PreCalculus 2 nd Nine-Weeks Scope and Sequence

Transcription

1 PreCalculus 2 nd Nine-Weeks Scope and Sequence Topic 2: Polynomial, Power, and Rational Functions (40 45 days) (Continued from 1 st Nine-Weeks) A) Determines the characteristics of the polynomial functions of any degree, general shape, number of real and nonreal (real and nonreal), domain and range, and end behavior, and finds real and nonreal zeros. B) Identifies power functions and direct and inverse variation. C) Describes and compares the characteristics of rational functions; e.g., general shape, number of zeros (real and nonreal), domain and range, asymptotic behavior, and end behavior. D) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and make predictions. Topic 3: Exponential, Logarithmic, and Logistic Functions (25 30 days) (Continued in 3 rd Nine-Weeks) A) Identify exponential, logarithmic, and logistic functions. B) Describe and compare the characteristics of exponential, logarithmic, and logistic functions: e.g. general shape, number of roots, domain and range, asymptotic and end behavior, extrema, local and global behavior. C) Solve exponential, logarithmic, and logistic equations graphically and algebraically. D) Create a scatterplot of bivariate data and identify an exponential, logarithmic, or logistic function to model the data and make predictions.

2 COLUMBUS PUBLIC SCHOOLS MATHEMATICS CURRICULUM GUIDE GRADE LEVEL STATE STANDARD 4 and 5 TIME RANGE GRADING PreCalculus Patterns, Functions, and Algebra days PERIOD Data Analysis and Probability 2-3 MATHEMATICAL TOPIC 3 Exponential, Logarithmic, and Logistic Functions CPS LEARNING GOALS A) Identifies exponential, logarithmic, and logistic functions. B) Describes and compares the characteristics of exponential, logarithmic, and logistic functions: e.g. general shape, number of roots, domain and range, asymptotic and end behavior, extrema, local and global behavior. C) Solves exponential, logarithmic, and logistic equations graphically and algebraically. D) Creates a scatterplot of bivariate data and identifies an exponential, logarithmic, or logistic function to model the data and make predictions. COURSE LEVEL INDICATORS Course Level (i.e., How does a student demonstrate mastery?): Describes how a change in the value of a constant in an exponential, logarithmic, or logistic equation affects the graph of the equation. Math A:11-A:11 Identifies exponential and logarithmic functions to any base including e and natural log. Math A:12-A:03 Identifies logarithmic and exponential equations as inverses and uses the relationship to solve equations. Math A:12-A:04 x Recognizes e as. Math A:12-A:10 lim 1+ 1 x x Sketches the graph of exponential, logistic, and logarithmic functions (including base e and natural log) and their transformations. Math A:12-A:03 Models real world data with exponential, logarithmic, and logistic functions. Math D:11-A:04 Identifies the inverse of exponential and logarithmic functions including e and natural log. Math A:12-A:04 Solves problems involving compound interest, annuities, growth, and decay. Math D:11-A:03 and Math A:11-A:04 Solves exponential and logarithmic equations to any base including e and natural log. Math A:11-A:03 Applies the rules of logarithms. Math A:11-A:03 Transforms bivariate data so it can be modeled by a function; e.g., uses logarithms to allow nonlinear relationships to be modeled by linear functions. Math D:12-A:02 Previous Level: Performs computations using the rules of logarithms and exponents. Math N:11-C:08 Describes the characteristics of quadratics with complex roots. Math A:11-B:03 Identifies the inverse of exponential and logarithmic functions. Math A:11-A:06 Applies the rules of exponents. Math N:11-C:08 Page 1 of 131 Columbus Public Schools 7/20/05

3 The description from the state, for the Patterns, Functions, and Algebra Standard says: Students use patterns, relations, and functions to model, represent, and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs, and equations. The grade-band benchmark from the state, for this topic in the grade band is: A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local and global behavior. The description from the state, for the Data Analysis and Probability Standard says: Students pose questions and collect, organize represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are abased on data. The grade-band benchmark from the state, for this topic in the grade band is: A. Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators. The description from the state, for the Mathematical Processes Standard says: Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The grade-band benchmark from the state, for this topic in the grade band is: H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations. Page 2 of 131 Columbus Public Schools 7/20/05

4 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - A Which of the following gives the equation of an exponential function? A. y = 2x 7x B. y = 3 x C. y = 2 7x D. 2 y = 0.3 x Which of the following gives the equation of a logarithmic function? A. f ( x) = ln( x) B. f( x ) = 6 x C. f ( x) = cos( x) D. f ( x) = 2 x+ 1 Page 3 of 131 Columbus Public Schools 7/20/05

5 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - A Low Complexity Which of the following gives the equation of an exponential function? A. y = 2x 7x B. y = 3 x C. y = 2 7x D. 2 y = 0.3 x Answer: D Moderate Complexity Which of the following gives the equation of a logarithmic function? A. f ( x) = ln( x) B. f( x ) = 6 x C. f ( x) = cos( x) D. f ( x) = 2 x+ 1 Answer: A Page 4 of 131 Columbus Public Schools 7/20/05

6 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - A Which of the following graphs represents a logistic function? A. 10 B. 10 C. 10 D Describe the difference between the graph of an exponential function and the graph of a logistic function. Page 5 of 131 Columbus Public Schools 7/20/05

7 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - A High Complexity Which of the following graphs represents a logistic function? A. 10 B. 10 C. 10 D Answer: A Short Answer/Extended Response Describe the difference between the graph of an exponential function and the graph of a logistic function. Answer: An exponential graph has a single horizontal asymptote while a logistic graph has two horizontal asymptotes. A 2-point response provides a clear description of the difference between the two graphs. A 1-point response provides a description of either the exponential graph or the logistic graph but does not describe the difference between the two graphs. A 0-point response shows no mathematical understanding. Page 6 of 131 Columbus Public Schools 7/20/05

8 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - B How does the graph of y = 3e x+1 2 compare to the graph of y = e x? A. y = 3e x+1 2 is steeper, moved one unit to the left and two units down. B. y = 3e x+1 2 is steeper, moved one unit to the right and two units down. C. y = 3e x+1 2 is less steep, moved one unit to the left and two units down. D. y = 3e x+1 2 is less steep, moved one unit to the right and two units down. How does the graph of y = a x compare to the graph of y = b x if 0 < a < b? A. y = a x is steeper and has the same y-intercept. B. y = a x is steeper and has y-intercept a + b. C. y = a x is less steep and has the same y-intercept. D. y = a x is less steep and has y-intercept a b Page 7 of 131 Columbus Public Schools 7/20/05

9 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - B Low Complexity How does the graph of y = 3e x+1 2 compare to the graph of y = e x? A. y = 3e x+1 2 is steeper, moved one unit to the left and two units down. B. y = 3e x+1 2 is steeper, moved one unit to the right and two units down. C. y = 3e x+1 2 is less steep, moved one unit to the left and two units down. D. y = 3e x+1 2 is less steep, moved one unit to the right and two units down. Answer: A Moderate Complexity How does the graph of y = a x compare to the graph of y = b x if 0 < a < b? A. y = a x is steeper and has the same y-intercept. B. y = log a x is steeper and has y-intercept a + b. C. y = a x is less steep and has the same y-intercept. D. y = a x is less steep and has y-intercept a b Answer: C Page 8 of 131 Columbus Public Schools 7/20/05

10 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - B How does the graph of y = log a x compare to the graph of y = log 3a x for a > 0? A. y = log a x is steeper and has the same x-intercept. B. y = log a x is steeper and has x-intercept a + b. C. y = log a x is less steep and has the same x-intercept. D. y = log a x is less steep and has x-intercept a b. Find the y-intercept and horizontal asymptotes of f (x) = 8. Justify your answers. x 1+ 3(.7) Page 9 of 131 Columbus Public Schools 7/20/05

11 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - B High Complexity How does the graph of y = log a x compare to the graph of y = log 3a x for a > 0? A. y = log a x is steeper and has the same y-intercept. B. y = log a x is steeper and has y-intercept a + b. C. y = log a x is less steep and has the same y-intercept. D. y = log a x is less steep and has y-intercept a b. Short Answer/Extended Response Answer: A Find the y-intercept and horizontal asymptotes of f (x) = 8. Justify your answers. x 1+ 3(.7) Answer: The horizontal asymptote is y = 8, because the numerator of the logistic is 8. 8 The y-intercept is found evaluating f(0). f (0) = 1 + 3(.7) = 8 = 2. The point is (0, 2). 0 4 A 2-point response contains the correct y-intercept and horizontal asymptote and the supporting justification for each statement. A 1-point response contains at least two of the four components above. A 0-point response shows no mathematical understanding. Page 10 of 131 Columbus Public Schools 7/20/05

12 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - B Given f (x) = e 7 x, which statement describes the function and its end behavior? A. Exponential growth function; lim x f (x) = ; lim x f (x) = 0 B. Exponential growth function; lim f (x) = 0 ; lim f (x) = x x C. Exponential decay function; lim x f (x) = ; lim x f (x) = 0 D. Exponential decay function; lim f (x) = 0 ; lim f (x) = x x 6 x Which statement is equivalent to log 7 y? A..5(log 7 6)log 7 x log 7 y B. log 7 x.5 log log 7 y C. log log x 7 log 7 y D..5log 7 (6x) log 7 y Page 11 of 131 Columbus Public Schools 7/20/05

13 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - B Low Complexity Given f (x) = e 7 x, which statement describes the function and its end behavior? A. Exponential growth function; lim f (x) = ; lim f (x) = 0 x x B. Exponential growth function; lim x f (x) = 0 ; lim x f (x) = C. Exponential decay function; lim f (x) = ; lim f (x) = 0 x x D. Exponential decay function; lim f (x) = 0 ; lim f (x) = x x Moderate Complexity 6 x Which statement is equivalent to log 7 y A..5(log 7 6)log 7 x log 7 y B. log 7 x.5 log log 7 y C. log log x 7 log 7 y D..5log 7 (6x) log 7 y Answer: C? Answer: C Page 12 of 131 Columbus Public Schools 7/20/05

14 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - B x Which answer contains the correct way to enter y = log8 4 in order to graph it? into the Y= menu of the calculator A. B. log x 4log8 log x log 32 C. 8logx log 4 D. log x log 4 log 8 8 Use the function f (x) =. Give the domain, range, asymptote(s), any symmetry x 1+ 3(.7) displayed, continuity, and end behavior. Page 13 of 131 Columbus Public Schools 7/20/05

15 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - B High Complexity x Which answer contains the correct way to enter y = log8 4 calculator in order to graph it? log x A. 4log8 into the Y= menu of the B. log x log 32 C. 8logx log 4 D. log x log 4 log 8 Answer: D Short Answer/Extended Response 8 Use the function f (x) =. Give the domain, range, asymptote(s), any symmetry x 1+ 3(.7) displayed, continuity, and end behavior. Answer: The domain is the set of all real numbers, the range is (0, 8), and y = 0, and y = 8 are horizontal asymptotes. The graph is symmetric about the point (0,.5) and is always continuous. The end behavior is described by lim f( x ) = 0 and lim f( x ) = 8. x A 2-point response correctly gives all of the answers. A 1-point response correctly gives at least four of the answers. A 0-point response gives less than four correct answers. x Page 14 of 131 Columbus Public Schools 7/20/05

16 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - C What is the solution to the equation log x 256 = 8? A. 2 B. 5 C. 8 D. 32 The number of bacteria in a Petri dish doubles every 3 hours. If there are originally 10 bacteria in the dish, after how many hours will the number of bacteria be equal to 200? A hours B hours C hours D hours Page 15 of 131 Columbus Public Schools 7/20/05

17 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - C Low Complexity What is the solution to the equation log x 256 = 8? A. 2 B. 5 C. 8 D. 32 Moderate Complexity Answer: A The number of bacteria in a Petri dish doubles every 3 hours. If there are originally 10 bacteria in the dish, after how many hours will the number of bacteria be equal to 200? A hours B hours C hours D hours Answer: B Page 16 of 131 Columbus Public Schools 7/20/05

18 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - C Some especially sour vinegar has a ph of 2.4 and a box of Leg and Sickle baking soda has a ph of 8.4. How many times greater is the hydrogen-ion concentration of the vinegar than that of the baking soda? Hint: ph is based on a logarithmic scale. A B. 6 C D A hard-boiled egg at temperature 96 C is placed in 16 C water to cool. Four minutes later the temperature of the egg is 45 C. Use Newton's Law of Cooling to determine when the temperature of the egg will be 20 C. Show your solution. kt Newton s Law of Cooling: Tt () = T + ( T0 T) e m m Page 17 of 131 Columbus Public Schools 7/20/05

19 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - C High Complexity Some especially sour vinegar has a ph of 2.4 and a box of Leg and Sickle baking soda has a ph of 8.4. How many times greater is the hydrogen-ion concentration of the vinegar than that of the baking soda? Hint: ph is based on a logarithmic scale. A B. 6 C D Short Answer/Extended Response Answer: D A hard-boiled egg at temperature 96 C is placed in 16 C water to cool. Four minutes later the temperature of the egg is 45 C. Use Newton's Law of Cooling to determine when the temperature of the egg will be 20 C. Show your solution. kt Newton s Law of Cooling: Tt () = Tm + ( T0 Tm) e Answer: T 0 =96 and T M =16, T 0 T M =80 and T(t) = T m + (T 0 T m )e kt = e kt. So, kt 20 = e kt 45 = e 4 kt = 4e ln = 4k ln = kt 29 ln k = 4 ln 80 4 t = k A 4-point response correctly applies Newton's Law of Cooling, solves for k, and finds the correct amount of time. A 3-point response correctly applies Newton's Law of Cooling, solves for k, and finds a time making computational errors. A 2-point response correctly applies Newton's Law of Cooling and sets up the two equations to be solved. A 1-point response correctly applies Newton's Law of Cooling. A 0-point response demonstrates no mathematical understanding. Page 18 of 131 Columbus Public Schools 7/20/05

20 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - D Use the scatterplot below: Which statement best describes a function that would model the data? A. The model should be of the form y = ax b, where a and b are positive B. The model should be of the form y = ax b, where a is positive and b is negative. C. The model should be of the form y = alog b x, where a and b are positive. D. The model should be of the form y = alog b x, where a is positive and b is negative. E Given the table below: x y Which equation is a logarithmic model for the data? A. y = 0.86(1.38) x B. y = 0.79(1.45) x C. y = ln x D. y = ln x Page 19 of 131 Columbus Public Schools 7/20/05

21 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - D Low Complexity Use the scatterplot below: Which statement best describes a function that would model the data? A. The model should be of the form y = ax b, where a and b are positive. B. The model should be of the form y = ax b, where a is positive and b is negative.. C. The model should be of the form y = alog b x, where a and b are positive D. The model should be of the form y = alog b x, where a is positive and b is negative. Moderate Complexity Given the table below: Answer: A x y Which equation is a logarithmic model for the data? A. y = 0.86(1.38) x B. y = 0.79(1.45) x C. y = ln x D. y = ln x Answer: C Page 20 of 131 Columbus Public Schools 7/20/05

22 PRACTICE ASSESSMENT ITEMS Exponent, Log, & Logistic Fctn - D Given the table below showing the US population (in millions) for the decades from 1900 to 2000: Year Population If the trend continued, what is the best estimate of the population in the year 2020? A million B million C million D million The graph below shows the graphs of one exponential function and one logistic function, modeling the growth of bacteria in a Petri dish over a ten-hour period Number of Bacteria y=g(x) y=f(x) Hour Identify which function (f(x) or g(x)) is logistic and which is exponential. Discuss the rates of growth of each function and give examples of situations that would cause the two different models of growth of bacteria over ten hours. Page 21 of 131 Columbus Public Schools 7/20/05

23 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Exponent, Log, & Logistic Fctn - D High Complexity Given the table below showing the US population (in millions) for the decades from 1900 to 2000: Year Population If the trend continued, what is the best estimate of the population in the year 2020? A million B million C million D million Answer: D Short Answer/Extended Response The graph below shows the graphs of one exponential function and one logistic function, modeling the growth of bacteria in a Petri dish over a ten-hour period Number of Bacteria y=f(x) y=g(x) Hour Identify which function (f(x) or g(x)) is logistic and which is exponential. Discuss the rates of growth of each function and give examples of situations that would cause the two different models of growth of bacteria over ten hours. Answer: g(x) is exponential and f(x) is logistic. Although the growth is approximately the same for the first few hours, the growth of the logistic function begins to slow and eventually stops, with the population stabilizing at about In the exponential population, the bacteria had not experienced a shortage of food or space over the first ten hours. In the logistic population, the population conditions would not support such growth and the growth leveled off. A 2-point response correctly identifies both graphs and provides a logical explanation. A 1-point response correctly identifies both graphs but does not provide an explanation. A 0-point response demonstrates no mathematical understanding. Page 22 of 131 Columbus Public Schools 7/20/05

24 Teacher Introduction Exponential, Logarithmic, and Logistic Functions The nature of the topic requires that the learning goals be integrated. The pacing guide and correlations demonstrate this. The students will probably have had limited exposure to exponential functions and no exposure to either logarithmic or logistic functions. The strategies and activities section of learning goal A refer to teacher notes (included in this Curriculum Guide) that provide you, the teacher, with a method of introducing these three functions. Page 23 of 131 Columbus Public Schools 7/20/05

25 Core: TEACHING STRATEGIES/ACTIVITIES Vocabulary: logarithm, logistic function, exponential function, logarithmic function, scatterplot, asymptotes, domain, range, root, end behavior, extrema, asymptotic behavior, Newton s Law of Cooling, local behavior, global behavior, natural logarithm, growth, decay, base e, compound interest, annuity. Learning Goal A: Identify exponential, logarithmic, and logistic functions. 1. Introduce an exponential function by having students complete the Million Dollar Mission (included in this Curriculum Guide). 2. Continue the introduction of exponential functions by using the teacher notes (included in this Curriculum Guide). 3. Students will learn the behaviors of different types of exponential bases by completing the activity Comparing Exponential Effects (included in this Curriculum Guide). 4. Develop the concept of fractional exponents by comparing values obtained by raising numbers to fractional exponents versus whole number exponents. This activity clarifies how fractional exponents behave. Specifically, students will make observations such as lies somewhere between 2 2 and 2 3. Students complete the What Are Fractional Exponents Really? activity (included in this Curriculum Guide), answering all questions. Regroup as a whole class to discuss the questions at the end of each section. 5. Have the students explore, using the calculator, what happens to the exponential function, f(x) = Na cx, as the values of N, a, and c change. The students should explore changing one value at a time and write down their conclusion. Once they have written the effect of each value, have the students share their findings with a partner. If their conclusions do not agree, encourage them to explore more to come to a clearer understanding. Wrap up this exploration by summarizing the basic features of an exponential function. See teacher notes for summarization of the key features of exponential functions (included in this Curriculum Guide). 6. Have students complete the activity Constant Effects? (included in this Curriculum Guide) to have the students discover how adding or subtracting a number to the exponent and adding or subtracting a number from the function will affect the graph of the function. 7. Introduce logarithmic functions by using the teacher notes (included in this Curriculum Guide). Learning Goal B: Describe and compare the characteristics of exponential, logarithmic and logistic functions: e.g. general shape, number of roots, domain and range, asymptotic and end behavior, extrema, local and global behavior. 1. Have the students complete the Paper Folding Activity (included in this Curriculum Guide) as a starting point for describing the characteristics of exponential graphs. 2. Students will complete the M&M s activity (included in this Curriculum Guide) as a follow-up to the Paper Folding activity and to further solidify the exponential graph characteristics. 3. Relating negative exponents to what they know already about positive exponents, students will take expressions like 2 3x 1 2 x and ( 3) and convert them to expressions without negative exponents. They will describe and sketch the resulting function found in the activity. Negative and Positive Exponential Function Graphs (included in this Curriculum Guide). Page 24 of 131 Columbus Public Schools 7/20/05

26 4. Apply the concept of rate of change to different exponential functions. Students complete the activity Rate of Change of Exponential Functions (included in this Curriculum Guide) in order to find that a larger growth factor does translate to a greater rate of change. 5. Students will be able to see the inverse relationship between a logarithmic and an exponential function by completing Let s Graph Logarithmic Functions (included in this Curriculum Guide). Learning Goal C: Solve exponential, logarithmic, and logistic equations graphically and algebraically. 1. After introducing exponential growth and decay, discuss the ways to solve exponential functions. The student will be able to complete the worksheet Exponential Growth and Decay (included in this Curriculum Guide). 2. Introduce the generalizations necessary to be able to simplify logarithmic expressions. Students should already know how to evaluate logarithms on their calculators (including logarithms of bases other than 10 or e using the change of base rule ). They will be able to use their calculators to carry out the computations in the worksheet Discovering Logarithmic Identities (included in this Curriculum Guide) in order to discover some of the more basic logarithm combination laws. Upon completion, students should compare their conclusions with each other. 3. Introduce the three fundamental logarithm combination laws (log of a product, log of a quotient, log of a power) by observing numerical patterns. Logarithm Combination Laws (included in this Curriculum Guide). Follow up with lots of practice combining and taking apart logarithmic expressions. 4. Have students practice solving logarithmic equations by using the Logarithmic cut-out puzzle (included in this Curriculum Guide). 5. Students will develop a better understanding of how to solve logarithmic equations by completing the Properties of Logarithms (included in this Curriculum Guide). 6. Student can proceed from solving a logarithmic equation in basic definition form to solving logarithmic equations which require more logical thought using the worksheet Exercises with Logarithms (included in this Curriculum Guide). 7. Extend knowledge of exponential curves to an application where students derive information from the graph itself. The activity Time Estimation (included in this Curriculum Guide) has students use the trace function on the TI-83 graphing calculator to find an x-value (time) which produces a given y-value (amount). 8. Practice solving the exponential equations by using logarithms. Students complete the activity Why Settle for Time Estimation When You can be Exact (included in this Curriculum Guide) which is built off of the Time Estimation Activity. 9. Another application of using logarithms to solve an equation is done by using the activity, The Rule of 72 (included in this Curriculum Guide). Learning Goal D: Create a scatterplot of bivariate data and identify an exponential, logarithmic, or logistic function to model the data and make predictions. 1. Complete the activity USA TODAY Snapshot More of U.S. (included in this Curriculum Guide) to show a set of data that will provide an exponential equation. 2. Use a graph to solve an exponential equation where the variable to solve for is in the exponent. Students complete the activity Determining the Half-Life of Hydrogen-3 (included in this Curriculum Guide) using the substitutions to obtain a graph which they use Page 25 of 131 Columbus Public Schools 7/20/05

27 to estimate the half-life. (This involves the use of logarithms to solve for a variable in the exponent). 3. Have students complete the activity "Not Just a Good Idea, It's the Law" (included in this Curriculum Guide). Here students trace the temperature of a microwaved potato as it gets closer and closer to room temperature. The measuring time will be approximately 200 minutes, depending on the size of the potato and other factors, so you need to plan accordingly. Students get to use Newton's Law of Cooling in order to create an exponential equation to model their data. They also get to consider some of the limits of the regression function of the TI-83 calculator and how to get around these. 4. In Ball Bounce Revisited (included in this Curriculum Guide) the students will have the opportunity to see that data that was collected and analyzed using parabolas can be used to produce an exponential function as well. 5. In the Population Growth Investigation (included in this Curriculum Guide,) students summarize their knowledge of exponential growth and take a first look at logistic growth. The teacher notes preceding the activity describe the use of the activity for extending exponential growth into an intuitive understanding of logistic growth. They also describe the next day discussion of the activity which is essential for students to understand logistic functions. 6. The in-class simulation Rumors, described in the teacher notes of this Curriculum Guide is the culminating activity fro the Population Growth Investigation. Students are actively involved in creating and modeling a logistic scenario. 7. Complete the activity Breast Cancer Risks (included in this Curriculum Guide) to show a set of data that will provide a logistic equation. 8. The student will complete the Comparison of Curve-Fittings (included in this Curriculum Guide) as a way to compare the different types of curves that fit a given set of data. Reteach: 1. To illustrate the relationship between decimal exponents and integer exponents, students will consider the continuity aspect of exponential functions; ( i.e., there are points between any two points found on a graph). Students will examine various pieces of an exponential graph by completing the activity Decimal Exponents (included in this Curriculum Guide). 2. Apply concepts of exponential decay to the real-world situation of depreciation. Students are guided through this process in the Car Value Depreciation worksheet (included in this Curriculum Guide). Students work individually and compare answers at the end. The question may also arise of how much error is acceptable, and how do we check our final answer? 3. The student will use their calculator to reinforce the relationships of common logarithms in Exploring Common Logarithms (included in this Curriculum Guide). 4. Additional practice in the relationship between logarithms and exponentials is provided by the worksheet Logarithmic & Exponential Form (included in this Curriculum Guide). 5. Investigating Compound Interest (included in this Curriculum Guide) will allow the student to explore the difference between simple and compound interest. 6. Some students will have a difficult time understanding the base of natural logarithms, e. Their understanding can be increased by using A Number Called e (included in this Curriculum Guide). Page 26 of 131 Columbus Public Schools 7/20/05

28 RESOURCES Learning Goal A: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp ; Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp Learning Goal B: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp Learning Goal C: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp Learning Goal D: Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp ; Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004): Resource Manual pp ; Page 27 of 131 Columbus Public Schools 7/20/05

29 The Million Dollar Mission Exponent, Log, & Logistic Fctn - A Name You re sitting in math class, minding your own business, when in walks a Bill Gates kind of guy - the real success story of your school. He's made it big, and now he has a job offer for you. He doesn't give too many details, mumbles something about the possibility of danger. He's going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?) And you've got to make sure your passport is current. (Get real, Bill, this isn t Paris). But do you ever sit up at the next thing he says: You'll have your choice of two payment options: 1. One cent on the first day, two cents on the second day, and double your salary every day thereafter for the thirty days; or 2. Exactly $1,000,000. (That's one million dollars!) You jump up out of your seat at that. You've got your man, Bill, right here. You'll take that million. You are there. And off you go on this dangerous million-dollar mission. So how smart was this guy? Did you make the best choice? Before we decide for sure, let's investigate the first payment option. Complete the table for the first week's work. First Week First Option Day No. Pay for that Day Total Pay (In Dollars) So, after a whole week you would have only made. That's pretty awful, all right. There's no way to make a million in a month at this rate. Right? Let's check out the second week. Complete the second table. Second Week First Option Day No. Pay for that Day Total Pay (In Dollars) Page 28 of 131 Columbus Public Schools 7/20/05

30 Well, you would make a little more the second week; at least you would have made. But there's still a big difference between this salary and $1,000,000. What about the third week? Third Week First Option Day No. Pay for that Day Total Pay (In Dollars) We're getting into some serious money here now, but still nowhere even close to a million. And there's only 10 days left. So it looks like the million dollars is the best deal. Of course, we suspected that all along. Fourth Week First Option Day No. Pay for that Day Total Pay (In Dollars) Exponent, Log, & Logistic Fctn - A Hold it! Look what has happened. What's going on here? This can't be right. This is amazing. Look how fast this pay is growing. Let's keep going. I can't wait to see what the total will be. Last 2 Days First Option Day No. Pay for that Day Total Pay (In Dollars) In 30 days, it increases from 1 penny to over is absolutely amazing. dollars. That Page 29 of 131 Columbus Public Schools 7/20/05

31 The Million Dollar Mission Answer Key Exponent, Log, & Logistic Fctn - A You're sitting in math class, minding your own business, when in walks a Bill Gates kind of guy - the real success story of your school. He's made it big, and now he has a job offer for you. He doesn't give too many details, mumbles something about the possibility of danger. He's going to need you for 30 days, and you'll have to miss school. (Won't that just be too awful?) And you've got to make sure your passport is current. (Get real, Bill, this isn t Paris). But do you ever sit up at the next thing he says: You'll have your choice of two payment options: 1. One cent on the first day, two cents on the second day, and double your salary every day thereafter for the thirty days; or 2. Exactly $1,000,000. (That's one million dollars!) You jump up out of your seat at that. You've got your man, Bill, right here. You'll take that million. You are there. And off you go on this dangerous million-dollar mission. So how smart was this guy? Did you make the best choice? Before we decide for sure, let's investigate the first payment option. Complete the table for the first week's work. First Week First Option Day No. Pay for that Day Total Pay (In Dollars) So after a whole week you would have only made $1.27. That's pretty awful, all right. There's no way to make a million in a month at this rate. Right? Let's check out the second week. Complete the second table. Second Week First Option Day No. Pay for that Day Total Pay (In Dollars) Page 30 of 131 Columbus Public Schools 7/20/05

32 Exponent, Log, & Logistic Fctn - A Well, you would make a little more the second week; at least you would have made $ But there's still a big difference between this salary and $1,000,000. What about the third week? Third Week First Option Day No. Pay for that Day Total Pay (In Dollars) , , , , , , , , , We're getting into some serious money here now, but still nowhere even close to a million. And there's only 10 days left. So it looks like the million dollars is the best deal. Of course, we suspected that all along. Fourth Week First Option Day No. Pay for that Day Total Pay (In Dollars) 22 20, , , , , , , , , , , ,342, ,342, ,684, Hold it! Look what has happened. What's going on here? This can't be right. This is amazing. Look how fast this pay is growing. Let's keep going. I can't wait to see what the total will be. Last 2 Days First Option Day No. Pay for that Day Total Pay (In Dollars) 29 2,684, ,368, ,368, ,737, In 30 days, it increases from 1 penny to over 10 million dollars. That is absolutely amazing. Page 31 of 131 Columbus Public Schools 7/20/05

33 Teacher Notes - A Introduction to the Exponential Function Think of a number, any number. Double it. Double it again. Now think about how fast your number grew at each stage. The larger your number grew, the faster it grew, right? Do it again, or keep doubling it a few more times, to get the feel of this fact. The student can see this process very quickly by using the calculator. The student inputs their numbers into the calculator times 2, and then press ENTER. Continuing to use the calculator, the student would input times 2 (Ans*2) so that the student has taken the previous answer times 2. At this point, the student can continue hitting the ENTER key to see how the number has increased. This is the crucial property that makes an exponential function different from any other function: it increases (or decreases) faster if its value is larger because its growth rate is directly proportional to its value. Going back to your doubling exercise, let's say you chose the number 5. Then doubling it gives 5 x 2 = 10. Doubling again gives 5 x 2 x 2 = 20. If you doubled again you'd get 5 x 2 x 2 x 2 = 40. So, for the first doubling, when the value was 5, the process of doubling increased it by 5. But later when its value had increased to 20, the same process of doubling increased it by 20. This shows the growth rate depending on the value. Now looking back at the equations above, we see that we can write them more briefly. At the first step we have 5 x 2 = 10. At the second step we have 5 x 2 2 = 20. And then at the third step we have 5 x 2 3 = 40. Can you see a general rule? Of course we can do the same thing with tripling instead of doubling, or we could halve the number each time instead (i.e. multiply by a half). In fact we can choose any number (integer, fraction, decimal, irrational... any number!) and just keep multiplying by that number. Suppose we call this number p. Then what we have found above is that after the n th step the value we get is the original number, multiplied by p n. In this way we are looking at our result as a function of n (the step). If we know n we can use the above formula to work out the result. Finally, let's go back to the "doubling 5" example and suppose we want to increase our number gradually, continuously, instead of in jumps or steps. This is more useful in some real problems, for example if we know something is growing steadily so that it doubles every minute, but we want to know how much it has grown after a minute and a half. Then between the first and second steps we'd want to get a result that's bigger than the first result, 5 x 2 = 10, but not as big as the second result, 5 x 2 2 = 20. So instead of 5 x 2 1 (after 1 minute) or 5 x 2 2 (after 2 minutes) we just use 5 x to find the value after 1.5 minutes. Therefore, we can have any number as a power, so in this way we can calculate the value at any time t as 5 x 2 t. This is called an exponential function of t, because the t is the exponent (which means the power). Page 32 of 131 Columbus Public Schools 7/20/05

34 Comparing Exponential Effects Exponent, Log, & Logistic Fctn - A Name Directions: Using a calculator, fill in the chart below. For the observations column, indicate if the numbers in that row appear to be increasing, decreasing, or no change. n n 2 n 3 n 4 n 5 Observations How can you predict whether the numbers in each row are increasing or decreasing? Page 33 of 131 Columbus Public Schools 7/20/05

35 Comparing Exponential Effects Answer Key Exponent, Log, & Logistic Fctn - A Directions: Using a calculator, fill in the chart below. For the observations column, indicate if the numbers in that row appear to be increasing, decreasing, or no change. n n 2 n 3 n 4 n 5 Observations Increasing Increasing Decreasing No change No change Decreasing Increasing Decreasing Increasing How can you predict whether the numbers in each row are increasing or decreasing? 0 < n < 1 means numbers get smaller or are decreasing. n > 1 means the numbers get bigger or are increasing. Page 34 of 131 Columbus Public Schools 7/20/05

36 What Are Fractional Exponents Really? Exponent, Log, & Logistic Fctn - A Name 1. In the space between each pair of integers of the top row (1&2, 2&3, 3&4), write a mixed number (in fraction or decimal form) whose value lies somewhere between them. 2. Use your calculator to raise the given integer (in the leftmost column) to the power given along the top. Write your result in the grid. x = x 3 x 4 x 5 x 3. What do you observe about the numbers in each row as you move from left-to-right? 4. What does this tell you about the nature of your in-between exponents? 5. The following fractions all lie somewhere between 0 and 1. Order them from least to greatest, then place them in the first row of the table below, in least-to-greatest order: Then complete the table as above. x = x 2 0 = = 2 3 x 3 0 = = 3 4 x 4 0 = = 4 5 x 5 0 = = Hint: To type 5 into the TI-83 graphing calculator, you will need to treat the fractional exponent as division and enclose it in parentheses. This example would look like 5^(2 / 3). 6. What do you notice about the values in each row as you go from left to right? Page 35 of 131 Columbus Public Schools 7/20/05

37 What Are Fractional Exponents Really? Exponent, Log, & Logistic Fctn - A Name 1. In the space between each pair of integers of the top row (1&2, 2&3, 3&4), write a mixed number (in fraction or decimal form) whose value lies somewhere between them. 2. Use your calculator to raise the given integer (in the leftmost column) to the power given along the top. Write your result in the grid. Note: Answers to the blank columns will vary with student input values for x. x = x x x x What do you observe about the numbers in each row as you move from left-to-right? The numbers get progressively larger. 4. What does this tell you about the nature of your in-between exponents? They give values in-between the values from the whole number exponents. 5. The following fractions all lie somewhere between 0 and 1. Order them from least to greatest, then place them in the first row of the table below, in least-to-greatest order: Note: 1 =.5 1 =.33 2 =.66 1 =.25 1 =.2 3 =.75 3 = Then complete the table as above. x = x 2 0 = = 2 3 x 3 0 = = 3 4 x 4 0 = = 4 5 x 5 0 = = Hint: To type 5 into the TI-83 graphing calculator, you will need to treat the fractional exponent as division and enclose it in parentheses. This example would look like 5^(2 / 3). 6. What do you notice about the values in each row as you go from left to right? They get progressively larger. Page 36 of 131 Columbus Public Schools 7/20/05

38 Basic Features of Exponential Functions Teacher Notes - A f(x) = Na cx 1. When x = 0, a 0 = The function a x is never zero for any value of x when a For a > 1, the exponential function will increase. 4. For 0 < a < 1, the exponential function will decrease, but will never reaches zero. 5. As c > 1, the exponent enhances the growth property, making the growth faster. 6. As 0 < c < 1, the exponent inhibits the growth property, making the growth slower. 7. For c < 0, the graph is reflected about the y-axis. 8. For a > 0, the choice of a affects how rapidly the function grows or decays as you leave x = 0: for larger values of a, the growth or decay is faster. 9. The constant N multiplies the function by N everywhere, in particular it gives f (0) = N. 10. For N > 0, the value of N represents the starting value. 11. For N < 0, the graph is a reflection about the x-axis. Page 37 of 131 Columbus Public Schools 7/20/05

39 Constant Effects? Exponent, Log, & Logistic Fctn - A Name In this activity, you will explore how the constants h and k affect the graph of the exponential x h function f ( x) = b + k. 1. Sketch the graph of f( x ) = 3 x below. 2. Sketch the graph of gx ( ) 3 x transformed from the graph of f( x ) = 3 x 3 = on the same coordinate plane above. How is this graph? Is this what you had expected? 3. Sketch the graph of hx ( ) 3 x transformed from the graph of f( x ) = 3 x + 2 = on the same coordinate plane above. How is this graph? Is this what you had expected? 4. In the exponential function f ( x) x f ( x) = b? Be specific in your answer. x h = b, how does the constant h affect the graph of Page 38 of 131 Columbus Public Schools 7/20/05

40 5. Sketch the graph of f( x ) = 4 x below. Exponent, Log, & Logistic Fctn - A x 6. Sketch the graph of gx= ( ) on the same coordinate plane above. How is this graph transformed from the graph of f( x ) = 4 x? Is this what you had expected? x 7. Sketch the graph of hx ( ) = 4 2 on the same coordinate plane above. How is this graph transformed from the graph of f( x ) = 4 x? Is this what you had expected? x 8. In the exponential function f ( x) = b + k, how does the constant k affect the graph of x f ( x) = b? Be specific in your answer. 9. How do you think the graph of f( x ) = 2 x? kx x+ 2 ( ) 2 5 = + is transformed from the graph of Page 39 of 131 Columbus Public Schools 7/20/05

41 Constant Effects? Answer Key Exponent, Log, & Logistic Fctn - A In this activity, you will explore how the constants h and k affect the graph of the exponential x h function f ( x) = b + k. 1. Sketch the graph of f( x ) = 3 x below. 2. Sketch the graph of gx ( ) 3 x transformed from the graph of f( x ) = 3 x This is a shift right 3 units. 3 = on the same coordinate plane above. How is this graph? Is this what you had expected? 3. Sketch the graph of hx ( ) 3 x transformed from the graph of f( x ) = 3 x This is a shift left 2 units. + 2 = on the same coordinate plane above. How is this graph? Is this what you had expected? 4. In the exponential function f ( x) x h = b, how does the constant h affect the graph of x f ( x) = b? Be specific in your answer. If h is positive then the graph will shift right h units. If h is negative then the graph will shift left h units. Page 40 of 131 Columbus Public Schools 7/20/05

42 5. Sketch the graph of f( x ) = 4 x below. Exponent, Log, & Logistic Fctn - A x 6. Sketch the graph of gx= ( ) on the same coordinate plane above. How is this graph transformed from the graph of f( x ) = 4 x? Is this what you had expected? It is a shift up 8 units. x 7. Sketch the graph of hx ( ) = 4 2 on the same coordinate plane above. How is this graph transformed from the graph of f( x ) = 4 x? Is this what you had expected? It is a shift down 2 units. x 8. In the exponential function f ( x) = b + k, how does the constant k affect the graph of x f ( x) = b? Be specific in your answer. If k is positive then the graph will shift up k units. If k is negative then the graph will shift down k units. 9. How do you think the graph of kx x+ 2 ( ) 2 5 = + is transformed from the graph of f( x ) = 2 x? The graph of k(x) will shift left 2 units and up 5 units from the graph of f(x). Page 41 of 131 Columbus Public Schools 7/20/05

43 Introduction to the Logarithm Function Teacher Notes - A The value of the log function for an input x is the power you have to raise something to, to get x. The "something" is called the base and it can be any number you choose. If we choose the base to be 2, then the log function is called the "log base 2" function. So the value of log base 2 for input x, is the power you have to raise 2 to, to get x. Just remember that the log is the power, the index, the exponent, the little number up in the air. It is inconvenient to write log base 2 all the time, so we write it more concisely as below, with the base written as a subscript. log 2 (x) If there's no base shown as a subscript, the function is just log(x), this means that the base is 10. Another common base is "e". This turns out to be such a useful base that the function "log-tobase-e" is known as the "natural log" function. It is very often written as ln(x), rather than log e (x). The "base" is the number that's raised to the power. The other number is what we want to get when we raise the base to the power. So the expression "log base 5 of 125" means: "the power we have to raise 5 to, to get 125" or log = 3. Although logarithms may seem awkward, they turn a problem that involves powers into a problem that does not involve powers, thus easier to solve. The second purpose for logs is to look at different scales at the same time. The main thing to remember when trying to find the log of something: it's the power that we're looking for. Page 42 of 131 Columbus Public Schools 7/20/05

44 The Paper Folding Activity Exponent, Log, & Logistic Fctn - B Name Part I. Number of Sections 1. Fold an 8.5 x 11 sheet of paper in half and determine the number of sections the paper has after you have made the fold. 2. Record the data in the table and continue in the same manner until it becomes too hard to fold the paper. Number of Folds Number of Sections 3. Make a scatterplot of your data. Page 43 of 131 Columbus Public Schools 7/20/05

45 Exponent, Log, & Logistic Fctn - B 4. Determine a mathematical model that represents this data by examining the patterns in the table. 5. What might be different if you tried this experiment with an 8.5 x 11 sheet of wax paper or tissue paper? Part II: Area of Smallest Section 6. Fold an 8.5 x 11 sheet of paper in half and determine the fractional part of the smallest section after you have made the fold. 7. Record the data in the table and continue in the same manner until it becomes too hard to fold the paper. Number of Folds Fractional Part of Smallest Section 0 1 Page 44 of 131 Columbus Public Schools 7/20/05

46 Exponent, Log, & Logistic Fctn - B 8. Make a scatterplot of your data. 9. Determine a mathematical model that represents the data by examining the patterns in the table. Page 45 of 131 Columbus Public Schools 7/20/05

47 The Paper Folding Activity Answer Key Exponent, Log, & Logistic Fctn - B Part I. Number of Sections 1. Fold an 8.5 x 11 sheet of paper in half and determine the number of sections the paper has after you have made the fold. 2. Record the data in the table and continue in the same manner until it becomes too hard to fold the paper. Number of Folds Number of Sections Make a scatterplot of your data. Page 46 of 131 Columbus Public Schools 7/20/05

48 4. Determine a mathematical model that represents this data by examining the patterns in the table. y = 2 x Exponent, Log, & Logistic Fctn - B 5. What might be different if you tried this experiment with an 8.5 x 11 sheet of wax paper or tissue paper? The results would be the same, but you would be able to make more folds and collect more data because the paper would be thinner. Part II: Area of Smallest Section 6. Fold an 8.5 x 11 sheet of paper in half and determine the fractional part of the smallest section after you have made the fold. 7. Record the data in the table and continue in the same manner until it becomes too hard to fold the paper. Number of Folds Fractional Part of Smallest Section /2 2 1/4 3 1/8 4 1/16 5 1/32 6 1/64 Page 47 of 131 Columbus Public Schools 7/20/05

49 Exponent, Log, & Logistic Fctn - B 8. Make a scatterplot of your data. Area of Smallest Section 9. Determine a mathematical model that represents the data by examining the patterns in the table. y = (1/2) x Page 48 of 131 Columbus Public Schools 7/20/05

50 The M&M Investigation Exponent, Log, & Logistic Fctn - B Name Part I: Collecting Data 1. Empty your bag of M&M s and count them. Then place them back in the bag, and mix them well. Pour them out on the desk, count the number that show an m and place these back in the bag. The others may be eaten or removed. Record the number that show an m in your data table then repeat this procedure. Continue until the number of M&M s remaining is less than 5 but greater than 0. Record your data in the data table. Trial Number Number of M&M s Remaining Part II: Graphing and Determining the Exponential Model 2. Use a graphing calculator to make a scatterplot of your data. Copy your scatterplot onto the grid below. Then use the graphing calculator to determine an exponential model, and graph the equation. Sketch in the graph, and write your exponential equation. Equation: Page 49 of 131 Columbus Public Schools 7/20/05

51 Part III: Interpreting the Data Exponent, Log, & Logistic Fctn - B 3. In your model, y = a(b) x, what value do you have for a? What does that seem to relate to when you consider your data? When x = 0, what is your function value? Compare this to the values in your data table. 4. What is the value for b in your exponential model? How does this value relate to the data collection process? 5. How does the M&M experiment compare to the paper folding activity? How are they alike and how are they different? Page 50 of 131 Columbus Public Schools 7/20/05

52 Part I: Collecting Data The M&M Investigation Answer Key Exponent, Log, & Logistic Fctn - B 1. Empty your bag of M&M s and count them. Then place them back in the bag, and mix them well. Pour them out on the desk, count the number that show an m and place these back in the bag. The others may be eaten or removed. Record the number that show an m in your data table then repeat this procedure. Continue until the number of M&M s remaining is less than 5 but greater than 0. Record your data in the data table. Answers will vary. A sample set of data has been included on this sheet for your reference. Trial Number Number of M&M s Remaining Part II: Graphing and Determining the Exponential Model 2. Use a graphing calculator to make a scatterplot of your data. Copy your scatterplot onto the grid below. Then use the graphing calculator to determine an exponential model, and graph the equation. Sketch in the graph, and write your exponential equation. Equation: y = 140.1(0.54) x Number Remaining Page 51 of 131 Columbus Public Schools 7/20/05

53 Part III: Interpreting the Data Exponent, Log, & Logistic Fctn - B 3. In your model, y = a(b) x, what value do you have for a? What does that seem to relate to when you consider your data? initial number of m & m s When x = 0, what is your function value? Compare this to the values in your data table. Very close to the original number of M&M s in the bag 4. What is the value for b in your exponential model? b = 0.54 How does this value relate to the data collection process? This is close to the probability that an M&M candy will land with the m showing. As the data is collected, the number of M&M s decreases by almost half. 5. How does the M&M experiment compare to the paper folding activity? The M&M data pattern is close to the data pattern for the paper folding activity when the student recorded the trial number and the area of the smallest section. How are they alike and how are they different? The starting numbers are different, but the pattern of decrease is almost the same - each term is about half of the previous term. in the case of the paper, it is exactly half. In the case of the candy, it is approximately half. Page 52 of 131 Columbus Public Schools 7/20/05

54 Negative and Positive Exponential Function Graphs Exponent, Log, & Logistic Fctn - B Name Function y = 3 2 x Function Rewritten with Positive Exponents Graph of Function y 1 = 3 x 2 y = 5 2x y = 1.25 x y = 6 x Page 53 of 131 Columbus Public Schools 7/20/05

55 Negative and Positive Exponential Function Graphs Answer Key Exponent, Log, & Logistic Fctn - B Function Function Rewritten with Positive Exponents Graph of Function y = 3 2 x 3x y= 1 or y= x y 1 = 3 x x y = x 2 x 2 y = y= or y= 2 4 x y = 1.25 x y = 1 = x x y = 6 x x 1 y = Page 54 of 131 Columbus Public Schools 7/20/05

56 Rate of Change of Exponential Functions Exponent, Log, & Logistic Fctn - B Name Let s compare the properties of some exponential function graphs. Fill in the following table. Use your calculator if you need to: 1. f( x ) = 3 x f(0) = (y-intercept) f(1) = f(2) = f(3) = gx= ( ) 3.5 x g(0) = (y-intercept) g(1) = g(2) = g(3) = hx ( ) = 2 x h(0) = (y-intercept) h(1) = h(2) = h(3) = jx ( ) = 4 x j(0) = (y-intercept) j(1) = j(2) = j(3) = To compare the rate of change of these functions, we will find the slopes between points on these graphs as follows: Average Rate of Change From 0 To 1 Average Rate of Change From 1 To 2 Average Rate of Change From 2 To 3 For f: For g: For h: For j: f (1) f (0) = 1 0 g(1) g(0) = 1 0 h(1) h(0) = 1 0 j(1) j(0) = 1 0 f (2) f (1) = 2 1 g(2) g(1) = 2 1 h(2) h(1) = 2 1 j(2) j(1) = 2 1 Arrange the four functions from steepest (increasing most quickly) to shallowest. f (3) f(2) = 3 2 g(3) g(2) = 3 2 h(3) h(2) = 3 2 j(3) j(2) = 3 2 Would you expect these patterns to continue for higher values of x? Why or why not? Page 55 of 131 Columbus Public Schools 7/20/05

57 Rate of Change of Exponential Functions Answer Key Let s compare the properties of some exponential function graphs. Fill in the following table. Use your calculator if you need to: 1. f( x ) = 3 x f(0) = 1 (y-intercept) f(1) = 3 f(2) = 9 f(3) = gx= ( ) 3.5 x g(0) = 1 (y-intercept) g(1) = 3.5 g(2) = g(3) = Exponent, Log, & Logistic Fctn - B hx ( ) = 2 x h(0) = 1 (y-intercept) h(1) = 2 h(2) = 4 h(3) = jx ( ) = 4 x j(0) = 1 (y-intercept) j(1) = 4 j(2) = 16 j(3) = To compare the rate of change of these functions, we will find the slopes between points on these graphs as follows: Average Rate of Change From 0 To 1 Average Rate of Change From 1 To 2 Average Rate of Change From 2 To 3 For f: For g: For h: For j: f (1) f (0) = 1 0 g(1) g(0) = 1 0 h(1) h(0) = 1 0 j(1) j(0) = f (2) f (1) = 2 1 g(2) g(1) = 2 1 h(2) h(1) = 2 1 j(2) j(1) = Arrange the four functions from steepest (increasing most quickly) to shallowest. j, g, f, h 2 12 f (3) f(2) = 3 2 g(3) g(2) = 3 2 h(3) h(2) = 3 2 j(3) j(2) = 3 2 Would you expect these patterns to continue for higher values of x? Why or why not? Yes, because you are multiplying bases of numbers greater than Page 56 of 131 Columbus Public Schools 7/20/05

58 Let s Graph Logarithmic Functions Exponent, Log, & Logistic Fctn - B Name 1. Given the function f( x ) = 2 x, a) Complete the table below for f(x). b) Graph f(x) using the ordered pairs from (a). x y c) Graph the inverse of f(x) on the same coordinate plane as f(x). Write down the five new ordered pairs on the inverse graph. x y The inverse of the exponential function that you have just graphed is called a Logarithmic Function. We will now find the equation for this inverse of f(x). d) Using y = 2 x, what is the first thing you have to do to find the inverse? Write down your new equation. Write this new equation in logarithmic form. This equation is the inverse of f(x). Use the ordered pairs from part (c) to check that your equation is correct. e) What is the domain of f(x)? What is the range of f(x)? What is the domain of the inverse of f(x)? What is the range of the inverse of f(x)? Page 57 of 131 Columbus Public Schools 7/20/05

59 2. Given the function f( x ) = 3 x, a) Complete the table below for f(x). b) Graph f(x) using the ordered pairs from (a). x y c) Graph the inverse of f(x) on the same coordinate plane as f(x). Write down the four new ordered pairs on the inverse graph. x y Exponent, Log, & Logistic Fctn - B d) Using y = 3 x, what is the first thing you have to do to find the inverse? Write down your new equation. Write this new exponential equation in logarithmic form. Use the ordered pairs from part (c) to check that your equation is correct. e) What is the domain of f(x)? What is the range of f(x)? What is the domain of the inverse of f(x)? What is the range of the inverse of f(x)? 3. Based on #1 and #2, if the exponential function f(x) = b x is given, what is the inverse function of f(x)? Page 58 of 131 Columbus Public Schools 7/20/05

60 Let s Graph Logarithmic Functions Answer Key 1. Given the function f( x ) = 2 x, a) Complete the table below for f(x). b) Graph f(x) using the ordered pairs from (a). x y - 1 ½ c) Graph the inverse of f(x) on the same coordinate plane as f(x). Write down the five new ordered pairs on the inverse graph. x y ½ The inverse of the exponential function that you have just graphed is called a Logarithmic Function. We will now find the equation for this inverse of f(x). d) Using y = 2 x, what is the first thing you have to do to find the inverse? Write down your new equation. Switch x and y. x = 2 y Write this new exponential equation in logarithmic form. This equation is the inverse of f(x). Use the ordered pairs from part (c) to check that your equation is correct. y = log 2 x e) What is the domain of f(x)? What is the range of f(x)? (-, ) (0, ) Exponent, Log, & Logistic Fctn - B What is the domain of the inverse of f(x)? What is the range of the inverse of f(x)? (0, ) (-, ) Page 59 of 131 Columbus Public Schools 7/20/05

61 2. Given the function f( x ) = 3 x, a) Complete the table below for f(x). b) Graph f(x) using the ordered pairs from (a). x y - 1 1/ c) Graph the inverse of f(x) on the same coordinate plane as f(x). Write down the four new ordered pairs on the inverse graph. x y 1/ d) Using y = 3 x, what is the first thing you have to do to find the inverse? Write down your new equation. Switch x and y. x = 3 y Write this new equation in logarithmic form. Use the ordered pairs from part (c) to check that your equation is correct. y = log 3 x Exponent, Log, & Logistic Fctn - B e) What is the domain of f(x)? What is the range of f(x)? (-, ) (0, ) What is the domain of the inverse of f(x)? (0, ) What is the range of the inverse of f(x)? (-, ) 3. Based on #1 and #2, if the exponential function f(x) = b x is given, what is the inverse function of f(x)? y =f -1 (x) = log b x Page 60 of 131 Columbus Public Schools 7/20/05

62 Exponential Growth and Decay Exponent, Log, & Logistic Fctn - C Name The processes of uninhibited population growth and radioactive decay both follow the exponential model P(t) = P o (e kt ) where P o is the initial amount and P(t) is the amount after time t, k is positive for processes which grow over time, and is negative for processes which decay. Complete the following of growth or decay: 1. If a city has a population of 340 people, and if the population grows continuously at an annual rate of 2.3%, what will the population be in 6 years? 2. A certain radioactive isotope has a half-life of 37 years. How many years will it take for 100 grams to decay to 64 grams? 3. If a small island has a population of 420 people, and if the population doubles every 9 years, what will the population be in 7 years? Page 61 of 131 Columbus Public Schools 7/20/05

63 Exponent, Log, & Logistic Fctn - C 4. A certain radioactive isotope decays from 70 grams to 54 grams in 27 years. What is the half-life of the isotope? 5. A certain radioactive isotope has a half-life of 16 days. If one starts with 15 grams of the isotope, how much is left after 4 days? 6. A certain radioactive isotope has a half-life of 14 years. If 19 grams of the isotope are left after 5 years, how much was present at the beginning? Page 62 of 131 Columbus Public Schools 7/20/05

64 Exponential Growth and Decay Answer Key The processes of uninhibited population growth and radioactive decay both follow the exponential model P(t) = P o (e kt ) where P o is the initial amount and P(t) is the amount after time t, k is positive for processes which grow over time, and is negative for processes which decay. Complete the following of growth or decay: 1. If a city has a population of 340 people, and if the population grows continuously at an annual rate of 2.3%, what will the population be in 6 years? P(t) = P o (e kt ) P(6) = 340(e 0.023(6) ) Exponent, Log, & Logistic Fctn - C P o = 340 P(6) = 390 people k = t = 6 2. A certain radioactive isotope has a half-life of 37 years. How many years will it take for 100 grams to decay to 64 grams? P(t) = P o (e kt ) P(t) = P o (e kt ) Po 37 ( ) 2 = k Po e 64 = 100(e kt ) = k 64 e 100 = e kt 1 37 ln = k 64 ln( e ) ln = kt ln( e ) ln k = ln t = ln = = k ln 2 years 3. If a small island has a population of 420 people, and if the population doubles every 9 years, what will the population be in 7 years? P(t) = P o (e kt ) P(t) = P o (e kt ) 840 = 420(e 9k ) P(7) = 420(e 7k ) 2 = e 9k P(7) = 720 people ln 2 k = 9 Page 63 of 131 Columbus Public Schools 7/20/05

65 Exponent, Log, & Logistic Fctn - C 4. A certain radioactive isotope decays from 70 grams to 54 grams in 27 years. What is the halflife of the isotope? 27k 1 54 = 70e 2 = e kt k = ln ln = kt ln( e ) kt = ln 2 t = years 5. A certain radioactive isotope has a half-life of 16 days. If one starts with 15 grams of the isotope, how much is left after 4 days? 15 15e 16 2 = k k = ln 2 16 P(4) = 15e 4k = grams 6. A certain radioactive isotope has a half-life of 14 years. If 19 grams of the isotope are left after 5 years, how much was present at the beginning? 1 e 14 2 = k k = ln = P o e 5k P o = grams Page 64 of 131 Columbus Public Schools 7/20/05

66 Discovering Logarithmic Identities Exponential/Logarithmic Functions C Name Each of the following equations can be solved by turning the logarithmic equation into an equivalent exponential equation. In each set, see if you can determine what the general rule is supposed to be before you get to the end: A. log 2 2 = log 7 7 = log 5 5 = log 8 8 = General Rule: log a = a B. log 4 1 = 1 = log 3 1 = log 5 log 2 1 = General Rule: log b 1 = 2 C. log (3 ) = 3 4 log (5 ) = 5 7 log (2 ) = 2 3 log ( 8 ) = 8 30 log = 5 log(10 ) = g General Rule: log ( ) = b b Page 65 of 131 Columbus Public Schools 7/20/05

67 Discovering Logarithmic Identities Answer Key Each of the following equations can be solved by turning the logarithmic equation into an equivalent exponential equation. In each set, see if you can determine what the general rule is supposed to be before you get to the end: A. log 2 2 = x 2 x = 2 x = 1 log 7 7 = x 7 x = 7 x = 1 log 5 5 = x 5 x = 5 x = 1 log 8 8 = x 8 x = 8 x = 1 General Rule: log a = 1 a x = a x = 1 a Exponent, Log, & Logistic Fctn - C B. log 4 1 = x 4 x = 1 x = 0 log 3 1 = x 3 x = 1 x = 0 log 5 1 = x 5 x = 1 x = 0 log 2 1 = x 2 x = 1 x = 0 General Rule: log b 1 = x b x = 1 x = 0 2 C. log3(3 ) = x 3 x = 3 2 x = 2 4 log5(5 ) = x 5 x = 5 4 x = 4 7 log2(2 ) = x 2 x = 2 7 x = 7 3 log ( 8 ) = x 8 x = 8 3 x = log = x 14 x = x = 30 5 log(10 ) = x 10 x = 10 5 x = 5 g General Rule: logb( b ) = x b x = b g x = g Page 66 of 131 Columbus Public Schools 7/20/05

68 Logarithm Combination Laws Exponent, Log, & Logistic Fctn - C Name Using the TI-83 graphing calculator, answer all parts for each of the following: A. If 21 = 7 x, then what does x have to be? log(21) = log(7) = log(x) = What is the relationship between these last three answers? B. If 30 = y 5, then what does y have to be? log(30) = log(y) = log(5) = What is the relationship between these last three answers? C. If 72 8 = x, then what is x? log(72)= log(8)= log(x) = What is the relationship between these three last answers? D. If 243 = 3 x, then what value of x will make this true? log(243) = log(3) = What relationship do you see between these last two answers and x? E. If 7 = 49 x, then what is x? log(7) = log(49) = What relationship do we see now? (Your answer should be related to part D above). Page 67 of 131 Columbus Public Schools 7/20/05

69 Logarithm Combination Laws Answer Key Exponent, Log, & Logistic Fctn - C Using the TI-83 graphing calculator, answer all parts for each of the following. A. If 21 = 7 x, then what does x have to be? x = 3 log(21) = 1.32 log(7) =.85 log(3) =.47 What is the relationship between these last three answers? log 21 = log 7 + log 3 B. If 30 = y 5, then what does y have to be? y = 6 log(30) = log(6) =.7782 log(5) =.6990 What is the relationship between these last three answers? log 30 = log 6 + log 5 C. If 72 8 = x, then what is x? x = 9 D. If log(72) = log(8) =.9030 log(9) =.9542 What is the relationship between these three last answers? log 72 = log 8 + log 9 or log 72 log 8 = log 9 x 243 = 3, then what value of x will make this true? x = 5 log(243) = log (3) = What relationship do you see between these last two answers and x? The first log is 5 times the second. E. If 7 = 49 x, then what is x? x = 1 2 log(7) =.8451 log(49) = What relationship do we see now? (Your answer should be related to part D above). The second is double the first. Page 68 of 131 Columbus Public Schools 7/20/05

70 Log Cut Out Puzzle Exponent, Log, & Logistic Fctn - C Name All bases are positive. Cut out the squares. Arrange them so that touching edges are equivalent equations. Page 69 of 131 Columbus Public Schools 7/20/05

71 Log Cut Out Puzzle Answer Key Exponent, Log, & Logistic Fctn - C Note: This is only one possible answer. There may be others Page 70 of 131 Columbus Public Schools 7/20/05

72 Properties of Logarithms Exponent, Log, & Logistic Fctn - C Name Part I: 1. Evaluate log log Think of a logarithm expression with base 2 whose value is the same as the value in #1. 3. Evaluate log log Think of a logarithm expression with base 3 whose value is the same as the value in #3. 5. Evaluate log1 + log Think of a common logarithm expression whose value is the same as the value in #5. 7. What pattern do you see in the problems above? How can you write log b u + log b v as a single logarithm? Part II: 8. Evaluate log 2 64 log Think of a logarithm expression with base 2 whose value is the same as the value in # Evaluate log 3 81 log Think of a logarithm expression with base 3 whose value is the same as the value in # Evaluate log10,000 log10. Page 71 of 131 Columbus Public Schools 7/20/05

73 Exponent, Log, & Logistic Fctn - C 13. Think of a common logarithm expression whose value is the same as the value in # What pattern do you see in the problems above? How can you write log b u log b v as a single logarithm? Part III: 15. Evaluate log Think of a logarithm expression with base 2 (besides log 2 64) whose value is the same as the value in # Evaluate log Think of a logarithm expression with base 3 (besides log 3 729) whose value is the same as the value in # Evaluate log Think of a common logarithm expression (besides log10,000) whose value is the same as the value in # What pattern do you see in the problems above? How else can you write log b u k? Page 72 of 131 Columbus Public Schools 7/20/05

74 Name Part I: 1. Evaluate log log = 5 Properties of Logarithms Answer Key Exponent, Log, & Logistic Fctn - C 2. Think of a logarithm expression with base 2 whose value is the same as the value in #1. log Evaluate log log = 3 4. Think of a logarithm expression with base 3 whose value is the same as the value in #3. log Evaluate log1 + log = 3 6. Think of a common logarithm expression whose value is the same as the value in #5. log What pattern do you see in the problems above? The logarithm of a product is the sum of the logarithms of its factors. How can you write log b u + log b v as a single logarithm? log b (uv) = log b u + log b v Part II: 8. Evaluate log 2 64 log = 3 9. Think of a logarithm expression with base 2 whose value is the same as the value in #8. log Evaluate log 3 81 log = Think of a logarithm expression with base 3 whose value is the same as the value in #10. log Evaluate log10,000 log = 3 Page 73 of 131 Columbus Public Schools 7/20/05

75 Exponent, Log, & Logistic Fctn - C 13. Think of a common logarithm expression whose value is the same as the value in #12. log What pattern do you see in the problems above? The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. How can you write log b u log b v as a single logarithm? log u b = log b u - log b v v Part III: 15. Evaluate log Think of a logarithm expression with base 2 (besides log 2 64) whose value is the same as the value in #15. 3log Evaluate log Think of a logarithm expression with base 3 (besides log 3 729) whose value is the same as the value in #17. 2log Evaluate log Think of a common logarithm expression (besides log10,000) whose value is the same as the value in #12. 4log What pattern do you see in the problems above? The exponent can be written as a coefficient of the logarithm expression. The logarithm of a power is the product of the logarithm and the exponent. How else can you write log b u k? klog b u Page 74 of 131 Columbus Public Schools 7/20/05

76 Exercises with Logarithms Exponent, Log, & Logistic Fctn - C Name 1. Which numbers x satisfy the equation: (log 3 x)(log x 5) = log 3 5? 2. Suppose that the Canadian dollar loses 5% of its value each year. How many years are needed in order that the Canadian dollar to lose 90% of its value? (That is, the future value of the dollar to become the present value of a dime.) 3. Simplify the product: P = (log 2 3)(log 3 4)(log 4 5)... (log 31 32) 2 logb (log a a ) 4. If p = find a p. log a b 5. If log b (xy) = 11 and log b (x/y) = 5, what is log b x? Page 75 of 131 Columbus Public Schools 7/20/05

77 6. Positive integers A, B, and C, with no common factor greater than 1, exist such that A log B log = C. What is A + B + C? Exponent, Log, & Logistic Fctn - C log 7. What is the value of ? 8. A computer manufacturer finds that when x millions of dollars are spent on research, the profit, P(x), in millions of dollars, is given by Px ( ) = log3 ( x+ 3). How much should be spent on research to make a profit of 40 million dollars? 9. Solve the system of equations y = log2 2x and y = log4 x for all x. 10. Solve the equation log 3 (x - 2) + log 3 10 = log 3 (x 2 + 3x 10) 11. log 2 (9-2 x ) = 3 x Page 76 of 131 Columbus Public Schools 7/20/05

78 Exercises with Logarithms Answer Key Exponent, Log, & Logistic Fctn - C 1. Which numbers x satisfy the equation: (log 3 x)(log x 5) = log 3 5? All x > 0, x 1 2. Suppose that the Canadian dollar loses 5% of its value each year. How many years are needed in order that the Canadian dollar to lose 90% of its value? (That is, the future value of the dollar to become the present value of a dime.) About 45 years 3. Simplify the product: P = (log 2 3)(log 3 4)(log 4 5)... (log 31 32) P = 5 2 logb (log a a ) 4. If p = find a p. log a b a p = 2 5. If log b (xy) = 11 and log b (x/y) = 5, what is log b x? 8 Page 77 of 131 Columbus Public Schools 7/20/05

79 6. Positive integers A, B, and C, with no common factor greater than 1, exist such that A log B log = C. What is A + B + C? Exponent, Log, & Logistic Fctn - C 6 7. What is the value of 2 log5 2 25? 8. A computer manufacturer finds that when x millions of dollars are spent on research, the profit, P(x), in millions of dollars, is given by Px ( ) = log3 ( x+ 3). How much should be spent on research to make a profit of 40 million dollars? 78 million 9. Solve the system of equations y = log2 2x and y = log4 x. (.25, -1) 10. Solve the equation log 3 (x - 2) + log 3 10 = log 3 (x 2 + 3x 10) log 2 (9-2 x ) = 3 x 0, 3 Page 78 of 131 Columbus Public Schools 7/20/05

80 Time Estimation Exponent, Log, & Logistic Fctn - C Name For this activity, we will be looking at the compound interest formula from another angle. Recall the compound interest formula: A = P(1+ r n )n t 1. Determine what the compound interest formula would look like if we are compounding annually at 11% interest. Assume $100 to start. You should have a function for A in terms of t. 2. Put this formula into the TI-83 graphing calculator (you ll be using x instead of t for this. What x- and y-windows would make sense here? Decide how many years you wish to consider, and sketch the graph you get. 3. Using the trace function on the calculator, determine approximately how many years it would take for our investment to: a) double (this happens when y reaches 200) b) triple c) quadruple d) multiply by If our interest is 8% instead of 11%, would you expect it to take more or less time? Graph the relevant amount-vs.-time function, check your hypothesis, and justify your answer. Page 79 of 131 Columbus Public Schools 7/20/05

81 Time Estimation Answer Key Exponent, Log, & Logistic Fctn - C For this activity, we will be looking at the compound interest formula from another angle. Recall the compound interest formula: A = P(1+ r n )n t 1. Determine what the compound interest formula would look like if we are compounding annually at 11% interest. Assume $100 to start. You should have a function for A in terms of t. A = 100(1 +.11) t 2. Put this formula into the TI-83 graphing calculator (you ll be using x instead of t for this. What x- and y-windows would make sense here? Decide how many years you wish to consider, and sketch the graph you get Using the trace function on the calculator, determine approximately how many years it would take for our investment to: a) double (this happens when y reaches 200) 6.64 years b) triple years c) quadruple years d) multiply by years 4. If our interest is 8% instead of 11%, would you expect it to take more or less time? Graph the relevant amount-vs.-time function, check your hypothesis, and justify your answer. More time, the graph is not as steep, the values increase more slowly Page 80 of 131 Columbus Public Schools 7/20/05

82 Why Settle for Time Estimation When You Can Be Exact? Exponent, Log, & Logistic Fctn - C Name Luckily, logarithms allow us to solve for the t-variable in the compound interest formula. Let s see how we can use this fact to find out how long it takes any amount of money to double, triple, etc. 1. Rewrite the compound interest formula with the appropriate substitutions to indicate 11% interest, $100 to start, and annual compounding. Don t forget to substitute the correct value for A also (how much should it be if we re doubling our investment?). What does the formula now look like? 2. Solve this formula for t. See how this compares with your approximation in exercise 3a of the last activity. 3. Now create and solve an appropriate equation to represent tripling of our investment. Page 81 of 131 Columbus Public Schools 7/20/05

83 4. Do the same for quadrupling. Exponent, Log, & Logistic Fctn - C 5. How long, in exact terms, should it take to multiply our initial investment by 10? 6. Observe the relationship between each problem above and the exact answer (in terms of logarithms). Without going through all the steps of solving an exponential equation, how long should it take our investment to double if the interest rate were 8%? How should our numerical answer compare with number 2 above? Be sure to justify your answers. Page 82 of 131 Columbus Public Schools 7/20/05

84 Why Settle for Time Estimation When You Can Be Exact? Answer Key Luckily, logarithms allow us to solve for the t-variable in the compound interest formula.. Let s see how we can use this fact to find out how long it takes any amount of money to double, triple, etc. 1. Rewrite the compound interest formula with the appropriate substitutions to indicate 11% interest, $100 to start, and annual compounding. Don t forget to substitute the correct value for A also (how much should it be if we re doubling our investment?). What does the formula now look like? Answer: 200 = 100( ) t Exponent, Log, & Logistic Fctn - C 2. Solve this formula for t. See how this compares with your approximation in exercise 3a of the last activity. Answer: 200 = ( ) t ( ) t 2= 1.11 log 2= t log1.11 log 2 t = log1.11 This evaluates to 6.64 years, just like the graph. 3. Now create and solve an appropriate equation to represent tripling of our investment. Answer: 300 = ( ) t ( ) 3= 1.11 log 3 = log1.11 log 3 t = log1.11 t This evaluates to years, just like the graph. Page 83 of 131 Columbus Public Schools 7/20/05

85 4. Do the same for quadrupling. Exponent, Log, & Logistic Fctn - C Answer: 400 = ( ) t ( ) 4 = 1.11 log 4= t log1.11 log 4 t = log1.11 t This evaluates to years, just like the graph. 5. How long, in exact terms, should it take to multiply our initial investment by 10? Answer: 1000 = ( ) t ( ) 10 = 1.11 log10 = t log1.11 log10 t = log1.11 t This evaluates to years, just like the graph. 6. Observe the relationship between each problem above and the exact answer (in terms of logarithms). Without going through all the steps of solving an exponential equation, how long should it take our investment to double if the interest rate were 8%? How should our numerical answer compare with number 2 above? Be sure to justify your answers. Answer: t = log 2 log1.08 Since log 1.08 is less than log 1.11, t should be greater. Page 84 of 131 Columbus Public Schools 7/20/05

86 The Rule of 72 Exponent, Log, & Logistic Fctn - C Name Directions: For each row of the chart below, we are trying to double our investment. To do this, you will need to calculate the final amount A (double our investment) and then use the formula for continuous compound interest A = Pe rt to find the missing variable (round to 4 decimal places). P A r t $ $ $40.13 $70 12 years $900 7 years $13, years $50,000 9 years $250,000 5 years What is the relationship you observe between our values for r and t? Solve the continuous compound interest formula for the expression rt. How does this compare with our last observation? This relationship is referred to as the "Rule of 72". Why do you think the number 72 is used here? Page 85 of 131 Columbus Public Schools 7/20/05

87 The Rule of 72 Answer Key Exponent, Log, & Logistic Fctn - C Directions: For each row of the chart below, we are trying to double our investment. To do this, you will need to calculate the final amount A (double our investment) and then use the formula for continuous compound interest A = Pe rt to find the missing variable (round to 4 decimal places). P A r t $500 $ years $2000 $ years $40 $ years $70 $ years $900 $ years $6500 $13, years $25,000 $50, years $125,000 $250, years What is the relationship you observe between our values for r and t? They multiply to Solve the continuous compound interest formula for the expression rt. How does this compare with our last observation? A A = Pe rt P = ert 2 = e rt ln2 = rt This agrees because ln (2) This relationship is referred to as the "Rule of 72". Why do you think the number 72 is used? 72 has many more factors than 69, and it gives approximate results, though not as accurate as if we said rate times time. Page 86 of 131 Columbus Public Schools 7/20/05

88 USA TODAY Snapshot More of U.S. Exponent, Log, & Logistic Fctn - D Name The USA TODAY Snapshot - "More of U.S." shows the population (in millions) of the United States from 1940 through You will use an exponential function to model the growth of the population over time. Population growth often is restricted while the exponential growth model is not restricted. However, the population data often behaves as an exponential function over a limited time period. You will use the model to make a prediction about the population for a known year and then compare this value with the actual population. Finally, you will be given a population and determine the year this population figure was attained. MATH TODAY STUDENT EDITION Focus Questions: 1. Assume that the population of the United States is growing exponentially. What is the exponential function that best models the data provided? 2. What is the projected population in this Snapshot for 1997? What is the percent error in the estimated population compared to the actual population? 3. Determine when the U.S. population reached 100 million. MATH TODAY STUDENT EDITION PAGE 2 Data Source: U.S. Census Bureau Activity 1: Assume that the population of the United States is growing exponentially. What is the exponential function that best models the data provided? A. Use the data in the table below to create a scatterplot on the handheld. Let 0 represent 1900, 10 represent 1910, and so forth. Year Pop. U.S. (millions) Population (millions) Year Page 87 of 131 Columbus Public Schools 7/20/05

89 B. Use the regression capabilities of the handheld to determine the exponential regression model for the data set. Exponential regression model: Activity 2: What is the projected population for 1997? What is the percent error in the estimated population compared to the actual population? A. Graph the scatterplot and the exponential regression model in the same window. Population (millions) Year B. Trace the regression model to find the projected population for Projected population for 1997: C. Compare the projected population to the listed population in the Snapshot "More of U.S." for What is the percent error in your projected population? Note: The population at the end of 1997 was 269 million. Activity 3: Determine when the U.S. population reached 100 million. When did the U.S. population reach 100 million? U.S. Population will reach 100 million in: Page 88 of 131 Columbus Public Schools 7/20/05

90 USA TODAY Snapshot More of U.S. Answer Key THE NATION S NEWSPAPER The USA TODAY Snapshot - "More of U.S." shows the population (in millions) of the United States from 1940 through You will use an exponential function to model the growth of the population over time. Population growth often is restricted while the exponential growth model is not restricted. However, the population data often behaves as an exponential function over a limited time period. You will use the model to make a prediction about the population for a known year and then compare this value with the actual population. Finally, you will be given a population and determine the year this population figure was attained. MATH TODAY STUDENT EDITION Focus Questions: 1. Assume that the population of the United States is growing exponentially. What is the exponential function that best models the data provided? 2. What is the projected population in this Snapshot for 1997? What is the percent error in the estimated population compared to the actual population? 3. Determine when the U.S. population reached 100 million. MATH TODAY STUDENT EDITION PAGE 2 Data Source: U.S. Census Bureau Activity 1: Assume that the population of the United States is growing exponentially. What is the exponential function that best models the data provided? A. Use the data in the table below to create a scatterplot on the handheld. Let 0 represent 1900, 10 represent 1910, and so forth. Year Pop. U.S. (millions) Exponent, Log, & Logistic Fctn - D Population (Millions) Years Page 89 of 131 Columbus Public Schools 7/20/05

91 B. Use the regression capabilities of the handheld to determine the exponential regression model for the data set. Exponential regression model: y = ( ^x) Exponent, Log, & Logistic Fctn - D Activity 2: What is the projected population for 1997? What is the percent error in the estimated population compared to the actual population? A. Graph the scatterplot and the exponential regression model in the same window. Population (Millions) Years B. Trace the regression model to find the projected population for Projected population for 1997: million C. Compare the projected population to the listed population in the Snapshot "More of U.S." for What is the percent error in your projected population? Note: The population at the end of 1997 was 269 million. 3.7 % Activity 3: Determine when the U.S. population reached 100 million. When did the U.S. population reach 100 million? U.S. Population will reach 100 million in: 1917 Page 90 of 131 Columbus Public Schools 7/20/05

92 Determining the Half-Life of Hydrogen-3 Exponent, Log, & Logistic Fctn - D Name 1. Hydrogen-3, an isotope of Hydrogen, decays at the rate of 5.59% per year. If we start with 10 grams of Hydrogen-3, how much would be left at the end of one year? What percent of the original amount remains at the end of one year? Record your answers for years one through six in the table below. Number of Years Amount of Hydrogen-3 Remaining (in grams) Percent of Original Hydrogen-3 Amount Remaining % The time required for half of a radioactive substance to decay is called the half-life of that substance. Add rows to the table above until the percentage of the original amount goes below 50%. Complete the extended table. Approximately how many years did it take to reach this 50% mark? This is our approximate half-life for hydrogen The general formula for using half-life to determine the amount of radioactive decay is: t S = I (.) 05 h where I = the initial amount of hydrogen-3, S = the remaining amount of hydrogen-3, and t = amount of time. Use one of your rows of data from the table above to substitute values (number of years, amount remaining, and our initial amount of 10g) until there is only one variable left, h. 4. Graph the equation on a TI-83 graphing calculator. Use your graph to determine what x-value would represent the half-life of hydrogen-3. Explain why you chose this particular place on the graph, and what that point represents. Page 91 of 131 Columbus Public Schools 7/20/05

93 Determining the Half-Life of Hydrogen-3 Answer Key 1. Hydrogen-3, an isotope of Hydrogen, decays at the rate of 5.59% per year. If we start with 10 grams of Hydrogen-3, how much would be left at the end of one year? What percent of the original amount remains at the end of one year? Record your answers for years one through six in the table below. Number of Years Amount of Hydrogen-3 Remaining (in grams) Percent of Original Hydrogen-3 Amount Remaining % % % % % % % 2. The time required for half of a radioactive substance to decay is called the half-life of that substance. Add rows to the table above until the percentage of the original amount goes below 50%. Complete the extended table. Approximately how many years did it take to reach this 50% mark? This is our approximate half-life for hydrogen years 3. The general formula for using half-life to determine the amount of radioactive decay is: t Exponent, Log, & Logistic Fctn - D h S = I (0.5) where I = the initial amount of hydrogen-3, S = the remaining amount of hydrogen-3, and t = amount of time. Use one of your rows of data from the table above to substitute values (number of years, amount remaining, and our initial amount of 10g) until there is only one variable left, h. Student answers will vary depending on which line of the table the student uses. For example, when t = 2 years, you get S = 10(.5) (2/h). 4. Graph the equation on a TI-83 graphing calculator. Use your graph to determine what x-value would represent the half-life of hydrogen-3. Explain why you chose this particular place on the graph, and what that point represents. 10 x = Using y = 10(.5) (2/h) and the trace function when y = , then x is our half-life Page 92 of 131 Columbus Public Schools 7/20/05

94 Not Just a Good Idea It's The Law! Exponent, Log, & Logistic Fctn - D Name Directions: 1. Use a fork to stick holes in your potato (4 should be enough). Make one hole large enough to be able to insert a thermometer later. 2. Microwave the potato on high for 5 minutes. 3. Remove the potato and start the stopwatch or timer. 4. Immediately insert the thermometer and record the temperature, as well as the time, in minutes. 5. Every 5 minutes, re-record the temperature and time on the recording sheet. 6. Keep repeating step 5 until the you get the same temperature readings for 2 consecutive readings. 7. Determine the room temperature and make a note of it. time (minutes) temperature (Fahrenheit or Celsius) 8. Use your TI-83 to graph these points (put time in L1, temperature in L2). Sketch that graph on the axis below, then try to get an appropriate regression equation to fit the data. (First determine which type of regression might be the most appropriate). 9. Now for the fun part. Using the information you gathered and Newton s Law of Cooling, derive an equation which should model this data. 10. Graph both of the equations from questions 8 and 9 along with a scatterplot of the data. Which function more closely represents our data? Can you figure out why this is the case? 11. How might you get the exponential regression to work better? Page 93 of 131 Columbus Public Schools 7/20/05

95 Not Just a Good Idea It's The Law! Answer Key Directions: 1. Use a fork to stick holes in your potato (4 should be enough). Make one hole large enough to be able to insert a thermometer later. 2. Microwave the potato on high for 5 minutes. 3. Remove the potato and start the stopwatch or timer. 4. Immediately insert the thermometer and record the temperature, as well as the time, in minutes. 5. Every 5 minutes, re-record the temperature and time on the recording sheet. 6. Keep repeating step 5 until the you get the same temperature readings for 2 consecutive readings. 7. Determine the room temperature and make a note of it. Answers will vary. This is one possible result. time (minutes) temperature ( o Fahrenheit or Celsius) ºF Exponent, Log, & Logistic Fctn - D Page 94 of 131 Columbus Public Schools 7/20/05

96 Exponent, Log, & Logistic Fctn - D 8. Use your TI-83 to graph these points (put time in L1, temperature in L2). Sketch that graph on the axis below, then try to get an appropriate regression equation to fit the data. (First determine which type of regression might be the most appropriate). Exponential Regression Equation: y = ( x ) 9. Now for the fun part. Using the information you gathered and Newton s Law of Cooling, derive an equation which should model this data. T t = T m + (T 0 T m )e kt T t = 74 + (208 74)e kt T t = e kt use the fact that at t=20 minutes, T=161: 161 = e 20k = e 20k ln k = 20 k This makes our equation (courtesy of Newton's Law): T t = e t Page 95 of 131 Columbus Public Schools 7/20/05

97 10. Graph both of the equations from questions 8 and 9 along with a scatterplot of the data. Which function more closely represents our data? Can you figure out why this is the case? Exponent, Log, & Logistic Fctn - D This equation gives us a much closer fit than the calculator's attempt at regression because the TI doesn't do vertical shifts. It will only give you exponentials of the forma bx. 11. How might you get the exponential regression to work better? Here's one way: In your Stats editor, define L3 to have the values of whatever your room temperature is taken away from L2's values. Then do an exponential regression of L1 versus L3; put the results into Y1(X). Translate L1 upward by whatever your room temperature is (add you room temperature to the results from your regression). Page 96 of 131 Columbus Public Schools 7/20/05

98 Name Ball Bounce Revisited Exponent, Log, & Logistic Fctn - D In the Ball Bounce activity, you found that the graph of time after the ball released vs. the height of the ball from the ground formed a series of parabolas. Your graph should have looked something like this. In this activity, we are going to investigate how the maximum height of each parabola changes from bounce to bounce. From your ball bounce data, complete the chart, giving the maximum height of each bounce. (Ignore any extra spaces; not every group got the same number of bounces. Bounce Number 1 2 Height (in ft.) Make a scatterplot of the data in your chart. What kind of function do you think would model this data? Complete the appropriate regression, and add the curve to your scatterplot. Give your regression equation and sketch the graph below You probably found an exponential regression, an equation of the form y = ab x. What might a stand for in the real world situation? What might b stand for in the real-world situation? Page 97 of 131 Columbus Public Schools 7/20/05

99 Ball Bounce Revisited Answer Key In the Ball Bounce activity, you found that the graph of time after the ball released vs. the height of the ball from the ground formed a series of parabolas. Your graph should have looked something like this. Exponent, Log, & Logistic Fctn - D In this activity, we are going to investigate how the maximum height of each parabola changes from bounce to bounce. From your ball bounce data, complete the chart, giving the maximum height of each bounce. (Ignore any extra spaces; not every group got the same number of bounces. Bounce Number 1 Height (in ft.) Make a scatterplot of the data in your chart. What kind of function do you think would model this data? Complete the appropriate regression, and add the curve to your scatterplot. Give your regression equation and sketch the graph below. This data is from the sample data given in the Ball Bounce Activity in Topic 2-Polynomial, Power, and Rational Functions You probably found an exponential regression, an equation of the form y = ab x. Compare your values with those of other groups. What might a stand for in the real world situation? What might b stand for in the real-world situation? The variable, a, could stand for the initial height from which the ball was dropped. The variable, b, is probably some sort of measure of elasticity or bounciness. Page 98 of 131 Columbus Public Schools 7/20/05

100 Logistic Growth Teacher Notes - D The logistic growth investigation will probably take three classes. On the first day, the class should visit the computer lab to explore the website and collect the data. This activity has usually been available on the website but has occasionally moved. If it is not there, a Google search using the parameters habitat fish "logistic growth" applet should find it. It is easily recognizable because clicking Run applet produces a screen that simulates the number of fish in pond with pictures of fish. The first part of the activity reinforces the students previous knowledge of exponential growth. The second part is an introduction to logistic growth. Each part requires that the teacher assign each student a particular value to record data for. It is necessary for students to complete the entire exploration and collect their assigned data for the lesson to be a success. The internet portion of the investigation should be completed in one class. This is possible so long as the students remain on task. The next day, the class should discuss logistic behavior and the constraints that caused the behavior exhibited. These should include food supply, oxygen supply, disease, predators, and probably many other factors the students will come up with. Spend some time looking at the graph of the 1.5 birth rate graph. Discuss the concavity of the graph, where the population is increasing rapidly, and where the rate of increase begins to slow down. Estimate the generation at which the change occurs. Complete a logistic regression on the data. The calculator should give approximately y = , which is of the form x e y = c. While this is the classic form of a logistic equation in calculus, it isn t very kx 1 + ae helpful for PreCalculus students to gain insight into the problem. Another way to look at the equation is y = c. Either of these equations can be obtained mx 1 + b from the other, but the second allows the students to approximate the equation without regression by identifying important parts of the problem situation. After looking at the calculator regression, most students will guess that c should be a number close the carrying capacity. Also, because the graph appears to be exponential at the beginning, the use of 1.5, the birth rate for b. When y = 1000 is graphed with data, the graph looks like x the graph below. Looking at the graph and remembering that the concavity changed at approximately the 15 th generation, a horizontal shift of about 15 seems appropriate. Page 99 of 131 Columbus Public Schools 7/20/05

101 1000 y = ( x ) Teacher Notes - D By adjusting the equation slightly, you can come up with a good fit with y = ( x. ) The next day, the class should follow up with the Rumors activity, the next section in the teacher notes. Page 100 of 131 Columbus Public Schools 7/20/05

102 Population Growth Investigation Exponent, Log, & Logistic Fctn - D Name Go to and select Exponential Growth from the buttons at the top of the page. Click Reset All to clear any past data. If this website is not available, do a Google search using habitat fish "logistic growth" applet. Run the simulation for birth rates 1, 1.2, 1.4, 1.6, 1.8, and 2 for 20 generations. Click Reset (not Reset all) between each simulation. Your teacher will assign you one of the birth rates to keep track of the data on the chart. Be sure to record the data for your assigned birth rate on the charts provided. We will share the data on calculators later. Sketch the composite graph below. Explain how each the graphs of each birth rate are different or similar. How does increasing the birth rate change the graph? Generation Population Now click Reset All to clear any past data. Set the average birth rate to be 1.5 and Step the population through 15 generations. Now, without clicking either reset button, change the birth rate to 0.8. This change means that each individual is now only producing (on average) 0.8 individuals in the next generation. Predict what the graph will look like up through 30 generations. Page 101 of 131 Columbus Public Schools 7/20/05

103 Sketch the actual graph below. Exponent, Log, & Logistic Fctn - D What must be true of the birth rate for the population to remain constant to increase? to decrease? How would you change the birth rate to make the population increase very rapidly? decrease very rapidly? Test your theories using the fish website. Use the exponential regression function of your calculator to model a function that describes the growth of the fish population as a function of time for the birth rate you were assigned. Collect the results from the class for the other birth rates. Describe the results in terms of the general form of an exponential equation f(x)=ab x? Birth Rate Function What equation would model a population that begins with 1000 fish and whose birth rate is 0.8? Graph it on your calculator and sketch it here. Page 102 of 131 Columbus Public Schools 7/20/05

104 Return to the website to and select Logistic Growth from the buttons at the top of the page. Click Reset All to clear any past data. Make sure the birth rate is set to 1.5. Now do a series of simulations using carrying capacities of 200, 400, 600, 800, and Step the population for 30 generations with each value of carrying capacity. Click the Reset (Not the Reset All) button between each simulation. Sketch your composite graph below. Describe the general shape of these graphs. Do they resemble the exponential functions in any way? Generation Population Now set the carrying capacity at 1000 and use these birth rates: 1.5, 2, 2.5, 3.5, and 4 Click the Reset (Not the Reset All) button between each simulation. Your teacher will assign you a growth rate to record. Be sure to record the ` Exponent, Log, & Logistic Fctn - D data for your assigned birth rate on the charts provided. Sketch the composite graph below. Describe the general shape of these graphs. How did changing the birth rates change the graphs? Page 103 of 131 Columbus Public Schools 7/20/05

105 Exponent, Log, & Logistic Fctn - D Repeat the simulation using birth rates of 2, 4, and 6 and a carrying capacity of Sketch your graphs below. Why do these birth rates create such different graphs? Explain this in terms of what could be happening in the pond. Can you find the minimum birth rate which will cause the population to become extinct? Page 104 of 131 Columbus Public Schools 7/20/05

106 Exponent, Log, & Logistic Fctn - D Population Growth Investigation Answer Key Go to and select Exponential Growth from the buttons at the top of the page. Click Reset All to clear any past data. If this website is not available, do a Google search using habitat fish "logistic growth" applet. Run the simulation for birth rates 1, 1.2, 1.4, 1.6, 1.8, and 2 for 20 generations. Click Reset (not Reset all) between each simulation. Your teacher will assign you one of the birth rates to keep track of the data on the chart. Be sure to record the data for your assigned birth rate on the charts provided. We will share the data on calculators later. Sketch the composite graph below. Explain how each the graphs of each birth rate are different or similar. How does increasing the birth rate change the graph? Generation 0ANSWERS 1WILL 2VARY Population Now click Reset All to clear any past data. Set the average birth rate to be 1.5 and Step the population through 15 generations. Now, without clicking either reset button, change the birth rate to 0.8. This change means that each individual is now only producing (on average) 0.8 individuals in the next generation. Predict what the graph will look like up through 30 generations. Page 105 of 131 Columbus Public Schools 7/20/05

107 Sketch the actual graph below. Exponent, Log, & Logistic Fctn - D What must be true of the birth rate for the population to remain constant to increase? to decrease? How would you change the birth rate to make the population increase very rapidly? decrease very rapidly? Test your theories using the fish website. Answers will vary. Constant: Birth rate = 1; Decrease: Birth rate between 0 and 1; Increase Birth rate; Decrease Birth rate Use the exponential regression function of your calculator to model a function that describes the growth of the fish population as a function of time for the birth rate you were assigned. Collect the results from the class for the other birth rates. Describe the results in terms of the general form of an exponential equation f(x)=ab x? Birth Rate Function 1 y=2 1.2 y=2(1.2) x 1.4 y=2(1.4) x 1.6 y=2(1.6) x 1.8 y=2(1.8) x 2.0 y=2(2.0) x What equation would model a population that begins with 1000 fish and whose birth rate is 0.8? Graph it on your calculator and sketch it here. y=1000(.8) x Page 106 of 131 Columbus Public Schools 7/20/05

108 Return to the website to and select Logistic Growth from the buttons at the top of the page. Click Reset All to clear any past data. Make sure the birth rate is set to 1.5. Now do a series of simulations using carrying capacities of 200, 400, 600, 800, and Step the population for 30 generations with each value of carrying capacity. Click the Reset (Not the Reset All) button between each simulation. Sketch your composite graph below. Describe the general shape of these graphs. Do they resemble the exponential functions in any way? Now set the carrying capacity at 1000 and use these birth rates: 1.5, 2, 2.5, 3.5, and 4 Click the Reset (Not the Reset All) button between each simulation. Be sure to record the data for your assigned birth rate on the charts Generation 0ANSWERS 1WILL 2VARY ` Exponent, Log, & Logistic Fctn - D Population 26 provided. Sketch the composite graph below. Describe the general shape of these graphs. How did changing the birth rates change the graphs? Answers will vary. Page 107 of 131 Columbus Public Schools 7/20/05

109 Exponent, Log, & Logistic Fctn - D Repeat the simulation using birth rates of 2, 4, and 6 and a carrying capacity of Sketch your graphs below. Why do these birth rates create such different graphs? Explain this in terms of what could be happening in the pond. Can you find the minimum birth rate which will cause the population to become extinct? Answers will vary. Food supply may run out, the oxygen may run out, etc. Birth rate greater than 4 causes extinction. Page 108 of 131 Columbus Public Schools 7/20/05

110 Rumors Teacher Notes - D In the Rumor activity, the situation is posed that one class member knows a rumor, and comes to class and tells one person. The next day each of those students tells another student and so on. Eventually students who already know the rumor are told rumor again, but eventually everyone will have heard the rumor. To be effective, there should be more than 25 students in the class. If the class is small, each student should represent two students. To begin assign each student a whole number between 1 and the number of students in the class. (If it is a small class, the upper number would be twice the number of students.). To select the original student who knows the rumor, on the calculator choose a random number between 1 and the total number of students. For this example, the total number of students will be 30. On the MATH menu, choose #5. The syntax is (lowest number, highest number, how many) This gives us one random number between 1 and 30. To get the next person, repeat the command, (2 nd ENTER.) Each time, the number at the end should be the number of people who know the rumor. To keep track of the students who know the rumor, begin with them all standing and have them sit down when they know the rumor. (If you had to double the number of students, use raised hands.) Keep a chart of the number of students who know the rumor and continue until all students have heard the rumor. Remember to eliminate repeated numbers. At the end, you should have data that is approximately logistic. It might look like: Generation Total number knowing rumor The equation y = 30 is a fairly close model ( x. ) to the graph. (See population growth) Page 109 of 131 Columbus Public Schools 7/20/05