.7 Modeling with Eponential and Logarithmic Functions Essential Question How can ou recognize polnomial, eponential, and logarithmic models? Recognizing Different Tpes of Models Work with a partner. Match each tpe of model with the appropriate scatter plot. Use a regression program to find a model that fits the scatter plot. a. linear (positive slope) b. linear (negative slope) c. quadratic d. cubic e. eponential f. logarithmic A. B. C. D. E. F. 8 USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. Eploring Gaussian and Logistic Models Work with a partner. Two common tpes of functions that are related to eponential functions are given. Use a graphing calculator to graph each function. Then determine the domain, range, intercept, and asmptote(s) of the function. 1 a. Gaussian Function: f () = e b. Logistic Function: f () = 1 + e Communicate Your Answer 3. How can ou recognize polnomial, eponential, and logarithmic models?. Use the Internet or some other reference to find real-life data that can be modeled using one of the tpes given in Eploration 1. Create a table and a scatter plot of the data. Then use a regression program to find a model that fits the data. Section.7 Modeling with Eponential and Logarithmic Functions 31
.7 Lesson What You Will Learn Core Vocabular Previous finite differences common ratio point-slope form Classif data sets. Write eponential functions. Use technolog to find eponential and logarithmic models. Classifing Data You have analzed fi nite differences of data with equall-spaced inputs to determine what tpe of polnomial function can be used to model the data. For eponential data with equall-spaced inputs, the outputs are multiplied b a constant factor. So, consecutive outputs form a constant ratio. Classifing Data Sets Determine the tpe of function represented b each table. a. 1 0 1 3 0.5 1 8 1 3 b. 0 8 10 0 8 18 3 50 a. The inputs are equall spaced. Look for a pattern in the outputs. 1 0 1 3 0.5 1 8 1 3 As increases b 1, is multiplied b. So, the common ratio is, and the data in the table represent an eponential function. REMEMBER First differences of linear functions are constant, second differences of quadratic functions are constant, and so on. b. The inputs are equall spaced. The outputs do not have a common ratio. So, analze the finite differences. 0 8 10 0 8 18 3 50 10 1 18 first differences second differences The second differences are constant. So, the data in the table represent a quadratic function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine the tpe of function represented b the table. Eplain our reasoning. 1. 0 10 0 30. 0 15 1 9 7 9 3 1 3 Chapter Eponential and Logarithmic Functions
Writing Eponential Functions You know that two points determine a line. Similarl, two points determine an eponential curve. Writing an Eponential Function Using Two Points Write an eponential function = ab whose graph passes through (1, ) and (3, 5). Step 1 Substitute the coordinates of the two given points into = ab. = ab 1 Equation 1: Substitute for and 1 for. 5 = ab 3 Equation : Substitute 5 for and 3 for. REMEMBER You know that b must be positive b the definition of an eponential function. Step Solve for a in Equation 1 to obtain a = and substitute this epression for a b in Equation. 5 = ( b ) b3 Substitute for a in Equation. b 5 = b Simplif. 9 = b Divide each side b. 3 = b Take the positive square root because b > 0. Step 3 Determine that a = b = 3 =. So, the eponential function is = (3 ). Data do not alwas show an eact eponential relationship. When the data in a scatter plot show an approimatel eponential relationship, ou can model the data with an eponential function. Finding an Eponential Model Year, Number of trampolines, 1 1 1 3 5 3 5 50 7 7 9 A store sells trampolines. The table shows the numbers of trampolines sold during the th ear that the store has been open. Write a function that models the data. Step 1 Make a scatter plot of the data. The data appear eponential. Step Choose an two points to write a model, such as (1, 1) and (, 3). Substitute the coordinates of these two points into = ab. 1 = ab 1 3 = ab Solve for a in the first equation to obtain a = 1 b. Substitute to obtain b = 3 3 1. and a = 1 8.3. 3 3 So, an eponential function that models the data is = 8.3(1.). Number of trampolines Trampoline Sales 80 0 0 0 0 0 Year Section.7 Modeling with Eponential and Logarithmic Functions 33
A set of more than two points (, ) fits an eponential pattern if and onl if the set of transformed points (, ln ) fits a linear pattern. Graph of points (, ) Graph of points (, ln ) = 1 (, 3 ( 1 ( 1, 1 ( 1 (0, 1) 1 (1, ) The graph is an eponential curve. ln ln = (ln ) 3 ( 1, 0.9) (0, 0) (1, 0.9) 1 1 (, 1.39) The graph is a line. LOOKING FOR STRUCTURE Because the aes are and ln, the point-slope form is rewritten as ln ln 1 = m( 1 ). The slope of the line through (1,.8) and (7,.5) is.5.8 0.35. 7 1 Writing a Model Using Transformed Points Use the data from Eample 3. Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. Step 1 Create a table of data pairs (, ln ). 1 3 5 7 ln.8.77 3. 3.58 3.91.0.5 Step Plot the transformed points as shown. The points lie close to a line, so an eponential model should be a good fit for the original data. Step 3 Find an eponential model = ab b choosing an two points on the line, such as (1,.8) and (7,.5). Use these points to write an equation of the line. Then solve for. ln.8 = 0.35( 1) Equation of line ln = 0.35 +.13 Simplif. = e 0.35 +.13 Eponentiate each side using base e. = e 0.35 (e.13 ) = 8.1(1.) Use properties of eponents. Simplif. So, an eponential function that models the data is = 8.1(1.). ln 8 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an eponential function = ab whose graph passes through the given points. 3. (, 1), (3, ). (1, ), (3, 3) 5. (, 1), (5, ). WHAT IF? Repeat Eamples 3 and using the sales data from another store. Year, 1 3 5 7 Number of trampolines, 15 3 0 5 80 105 10 3 Chapter Eponential and Logarithmic Functions
Using Technolog You can use technolog to find best-fit models for eponential and logarithmic data. Finding an Eponential Model Use a graphing calculator to find an eponential model for the data in Eample 3. Then use this model and the models in Eamples 3 and to predict the number of trampolines sold in the eighth ear. Compare the predictions. Enter the data into a graphing calculator and perform an eponential regression. The model is = 8.(1.). Substitute = 8 into each model to predict the number of trampolines sold in the eighth ear. Eample 3: = 8.3(1.) 8 15 Eample : = 8.1(1.) 8 139 Regression model: = 8.(1.) 8 10 EpReg =a*b^ a=8.57377971 b=1.188803 r =.9975053 r=.9981303 The predictions are close for the regression model and the model in Eample that used transformed points. These predictions are less than the prediction for the model in Eample 3. Finding a Logarithmic Model The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is 1 atmosphere (1.0337 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers). Use a graphing calculator to find a logarithmic model of the form h = a + b ln p that represents the data. Estimate the altitude when the pressure is 0.75 atmosphere. Air pressure, p 1 0.55 0.5 0.1 0.0 0.0 Altitude, h 0 5 10 15 0 5 Weather balloons carr instruments that send back information such as wind speed, temperature, and air pressure. Enter the data into a graphing calculator and perform a logarithmic regression. The model is h = 0.8.5 ln p. Substitute p = 0.75 into the model to obtain h = 0.8.5 ln 0.75.7. So, when the air pressure is 0.75 atmosphere, the altitude is about.7 kilometers. Monitoring Progress LnReg =a+bln a=.8578705 b=-.738985 r =.9955887 r=-.9971 Help in English and Spanish at BigIdeasMath.com 7. Use a graphing calculator to find an eponential model for the data in Monitoring Progress Question. 8. Use a graphing calculator to find a logarithmic model of the form p = a + b ln h for the data in Eample. Eplain wh the result is an error message. Section.7 Modeling with Eponential and Logarithmic Functions 35
.7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE Given a set of more than two data pairs (, ), ou can decide whether a(n) function fits the data well b making a scatter plot of the points (, ln ).. WRITING Given a table of values, eplain how ou can determine whether an eponential function is a good model for a set of data pairs (, ). Monitoring Progress and Modeling with Mathematics In Eercises 3, determine the tpe of function represented b the table. Eplain our reasoning. (See Eample 1.) 3.. 5.. 0 3 9 1 15 0.5 1 1 5 3 1 0 1 1 8 1 1 1 5 10 15 0 5 30 3 7 1 30 9 3 1 5 9 13 8 3 1 5 3 In Eercises 7 1, write an eponential function = ab whose graph passes through the given points. (See Eample.) 7. (1, 3), (, 1) 8. (, ), (3, 1) 9. (3, 1), (5, ) 10. (3, 7), (5, 3) 11. (1, ), (3, 50) 1. (1, 0), (3, 0) 13. ( 1, 10), (, 0.31) 1. (,.), (5, 09.) 15. (, ) (1, 0.5) 1. ( 3, 10.8) (, 3.) 1 8 ERROR ANALYSIS In Eercises 17 and 18, describe and correct the error in determining the tpe of function represented b the data. 17. 18. 0 1 3 1 9 1 3 1 3 9 3 3 3 3 The outputs have a common ratio of 3, so the data represent a linear function. 1 1 3 1 8 The outputs have a common ratio of, so the data represent an eponential function. 19. MODELING WITH MATHEMATICS A store sells motorized scooters. The table shows the numbers of scooters sold during the th ear that the store has been open. Write a function that models the data. (See Eample 3.) 1 9 1 3 19 5 5 37 53 7 71 3 Chapter Eponential and Logarithmic Functions
0. MODELING WITH MATHEMATICS The table shows the numbers of visits to a website during the th month. Write a function that models the data. Then use our model to predict the number of visits after 1 ear. 1 3 5 7 39 70 1 7 08 735 In Eercises 1, determine whether the data show an eponential relationship. Then write a function that models the data. 1.. 3.. 1 11 1 1 1 8 7 190 50 3 1 1 3 5 7 8 19 0 10 0 30 0 50 0 58 8 31 1 0 13 1 8 15 5 19 1 11 8 5. MODELING WITH MATHEMATICS Your visual near point is the closest point at which our ees can see an object distinctl. The diagram shows the near point (in centimeters) at age (in ears). Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. (See Eample.) Visual Near Point Distances Age 0 1 cm Age 30 15 cm Age 0 5 cm Age 50 0 cm Age 0 100 cm. MODELING WITH MATHEMATICS Use the data from Eercise 19. Create a scatter plot of the data pairs (, ln ) to show that an eponential model should be a good fit for the original data pairs (, ). Then write an eponential model for the original data. In Eercises 7 30, create a scatter plot of the points (, ln ) to determine whether an eponential model fits the data. If so, find an eponential model for the data. 7. 8. 9. 30. 1 3 5 18 3 7 1 88 1 7 10 13 3.3 10.1 30. 9.7 80.9 13 1 8 15 9.8 1. 15. 19 3.8 8 5 1 1. 1.7 5.3.1 7.97 31. USING TOOLS Use a graphing calculator to find an eponential model for the data in Eercise 19. Then use the model to predict the number of motorized scooters sold in the tenth ear. (See Eample 5.) 3. USING TOOLS A doctor measures an astronaut s pulse rate (in beats per minute) at various times (in minutes) after the astronaut has finished eercising. The results are shown in the table. Use a graphing calculator to find an eponential model for the data. Then use the model to predict the astronaut s pulse rate after 1 minutes. 0 17 13 110 9 8 8 10 78 1 75 Section.7 Modeling with Eponential and Logarithmic Functions 37
33. USING TOOLS An object at a temperature of 10 C is removed from a furnace and placed in a room at 0 C. The table shows the temperatures d (in degrees Celsius) at selected times t (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form t = a + b ln d that represents the data. Estimate how long it takes for the object to cool to 50 C. (See Eample.) d 10 90 5 38 9 t 0 1 3 5 3. HOW DO YOU SEE IT? The graph shows a set of data points (, ln ). Do the data pairs (, ) fit an eponential pattern? Eplain our reasoning. (, 1) (, 3) ln (0, 1) (, 3) 3. USING TOOLS The f-stops on a camera control the amount of light that enters the camera. Let s be a measure of the amount of light that strikes the film and let f be the f-stop. The table shows several f-stops on a 35-millimeter camera. Use a graphing calculator to find a logarithmic model of the form s = a + b ln f that represents the data. Estimate the amount of light that strikes the film when f = 5.57. f s 1.1 1.000.88 3.000 11.31 7 37. MAKING AN ARGUMENT Your friend sas it is possible to find a logarithmic model of the form d = a + b ln t for the data in Eercise 33. Is our friend correct? Eplain. 38. THOUGHT PROVOKING Is it possible to write as an eponential function of? Eplain our reasoning. (Assume p is positive.) 1 p p 3 p 8p 5 1p 35. DRAWING CONCLUSIONS The table shows the average weight (in kilograms) of an Atlantic cod that is ears old from the Gulf of Maine. Age, 1 3 5 Weight, 0.751 1.079 1.70.198 3.38 a. Show that an eponential model fits the data. Then find an eponential model for the data. b. B what percent does the weight of an Atlantic cod increase each ear in this period of time? Eplain. Maintaining Mathematical Proficienc Tell whether and are in a proportional relationship. Eplain our reasoning. (Skills Review Handbook) 0. = 1. = 3 1. = 5 39. CRITICAL THINKING You plant a sunflower seedling in our garden. The height h (in centimeters) of the seedling after t weeks can be modeled b the logistic function 5 h(t) = 1 + 13e 0.5t. a. Find the time it takes the sunflower seedling to reach a height of 00 centimeters. b. Use a graphing calculator to graph the function. Interpret the meaning of the asmptote in the contet of this situation. Reviewing what ou learned in previous grades and lessons 3. = Identif the focus, directri, and ais of smmetr of the parabola. Then graph the equation. (Section.3). = 1 8 5. =. = 3 7. = 5 38 Chapter Eponential and Logarithmic Functions