Laurie s Notes. Overview of Section 6.3

Transcription

1 Overview of Section.3 Introduction In this lesson, eponential equations are defined. Students distinguish between linear and eponential equations, helping to focus on the definition of each. A linear function has a constant rate of change, and an eponential function has a constant percent rate of change. The constant percent rate of change is the constant factor that is being multiplied b over equal intervals of the -values. Once eponential equations are evaluated, the are graphed. The parent functions = and = ( ) are used to eplore transformations. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations Resources The graphing calculator can be used to graph the eponential equations, giving students the abilit to eplore a series of graphs. Transformations of eponential equations can also be eplored graphicall, as well as numericall. Formative Assessment Tips Refer to page T-9 for a description of Response Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use our judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. For a beginning list of writing stems, see page T-9. Another Wa MP Look For and Epress Regularit in Repeated Reasoning: Show our students another wa to rewrite the epression (.5) in Eample (b). (.5) = (.5) ()( ) = [ ( ) ] = ( ) = After a few eamples, students learn the pattern and quit using all the steps. Pacing Suggestion Complete at least the first two eplorations and then start the formal lesson b defining eponential functions. Section.3 T-3

2 Common Core State Standards HSA-CED.A. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. HSF-IF.B. For a function that models a relationship between two quantities, interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. HSF-IF.C.7e Graph eponential functions, showing intercepts and end behavior, HSF-IF.C.9 Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). HSF-BF.A.a Determine an eplicit epression, or steps for calculation from a contet. HSF-BF.B.3 Identif the effect on the graph of replacing f() b f() + k, kf(), and f( + k) for specific values of k (both positive and negative); find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph HSF-LE.A.a Prove that linear functions grow b equal differences over equal intervals, and that eponential functions grow b equal factors over equal intervals. HSF-LE.A. Construct eponential functions, given a graph, or two input-output pairs (include reading these from a table). Eploration Motivate Pose this scenario. You give a test and scores range from % to %. To help our students, ou decide to increase each score b % of the score (% scaling). Does everone s score increase? es Does everone s score increase the same amount? no Which students receive the least benefit? the greatest benefit? Eplain. A % test score receives the least benefit ( points). A % test score receives the greatest benefit ( points). Without asking about fairness, discuss that the percent of change is the same (%) for all tests, but the amount of change is not. In a linear function, the amount of change is the same, so this function is not linear. Eplain that this section will eplore this idea further. Eploration Fact-First Questioning: Onl is being raised to the. Wh is this true? Listen for an understanding that this is not a power of a product. The constant would need to be inside the parentheses in order to raise it to the : ( ). Students should quickl complete the tables of values. What did ou notice about the values of? increases b in the first table and increases b in the second table. What did ou notice about the values of? increases b a factor of in the first table and increases b a factor of in the second table. Eploration Students should quickl complete the tables of values. What did ou notice about the values of? increases b in the first table and increases b in the second table. What did ou notice about the values of? decreases b a factor of in the first table and decreases b a factor of in the second table. MP3 Construct Viable Arguments and Critique the Reasoning of Others: Students abilit to offer justification to the question posed will var. Improving the abilit to justif their thinking is a goal that students should be working on. Eploration 3 Students could do a quick sketch on graph paper, or ou might want them to enter the data points in their calculator to get a more accurate graph. The Big Idea is certainl that two of the graphs are increasing and two are decreasing. Students should mention that all of the graphs are located in the first and second quadrants, meaning that >. Communicate Your Answer You might consider a jigsaw approach to Question 5 to differentiate or when time is short. Connecting to Net Step Students have now developed a sense of how eponential equations increase (or decrease) and what their graphs look like. The formal definition follows in the lesson, along with transformations of the parent function. T-35 Chapter

3 .3 Eponential Functions Essential Question What are some of the characteristics of the graph of an eponential function? Eploring an Eponential Function Work with a partner. Cop and complete each table for the eponential function = (). In each table, what do ou notice about the values of? What do ou notice about the values of? 3 = () = () Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations., 3,,, 5, 5;,, 5,, 9,,3; Each value of increases b the same amount; Each value of is multiplied b the same factor..,,,,, ;,,,,, ; es; As the eponent increases b a constant amount, the base is multiplied b itself the same number of additional times. 3.,, JUSTIFYING CONCLUSIONS To be proficient in math, ou need to justif our conclusions and communicate them to others. 5 Eploring an Eponential Function Work with a partner. Repeat Eploration for the eponential function = ( ). Do ou think the statement below is true for an eponential function? Justif our answer. As the independent variable changes b a constant amount, the dependent variable is multiplied b a constant factor. Graphing Eponential Functions Work with a partner. Sketch the graphs of the functions given in Eplorations and. How are the graphs similar? How are the different? Communicate Your Answer. What are some of the characteristics of the graph of an eponential function? 5. Sketch the graph of each eponential function. Does each graph have the characteristics ou described in Question? Eplain our reasoning. a. = b. = (3) c. = 3(.5) d. = ( ) e. = 3 ( ) f. = ( 3 ) Section.3 Eponential Functions 35 = () = ( ) Both are curved and do not intersect the -ais; The graph from Eploration is increasing, the graph from Eploration is decreasing.. Sample answer: curved shape, does not intersect the -ais 5. a. 5. b. c. 3 = 3(.5) = = (3) 5d f. See Additional Answers. Section.3 35

4 Differentiated Instruction Kinesthetic/Visual Some students ma benefit from creating their own function tables. Have students work in pairs. Each student creates a table to represent a linear or eponential function and then echanges tables with his or her partner. Students determine which tpe of function the table represents, and then eplain their reasoning to their partners. Etra Eample Does each table represent a linear or an eponential function? Eplain. a. The function is linear. As increases b, increases b. The rate of change is constant. b The function is eponential. As increases b, is multiplied b 3. Etra Eample Evaluate each function for the given value of. a. = 3() ; = b. = (.5) ; = 3 MONITORING PROGRESS. eponential; As increases b, is multiplied b.. linear; As increases b, decreases b. The rate of change is constant. 3. ; ;..375;.5; about..3 Lesson What You Will Learn Core Vocabular eponential function, p. 3 Previous independent variable dependent variable parent function STUDY TIP In Eample b, consecutive -values form a constant ratio. =, 3 =, = Identif and evaluate eponential functions. Graph eponential functions. 3 Chapter Eponential Functions and Sequences Solve real-life problems involving eponential functions. Identifing and Evaluating Eponential Functions An eponential function is a nonlinear function of the form = ab, where a, b, and b >. As the independent variable changes b a constant amount, the dependent variable is multiplied b a constant factor, which means consecutive -values form a constant ratio. Identifing Functions Does each table represent a linear or an eponential function? Eplain. a. 3 b. 3 3 a As increases b, increases b. The rate of change is constant. So, the function is linear. b As increases b, is multiplied b. So, the function is eponential. Evaluating Eponential Functions Evaluate each function for the given value of. a. = (5) ; = 3 b. = 3(.5) ; = a. = (5) Write the function. b. = 3(.5) = (5) 3 Substitute for. = 3(.5) = (5) Evaluate the power. = 3() = 5 Multipl. = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Does the table represent a linear or an eponential function? Eplain.. 3 Evaluate the function when =,, and. 3. = (9). =.5() Teacher Actions. Write the definition of an eponential function. Do not dwell on the constraints. Instead, return to the constraints (a, b, and b > ) after students have a sense of how the function behaves. Big Idea: Eponential functions change b equal factors over equal intervals. Connection: A linear function has a constant rate of change, and an eponential function has a constant percent rate of change. COMMON ERROR Evaluating (5) 3, students ma multipl first and evaluate ( ) 3. Remind students that onl 5 is being cubed. 3 Chapter

5 STUDY TIP The graph of = ab approaches the -ais but never intersects it. Graphing Eponential Functions The graph of a function = ab is a vertical stretch or shrink b a factor of a of the graph of the parent function = b. When a <, the graph is also reflected in the -ais. The -intercept of the graph of = ab is a. Core Concept Graphing = ab When b > Graphing = ab When < b < a > (, a) (, a) a < a > a < (, a) (, a) English Language Learners Pair Activit Pair each English learner with an English speaker to graph eponential functions. Ask one student to make the table of values and the other student to plot the ordered pairs and draw a smooth curve through the points. Have the pair compare the graph to the graph of the parent function and describe the domain and range. Graphing = ab When b > f() = ( ) g() = Graph f () = (). Compare the graph to the graph of the parent function. Describe the domain and range of f. Step Make a table of values. Step Plot the ordered pairs. f ( ) Step 3 Draw a smooth curve through the points. The parent function is g() =. The graph of f is a vertical stretch b a factor of of the graph of g. The -intercept of the graph of f,, is above the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is >. Graphing = ab When < b < Etra Eample 3 Graph f() = 5(). Compare the graph to the graph of the parent function. Describe the domain and range of f. f() = 5() g() = g() = ( ) f() = ( ) Graph f () = ( ). Compare the graph to the graph of the parent function. Describe the domain and range of f. Step Make a table of values. Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. f ( ) The parent function is g() = ( ). The graph of f is a reflection in the -ais of the graph of g. The -intercept of the graph of f,, is below the -intercept of the graph of g,. From the graph of f, ou can see that the domain is all real numbers and the range is <. The parent function is g() =. The graph of f is a vertical stretch b a factor of 5 of the graph of g. The domain of f is all real numbers and the range is >. Etra Eample Graph f() = (.5). Compare the graph to the graph of the parent function. Describe the domain and range of f. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. 5. f () = (). f () = ( ) g() =.5 Section.3 Eponential Functions 37 f() = (.5) Teacher Actions Students should have an understanding of what the graphs of f() = and f() = ( ) look like before performing an transformations. Connection: Transformations were presented in Section 3.. It is hard to see a vertical stretch or shrink of an eponential equation. Probing Question: Is there a wa to determine the -intercept from the equation as with linear equations? Answers will var. Response Logs: I was reall surprised when The parent function is g() = (.5). The graph of f is a reflection in the -ais of the graph of g. The domain of f is all real numbers and the range is <. MONITORING PROGRESS 5. See Additional Answers. Section.3 37

6 Etra Eample 5 Graph = 3() 3. Describe the domain and range. = 3() MONITORING PROGRESS 7. = 3() 3 The domain is all real numbers and the range is > 3. Etra Eample An eponential function g models a relationship in which the dependent variable is multiplied b.5 for ever unit the independent variable increases. Graph g when g() =. Compare g and the function f from Eample 3 over the interval = to =. 3 Both functions have the same value when =, but the value of f is less than the value of g over the rest of the interval. g f = () STUDY TIP = () 3 + Note that f is increasing faster than g to the right of =. To graph a function of the form = ab h + k, begin b graphing = ab. Then translate the graph horizontall h units and verticall k units. Graphing = ab h + k Graph = () 3 +. Describe the domain and range. Step Graph = (). This is the same function that is in Eample 3, which passes through (, ) and (, ). Step Translate the graph 3 units right and units up. The graph passes through (3, ) and (, ). Notice that the graph approaches the line = but does not intersect it. From the graph, ou can see that the domain is all real numbers and the range is >. Comparing Eponential Functions An eponential function g models a relationship in which the dependent variable is multiplied b.5 for ever unit the independent variable increases. Graph g when g() =. Compare g and the function f from Eample 3 over the interval = to =. You know (, ) is on the graph of g. To find points to the right of (, ), multipl g() b.5 for ever unit increase in. To find points to the left of (, ), divide g() b.5 for ever unit decrease in. Step Make a table of values. 3 g() Step Plot the ordered pairs. Step 3 Draw a smooth curve through the points. Both functions have the same value when =, but the value of f is greater than the value of g over the rest of the interval. Monitoring Progress f Help in English and Spanish at BigIdeasMath.com Graph the function. Describe the domain and range. 7. = (3) +. f () = (.5) WHAT IF? In Eample, the dependent variable of g is multiplied b 3 for ever unit the independent variable increases. Graph g when g() =. Compare g and the function f from Eample 3 over the interval = to =. g = (3) + 3 Chapter Eponential Functions and Sequences 3 Teacher Actions domain: all real numbers, range: < 9. See Additional Answers. In the equation = () 3 +, what is the parent function? = What do the, 3, and do to the parent function? The parent function is stretched verticall b a factor of ; this function is shifted 3 units right and units up. Pose Eample. Ask, What is known in this problem? the -intercept and the factor of increase Have students compare the bases of and.5. Think-Pair-Share: Have students answer Questions 7 9, and then share and discuss as a class. 3 Chapter

7 Population Bacterial Population (, 7) 7 5 (, 3) 3 (, ) (3, 9) (, ) Da Solving Real-Life Problems For an eponential function of the form = ab, the -values change b a factor of b as increases b. You can use this fact to write an eponential function when ou know the -intercept, a. The table represents the eponential function = (5). Modeling with Mathematics The graph represents a bacterial population after das. a. Write an eponential function that represents the population. b. Find the population after hours and after 5 das Understand the Problem You have a graph of the population that shows some data points. You are asked to write an eponential function that represents the population and find the population after different amounts of time.. Make a Plan Use the graph to make a table of values. Use the table and the -intercept to write an eponential function. Then evaluate the function to find the populations. 3. Solve the Problem a. Use the graph to make a table of values The -intercept is 3. The -values increase b a factor of as increases b Etra Eample 7 The graph represents a bacterial population after das. Population Bacterial Population Da (5, 79) (, 3) (, 9) (, 3) (3, ) (, 7) a. Write an eponential function that represents the population. The population can be modeled b = 3(3). b. Find the population after hours and after das. There are about 5 bacteria after hours and 7 bacteria after das. So, the population can be modeled b = 3(). b. Population after hours Population after 5 das hours = da = 3() Write the function. = 3() = 3() / Substitute for. = 3() 5 = 3() Evaluate the power. = 3() = Multipl. = 37 There are bacteria after hours and 37 bacteria after 5 das.. Look Back The graph resembles an eponential function of the form = ab, where b > and a >. So, the eponential function = 3() is reasonable. MONITORING PROGRESS ANSWER. a. = () b. bacteria c. no; The bacteria population in Eample 7 is growing b a factor of, and the bacteria population in this problem is onl growing b a factor of. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. A bacterial population after das can be represented b an eponential function whose graph passes through (, ) and (, ). (a) Write a function that represents the population. (b) Find the population after das. (c) Does this bacterial population grow faster than the bacterial population in Eample 7? Eplain. Section.3 Eponential Functions 39 Teacher Actions Teaching Tip: Have students use their calculators to graph =, = 3(), and = (). Have them use the trace or table feature to determine the -intercepts of the graphs. This will help students in the net eample where a = 3, so the equation is = 3(b). What information can be told from the graph? The -intercept is 3, and -values are increasing b a factor of. Closure Response Logs: Select from I made progress with or Tomorrow I need to find out or Right now I know. Section.3 39

8 Assignment Guide and Homework Check ASSIGNMENT Basic:, 5 35 odd, 5 odd, 5, 5, 7 Average: 3, 5 even, 5, 5 5 even, 7 Advanced:, even, even, 3 even, 5 7 HOMEWORK CHECK Basic:, 5, 5, 3, 5 Average:,,, 3, 5 Advanced:,,, 3, 5. Sample answer:. The -intercept occurs when =. So, = ab = a = a. 3. The graph of = (5) is a vertical stretch b a factor of of the graph of = 5. The -intercept of = (5),, is above the -intercept of = 5,.. f () = ( 3) ; It is the onl one that is not an eponential function. 5. es; It fits the pattern = ab.. no; It is a linear function. 7. no; The eponent is a constant.. es: It fits the pattern = ab. 9. no; Although it fits the pattern = ab, b cannot be negative.. no; Although it fits the pattern = ab, b cannot be.. linear; As increases b, increases b. The rate of change is constant.. eponential; As increases b, is multiplied b. 3. eponential; As increases b, is multiplied b.. linear; As increases b 3, decreases b 9. The rate of change is constant Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. OPEN-ENDED Sketch an increasing eponential function whose graph has a -intercept of.. REASONING Wh is a the -intercept of the graph of the function = ab? 3. WRITING Compare the graph of = (5) with the graph of = 5.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. In Eercises 5, determine whether the equation represents an eponential function. Eplain. 5. = (7). = 7. = 3. = 3 9. = 9( 5). = () In Eercises, determine whether the table represents a linear or an eponential function. Eplain. (See Eample.) In Eercises 5, evaluate the function for the given value of. (See Eample.) 5. = 3 ; =. f () = 3() ; = 7. = (5) ; =. f () =.5 ; = 3 9. f () = 3 () ; = 3. = () ; = 3 = 3 f () = () f () = ( 3) = 5(3) Monitoring Progress and Modeling with Mathematics 3 Chapter Eponential Functions and Sequences C. B 3. A. D 5. f() = 3(.5) USING STRUCTURE In Eercises, match the function with its graph.. f () = (.5). = (.5) 3. = (). f () = () A. C. B. D In Eercises 5 3, graph the function. Compare the graph to the graph of the parent function. Describe the domain and range of f. (See Eamples 3 and.) 5. f () = 3(.5). f () = 7. f () = (7). f () = ( ) 3 9. f () = () 3. f () = 3 (.5) In Eercises 3 3, graph the function. Describe the domain and range. (See Eample 5.) 3. f () = 3 3. f () = + 3 The graph of f is a vertical stretch b a factor of 3 of the graph of g() =.5. The -intercept of the graph of f, 3, is above the -intercept of the graph of g, ; domain: all real numbers, range: > 3. See Additional Answers. 3 Chapter

9 33. = = ( ) = (.75) + 3. f () = 3() 5 In Eercises 37, compare the graphs. Find the value of h, k, or a g() = a() 3. f() = g() = 3 h. f() = 3 g() =.5 + k f() =.5 f() = () 3 g() = () 3 h. ERROR ANALYSIS Describe and correct the error in evaluating the function. g() = (.5) ; = g( ) = (.5) = 3 = 9. ERROR ANALYSIS Describe and correct the error in finding the domain and range of the function. The domain is all real numbers, and the range is <. g() = (.5) In Eercises 3 and, graph the function with the given description. Compare the function to f () =.5() over the interval = to =. (See Eample.) 3. An eponential function g models a relationship in which the dependent variable is multiplied b.5 for ever unit the independent variable increases. The value of the function at is.. An eponential function h models a relationship in which the dependent variable is multiplied b for ever unit the independent variable increases. The value of the function at is MODELING WITH MATHEMATICS You graph an eponential function on a calculator. You zoom in repeatedl to 5% of the screen size. The function =.5 represents the percent (in decimal form) of the original screen displa that ou see, where is the number of times ou zoom in. a. Graph the function. Describe the domain and range. b. Find and interpret the -intercept. c. You zoom in twice. What percent of the original screen do ou see?. MODELING WITH MATHEMATICS A population of cootes in a national park triples ever ears. The function = 5(3) represents the population, where is the number of -ear periods. a. Graph the function. Describe the domain and range. b. Find and interpret the -intercept. c. How man cootes are in the national park in ears? In Eercises 7 5, write an eponential function represented b the table or graph. (See Eample 7.) (, ) (,.5) 5. (, ) (, ) (3, ) (, ) (, ) (3, ) Section.3 Eponential Functions 3 Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations = domain: all real numbers, range: > 7 3. = ( ) + 3 domain: all real numbers, range: < = (.75) + domain: all real numbers, range: < 3. hsnb_alg_pe_3.indd 3. The graph of g is a horizontal translation units left of the graph of f; h =. need to simplif the power before multipling; (.5) = () =. The graph approaches the line =, not = ; The range is <. 3. /5/5 7:9 AM The value of f is less than the value of g over the entire interval. 5. See Additional Answers. f() = 3() 5 domain: all real numbers, range: > The graph of g is a vertical shrink b a factor of of the graph of f; a = 3. The graph of g is a vertical translation 3 units up of the graph of f; k = The graph of g is a horizontal translation units right of the graph of f; h = Section.3 3

10 5. a. = ( 3 ) b. about 3 visitors 5. about grills; = 33(.) See Additional Answers. Mini-Assessment. Does the table represent a linear or an eponential function? Eplain. 3 5 The function is eponential. As increases b, is multiplied b 3.. Evaluate the function = 5(3) for = Graph f() =.5(). Compare the graph to the graph of the parent function. Describe the domain and range of f. f() =.5() g() = The parent function is g() =. The graph of f is a vertical stretch b a factor of.5 of the graph of g. The domain is all real numbers and the range is >.. The graph represents the number of customers at a new bookstore after weeks. Book Store 5. MODELING WITH MATHEMATICS The graph represents the number of visitors to a new art galler after months. Visitors Art Galler (3, 35) 75 (, 9) 5 (, ) 5 (, ) 3 5 Month a. Write an eponential function that represents this situation. b. Approimate the number of visitors after 5 months. 5. PROBLEM SOLVING A sales report shows that 33 gas grills were purchased from a chain of hardware stores last ear. The store epects grill sales to increase % each ear. About how man grills does the store epect to sell in Year? Use an equation to justif our answer. 53. WRITING Graph the function f () =. Then graph g() = 3. How are the -intercept, domain, and range affected b the translation? 5. MAKING AN ARGUMENT Your friend sas that the table represents an eponential function because is multiplied b a constant factor. Is our friend correct? Eplain WRITING Describe the effect of a on the graph of = a when a is positive and when a is negative. 5. OPEN-ENDED Write a function whose graph is a horizontal translation of the graph of h() =. 57. USING STRUCTURE The graph of g is a translation units up and 3 units right of the graph of f () = 5. Write an equation for g. Maintaining Mathematical Proficienc 3 Chapter Eponential Functions and Sequences 5. HOW DO YOU SEE IT? The eponential function = V() represents the projected value of a stock weeks after a corporation loses an important legal battle. The graph of the function is shown. Stock price (dollars) Stock Week a. After how man weeks will the stock be worth $? b. Describe the change in the stock price from Week to Week USING GRAPHS The graph represents the eponential function f. Find f (7).. THOUGHT PROVOKING Write a function of the form = ab that represents a real-life population. Eplain the meaning of each of the constants a and b in the real-life contet.. REASONING Let f () = ab. Show that when is f ( + k) increased b a constant k, the quotient is f () alwas the same regardless of the value of.. PROBLEM SOLVING A function g models a relationship in which the dependent variable is multiplied b for ever units the independent variable increases. The value of the function at is 5. Write an equation that represents the function. 3. PROBLEM SOLVING Write an eponential function f so that the slope from the point (, f ()) to the point (, f ()) is equal to. Reviewing what ou learned in previous grades and lessons Write the percent as a decimal. (Skills Review Handbook). % 5. 35%. % 7. 5% (,.5) (, 3) (, ) Customers (, ) (, ) (, 3) (3, ) (, ) Week a. Write an eponential function that represents the number of customers. = () b. Find the number of customers after 5 weeks. customers If students need help... Resources b Chapter Practice A and Practice B Puzzle Time Student Journal Practice Differentiating the Lesson Skills Review Handbook If students got it... Resources b Chapter Enrichment and Etension Cumulative Review Start the net Section 3 Chapter