Mathematics

A. CHARACTERISTICS OF FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of functions, their representations, and their inverses, and makeconnections between the algebraic and graphical representations of functions using transformations;2. determine the zeros and the maximum or minimum of a quadratic function, and solve problemsinvolving quadratic functions, including problems arising from real-world applications;3. demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, andrational expressions.FunctionsSPECIFIC EXPECTATIONS1. Representing FunctionsBy the end of this course, students will:1.1 explain the meaning of the term function, anddistinguish a function from a relation that isnot a function, through investigation of linearand quadratic relations using a variety of representations(i.e., tables of values, mapping diagrams,graphs, function machines, equations)and strategies (e.g., identifying a one-to-oneor many-to-one mapping; using the verticallinetest)Sample problem: Investigate, using numericand graphical representations, whether the2relation x = y is a function, and justify yourreasoning.1.2 represent linear and quadratic functions usingfunction notation, given their equations, tablesof values, or graphs, and substitute into andevaluate functions [e.g., evaluate f (1 ) , given22f(x) = 2x + 3x – 1]1.3 explain the meanings of the terms domainand range, through investigation using numeric,graphical, and algebraic representations of2the functions f(x) = x, f(x) = x , f(x) = √x,1and f(x) = ; describe the domain and range ofx2a function appropriately (e.g., for y = x + 1,the domain is the set of real numbers, and theSample problem: A quadratic function representsthe relationship between the heightrange is y ≥ 1); and explain any restrictions onthe domain and range in contexts arising fromof a ball and the time elapsed since the ballreal-world applicationswas thrown. What physical factors willSample problem: A quadratic function representsthe relationship between the heightof a ball and the time elapsed since the ballwas thrown. What physical factors willrestrict the domain and range of the quadraticfunction?1.4 relate the process of determining the inverseof a function to their understanding ofreverse processes (e.g., applying inverseoperations)1.5 determine the numeric or graphical representationof the inverse of a linear or quadraticfunction, given the numeric, graphical, oralgebraic representation of the function, andmake connections, through investigationusing a variety of tools (e.g., graphing technology,Mira, tracing paper), between thegraph of a function and the graph of itsinverse (e.g., the graph of the inverse is thereflection of the graph of the function in theline y = x)Sample problem: Given a graph and a table ofvalues representing population over time,produce a table of values for the inverse andgraph the inverse on a new set of axes.1.6 determine, through investigation, the relationshipbetween the domain and range of a functionand the domain and range of the inverserelation, and determine whether or not theinverse relation is a functionSample problem: Given the graph of f(x) = x ,graph the inverse relation. Compare the domainand range of the function with the domain2MCR3UCHARACTERISTICS OF FUNCTIONS45