MHF4U Advanced Functions - OAME
MHF4UAdvanced FunctionsUniversity Preparation
Advanced Functions: Content and Reporting TargetsMathematical Processes across all strands: Problem Solving, Reasoning and Proving, Reflecting, Selecting Tools and ComputationalStrategies, Connecting, Representing, and CommunicatingIntroductoryUnit Sketch graphsfromdescriptionsof a set ofproperties,from adescription ofa scenario,using priorknowledge of:FunctionnotationVertical linetest Keyproperties offunctions:- average rateof change- instantaneous rate ofchange- zeros- y-intercept/initialcondition- domain andrangeInversefunctionsTransformations offunctionsDifferencetablesMFH4U – OverviewUnit 1 PolynomialFunctions –Characteristicsembed averageandinstantaneousrates of changeSolvingEquations –embed intervalsof increase/decreaseTransformationsExplore endbehavioursEmbedinequalitiesUnit 2 RationalFunctionsEmbedaverage andinstantaneousrates haviouraroundasymptotesUnit 3 Radianmeasure andgraphingprimary andtheirreciprocaltrigfunctions(using keyproperties)Square scaleRate ofchangeUnit 4 emsIdentitiestransformationsUnit 5 taneous rates tionsUnit 6 Consolidatecharacteristicsof functions(compare andcontrast)Compositionof functionsCompoundfunctionsGeneralizefunctions1
RationaleInclusion of the Introductory unit: This introductory unit is not to be presented as a review unit. Rather, build conceptual understanding of general functionsbased on key concepts previously studied.Build a framework of key concepts that can be applied to all functions e.g., average rate of change, intervals ofincrease/decrease, domain/range, zerosEmbedding ‘average and instantaneous rates of change’ with each type of function: The different natures of the average and instantaneous rates of change of various types of functions can be appreciated moredeeply by linking them function by functionA gradual building of this key concept allows re-visiting it as students’ thinking maturesSplitting polynomial and rational functions into two units The framework of key concepts developed in the Introductory Unit is applied to familiar polynomial functionsConcepts of average rate of change, and end behaviours can be established with polynomial functions without the addedcomplexity of asymptotesPolynomial and rational functions have different properties.Concepts of rational functions can be built on known properties of polynomial functions.Positioning trigonometry after rational functions The tangent and reciprocal functions are introduced as applications of concepts of rational functionsSplitting trigonometry into two units The initial focus is on introducing radian measure, and revisiting all of the Grade 11 concepts using radians e.g., zeros,period, amplitude, domain/range, phase shiftRates of change are connected to graphs and numerical representations in the first trigonometric unit to consolidate graphicalpropertiesSome consolidation of basic trigonometric facts is needed before students are ready to pose and solve problems that can bemodelled by these functions.Using Unit 6 for Consolidating and Culminating Allows for a consolidation of characteristics of functions to compare and contrast the various types of functions in this courseUses the characteristics of functions as the basis for combining and composing functionsConcepts in this unit provides opportunities to generalize functions and properties of functionsMFH4U – Overview2
Advanced Functions – Planning ToolPNumber of pre-planned lessons (including instruction, diagnostic and formativeassessments, summative assessments other than summative performance tasks)JNumber of jazz days of time (instructional or assessment)TTotal number of daysSPSummative performance task (see Assessment – Grade 9 Applied)Cluster of CurriculumExpectationsUnit0 1 Revisit contexts studied in the Grade 11Functions course MCR3U usingsimplifying assumptions, addingprecision to the graphical models, anddiscussing key features of the graphsusing prior academic language (e.g.,domain, range, intervals ofincrease/decrease, intercepts, slope) and‘local maximum/minimum,’ ‘overallmaximum/minimum’Recognize that transformationspreviously applied to quadratic andtrigonometric functions also apply tolinear and exponential functions, and tofunctions in generalUse function notation to generalizerelationships between two functions thatare transformations of each other andwhose graphs are givenRepresent key properties of functionsgraphically and using function notationForm inverses of functions whosegraphs are given, and apply the verticalline test to determine whether or notthese inverses are functionsIdentify and use key features ofpolynomial functionsSolve problems using a variety of toolsand strategies related to polynomialfunctionsDetermine and interpret average andinstantaneous rates of change forpolynomial functionsOverall and SpecificExpectationsPJT61715217SPC1 identify and describe some key features ofpolynomial (linear, quadratic,trigonometric, exponential)* functions, andmake connections between the numeric,graphical, and algebraic representations ofpolynomial* functionsD1 demonstrate an understanding of averageand instantaneous** rate of change, anddetermine, numerically and graphically,and interpret the average rate of change of afunction over a given interval and theinstantaneous rate of change of a functionat a given point***reviews characteristics of functions alreadyknown**to be addressed in Units 1, 2, 3, 4, 5, and 6C1 identify and describe some key features ofpolynomial functions, and makeconnections between the numeric,graphical, and algebraic representations ofpolynomial functionsC3 solve problems involving polynomial andsimple rational* equations graphically andalgebraicallyC4 demonstrate an understanding of solvingpolynomial and simple rationalinequalities*D1 demonstrate an understanding of averageand instantaneous rate of change, anddetermine, numerically and graphically,and interpret the average rate of change of afunction over a given interval and theinstantaneous rate of change of a functionat a given point*to be addressed in Unit 2MFH4U – Overview3
Unit2 Cluster of CurriculumExpectationsOverall and SpecificExpectationsIdentify and use key features of rationalfunctionsSolve problems using a variety of toolsand strategies related to rationalfunctionsDetermine and interpret average andinstantaneous rates of change forrational functionsC2 identify and describe some key features ofthe graphs of rational functions, andrepresent rational functions graphicallyPJT617911011213SPC3 solve problems involving polynomial* andsimple rational equations graphically andalgebraicallyC4 demonstrate and understanding of solvingpolynomial* and simple rationalinequalitiesD1 demonstrate an understanding of averageand instantaneous rate of change, anddetermine, numerically and graphically,and interpret the average rate of change of afunction over a given interval and theinstantaneous rate of change of a functionat a given point3 4 Explore, define and use radian measureGraph primary trigonometric functionsand their reciprocals in radians andidentify key features of the functionsSolve problems using a variety of toolsand strategies related to trigonometricfunctionsDetermine and interpret average andinstantaneous rates of change fortrigonometric functions* addressed in Unit 1B1 demonstrate an understanding of themeaning an application of radian measureB2 make connections between trigonometricratios and the graphical and algebraicrepresentations of the correspondingtrigonometric functions and betweentrigonometric functions and theirreciprocals, and use these connections tosolve problemsD1 demonstrate an understanding of averageand instantaneous rate of change, anddetermine, numerically and graphically,and interpret the average rate of change of afunction over a given interval and theinstantaneous rate of change of a functionat a given pointB2 make connections between trigonometricratios and the graphical and algebraicrepresentations of the correspondingtrigonometric functions and betweentrigonometric functions and theirreciprocals, and use these connections tosolve problemsGraph and transform sinusoidalfunctions using radian measureIdentify domain, range, phase shift,period, amplitude, and vertical shift ofsinusoidal functions using radianmeasuresDevelop equations of sinusoidalfunctions from graphs and descriptionsexpressed in radian measureB3 solve problems involving trigonometricequations and prove trigonometricSolve problems graphically that can beidentitiesmodeled using sinusoidal functionsProve trigonometric identitiesSolve linear and quadratic trigonometricequations using radian measuresMake connections between graphic andalgebraic representations oftrigonometric relationshipsMFH4U – Overview4
Cluster of CurriculumExpectationsUnit5 6 Overall and SpecificExpectationsDevelop the understanding that theA1 demonstrate an understanding of thelogarithmic function is the inverse of therelationship between exponentialexponential functionexpressions and logarithmic expressions,Simplify exponential and logarithmicevaluate logarithms, and apply the laws ofexpressions using exponent ruleslogarithms to simplify numeric expressionsIdentify features of the logarithmicfunction including rates of changeA2 identify and describe some key features ofTransform logarithmic functionsthe graphs of logarithmic functions, makeconnections between the numeric,Evaluate exponential and logarithmicexpressions and equationsgraphical, and algebraic representations ofSolve problems that can be modeledlogarithmic functions, and solve relatedusing exponential or logarithmicproblems graphicallyfunctionsA3 solve problems involving exponential andlogarithmic equations algebraically,including problems arising from real-worldapplicationsD1 demonstrate an understanding of averageand instantaneous rate of change, anddetermine, numerically and graphically,and interpret the average rate of change of afunction over a given interval and theinstantaneous rate of change of a functionat a given pointD1 demonstrate an understanding of averageConsolidate understanding ofand instantaneous rate of change, andcharacteristics of functions (polynomial,determine, numerically and graphically,rational, trigonometric, and exponential)and interpret the average rate of change of aCreate new functions by adding,function over a given interval and thesubtracting, multiplying, or dividingfunctionsinstantaneous rate of change of a functionat a given pointCreate composite functionsDetermine key properties of the newfunctionsD2 determine functions that result from theGeneralize their understanding of aaddition, subtraction, multiplication, andfunctiondivision of two functions and from thecomposition of two functions, describesome properties of the resulting functions,and solve related problemsPJT122141121311485SPD3 compare the characteristics of functions,and solve problems by modeling andreasoning with functions, includingproblems with solutions that are notaccessible by standard algebraic techniquesSummative Performance TasksTotal Days70The number of prepared lessons represents the lessons that could be planned ahead based on the range of studentreadiness, interests, and learning profiles that can be expected in a class. The extra time available for “instructionaljazz” can be taken a few minutes at a time within a pre-planned lesson or taken a whole class at a time, as informedby teachers’ observations of student needs.The reference numbers are intended to indicate which lessons are planned to precede and follow each other. Actualday numbers for particular lessons and separations between terms will need to be adjusted by teachers.MFH4U – Overview5
Introductory Unit: Advanced FunctionsGrade 12Lesson OutlineBig PictureStudents will: revisit contexts studied in the Grade 11 Functions course (MHF4U) using simplifying assumptions, addingprecision to the graphical models, and discussing key features of the graphs using prior academic language(e.g., domain, range, intervals of increase/decrease, intercepts, slope) and ‘local maximum/minimum,’ ‘overallmaximum/minimum;’ recognize that transformations previously applied to quadratic and trigonometric functions also apply to linearand exponential functions, and to functions in general; use function notation to generalize relationships between two functions that are transformations of each otherand whose graphs are given; represent key properties of functions graphically and using function notation; form inverses of functions whose graphs are given, and apply the vertical line test to determine whether or notthese inverses are functions.DayLesson Title1–2 Adding Precision toGraphical Modelsand TheirDescriptions 3TransformationsAcross FunctionTypes 4Using FunctionNotation toGeneralizeRelationships MFH4U – Introductory UnitMath Learning GoalsExpectationsFrom initial simplifying assumptions about a context and the D1.1, D1.1, D3.1,and setting upcorresponding distance/time graph, introduce thecomplicating factors in the context and analyse adjustments C1.2needed in the graph e.g., swimming laps in a pool; riding abicycle up a hill, down a hill, on the flatUse the following academic language to describe changes:speed (rate of change), intervals of increase/decrease,domain/range, overall and local maximum, and overall andlocal minimumGraph corresponding speed/time graphsSetting up C1.6,Use function notation to generalize relationships betweenA2.3sets of two congruent functions e.g., h(x) f(x) 2 togeneralize a line and the line shifted 2 units, a parabola andthe parabola shifted 2 units up, an exponential function andthe exponential function shifted 2 unit up; f(x) g(x 3)Use graphical and numerical representations of the functionsIntroduce the concept that lines and exponential functionscan be seen through a transformational lens.Graph y f(x) 3 from any given y f(x)Setting up C1.6Use function notation to generalize relationships betweensets of two functions, one a single transformation of the othere.g., h(x) 2f(x) to generalize a sinusoidal function and thestretched sinusoidal function, a line and the stretched line, aparabola and the stretched parabola, an exponential functionand the stretched exponential function shifted 2 units up;f(x) g(x 3)Use graphical and numerical representations of the functions6
DayLesson Title5 Representing KeyProperties ofFunctionsGraphically andUsing FunctionNotationMath Learning Goals 67Forming Inversesand FunctionTestingJazz day tosummarizeMFH4U – Introductory Unit Interpret graphically, values shown in function notation e.g.,Graph y f(x) that has all of the following properties:f(1) 2, f(3) f(-1) 0, f(0) 4, f(x) 0 for x 0, andf(x) 0 for x 0, domain xєR, range –4 y 4Explore multiple solutions to each of the above, noting thelack of information for determining concavityRepresent critical points and key regions of the graph of afunction using functional notationForm inverses of given functions (graphical representations)and determine whether or not the inverse is a functionExpectationsSetting upC1.7,2.2, A2.1Setting up A1.1,2.2 7
Unit 1: Polynomial FunctionsGrade 12Lesson OutlineBig PictureStudents will: identify and use key features of polynomial functions; solve problems using a variety of tools and strategies related to polynomial functions; determine and interpret average and instantaneous rates of change for polynomial functions.DayLesson Title1 Ch Ch ChangesMath Learning Goals 2–3 “Secant” Best Is Not Good EnoughFreeze Frame 4Smooth CurvesPassing ThroughPoints5–6 Characteristics ofPolynomialFunctions 7In Factored Form 8 Make connections between verbal and graphical rates ofchange.Make connections between average velocity and slopes ofsecants.Calculate and interpret average rates of change of functionsarising from real-world applications.ExpectationsD1.2, 1.3, 1.4, 1.7D1.1, 1.3, 1.5-1.8Make connections between motion data and instantaneousrates of change.Make connections between instantaneous rates of change andthe tangent in context.Use technology to calculate slopes of secants and tangents atvarious points along a curve.Interpret slopes of secants and tangents in context.Solve problems involving average and instantaneous rates of D1.2, 1.9change at a point using numerical and graphical methods.Distinguish situations in which the rate of change is zero,constant, or changing by examining applications.C1.1, 1.2, 1.3, 1.4Investigate and summarize graphical characteristics, e.g.,zeros, finite differences, end behaviour, domain and range,increasing/decreasing behaviour, of polynomials functionsthrough numeric, graphical and algebraic representations.Compare these characteristics for linear, quadratic, cubic andquartic functions.Make connections between a polynomial function in factored C1.5, 1.7form and the x-intercepts of its graph.Sketch the graph of polynomial functions, expressed infactored form using the characteristics of polynomialfunctions.Determine the equation of a polynomial given a set ofconditions, e.g., zeros, end behaviour, and recognize theremay be more than one such functionInvestigate transformations applied to f(x) x3 and f(x) x4. C1.6, 1.9Investigate and compare the properties of odd and evenfunctions.MHF4U: Unit 1 – Polynomial Functions8
Day9–10Lesson TitleMath Learning Goals 11–12 13–14 15–1617Divide polynomials.Examine remainders of polynomial division and connect tothe remainder theorem.Make connections between the polynomial function f(x), thedivisor x –a, the remainder of the division f(x)/(x-a) and f(a)using technology.Identify the factor theorem as a special case of the remaindertheorem.Factor polynomial expressions in one variable of degree nohigher than four.ExpectationsC3.1, 3.2C1.8, 3.3, 3.4, 3.7Solve problems graphically and algebraically using theremainder and factor theorems.Solve polynomial equations by selecting an appropriatestrategy, and verify with technology.Make connections between the x-intercepts of a polynomialfunction and the real roots of the corresponding equation.Use properties of polynomials to fit a polynomial function toa graph or a given set of conditions.Determine the equation of a particular member of a family ofpolynomials given a set of conditions, e.g., zeros, endbehaviour, point on the graph [See homework and make useof graphs on Day 7.]C 4.1, 4.2, 4.3Understand the difference between the solution to anequation and the solution to an inequality.Solve polynomial inequalities and simple rationalinequalities by graphing with technology.Solve linear inequalities and factorable polynomialinequalities.Represent the solution to inequalities on a number line oralgebraically. Jazz days SummativeMHF4U: Unit 1 – Polynomial Functions9
Unit 1: Day 1: Ch Ch ChangesGrade 12Math Learning GoalsMake connections between verbal and graphical rates of change.Make connections between the average velocity and slope of the secant line.Investigate the average rates of change of two sprinters to determine who wasrunning the fastest during the race. Materials BLM 1.1.1, 1.1.275 minAssessmentOpportunitiesMinds On Think/Pair/Share Æ Matching ActivityStudents categorize the rates of change of the graphs as either: zero, constant orchanging (BLM 1.1.1). Ask: “Which graphs have a rate of change of zero? Howdo you know? Which graphs have a constant rate of change? How can you tell?Which graphs have a non-constant rate of changing? How do you know?”Students match verbal and graphical rates of change (BLM 1.1.1).Mathematical Process/Connec
Introductory Unit: Advanced Functions Grade 12 Lesson Outline Big Picture Students will: revisit contexts studied in the Grade 11 Functions course (MHF4U) using simplifying assumptions, adding precision to the graphical models, and discussing key features of the graphs using prior academic language
FINAL EXAMINATION - MHF4U JUNE 2015 Address: El Tagamoa El Khames, 4th District – Zone # 6 Katameya, New Cairo, Egypt Tel. /Fax: 202 –26174500 Email: infor@cise -eg.com –Website: www.cise eg.com Date: Monday, June 8th, 2015 Length: 8:50 – 11:20 (2.5 hours) Location: Room 46 Pages: 14 (including this one) Subject: Grade 12 Advanced Functions (MHF4U)
simple rational functions and equations [e.g., problems involving the factor theorem or remainder theorem. 6 hours A4. Solving Inequalities In this unit, students will learn how to explain, for polynomial and simple rational functions, the difference between the solution to an equation in one variable and the solution to an inequality in one .
When investigating special cases of functions, factor and reduce where possible. Indicate the restrictions on the variables in order to identify hidden discontinuities. When investigating new types of rational functions, consider what is different about the coefficients and the degree of the polynomials in the numerator and denominator. These
Grade 12 Advanced Functions (MHF4U) and Grade 12 Calculus and Vectors (MCV4U) Corequisite: MATA29H3 or MATA30H3 or MATA31H3 Exclusions: PHYA10H3, PHY131H, PHY135Y, PHY151H . the weight will be transferred to the final exam. PHYA11H3 Syllabus – Fall 2019 5 Resources
Aug 13, 2020 · exponential functions. Unit 5.1 –Exponential Functions & Their Graphs So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions –exponential funct
Advanced metering for SMEs The Impact of advanced metering for SMEs 0 Executive summary 02 Introduction to advanced metering 7.06 The potential benefits 06 .2 Use of advanced metering in businesses 06 .3 SupplierPrinciples of advanced metering 07 .4 Analysing advanced metering data 07 .5 Sources of energy savings 08 .6 Advanced metering technology 08 .7 Advanced metering services 09
Unit 1 - Chapter 2 Oracle Built in Functions There are two types of functions in Oracle. 1) Single Row Functions: Single row or Scalar functions return a value for every row that is processed in a query. 2) Group Functions: These functions group the rows of
Mata kulian Anatomi dan Fisiologi Ternak di fakultas Peternakan merupakan mata kuliah wajib bagi para mahasiswa peternakan dan m.k. ini diberikan pada semester 3 dengan jumlah sks 4 (2 kuliah dan 2 praktikum.Ilmu Anatomi dan Fisiologi ternak ini merupakan m.k. dasar yang harus dipahami oleh semua mahasiswa peternakan. Ilmu Anatomi dan Fisiologi Ternak ini yang mendasari ilmu-ilmu yang akan .
