Unit 1 Functions Overview: Power, Polynomial, Rational, Exponential, and Logarithmic

Transcription

1 Number f : 39 9/6/16 10/28/16 Unit Gals Stage 1 Unit Descriptin: In this unit, students extend their knwledge f functins and mdels. Students analyze functins and their prperties including dmain and range, cntinuity, symmetry, increasing and decreasing, extrema, and bundedness/limits. Parent functins are reviewed and applied t piece-wise functins. Students cmbine functins algebraically and determine inverses f nn-linear functins. Students review and extend their knwledge f the algebra and gemetry f transfrmatins. Pwer functins are cnnected t direct and inverse prprtinal relatinships. Students extend their knwledge f expnential and lgarithmic functins. The prperties f lgarithms are applied t slutins f expnential and lgarithmic equatins. The change f base rule is used t evaluate lgarithmic expressins. Real-wrld data is mdeled by expnential and lgarithmic functins. Materials: Graphing calculatrs, Desms Standards fr Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Make sense f prblems and persevere in slving them. Reasn abstractly and quantitatively. Cnstruct viable arguments and critique the reasning f thers. Mdel with mathematics. Use apprpriate tls strategically. Attend t precisin. Lk fr and make use f structure. Lk fr and express regularity in repeated reasning. Standards fr Mathematical Cntent Clusters Addressed Transfer Gals Students will be able t independently use their learning t Make sense f never-befre-seen prblems and persevere in slving them. Cnstruct viable arguments and critique the reasning f thers. Making Meaning UNDERSTANDINGS Students will understand that In the functin f(x) = a f(x b) + c, a, b, and c have the same effect n the shape f the graph in every graph family. Functins can be classified int different families, each with its wn characteristics. Representing a functin in different ways des nt change the functin itself, althugh different representatins may highlight different characteristics f that functin. Functins can be cmbined by adding, subtracting, multiplying, dividing and cmpsing them. Functins smetimes have inverses. Inverse functins und each ther. KNOWLEDGE Students will knw The basic shapes f parent functins. Hw the parameters a, b, and c, affect the graph f f(x) =a f(x b) + c. Ratinal expressins frm a system analgus t the ratinal numbers, clsed under additin, subtractin, multiplicatin, and divisin by a dn-zer ratinal expressin. Acquisitin ESSENTIAL QUESTIONS Students will keep cnsidering In the functin f(x) =a f(x b) +c, hw d a, b, and c effect the shape f the graph? Is this the same fr all graph families? What is the cnnectin between the characteristics f the graph f a functin and its equatin? Hw can graphing data help t analyze a real-wrld situatin and help in decisin making? SKILLS Students will be skilled at and/r be able t State the dmain f a functin. Identify even and dd functins. Use limits t determine the cntinuity f a functin and t describe a functin s end behavir. Describe the intervals n which a functin is increasing, decreasing, r remaining cnstant Psted 6/24/16

2 A-SSE.A Interpret the structure f expressins. A-APR.D Rewrite ratinal expressins. A-CED.A Create equatins that describe numbers r relatinships. F-IF.B Interpret functins that arise in applicatins in terms f the cntext. F-IF.C Analyze functins using different representatins. F-BF.A Build a functin that mdels a relatinship between tw quantities. F-BF.B Build new functins frm existing functins. Unit Gals Stage 1 Asympttes represent cnstraints n functins with pssible real-wrld relevance. The dmain and range f a functin can be determined algebraically and/r graphically. The best way t express a functin t highlight key features f a graph. Inverse functins und each ther. Expnential and lgarithmic functins are inverses f each ther. The Remainder and Factr Therems. The Fundamental Therem f Algebra. The Cnjugate Rt Therem. Determine the average rate f change f a functin r an interval f a functin. Identify, graph, analyze, and describe functin families. Identify and graph transfrmed functins. Perfrm peratins with and cmpsitins f functins. Determine whether a functin has an inverse functin. Find inverse functins algebraically and graphically. Verify algebraically and graphically that ne functin is the inverse f anther. Slve radical equatins. Divide plynmials using bth lng and synthetic divisin. Use the Remainder and Factr Therems. Find real and cmplex zers f plynmial functins. Slve ratinal equatins. Slve plynmial and ratinal inequalities. Evaluate, analyze and graph expnential functins. Slve prblems invlving expnential grwth and decay. Evaluate expressins invlving lgarithms. Apply prperties f lgarithms. Slve expnential and lgarithmic equatins. Mdel real-wrld data using nn-linear regressin Psted 6/24/16

3 Assessed Grade Level Standards Standards fr Mathematical Practice SMP 1 Make sense f prblems and persevere in slving them. SMP 2 Reasn abstractly and quantitatively. SMP 3 Cnstruct viable arguments and critique the reasning f thers. SMP 4 Mdel with mathematics. SMP 5 Use apprpriate tls strategically. SMP 6 Attend t precisin. SMP 7 Lk fr and make use f structure. SMP 8 Lk fr and express regularity in repeated reasning. Standards fr Mathematical Cntent A-SSE.A Interpret the structure f expressins. A-SSE.1 Interpret expressins that represent a quantity in terms f its cntext. b. Interpret cmplicated expressins by viewing ne r mre f their parts as a single entity. Fr example, interpret P(1+r)n as the prduct f P and a factr nt depending n P. A-SSE.2 Use the structure f an expressin t identify ways t rewrite it. Fr example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recgnizing it as a difference f squares that can be factred as (x 2 y 2 )(x 2 + y 2 ). A-APR.D Rewrite ratinal expressins. A-APR.6 Rewrite simple ratinal expressins in different frms; write a(x)/b(x) in the frm q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are plynmials with the degree f r(x) less than the degree f b(x), using inspectin, lng divisin, r, fr the mre cmplicated examples, a cmputer algebra system. A-APR.7 (+) Understand that ratinal expressins frm a system analgus t the ratinal numbers, clsed under additin, subtractin, multiplicatin, and divisin by a nnzer ratinal expressin; add, subtract, multiply, and divide ratinal expressins. A-CED.A Create equatins that describe numbers r relatinships. A-CED.1 Create equatins and inequalities in ne variable including nes with abslute value and use them t slve prblems. Include equatins arising frm linear and quadratic functins, and simple ratinal and expnential functins. CA A-CED.2 Create equatins in tw r mre variables t represent relatinships between quantities; graph equatins n crdinate axes with labels and scales. A-CED.3 Represent cnstraints by equatins r inequalities, and by systems f equatins and/r inequalities, and interpret slutins as viable r nnviable ptins in a mdeling cntext. Fr example, represent inequalities describing nutritinal and cst cnstraints n cmbinatins f different fds. A-CED.4 Rearrange frmulas t highlight a quantity f interest, using the same reasning as in slving equatins. Fr example, rearrange Ohm s law V = IR t highlight resistance R*. F-IF.B Interpret functins that arise in applicatins in terms f the cntext. F-IF.4 Fr a functin that mdels a relatinship between tw quantities, interpret key features f graphs and tables in terms f the quantities, and sketch graphs shwing key features given a verbal descriptin f the relatinship. Key features include: intercepts; intervals where the functin is increasing, decreasing, psitive, r negative; relative maximums and minimums; symmetries; end behavir; and peridicity. F-IF.5 Relate the dmain f a functin t its graph and, where applicable, t the quantitative relatinship it describes. Fr example, if the functin h(n) gives the number f persn-hurs it takes t assemble n engines in a factry, then the psitive integers wuld be an apprpriate dmain fr the functin Psted 6/24/16

4 F-IF.C F-IF.7 F-BF.A F-BF.1 F-BF.B F-BF.3 F-BF.4 F-BF.5 Assessed Grade Level Standards Analyze functins using different representatins. Graph functins expressed symblically and shw key features f the graph, by hand in simple cases and using technlgy fr mre cmplicated cases. d. (+) Graph ratinal functins, identifying zers and asympttes when suitable factrizatins are available, and shwing end behavir. e. Graph expnential and lgarithmic functins, shwing intercepts and end behavir, and trignmetric functins, shwing perid, midline, and amplitude. Build a functin that mdels a relatinship between tw quantities. Write a functin that describes a relatinship between tw quantities. a. Determine an explicit expressin, a recursive prcess, r steps fr calculatin frm a cntext. b. Cmbine standard functin types using arithmetic peratins. Fr example, build a functin that mdels the temperature f a cling bdy by adding a cnstant functin t a decaying expnential, and relate these functins t the mdel. c. (+) Cmpse functins. Fr example, if T(y) is the temperature in the atmsphere as a functin f height, and h(t) is the height f a weather balln as a functin f time, then T(h(t)) is the temperature at the lcatin f the weather balln as a functin f time. Build new functins frm existing functins. Identify the effect n the graph f replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) fr specific values f k (bth psitive and negative); find the value f k given the graphs. Experiment with cases and illustrate an explanatin f the effects n the graph using technlgy. Include recgnizing even and dd functins frm their graphs and algebraic expressins fr them. Find inverse functins. b. (+) Verify by cmpsitin that ne functin is the inverse f anther. c. (+) Read values f an inverse functin frm a graph r a table, given that the functin has an inverse. d. (+) Prduce an invertible functin frm a nn-invertible functin by restricting the dmain. (+) Understand the inverse relatinship between expnents and lgarithms and use this relatinship t slve prblems invlving lgarithms and expnents. Key: [m] = majr clusters; [s] = supprting clusters; [a] = additinal clusters Indicates a mdeling standard linking mathematics t everyday life, wrk, and decisin-making (+) Indicates standards included in curses intended t lead int a furth year f mathematics. CA Indicates a Califrnia-nly standard Psted 6/24/16

5 Assessment Evidence Evidence f Stage 2 Unit Assessment Students will cmplete selected respnse and cnstructed respnse items t indicate level f mastery/understanding f the unit standards as utlined in this guide. A-SSE.A Interpret the structure f expressins. Students will use the structure f an expressin t identify ways t rewrite the expressin. Students will interpret parts f an expressin in terms f a cntext. A-APR.D Rewrite ratinal expressins. Students will rewrite ratinal expressins in different frms. Students will use synthetic r lng divisin t rewrite a ratinal expressin f the frm a(x)/b(x) int the frm q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are plynmials with the degree f r(x) less than the degree f b(x) and b(x) is nt equal t 0. Students will add, subtract, and multiply ratinal expressins. A-CED.A Create equatins that describe numbers r relatinships. Students will create equatins and inequalities in ne r mre variables t represent relatinships in cntext. Students will graph equatins n crdinate axes with labels and scales. Students will represent cnstraints by equatins r inequalities and interpret slutins as viable r nnviable ptins in a real-wrld cntext. Students will rearrange frmulas t highlight a quantity f interest F-IF.B Interpret functins that arise in applicatins in terms f the cntext. Students will interpret key features f graphs and tables in terms f a cntext. Students will sketch graphs shwing key features given a verbal descriptin f a relatinship between tw quantities. Students will relate the dmain f a functin t its graph and, where applicable, t the quantitative relatinship it describes F-IF.C Analyze functins using different representatins. Students will graph functins expressed symblically and shw key features f the graph, by hand in simple cases and using technlgy fr mre cmplicated cases. Students will graph ratinal functins, identifying zers and asympttes when suitable factrizatins are available, and shwing end behavir. Students will graph expnential and lgarithmic functins, shwing intercepts and end behavir. Students will graph trignmetric functins, shwing perid, midline, and amplitude. F-BF.A Build a functin that mdels a relatinship between tw quantities. Students will determine an explicit expressin t describe a relatin between tw quantities given a cntext. Student will cmbine functins using arithmetic peratins. Students will cmpse functins Psted 6/24/16

6 Evidence f Stage 2 F-BF.B Build new functins frm existing functins. Students will Identify the effect n a graph caused when f(x) is replaced by f(x) + k, k f(x), f(kx), and f(x + k) fr specific values f k (bth psitive and negative. Students will find the value f k which transfrms the graph f a parent functin int the graph f a given functin. Students will find the inverse f a functin if that functin has an inverse. Students will verify that tw functins are inverses graphically, by cmpsitin, r by slving fr an inverse. Students will restrict the dmain f a nn-invertible functin t prduce an inverse functin. Students will use the inverse relatinship between expnents and lgarithms t slve prblems invlving lgarithms and expnents. Fr selected cntent, students will need t Slve cmplex prblems in pure and applied mathematics, making prductive use f knwledge and prblem slving strategies. Clearly and precisely cnstruct viable arguments t supprt their wn reasning and critique the reasning f thers. Other Evidence Frmative Assessment Opprtunities Opening Tasks Infrmal teacher bservatins Checking fr understanding using active participatin strategies Exit slips/summaries Mdeling Lessns (SMP 4) Tasks Frmative Assessment Lessns (FAL) Quizzes / Chapter Tests Big Ideas Math Perfrmance Tasks SBAC Interim Assessment Blcks Access Using Frmative Assessment fr Differentiatin fr suggestins. Lcated n the LBUSD website M Mathematics Curriculum Dcuments Psted 6/24/16

7 2 I will review the dmains, ranges, names and graphs f cmmn functins in the Opening Task. Suggested Sequence f Key Events and Instructin Expectatins OPENING TASK Match the Functins This Opening Task is a review f functins that were studied in Algebra 2. Begin the activity with a rund f graph aerbics. Students stand up. After the teacher calls ut an equatin, the students mdel that equatin with their arms. Fr further challenge, extend the activity t include transfrmatins and functins ther than the nes n the Graph Aerbics sheet. This is an activity that can be revisited each time a new functin is intrduced. After the aerbics review, prvide each grup (3 5) f students with a deck f Match the Functin cards. Students match the graph f a functin with its name, range and dmain. This game can be played with the rules as given r the rules may be changed t resemble Cncentratin, Fish, r any ther simple, familiar game. Play several runds f the game, changing the student grups between each rund. Students can be asked t add their wn transfrmed functins t the cards. (Activities and Lessns) Prcedural Skills and Fluency: Graph Aerbics Match the Functins Game Instructins Match the Functins Cards 5-7 I will differentiate between functin families by Identifying and evaluating functins. Identifying functin dmains. Using interval and set-builder ntatin t describe a dmain. Graphing functins t estimate functin values and find dmains, ranges, y-intercepts, and zers. Identifying functins as even r dd. Using limits t describe the end behavir f a functin. Applying the Intermediate Value Therem t cntinuus functins. Identifying pints f cntinuity. Determining key features f the graph f a functin: maxima and/r minima; intervals n which functins are increasing, cnstant r decreasing; and average rate f change ver an interval. (SMP 7) Lessn 1.1 Lessn 1.2 Lessn 1.3 Lessn 1.4 Lessn 1.5 Applicatin: Linear Functin: Stack f Cups Psted 6/24/16

8 Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Identifying, describing and graphing parent functins and transfrmatins f thse functins. (SMP 8) Answering questins such as Given a functin, hw des changing the value f x effect the value f y? What are the basic characteristics f each parent functin? Hw d the graphs f f(x) + k, k f(x), f(kx), and f(x + k) cmpare t the graph f the parent functin f(x)? What are ways t tell whether a functin is even r dd? 3-4 I will cmbine functins by Adding, subtracting, multiplying and dividing functins. Cmpsing functins. (SMP 7) Finding the inverse f an invertible functin algebraically and graphically. Limiting the dmain f a given functin in rder t find an inverse functin. (SMP 6) Apprximating the rate f change f a functin at a pint r ver a given interval. Answering questins such as Hw des cmbining and/r cmpsing functins affect their dmains? Is the cmpsitin f tw functins cmmutative? Always? Smetimes? Never? What is the advantage f knwing a functin s inverse? Hw d yu knw when tw functins are inverses? Lessn 1.6 Lessn 1.7 Cnnect t AP Calculus: Rate f Change at a Pint Cnceptual Understanding: Desms: Cmpsing Functins Explratin Applicatin: Flu n Campus Temperature Cnversins Temperatures in Degrees Fahrenheit and Celsius 2-3 I will identify the characteristics f pwer, radical and plynmial functins by Graphing and analyzing pwer and radical functins. Describing the dmain, range, intercepts, end behavir, cntinuity, and intervals ver which the functin is increasing r decreasing fr bth pwer and radical functins. (SMP 6) Slving radical equatins and checking fr extraneus slutins. Recgnizing and graphing plynmial functins. Lessn 2.1 Lessn 2.2 Cnceptual Understanding: FAL: Representing Plynmials Graphically Graphs f Pwer Functins Psted 6/24/16

9 Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Describing the end behavir, number f turning pints, and pssible number f zers given the leading term and degree f a plynmial. Determining the real zers f a plynmial. Recgnizing tangent pints n the graph f a plynmial. Using a graphing calculatr t mdel a functin t fit a set f data. (SMP 5) Answering questins such as Why is it imprtant t check fr extraneus slutins when yu are slving an equatin? Cmpare expnential and radical functins. What infrmatin d yu need t begin t describe the basic shape f the graph f a plynmial? What are ways t determine the zers f a plynmial? Can a plynmial functin have bth an abslute maximum and an abslute minimum? When wuld yu use these functins t mdel real-wrld situatins? Prcedural Skills and Fluency: Desms: Radical Transfrmatins Desms: Plynmial Functins: Degree, Intercepts, Extrema Applicatin: Quadratic Functin: Fence Prject 3-5 I will explre the key features f plynmial functins by Dividing plynmials using lng and synthetic divisin. (SMP 6) Using the Remainder Therem t evaluate a plynmial Using the Factr Therem t determine if a given binmial is a factr f a plynmial. Finding the real and/r cmplex zers f a plynmial functin. Using the Fundamental Therem f Algebra, alng with the degree f the plynmial, t determine the number f cmplex zers. (SMP 7) Using the Cnjugate Rt Therem t write a plynmial functin f least degree given a set f cmplex zers. (SMP 8) Finding all the cmplex zers f a plynmial given ne factr f that plynmial. Lessn 2.3 Lessn 2.4 Cnceptual Understanding: Desms: Building Plynmial Functins WODB: Plynmial Graphs Prcedural Skills and Fluency: Desms: Plynmial Functins: Degree, Intercepts, Extrema Applicatin: Quadratic Functin: Fence Prject Psted 6/24/16

10 1-2 I will analyze ratinal functins by Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Answering questins such as Can synthetic divisin always be used t divide plynmials? What is the advantage t knwing hw t divide plynmials? What infrmatin is prvided by the qutient and remainder, if there is ne, in a plynmial divisin prblem? Why des synthetic divisin wrk? What is the advantage f using lng divisin rather than synthetic divisin? May a plynmial with real cefficients have an dd number f imaginary zers? Always? Smetimes? Never? Graphing ratinal functins. Describing the dmain f a ratinal functin and finding the equatins f its vertical and hrizntal asympttes. (SMP 6) Slving a ratinal equatin and checking fr extraneus rts. Answering questins such as What purpse d asympttes serve? Hw d yu find the vertical asympttes f a ratinal functin? The hrizntal asympttes? Hw des the dmain f a ratinal functin differ frm the dmain f a plynmial functin? When d hles ccur in a ratinal functin? What d the hles indicate? Cmpare slving a fractin prblem with slving a ratinal equatin. When culd yu use a ratinal functin t mdel a realwrld situatin? Intrductin t Plynmials Cllege Fund 3-Act Lessn: Where Wuld the Angry Birds Have Landed? Angry Bird Resurce: Angry Birds with Grid Lessn 2.5 Cnceptual Understanding: Desms: Marbleslides: Ratinals Desms: Building Ratinal Functins WODB: Ratinal Graphs Prcedural Skills and Fluency: Ratinal Functins Desms: Plygraph: Ratinal Functins Applicatin: Field Trip: What s It Ging t Cst? Summer Intern MathematicsVisinPrject: The Gift Psted 6/24/16

11 1-2 I will slve plynmial and ratinal inequalities by Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) Using a sign chart t reveal where a plynmial functin is Lessn 2.6 psitive r negative. Cnnect t AP Using the leading cefficient and the degree f the Calculus: Area plynmial t determine end behavir. (SMP 7) Under a Curve Using vertical and hrizntal asympttes t begin t determine the sign changes f a ratinal functin. Graphing and shading plynmial and ratinal inequalities. Answering questins such as Explain why using a sign chart helps t slve a plynmial r a ratinal inequality. T slve fr the zers f an equatin, may we raise bth sides f a radical equatin t the same pwer r multiply bth sides f a ratinal equatin by the denminatr? Prcedural Skills and Fluency: Desms: Slving Plynmial Inequalities 2-3 I will identify the characteristics f and the relatinship between expnential and lgarithmic functins by Evaluating and graphing expnential and lgarithmic functins. Describing the dmain, range, intercepts, asympttes, end behavir and intervals ver which the expnential r lgarithmic functin is increasing r decreasing. (SMP 6) Describing a transfrmatin given an expnential r lgarithmic functin in the frm f(x) = af(x-b)+c where a, b, and c are real numbers. Differentiating between expnential grwth and decay. Slving prblems using cmpund interest and cntinuusly cmpunding interest. Deriving a lgarithmic functin frm an expnential functin. (SMP 3) Evaluating lgarithms. Differentiate between cmmn and natural lgarithms. Answering questins such as Cmpare transfrming an expnential r lgarithmic functin, f(x), t f(x) + k, k f(x), f(kx), r f(x + k) t transfrming any ther functin? What is the difference between simple, cmpund and cntinuus interest? Which is a better investment? Lessn 3.1 Lessn 3.2 Cnceptual Understanding: Desms: Marbleslides: Expnentials Expnentials and Lgarithms 1 Prcedural Skills and Fluency: Desms: Plygraph: Expnential and Lgarithmic Functins Desms: Lgarithmic Graphs Desms: Transfrmatins f the Lgarithmic Functin Psted 6/24/16

12 Suggested Sequence f Key Events and Instructin Expectatins (Activities and Lessns) What are real-wrld examples f expnential grwth r decay? Cmpare expnential and pwer functins. Hw des changing the base f a lgarithmic functin change the shape f a graph? In the functin y = lg bx, can b be negative? Cmpare the dmain, range, intercepts, asympttes, and end behavirs f the graphs f expnential and lgarithmic functins. 2-3 I will slve expnential and lgarithmic equatins by Deriving and applying the prperties f lgarithms. Evaluating lgarithms. Changing the base f a given lgarithmic expressin. Rewriting expnential equatins using a cmmn base. (SMP 7) Cnverting lgarithmic equatins t expnential equatins. Using algebraic prperties t islate a target variable. Checking fr extraneus slutins. Using expnential and lgarithmic prperties t slve realwrld prblems. Answering questins such as Cmpare the prperties f lgarithms with the prperties f expnents. Hw is slving a lgarithmic equatin similar t slving an algebraic equatin? Different? Lessn 3.3 Lessn 3.4 Cnceptual Understanding: Expnential Grwth vs. Plynmial Grwth Open Middle: Lgs 2 Open Middle: Laws f Lgarithms Rewriting Equatins Expnentials and Lgarithms 2 Applicatin: Just Hw Old Is It? Expnential Mdels 1 day I will slve fr an equatin t represent a given data set by Entering data int the calculatr and graphing the data pints. Deciding n the regressin mdel t use based n the shape f the data in the scatter plt. Finding an expnential, lgarithmic r lgistical regressin equatin using the calculatr. (SMP 5) Answering questins such as Des graphing data help t analyze a situatin? Lessn 3.5 Applicatin: Illuminatins: What s the Functin Mdeling Lndn s Ppulatin Psted 6/24/16

13 Suggested Sequence f Key Events and Instructin Expectatins What pattern shuld we see in the data t knw t use linear, quadratic, cubic, expnential r lgarithmic regressin n the calculatr? Hw can yu use the regressin equatin given by the calculatr t slve real-wrld prblems? Des an expnential curve d a gd jb illustrating ppulatin grwth? Can the graph f a lgistic functin ever have any intercepts? Hw d the parameters f an expnential r lgarithmic mdel relate t the situatin being mdeled? (Activities and Lessns) I will check my understanding f functins by participating in the FAL. I will prepare fr the unit assessment n linear and quadratic functins by... FORMATIVE ASSESSMENT LESSON Incrprating the Standards fr Mathematical Practice (SMPs) alng with the cntent standards t review the unit. Unit Assessment (LBUSD Math Intranet, Assessment) Cnceptual Understanding: FAL: Representing Functins f Everyday Situatins Psted 6/24/16