of g () = lo g b + k for each function. a. The graph of g () = lo g + 3 is a vertical translation of the graph of ƒ () = lo g [right/up/left/down] 3 units. b. The graph of g () = lo g - 3 is a vertical translation of the graph of ƒ () = lo g [right/up/left/down] 3 units. c. The graph of g () = log + is a vertical translation of the graph of ƒ () = log [right/up/left/down] units. d. The graph of g () = log - is a vertical translation of the graph of ƒ () = log [right/up/left/down] units. e. The graph of g () = ln + is a vertical translation of the graph of QUESTIONING STRATEGIES How can ou tell the difference between a logarithmic function that is a vertical translation of the parent function and one that is a horizontal translation of the parent function? It is a vertical translation if there is a number being added to or subtracted from the logarithmic epression. It is a horizontal translation if there is a number being added to or subtracted from. ƒ () = ln [right/left/up/down] unit. f. The graph of g () = ln - is a vertical translation of the graph of ƒ () = ln [right/left/up/down] unit. Check our predictions using a graphing calculator. C Predict the effect of the parameter a on the graph of g () = alo g b for each function. a. The graph of g () = lo g is a vertical stretch of the graph of ƒ () = lo g b a factor of. b. The graph of g () = - _ lo g is a vertical compression of the graph of _ ƒ () = lo g b a factor of reflection -ais as well as a across the. vertical stretch c. The graph of g () = -3 log is a of the graph of 3 ƒ () = log b a factor of as well as a reflection across the -ais. d. The graph of g () = _ 3 ƒ () = log b a factor of 3. log is a vertical compression of the graph of e. The graph of g () = ln is a vertical stretch of the graph of ƒ () = ln b a factor of. f. The graph of g () = - _ vertical compression ln is a of the graph of ƒ () = ln b a factor of reflection as well as a across the -ais. Check our predictions using a graphing calculator. Houghton Mifflin Harcourt Publishing Compan Module 5 8 Lesson DIFFERENTIATE INSTRUCTION Visual Cues Visual learners can benefit from using colored pencils to graph the intermediate stages when transforming the graph of the parent function, f () = log b, to the graph of g () = a log b ( - h) + k. For eample, one color can be used to graph a reflection across the -ais if a <, a different color can be used to graph an vertical stretch or compression of the reflected graph, and a third color can be used to graph the translation of the previous graph. Graphing Logarithmic Functions 8

5 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Discuss with students the fact that since a logarithm is an eponent, transformations of logarithmic functions are similar to transformations of eponential functions. Students should be epected to use what the have learned about the transformations of eponential functions to predict how the different tpes of transformations will affect the graph of logarithmic functions. D Complete the table b identifing which parameters (a, h, and/or k) affect the attributes of each logarithmic function listed in the table. When necessar, indicate specificall whether a positive or negative value of a parameter affects an attribute. (For end behavior, consider both increasing without bound and decreasing toward from the right.) Function g () = a lo g ( - h) + k g () = a log ( - h) + k g () = a ln ( - h) + k Parameters that affect the domain Parameters that affect the range Parameters that affect the end behavior Parameters that affect the vertical asmptote Parameters that affect whether the function is increasing or decreasing Parameters that affect the -intercept Parameters that affect the intervals where the function is positive or negative h None a, h; a < h a; a < h, k a, h; a < h None a, h; a < h a; a < h, k a, h; a < h None a, h; a < h a; a < h, k a, h; a < Reflect Houghton Mifflin Harcourt Publishing Compan. In Step D, do the effects of the parameters on the attributes of the logarithmic functions depend on the base of the function? Eplain. No, because all the bases in Step D are greater than and so all the functions have the same attributes. 3. In Step D, would our answers change if the logarithmic functions had bases between and instead of bases greater than? Eplain. No, because although logarithmic functions with bases between and have attributes that are different from logarithmic functions with bases greater than, each attribute is still affected b the same parameter(s) regardless of the base. Module 5 83 Lesson 83 Lesson 5.

6 Eplain Graphing Combined Transformations of f () = log b Where b > When graphing transformations of ƒ () = log b where b >, it helps to consider the effect of the transformations on the following features of the graph of ƒ () : the vertical asmptote, =, and two reference points, (, ) and (b, ). The table lists these features as well as the corresponding features of the graph of g () = a log b ( - h) + k. Function f () = log b g () = a log b ( - h) + k EXPLAIN Graphing Combined Transformations of f () = log b Where b > Eample Asmptote = = h Reference point (, ) ( + h, k) Reference point (b, ) (b + h, a + k) Identif the transformations of the graph of f () = log b that produce the graph of the given function g (). Then graph g () on the same coordinate plane as the graph of f () b appling the transformations to the asmptote = and to the reference points (, ) and (b, ). Also state the domain and range of g () using set notation. QUESTIONING STRATEGIES Wh is the equation of the asmptote = h? Because the asmptote of f () = log b is the vertical line =, and it is being translated h units in the horizontal direction. g () = - log ( - ) - The transformations of the graph of ƒ () = log that produce the graph of g () are as follows: a vertical stretch b a factor of a reflection across the -ais a translation of unit to the right and units down Note that the translation of unit to the right affects onl the -coordinates of points on the graph of ƒ (), while the vertical stretch b a factor of, the reflection across the -ais, and the translation of units down affect onl the -coordinates. Function f () = log g () = - log ( - ) - Asmptote = = Reference point (, ) ( +, - () - ) = (, -) Reference point (, ) ( +, - () - ) = (3, -) Houghton Mifflin Harcourt Publishing Compan Domain: > Range: < < + Module 5 8 Lesson Graphing Logarithmic Functions 8

7 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Discuss with students how a vertical compression affects the rate of change of the graph of a logarithmic function. Have them compare the steepness of the graphs of f () = log and g () = log in the intervals (, ) and (, + ). Students should observe that the graph of g () is steeper than the graph of f () in the first interval, and less steep in the second interval. B g () = log ( + ) + The transformations of the graph of ƒ () = log that produce the graph of g () are as follows: a vertical stretch b a factor of a translation of units to the left and units up Note that the translation of units to the left affects onl the -coordinates of points on the graph of ƒ (), while the vertical stretch b a factor of and the translation of units up affect onl the -coordinates. Function f () = log g () = log ( + ) + Asmptote = = - Reference point (, ) ( -, ( ) ) + = ( -, ) Reference point (, ) ( -, ( ) ) ( + = 8, 6 ) Domain: > - Houghton Mifflin Harcourt Publishing Compan Range: - < < + Module 5 85 Lesson 85 Lesson 5.

8 Your Turn Identif the transformations of the graph of ƒ () = lo g b that produce the graph of the given function g (). Then graph g () on the same coordinate plane as the graph of ƒ () b appling the transformations to the asmptote = and to the reference points (, ) and (b, ). Also state the domain and range of g () using set notation.. g () = 3 ln ( + ) Domain: > - Range: - < < + The transformations of the graph of f () = In that produce the graph of g () are as follows: a vertical stretch b a factor of 3 a translation of units to the left and units up Note that the translation of units to the left affects onl the -coordinates of points on the graph of f (), while the vertical stretch b a factor of 3 and the translation of units up affect onl the -coordinates. Function f () = In g () = 3 In ( + ) + Asmptote = = - Reference point (, ) ( -, 3 () + ) = (-3, ) Reference point (e, ) (e -, 3 () + ) (-.8, 5) 5. g () = lo g ( + ) Domain: > - Range: - < < + The transformations of the graph of f () = lo g that produce the graph of g () are as follows: a vertical compression b a factor of a translation of unit to the left and units up Houghton Mifflin Harcourt Publishing Compan Module 5 86 Lesson Graphing Logarithmic Functions 86

9 Note that the translation of unit to the left affects onl the -coordinates of points on the graph of f (), while the vertical compression b a factor of and the translation of units up affect onl the -coordinates. _ Function f () = lo g g () = l og ( + ) + Asmptote = = - Reference point (, ) ( _ -, () + ) = (, ) Reference point (, ) ( _ -, () + ) = ( _, ) 6. g () = -3 log ( - ) Domain: > Range: - < < + Houghton Mifflin Harcourt Publishing Compan The transformations of the graph of f () = log that produce the graph of g () are as follows: a vertical stretch b a factor of 3 a reflection across the -ais a translation of unit to the right and units down Note that the translation of unit to the right affects onl the -coordinates of points on the graph of f (), while the vertical stretch b a factor of 3, the reflection across the -ais, and the translation of units down affects onl the -coordinates. Function f () = log g () = -3 log ( - ) - Asmptote = = Reference point (, ) ( +, -3 () - ) = (, -) Reference point (, ) ( +, -3 () - ) = (, -7) Module 5 87 Lesson 87 Lesson 5.

10 Eplain Writing, Graphing, and Analzing a Logarithmic Model You can obtain a logarithmic model for real-world data either b performing logarithmic regression on the data or b finding the inverse of an eponential model if one is available. Eample A biologist studied a population of foes in a forest preserve over a period of time. The table gives the data that the biologist collected. Years Since Stud Began Fo Population From the data, the biologist obtained the eponential model P = 6 (.) t where P is the fo population at time t (in ears since the stud began). The biologist is interested in having a model that gives the time it takes the fo population to reach a certain level. EXPLAIN Writing, Graphing, and Analzing a Logarithmic Model QUESTIONING STRATEGIES How are the eponential model presented in the Eample and the logarithmic model obtained using regression related? The are inverses of each other. Wh do ou reverse the coordinates of the data points when entering the data into the calculator? because ou are finding the equation of the inverse function One wa to obtain the model that the biologist wants is to perform logarithmic regression on a graphing calculator using the data set but with the variables switched (that is, the fo population is the independent variable and time is the dependent variable). After obtaining the logarithmic regression model, graph it on a scatter plot of the data. Analze the model in terms of whether it is increasing or decreasing as well as its average rate of change from P = to P =, from P = to P = 3, and from P = 3 to P =. Do the model s average rates of change increase, decrease, or sta the same? What does this mean for the fo population? Using a graphing calculator, enter the population data into one list (L) and the time data into another list (L). Houghton Mifflin Harcourt Publishing Compan Image Credits: Roaltfree/Corbis Module 5 88 Lesson Graphing Logarithmic Functions 88

11 INTEGRATE MATHEMATICAL PROCESSES Focus on Technolog Students can use the table feature of a graphing calculator to compare the values of the two functions for particular values of the independent variable. Students should find that the functions produce values that are roughl equivalent. Perform logarithmic regression b pressing the STAT ke, choosing the CALC menu, and selecting 9:LnReg. Note that the calculator s regression model is a natural logarithmic function. So, the model is t = ln P. Graphing this model on a scatter plot of the data visuall confirms that the model is a good fit for the data. From the graph, ou can see that the function is increasing. To find the model s average rates of change, divide the change in t (the dependent variable) b the change in P (the independent variable): t Average rate of change = _ - t P - P Population Number of Years to Reach That Population Average Rate of Change Houghton Mifflin Harcourt Publishing Compan t = ln.3 t = ln.3 3 t = ln t = ln 6.3 _ _ - = 6. = = 3.5 = =.5 =.5 The model s average rates of change are decreasing. This means that as the fo population grows, it takes less time for the population to increase b another foes. Module 5 89 Lesson 89 Lesson 5.

12 B Another wa to obtain the model that the biologist wants is to find the inverse of the eponential model. Find the inverse model and compare it with the logarithmic regression model. In order to compare the inverse of the biologist s model, P = 6 (.) t, with the logarithmic regression model, ou must rewrite the biologist s model with base e so that the inverse will involve a natural logarithm. This means that ou want to find a constant c such that e c =.. Writing the eponential equation e c =. in logarithmic form gives c = ln., so c =.3 to the nearest thousandth. t Replacing. with e in the biologist s model gives P = 6 ( e ) find the inverse of this function. Write the equation. Divide both sides b 6. Write in logarithmic form P = 6e P_ 6 = e.3 t ln P_ 6 =.3 t Divide both sides b ln P_ 6 = t t, or P = 6 e t. Now So, the inverse of the eponential model is t = 8.85 ln P_. To compare this model with the logarithmic 6 regression model, use a graphing calculator to graph both = 8.85 ln _ and = ln. You observe that the graphs [roughl coincide/significantl diverge], so the models are [basicall equivalent/ver different]..3 Reflect 7. Discussion In a later lesson, ou will learn the quotient propert of logarithms, which states that log _ m b n = log b m - log b n for an positive numbers m and n. Eplain how ou can use this propert to compare the two models in Eample 3. You can use the quotient propert of logarithms to rewrite t = 8.85 In P_ P_ as follows: 6 t = 8.85 In = 8.85 (In P - In 6) 8.85 (In P -.3) 8.85 In P Comparing this result with the logarithmic regression model t = In P shows that the corresponding constants in the two equations are approimatel equal. Your Turn 8. Maria made a deposit in a bank account and left the mone untouched for several ears. The table lists her account balance at the end of each ear. Years Since the Deposit Was Made Account Balance $. $. $. 3 $6. Houghton Mifflin Harcourt Publishing Compan Module 5 85 Lesson Graphing Logarithmic Functions 85

13 ELABORATE QUESTIONING STRATEGIES Which transformations of f () = log b change the function s domain? horizontal translations Which transformations of f () = log b change the function s range? No transformations change the range. The range of logarithmic functions is all real numbers. SUMMARIZE THE LESSON How do the values of a, h, and k in the function g () = a log b ( - h) + k tell ou how to transform the graph of the function f () = log b? h tells ou how far to translate the graph horizontall; k tells ou how far to translate the graph verticall; a tells ou if there is a vertical stretch or compression and, if a <, that there is a reflection across the -ais. Houghton Mifflin Harcourt Publishing Compan a. Write an eponential model for the account balance as a function of time (in ears since the deposit was made). b. Find the inverse of the eponential model after rewriting it with a base of e. Describe what information the inverse gives. c. Perform logarithmic regression on the data (using the account balance as the independent variable and time as the dependent variable). Compare this model with the inverse model from part b. a. The ratio of the account balances for consecutive ears is., so an eponential model for the data is B = (.) t where B is the account balance at time t (in ears since the deposit was made). b. First, find the constant c such that e c =.. Rewriting e c =. in logarithmic form gives c = In... Then, find the inverse of B = ( e. ) t = e.t. Elaborate B = e.t B = e.t B In =.t B 5 In = t B So, the inverse of the eponential model is t = 5 In, which gives the time it takes for the account balance to reach a certain level. c. A graphing calculator gives the logarithmic regression model t = In B. Graphing this model and the inverse model from part b shows that the models are basicall equivalent. 9. Which transformations of ƒ () = log b () change the function s end behavior (both as increases without bound and as decreases toward from the right)? Which transformations change the location of the graph s -intercept? Reflections across the -ais change the end behavior. Horizontal and vertical translations affect the -intercept.. How are reference points helpful when graphing transformations of ƒ () = lo g b ()? Reference points provide a guide to the general shape of a transformed graph.. What are two was to obtain a logarithmic model for a set of data? Perform logarithmic regression on the data, or find the inverse of an eponential model if one is available.. Essential Question Check-In Describe the transformations ou must perform on the graph of ƒ () = lo g b () to obtain the graph of g () = a lo g b ( - h) + k. If a <, reflect the parent graph across the -ais. Then either stretch the graph verticall b a factor of a if a > or compress the graph verticall b a factor of a if a <. Finall, translate the graph h units horizontall and k units verticall. Module 5 85 Lesson LANGUAGE SUPPORT Communicating Math Have students work in pairs to complete a compare and contrast chart like the one below. Tpe of function Equation Graph Similarities and Differences Eponential Logarithmic 85 Lesson 5.

14 Evaluate: Homework and Practice. Graph the logarithmic functions ƒ () = log, ƒ () = log, and ƒ () = ln on the same coordinate plane. To distinguish the curves, label the point on each curve where the -coordinate is. Online Homework Hints and Help Etra Practice EVALUATE - - (, ) (e, ) (, ) 6 8. Describe the attributes that the logarithmic functions ƒ () = log, ƒ () = log, and ƒ () = ln have in common and the attributes that make them different. Attributes should include domain, range, end behavior, asmptotes, intercepts, intervals where the functions are positive and where the are negative, intervals where the functions are increasing and where the are decreasing, and the average rate of change on an interval. Attributes that the functions have in common: The domain of all three functions is >. The range of all three functions is - < < +. All three functions have the same end behavior: As +, f () +. As +, f () -. The graphs of all three functions have the same vertical asmptote: = (the -ais). The graphs of all three functions have the same -intercept, at =, and no -intercept. All three functions are increasing throughout their domains. All three functions are negative on the interval (, ) and positive on the interval (, + ). Attributes that make the functions different: Each function has a different average rate of change on a given interval. For instance, on the interval,, the average rate of change for f () = log is log - log = - =, the average rate of change for f () = log - log - log is.3 - =.3, and the average rate of change for f () = In - In - In is = Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Graphing and Analzing Parent Logarithmic Functions Eplore Predicting Transformations of the Graphs of Parent Logarithmic Functions Eample Graphing Combined Transformations of f () = log b Where b > Eample Writing, Graphing, and Analzing a Logarithmic Model QUESTIONING STRATEGIES Practice Eercises Eercises 3 5 Eercises 6 Eercises 3 Do translations of the graph of a logarithmic function affect the domain and/or the range of the function? Eplain. A horizontal translation affects the domain, but not the range. A vertical translation does not affect the domain or the range, because the range is alwas all real numbers. Module 5 85 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information.E Create and use representations Skills/Concepts.E Create and use representations 3 6 Skills/Concepts.F Analze relationships 7 Skills/Concepts.E Create and use representations 3 3 Strategic Thinking.A Everda life 6 3 Strategic Thinking.G Eplain and justif arguments Graphing Logarithmic Functions 85

15 AVOID COMMON ERRORS Students ma find that functions grow so quickl (eponential) or slowl (logarithmic) that it will appear that the functions are asmptotic when the are not. Remind students that eponential graphs have onl horizontal asmptotes, while logarithmic graphs have onl vertical asmptotes. 3. For each of the si functions, describe how its graph is a transformation of the graph of ƒ () = lo g. Also identif what attributes of ƒ () = lo g change as a result of the transformation. Attributes to consider are the domain, the range, the end behavior, the vertical asmptote, the -intercept, the intervals where the function is positive and where it is negative, and whether the function increases or decreases throughout its domain. a. g () = lo g - 5 The graph of g () is a vertical translation of the graph of f () down 5 units. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. b. g () = lo g The graph of g () is a vertical stretch of the graph of f () b a factor of. No attributes change. c. g () = lo g ( + 6) The graph of g () is a horizontal translation of the graph of f () left 6 units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. d. g () = - 3_ lo g The graph of g () is a vertical compression of the graph of f () b a factor of 3_ as well as a reflection across the -ais. The attributes that change are the end behavior and whether the function increases or decreases throughout its domain. e. g () = lo g + 7 The graph of g () is a vertical translation of the graph of f () up 7 units. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. f. g () = lo g ( - 8) The graph of g () is a horizontal translation of the graph of f () right 8 units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. Houghton Mifflin Harcourt Publishing Compan. For each of the si functions, describe how its graph is a transformation of the graph of ƒ () = log. Also identif what attributes of ƒ () = log change as a result of the transformation. Attributes to consider are the domain, the range, the end behavior, the vertical asmptote, the -intercept, the intervals where the function is positive and where it is negative, and whether the function increases or decreases throughout its domain. a. g () = -3 log The graph of g () is a vertical stretch of the graph of f () b a factor of 3 as well as a reflection across the -ais. The attributes that change are the end behavior and whether the function increases or decreases throughout its domain. b. g () = log ( - 5) The graph of g () is a horizontal translation of the graph of f () right 5 units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. Module Lesson 853 Lesson 5.

16 c. g () = log + The graph of g () is a vertical translation of the graph of f () up unit. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. d. g () = log ( + ) The graph of g () is a horizontal translation of the graph of f () left units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. e. g () =.5 log The graph of g () is a vertical compression of the graph of f () b a factor of.5. No attributes change. f. g () = log - The graph of g () is a vertical translation of the graph of f () down units. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. 5. For each of the si functions, describe how the graph is a transformation of the graph of ƒ () = ln. Also identif what attributes of ƒ () = ln change as a result of the transformation. Attributes to consider are the domain, the range, the end behavior, the vertical asmptote, the -intercept, the intervals where the function is positive and where it is negative, and whether the function increases or decreases throughout its domain. a. g () = ln ( + 6) The graph of g () is a horizontal translation of the graph of f () left 6 units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. b. g () = ln - The graph of g () is a vertical translation of the graph of f () down unit. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. c. g () = _ 3 ln The graph of g () is a vertical stretch of the graph of f () b a factor of 3_. No attributes change. d. g () = ln + 8 The graph of g () is a vertical translation of the graph of f () up 8 units. The attributes that change are the -intercept and the intervals where the function is positive and where it is negative. e. g () = - _ 3 ln The graph of g () is a vertical compression of the graph of f () b a factor of _ as well 3 as a reflection across the -ais. The attributes that change are the end behavior and whether the function increases or decreases throughout its domain. f. g () = ln ( - ) The graph of g () is a horizontal translation of the graph of f () right units. The attributes that change are the domain, the end behavior, the vertical asmptote, the -intercept, and the intervals where the function is positive and where it is negative. Houghton Mifflin Harcourt Publishing Compan Module 5 85 Lesson Graphing Logarithmic Functions 85

17 MULTIPLE REPRESENTATIONS Help students to recognize how, in each of the applications, the resulting equation represents the data. Use graphs and the table feature of the graphing calculator to ensure that students make connections among the various representations. Identif the transformations of the graph of ƒ () = lo g b that produce the graph of the given function g (). Then graph g () on the same coordinate plane as the graph of ƒ () b appling the transformations to the asmptote = and to the reference points (, ) and (b, ). Also state the domain and range of g () using set notation. 6. g () = - lo g ( + ) + The transformations of the graph of f () = lo g that produce the graph of g () are as follows: a vertical stretch b a factor of a reflection across the -ais a translation of units to the left and unit up Domain: > - Range: - < < g () = _ ln ( + ) - 3 The transformations of the graph of f () = In that produce the graph of g () are: as follows a vertical compression b a factor of _ a translation of units to the left and 3 units down Domain: > - Range: - < < g () = 3 log ( - ) - Houghton Mifflin Harcourt Publishing Compan The transformations of the graph of f () = log that produce the graph of g () are as follows: a vertical stretch b a factor of 3 a translation of unit to the right and unit down Domain: > Range: - < < Module Lesson 855 Lesson 5.

18 9. ƒ () = _ lo g ( - ) - The transformations of the graph of f () = lo g that produce the graph of g () are as follows: a vertical compression b a factor of _ a translation of unit to the right and units down Note that the translation of unit to the right affects onl the -coordinates of points on the graph of f (), while the vertical compression b a factor of _ and the translation of units down affect onl the -coordinates. Domain: > Range: - < < g () = - ln ( - ) + 3 The transformations of the graph of f () = In that produce the graph of g () are as follows: a vertical compression b a factor of a reflection across the -ais a translation of units to the right and 3 units up Domain: > Range: - < < g () = - log ( + ) + 5 The transformations of the graph of f () = log that produce the graph of g () are as follows: a vertical stretch b a factor of a reflection across the -ais a translation of units to the left and 5 units up Domain: > - Range: - < < Houghton Mifflin Harcourt Publishing Compan Module Lesson Graphing Logarithmic Functions 856

19 COMMUNICATING MATH Students ma find it helpful to write a verbal description of the transformations that are indicated b the function rule before attempting to draw the graph of the transformed function.. The radioactive isotope fluorine-8 is used in medicine to produce images of internal organs and detect cancer. It decas to the stable element ogen-8. The table gives the percent of fluorine-8 that remains in a sample over a period of time. Time (hours) Percent of Fluorine-8 Remaining Houghton Mifflin Harcourt Publishing Compan Image Credits: Scott Camazine/Photo Researchers/Gett Images a. Write an eponential model for the percent of fluorine-8 remaining as a function of time (in hours). The ratio of the percents for consecutive hours is about 68.5, so an eponential model for the data is p = (.685) t where p is the percent of fluorine-8 remaining at time t (in hours). b. Find the inverse of the eponential model after rewriting it with a base of e. Describe what information the inverse gives. First, find the constant c such that e c =.685. Rewriting e c =.685 in logarithmic form gives c = In Then, find the inverse of p = ( e ) t = e -.378t. p_ p_ p = e -.378t = e -.378t In = -.378t -.65 In p_ = t So, the inverse of the eponential model is t = -.65 In p, which gives the time it takes for the percent of fluorine-8 remaining to reach a certain level. c. Perform logarithmic regression on the data (using the percent of fluorine-8 remaining as the independent variable and time as the dependent variable). Compare this model with the inverse model from part b. A graphing calculator gives the logarithmic regression model t =. -.6 In p. Graphing this model and the inverse model from part b shows that the models are basicall equivalent. Module Lesson 857 Lesson 5.

20 3. During the period between, the average price of an ounce of gold doubled ever ears. In, the average price of gold was about $7 per ounce. Year Average Price of an Ounce of Gold $7. $ $ $9.7 5 $.7 6 $ $ $ $97.35 $.53 $57.5 PEER-TO-PEER DISCUSSION Ask students to work with a partner to determine the domain and range of the function f () = log b (-). The domain is all negative real numbers. The range is all real numbers. a. Write an eponential model for the average price of an ounce of gold as a function of time (in ears since ). An eponential model for the average price P of gold (in dollars per ounce) is P = 7 () t where t is the time in ears since. b. Find the inverse of the eponential model after rewriting it with a base of e. Describe what information the inverse gives. First, rewrite P = 7 () t as P = 7 ( ) t. Net, find the constant c such that e c =.9. Rewriting e c.9 in logarithmic form gives c In.9.7. Finall, find the inverse of P = 7 ( e.7 ) t = 7 e.7t. P_ P_ P = 7 e.7t 7 = e.7t In 7 =.7t 5.75 In P_ 7 = t So, the inverse of the eponential model is t = 5.75 In P 7, which gives the time it takes for the average price of gold to reach a certain level. c. Perform logarithmic regression on the data in the table (using the average price of gold as the independent variable and time as the dependent variable). Compare this model with the inverse model from part b. A graphing calculator gives the logarithmic regression model t = In P. Graphing this model and the inverse model from part b shows that the models are basicall equivalent. Houghton Mifflin Harcourt Publishing Compan Image Credits: Tetra Images/Corbis Module Lesson Graphing Logarithmic Functions 858

21 JOURNAL Have students describe how to use transformations of the graph of the parent logarithmic function f () = log to graph the function g () = alog ( - h) + k. H.O.T. Focus on Higher Order Thinking. Multiple Representations For the function g () = log ( - h), what value of the parameter h will cause the function to pass through the point (7, )? Answer the question in two different was: once b using the function s rule, and once b thinking in terms of the function s graph. Using the function s rule: g () = log ( - h) = log (7 - h) = 7 - h 3 = -h -3 = h Thinking in terms of the function s graph: The graph of g () = log ( - h) is the graph of f () = log translated h units horizontall. Since the graph of f () passes through the point (,), shifting the graph left 3 units causes it to pass through the point (7, ). A shift of 3 units to the left means that h = Eplain the Error A student drew the graph of g () = l og ( - ) as shown. Eplain the error that the student made, and draw the correct graph Houghton Mifflin Harcourt Publishing Compan The student overlooked the fact that the base of the logarithmic function is, not. When the function is rewritten with a base of, it becomes g () = - log ( - ), which means that in addition to verticall stretching the graph of f () = l og b a factor of and translating the graph units to the right, the student must reflect the graph across the -ais. 6. Construct Arguments Prove that l og = -lo g b for an positive value of b not b equal to. Begin the proof b setting l og equal to m and rewriting the equation in eponential form. Let l og = m. In eponential form, the equation is ( b b b) m =, so ( b - ) m = and b -m =. In logarithmic form, the equation b -m = is l og b = -m, or -l og b = m. B the transitive propert of equalit, l og = -lo g b. b Module Lesson 859 Lesson 5.

22 Lesson Performance Task Given the following data about the heights of chair seats and table tops for children, make separate scatterplots of the ordered pairs (age of child, chair seat height) and the ordered pairs (age of child, table top height). Eplain wh a logarithmic model would be appropriate for each data set. Perform a logarithmic regression on each data set, and describe the transformations needed to obtain the graph of the model from the graph of the parent function f () = ln. Chair Seat Height (inches) 6 8 Age of Child (ears) 6 Age of Child (ears) 8 Chair Seat Height (inches) Table Top Height (inches) Table Top Height (inches) Age of Child (ears) 8 AVOID COMMON ERRORS Students ma graph the age on the -ais and the other variables on the -ais. Eplain that the variable in common (in this case, the age) is usuall graphed on the -ais so the other variables can be more easil compared. Have students describe the resulting function if the age is graphed on the -ais. It is the inverse of the function being sought. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Have students discuss the differences between the two graphs. Have students eplain wh the table height graph is not simpl the chair height graph translated upward and how this could be related to the size of the child. In both graphs, the average rate of change between consecutive points is alwas positive and decreases as the independent variable (age of a child) increases. So, a logarithmic model is appropriate for both data sets. The logarithmic regression model for (age of child, chair seat height) is = In where is the age of a child and is the chair seat height. The graph of = In is the graph of f () = In verticall stretched b a factor of.59 and translated up.83 units. The logarithmic regression model for (age of child, table top height) is = In where is the age of a child and is the chair seat height. The graph of = In is the graph of f () = In verticall stretched b a factor of 5.9 and translated up.5. Houghton Mifflin Harcourt Publishing Compan Module 5 86 Lesson EXTENSION ACTIVITY Have students measure the chair and table heights for five classmates. Have students ask the age of each classmate, and then have th em plot this data on the graphs the drew in the Performance Task. Have students discuss whether the data the measured could be modeled b the same function the found for the ounger children, and wh or wh not. 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. Graphing Logarithmic Functions 86

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