Exponential Functions. Michelle Northshield

Transcription

1 Exponential Functions Michelle Northshield Introduction The exponential unction is probably the most recognizable graph to many nonmathematicians. The typical graph o an exponential growth model shows that the values have a slow initial growth leading to a rapid increase in value. The simple doubling unction,, 8, 6,... is probably the simplest case o an exponential but they can be used to model many situations: describing growth in populations, money investments, spread o disease and decay o radioactive substances or depreciation o value. Over the past ew years, I have noticed that my students can recognize graphs and tables that model exponential growth and decay when contrasted with a linear model, but lack the understanding o the algebraic properties o the exponential unctions. By discussing the basic concept o unction, ollowed by a look at the algebraic structure o the exponential unction, this unit will represent a dierent approach. We will examine the set o exponential unctions and discuss their algebraic properties. By developing this unit, I hope to develop a student population with a sense o understanding or the equation o the exponential unction. I will be presenting this unit to a irst year algebra class. Concept o Function in High School Mathematics Curriculum This unit will be presented to students in an introductory algebra class; as such these students have had a ew experiences with the concept o a unction. In this course, I will have exposed my students to the linear unction or ( x) mx b. They will have made y many tables and graphs and discussed the concept o a constant rate o change. They x will have been introduced to the concept o a unction as a machine that assigns one number usually called the input with a unique value known as the output. An Overview o the High School and Students My unit will be aimed at my algebra I class or the 0-0 school year. This is a college prep non honors introductory algebra course. At this time, my roster indicates ninth grade students. Although Conrad is a sixth through twelth grade building, a large number o our reshman class consists o students rom other public middle schools, private schools and parochial schools. Some o these students may have been introduced

2 to exponential growth problems through the integrated curriculum while other students will not have been exposed to unctions at all. This unit will help develop a better understanding o the structure o exponential unctions and their characteristics. Conrad Schools o Science is a magnet school under the umbrella o the Red Clay Consolidated School District. It is located in the Wilmington area and draws students rom New Castle to Newark. Students apply to Conrad by application, essay, interview, and science project. Each student receives a score on each admission measurement and all students meeting a certain percentage o a given total score have their name placed into a lottery system. As a result o the lottery system, we have a somewhat diverse group o academically perorming students, meaning, that students with A averages have an equally likely chance in the lottery as students with a C average. Our school is operated on an A B block schedule, which means I will have approximately 8 minutes with each class every other day. Each grade level houses approximately 60 students and I will have one section o college preparatory algebra I with approximately students in the class. Conrad has approximate % minority population with.6% o our students qualiying or ree or reduced lunch services. About 9% o our current juniors passed the state assessment in mathematics. This data was collected rom School Proiles and I have provided a snapshot o the data in igure and. Fall Enrollment Enrollment by Race/Ethnicity Other Student Characteristics 0-0- Grade Grade Grade Grade Grade 0 69 Grade 9 9 Grade 6 Total,08, 0-0- Arican American 0.%.% American Indian 0.% 0.% Asian.7%.9% Hispanic or Latino 0.% 8.9% White 66.0% 6.% Multi-Racial 0.6% 0.8% 0-0- English Language Learner 7.9% 7.7% Low Income 6.7%.6% Special Education.0%.9% Enrolled Full Year 98.9% N/A Figure

3 Figure Unit Objectives Students will be able to deine a unction using inputs and outputs Students will be able to deine the domain and range o a given unction Students will be able to describe a unction as either increasing, decreasing or constant Students will be able to determine i a given unction is injective or -, thus creating an inverse Students will be able to demonstrate their knowledge o the properties o the addition and multiplication operations with real numbers; commutativity, associativity, distributivity, identity element and inverse element. Students will be able to deine an exponent and use exponential notation to evaluate expressions Students will be able to simpliy expressions using the properties o exponents. Students will be able to deine an exponential unction with an initial value and a growth or decay rate Students will be able to observe using graphs, tables and equations the basic characteristics o an exponential unction. The Mathematics o the Unit I will begin the unit with the review o the deinition o a unction. In almost every aspect o our lives, we ind examples o situations where one quantity depends on another. For example, the wind chill depends on the speed o the wind; the area o a circle depends on its radius; and the amount earned by your investment depends on the interest rate. A unction, F, rom a set A to a set B is a relationship (rule) that associates each input value rom set A a unique output value rom set B. An example o a relation that is a unction is F = {(,), (,7), (,-), (9,), (8,), (6,)}. A unction can also be deined as a rule that assigns exactly one element in set B to each element in set A. Set A, the set o all irst coordinates, or the input values, is called the domain o the unction. Set B, the set o all second coordinates, or the output values, is called the range. In the

4 unction F above, the domain is the set o irst elements: Domain = {,,, 9, 8, 6} and the range is the set o second elements: Range = {, 7,,, }. The unction F can be represented in a mapping diagram as demonstrated below in igure. Figure The characteristic that distinguishes a unction rom any other relation (rule) is that there is only one output (y-value) or each input (x-value). Ater reviewing the concept o a unction, I will review the basic properties o exponents. My students will recall that the purpose o an exponent is to write repeated multiplication in short hand script. For example we know that where is deined to be the base and is deined to be the exponent. This type o short hand script makes writing large numbers easier, or example it is easier to record than it is to record the numerical value 9,67,,60,6. I can expand our deinition o an exponent to be The basic properties o exponents are Property Example ( a b m a b m m 0

5 Ater reviewing the concept o a unction and the properties o exponents, I will show the video The Greatest Shortcoming o the Human Race Is Our Inability to Understand the Exponential Function by Al Barlett on Growth and Sustainability. This video shows the importance o understanding mathematics in our daily lives. Ater showing the video, I would ask the students what inormation they could tell me relating to the video. I would create a list o things they know, things they do not know, and things they want to learn more about. Ater the video, I would deine the exponential unction. In this unit, my set will be deined as the set o all exponential unctions o the orm. where b is deined to be the base and x can represent any real number. In this deinition, x, the variable is now the exponent and the base is a ixed number. This idea will be new to my students, up until now, the base has been the variable, x in most cases, and the exponent was a ixed number. Beore I proceed into this topic, I would address the restrictions on b. I will place the restriction that b, or i b = 0 I would have and i b = I would have and these are constant unctions. They will not have all the same properties o general exponential unctions. We deine the domain o the exponential unction to be the set o real numbers, and the range o the exponential unction to be the set o positive numbers. Identiying the domain (set o inputs) and the range (set o outputs) can be shown by the graph o an exponential unction as shown in igure. For example value (input ) (output) Graph -9 9 ( 9) ( ). 8 - ( ). - ( ).0 0 (0) () () () 8 Figure () 6

6 From the graph o in igure, I can identiy the domain as (-, ) and the range as (0, ). I can also note that this graph starts with a slow initial growth leading to a rapid increase in value. This graph makes sense based on our inputs and our outputs. I we think about, I would expect the process to be the same with larger output values. I would expect the graph to have similarities to the graph o Thus I can conclude that I have a general idea o the shape o the exponential unction or base values that are positive whole numbers. What would the graph look like i b <? Remember the initial deinition claimed that b must be greater than zero. Would the graph have similarities to the graph above? Let s look at x x ) (, as pictured below in igure. Now that I have developed the irst graph o the exponential I will look at a smaller table to generate the graph. Ater I generate the graph or an exponential with a base value less than one, I will look or any similarities and dierences in the graphs. value (input ) (output) Graph ) ( - ) ( - ) ( 0 (0) 0 ) ( ) ( 8 ) ( As I examine both graphs we can conclude the common characteristics: The graph crosses the y-axis at (0,) or the y intercept is (0,) The domain is the set o all reals numbers (-, ) The range is the set o positive real numbers (0, ) Figure

7 The graph represents a unction, each input value is associated with a unique output value The graph is injective or -, which indicates there is an inverse When b >, the graph increases rom let to right When 0 < b <, the graph decreases rom let to right The graph is asymptotic to the x-axis which means the graph gets very, very close to the x-axis but does not touch it or cross it. Next I would want to examine what happens to the graph and equation i I multiply the unction by a constant number, k, where k > 0. The new unction would be deined as. Will the graph still retain the above characteristics? Let s examine the unctions and where k = in igure 6 and 7. x x ) ( value (input) (output) value (input) (output) ) ( ) ( - ) ( - 0 ) ( - ) ( - 0 ) ( 0 (0) 0 0 (0) 0 0 () () 0 () () 0 8 () 8 8 () Figure 6

8 As I examine both graphs I can conclude the common characteristics are relatively the same except or the y intercept and our output values have been multiplied by. The graph crosses the y-axis at (0, ) or the y intercept is (0, ). I can conclude that the initial y value o has been multiplied by the same constant value o, or that k has become our initial value The domain is still the set o all reals numbers (-, ) The range is still the set o positive real numbers (0, ) The graph still represents a unction, where each input value is associated with a unique output value The graph is still injective or - When b >, the graph increases rom let to right at a aster rate (ive times aster) thus the graph has been stretched When 0 < b <, the graph decreases rom let to right at a aster rate (ive times aster) thus the graph has been stretched The graph is asymptotic to the x-axis which means the graph gets very, very close to the x-axis but does not touch it or cross it. Thus, I can conclude that multiplying an exponential unction by a constant, k > 0 does not change the basic characteristics o the graph. What happens i I multiply by a constant k < 0? A quick look at and graphs in igure 8. Figure 7 and x ( x) yields the ollowing tables

9 ( x) x value (input) - - ( ) ( ) (output) value (input) (0) 0 () 0 () 0 () (output) ( ) ( ) (0) () () () Figure 8 Multiplying the exponential unction by a constant, k < 0, changed some o the characteristics o the exponential unction. The graph now crosses the y-axis at (0, -) or the y intercept is (0, -). I can conclude that the initial y value o has

10 been multiplied by the same constant value o -, or that k has become the initial value The domain is still the set o all reals numbers (-, ) The range is now the set o all negative real numbers (-, 0) The graph still represents a unction, where each input value is associated with a unique output value The graph is still injective or - When b >, the graph decreases rom let to right at a aster rate (ive times aster) thus the graph has been stretched When 0 < b <, the graph increases rom let to right at a aster rate (ive times aster) thus the graph has been stretched The graph is still asymptotic to the x-axis which means the graph gets very, very close to the x-axis but does not touch it or cross it. At this time, I can now expand my set to include all exponential unctions o the orm where k represents the initial value, or the multiplier. In the irst two examples our multiplier, k, would be, because Now let s look at what happens i I try to multiply two exponential unctions. Will the result still be an exponential unction? To start I will look at multiplying exponential unctions with no constant multiplier. Let s look at x = ( ) using the properties o exponents = which represents an exponential unction with base. So it will have all o the characteristics o a general exponential graph. What happens i I include some constant multipliers? Let s look at = = by applying the associative property o multiplication = by applying the commutative property o multiplication by applying the associative property o multiplication x = ( 0) ( ) by applying properties o exponents = which represents an exponential unction with base 6 and an initial value or y intercept o 0. Thus the set o exponential unctions o the orm is closed

11 under the operation o multiplication. In order to determine the algebraic structure o my set, I would need to check to see i my set has the property o associativity and since the operation is multiplication and multiplication is associative, then my set has associativity. I could make the same argument in regards to commutativity. Although I would like to deine the identity unction o my set to be, I realize that although this unction looks like it has the same algebraic structure we see that its graph is merely the horizontal line y = and does not have most o the characteristics o an exponential unction. Hence my set o exponential unctions does not have an identity unction, and without an identity unction I cannot discuss the concept o an inverse. So my set represents a commutative semigroup as deined in Abstract Algebra by Herstein. An Overview o the Activities Over the past ew years I have noticed that high school students enjoy making things, so I have read a lot about the interactive notebook and am using these approaches in my algebra class this year. It may be worth noting that my calculus students have commented why don t we get to do some o these? I ind it interesting that the smallest shit in approach can lead to such peaked interest. The irst item to be discussed in this unit is the concept o a unction. Students will cut out and decorate a unction versus non unction oldable to be placed in their notebook. This oldable will contain inormation regarding the deinition o a unction, a mapping diagram, a table and a graph. It will have examples showing what a unction would look like, contrasted with non-unction examples. A sample is given below in igure 9. Not Functions I it is not a unction, that means that an input value is paired with more than one output value Input is the domain Output is the range A unction means that each input value has exactly one output value. Input is the domain Output is the range Functions Domain/Range

12 input output input output Mapping Diagram Domain/Range Domain,, Range,,6 6 Domain,, Range,,6 6 Mapping Diagram Domain/Range Table Domain/Range Input Output X Y 7 9 Domain, 9 Range,,7 Input Output X Y 0 6 Domain,,6 Range 0,, Table Domain/Range Graph Domain/Range Vertical Line Test Fails Domain Range Domain Range Graph Domain/Range Vertical Line Test Passes Figure 9. Next I would review the basic properties o exponents. Again I would have the kids make a oldable or chart to place in their notebook Exponents rules and properties Rule name Rule Example Product rules a n a m = a n+m = + = 8

13 a n b n = (a b) n = ( ) = Quotient rules a n / a m = a n-m / = - = a n / b n = (a / b) n / = (/) = 8 (b n ) m = b n m ( ) = = 6 Power rules bn m = b(n m ) = ( ) = m (b n ) = b n/m ( 6 ) = 6/ = 8 b /n = n b 8 / = 8 = Negative exponents b -n = / b n - = / = 0. Zero rules One rules b 0 = 0 = 0 n = 0, or n>0 0 = 0 b = b = n = = Ater reviewing the concept o a unction and the properties o exponents, I will deine the exponential unction. At this point I would have my students a typical calculator exploration o the unction. We would graph a series o unctions that represent growth at dierent rates, and a series o unctions that represent decay at dierent rates. For each graph I would explore the characteristics o the graph, pointing out the y intercept, whether the unction was increasing or decreasing, the domain, the range, and that the graph is asymptotic to the x-axis. We would a chart similar to the one below. Equation Graph Table Description X Y Increasing 0 y intercept: (0,) Domain (-, )

14 8 - / - /8 Range (0, ) Asymptotic to x-axis Once the students had a chance to examine the graphs o each unction, I would review the properties o operations on real numbers such as Commutative property: When two numbers are multiplied together, the product is the same regardless o the order o the multiplicands. For example * = * Associative Property: When three or more numbers are multiplied, the product is the same regardless o the grouping o the actors. For example ( * ) * = * ( * ) And I would ask my students to ind the product o properties. using the above My last task or the students would be to cut and sort the activity cards below and to ill in the missing blanks. I am providing the answer key. The items that are in bold, would be the blanks that the students would need to add. Equation Table Graph Description

15 X Y X Y - 7 / 0 /9 X Y X Y X Y X Y Increasing y intercept (0,) Decreasing y intercept (0,) Decreasing y intercept (0,-) Increasing y intercept (0,) Decreasing y intercept (0,0) Decreasing y intercept (0,0) Appendix A: Standards The ollowing list o Common Core State Standards in mathematics will be directly addressed in this unit Primary Content Standards

16 CCSS.Math.Content.8.F.A. Understand that a unction is a rule that assigns to each input exactly one output. The graph o a unction is the set o ordered pairs consisting o an input and the corresponding output. CCSS.Math.Content.8.F.A. Compare properties o two unctions each represented in a dierent way (algebraically, graphically, numerically in tables, or by verbal descriptions). CCSS.Math.Content.8.F.A. Interpret the equation y = mx + b as deining a linear unction, whose graph is a straight line; give examples o unctions that are not linear, or example exponential unctions. 6 CCSS.Math.Content.8.EE.A. Know and apply the properties o integer exponents to generate equivalent numerical expressions. For example, - = - = / = /7. 7 CCSS.Math.Content.8.F.B. Describe qualitatively the unctional relationship between two quantities by analyzing a graph (e.g., where the unction is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative eatures o a unction that has been described verbally. 8 Secondary Content Standards CCSS.Math.Content.HSA.SSE.A..a Interpret parts o an expression, such as terms, actors, and coeicients. 9 CCSS.Math.Content.HSA.SSE.A..b Interpret complicated expressions by viewing one or more o their parts as a single entity. 0 CCSS.Math.Content.HSA.SSE.A. Use the structure o an expression to identiy ways to rewrite it. CCSS.Math.Content.HSA.SSE.B..c Use the properties o exponents to transorm expressions or exponential unctions. CCSS.Math.Content.HSF.IF.B. For a unction that models a relationship between two quantities, interpret key eatures o graphs and tables in terms o the quantities, and sketch graphs showing key eatures. CCSS.Math.Content.HSF.IF.B. Relate the domain o a unction to its graph and, where applicable, to the quantitative relationship it describes. CCSS.Math.Content.HSF.IF.C.7.e Graph exponential unctions, showing intercepts and end behavior. CCSS.Math.Content.HSF-IF.. Functions, Interpreting Functions: Interpret unctions that arise in applications in terms o the context. Relate the domain o a

17 unction to its graph and, where applicable, to the quantitative relationship it describes. 6 Primary Mathematical Practice Standards While I believe all eight mathematical practices are applicable to this unit, these ive are highlighted throughout the unit: MP-. Make sense o problems and persevere in solving them. 7 MP-. Reason abstractly and quantitatively. 8 MP-. Model with mathematics. 9 MP-6. Attend to precision. 0 MP-7. Look or and make use o structure. Bibliography ""The Greatest Shortcoming o the Human Race Is Our Inability to Understand the Exponential Function". Al Barlett on Growth and Sustainability /r/futurology." Reddit. Accessed January 6, 0. _o_the_human_race_is_our/. This site contains a video on exponential unctions Herstein, I. N. Abstract Algebra. New York: Macmillan, 986. A book on Abstract Alegbra useul or identiying dierent types o group structures "Introduction To Functions." Webalg Pre-Calculus. Accessed January 6, 0. A good reerence or deinitions and properties o unctions "School Proiles - School Proiles." School Proiles - School Proiles. Accessed January 6, &districtCode=. This website provides data and statistics on Delaware High Schools relating to enrollment and test scores "Standards or Mathematical Practice." Home. Accessed January 6, 0. This website provides the common core state standards and mathematical processes

18 Wagner, Patty. "Exploring Exponential Functions." Exploring Exponential Functions. Accessed January 6, 0. ml. This website provides many examples o working with exponential unctions. Weingarden, Michael. "Too Hot To Handle, Too Cold To Enjoy." Too Hot To Handle, Too Cold To Enjoy. Accessed January 6, 0. This website provides lessons on exponential unctions through laboratory exercises Notes rictcode=. _human_race_is_our/.html Herstein, I. N. Abstract Algebra, Ibid,8 6 Ibid,8 7 Ibid,8 8 Ibid,8 9 Ibid. 0 Ibid. Ibid. Ibid. Ibid. Ibid. Ibid. 6 Ibid. 7 Ibid. 8 Ibid. 9 Ibid. 0 Ibid.

19 Curriculum Unit Title Exponential Functions Author Michelle Northshield KEY LEARNING, ENDURING UNDERSTANDING, ETC. Exponential unctions and their characteristics ESSENTIAL QUESTION(S) or the UNIT What are the characteristics o an exponential unction? CONCEPT A CONCEPT B CONCEPT C Functions Exponential Functions Algebraic Structure ESSENTIAL QUESTIONS A ESSENTIAL QUESTIONS B ESSENTIAL QUESTIONS C What are the characteristics o a unction? What are the characteristics o an exponential unction? What does the product o two exponential unctions look like? VOCABULARY A VOCABULARY B VOCABULARY C mapping diagram, input, output, domain, range, vertical line test ADDITIONAL INFORMATION/MATERIAL/TEXT/FILM/RESOURCES ""The greatest shortcoming o the human race is our inability to understand the exponential unction". Al Barlett on Growth and Sustainability exponential, initial amount increasing, decreasing, domain, range, asymptotes associative, commutative, (accessed October, 0).