15.2 Graphing Logarithmic

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1 Name Class Date 15. Graphing Logarithmic Functions Essential Question: How is the graph of g () = a log b ( h) + k where b > and b 1 related to the graph of f () = log b? Resource Locker Eplore 1 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. - (, 1) (, ) (1, ) (8, 3) (.5, -1) Houghton Mifflin Harcourt Publishing Compan B 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 1.) 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 e.7 e 7.39 Module Lesson

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 End behavior > < < As +, f () +. As +, f () -. Vertical and horizontal asmptotes Vertical asmptote at = ; no horizontal asmptote Intervals where increasing or decreasing Increasing throughout its domain Intercepts -intercept at (1, ); no -intercepts Intervals where positive or negative Positive on (1, + ) ; negative on (, 1) Reflect 1. What similarities do ou notice about all logarithmic functions of the form ƒ () = lo g b where b > 1? What differences do ou notice? Houghton Mifflin Harcourt Publishing Compan Module Lesson

3 Eplain 1 Graphing Combined Transformations of f () = log b Where b > 1 When graphing transformations of ƒ () = log b where b > 1, 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, (1, ) and (b, 1). 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 Asmptote = = h Reference point (1, ) (1 + h, k) Reference point (b, 1) (b + h, a + k) Eample 1 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 (1, ) and (b, 1). Also state the domain and range of g () using set notation. g () = - log ( - 1) - 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 1 unit to the right and units down Note that the translation of 1 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 ( - 1) - Asmptote = = 1 Reference point (1, ) (1 + 1, - () - ) = (, -) Houghton Mifflin Harcourt Publishing Compan Reference point (, 1) ( + 1, - (1) - ) = (3, ) Domain: > 1 Range: < < + Module Lesson

4 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 (1, ) ( -, ( ) + ) = (, ) Reference point (1, 1) ( -, ( ) + ) = (, ) Domain: > Range: - < < Houghton Mifflin Harcourt Publishing Compan Module Lesson

5 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 (1, ) and (b, 1). Also state the domain and range of g () using set notation.. g () = 1 lo g ( + 1) Function f () = lo g g () = 1_ l og ( + 1) + Houghton Mifflin Harcourt Publishing Compan Module Lesson

6 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 (1.1) 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. 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 = 1 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 (L1) and the time data into another list (L). Houghton Mifflin Harcourt Publishing Compan Image Credits: Roaltfree/Corbis Module Lesson

7 Perform logarithmic regression b pressing the LIST ke, choosing the CALC menu, and STAT 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): Average rate of change = t - t 1 _ P - P 1 Population Number of Years to Reach That Population Average Rate of Change Houghton Mifflin Harcourt Publishing Compan 1 t = ln 1.3 t = ln t = ln t = ln 16.3 _ _ - 1 = 6. 1 = = = =.5 1 =.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 1 foes. Module Lesson

8 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 (1.1) 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 = 1.1. Writing the eponential equation e c = 1.1 in logarithmic form gives c = ln 1.1, so c = to the nearest thousandth. t Replacing 1.1 with e in the biologist s model gives P = 6 ( e ), or P = 6 e t. Now find the inverse of this function. Write the equation. P = 6 e t Divide both sides b 6. Write in logarithmic form. t P_ 6 = e ln P _ 6 = t Divide both sides b. ln P_ 6 = t So, the inverse of the eponential model is t = ln P_. To compare this model with the logarithmic 6 regression model, use a graphing calculator to graph both = ln _ and = ln. You observe that the graphs [roughl coincide/significantl diverge], so the models are [basicall equivalent/ver different]. Reflect 3. Discussion In a later lesson, ou will learn the quotient propert of logarithms, which states that log b m_ 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. Your Turn. 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 $1. Houghton Mifflin Harcourt Publishing Compan 1 $1. $1. 3 $161.1 Module Lesson

9 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. Elaborate 5. 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? Houghton Mifflin Harcourt Publishing Compan 6. How are reference points helpful when graphing transformations of ƒ () = lo g b ()? 7. What are two was to obtain a logarithmic model for a set of data? 8. 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. Module Lesson

10 Evaluate: Homework and Practice 1. 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 1. Online Homework Hints and Help Etra Practice 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. Houghton Mifflin Harcourt Publishing Compan Module Lesson

11 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 b. g () = lo g c. g () = lo g ( + 6) d. g () = - 3_ lo g e. g () = lo g + 7 f. g () = lo g ( - 8) Houghton Mifflin Harcourt Publishing Compan 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 (1, ) and (b, 1). Also state the domain and range of g () using set notation.. g () = lo g ( + ) Module Lesson

12 5. g () = 3 log ( - 1) ƒ () = 1_ lo g ( - 1) g () = ln ( - ) g () = - log ( + ) Houghton Mifflin Harcourt Publishing Compan Module Lesson

13 9. The radioactive isotope fluorine-18 is used in medicine to produce images of internal organs and detect cancer. It decas to the stable element ogen-18. The table gives the percent of fluorine-18 that remains in a sample over a period of time. Time (hours) Percent of Fluorine-18 Remaining a. Write an eponential model for the percent of fluorine-18 remaining as a function of time (in hours). b. Find the inverse of the eponential model after rewriting it with a base of e. Describe what information the inverse gives. Houghton Mifflin Harcourt Publishing Compan Image Credits: Scott Camazine/Photo Researchers/Gett Images c. Perform logarithmic regression on the data (using the percent of fluorine-18 remaining as the independent variable and time as the dependent variable). Compare this model with the inverse model from part b. Module Lesson

14 1. During the period between 1 11, the average price of an ounce of gold doubled ever ears. In 1, the average price of gold was about $7 per ounce. Year Average Price of an Ounce of Gold 1 $71. $ $ $9.7 5 $.7 6 $ $ $ $ $ $ a. Write an eponential model for the average price of an ounce of gold as a function of time (in ears since 1). 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 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. Houghton Mifflin Harcourt Publishing Compan Image Credits: Tetra Images/Corbis Module Lesson

15 H.O.T. Focus on Higher Order Thinking 11. Multiple Representations For the function g () = log ( - h), what value of the parameter h will cause the function to pass through the point (7, 1)? 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. 1. Eplain the Error A student drew the graph of g () = l og 1 ( - ) as shown. Eplain the error that the student made, and draw the correct graph Houghton Mifflin Harcourt Publishing Compan 13. Construct Arguments Prove that l og 1 = -lo g b for an positive value of b not b equal to 1. Begin the proof b setting l og 1 equal to m and rewriting the equation in b eponential form. Module Lesson

16 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. Age of Child (ears) Chair Seat Height (inches) Table Top Height (inches) Chair Seat Height (inches) Age of Child (ears) Table Top Height (inches) Age of Child (ears) Houghton Mifflin Harcourt Publishing Compan Module Lesson