Course: Mathematical Analysis (Honors)
Unit 1: Graphical Analysis of Functions
In this unit, students analyze the key features of graphs of functions and make connections between domain, range, relative extrema, intervals increasing/decreasing, end behavior, continuity, zeros, and symmetry. Students create an equation, graph, table, or verbal description given key properties including increasing or decreasing behavior, the sign of the leading coefficient, degree, and end behavior.
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 2: Matrices
In this unit, students are introduced to matrices. They use a matrix to represent and manipulate real world data using matrix addition and subtraction, matrices products, and multiplication of matrices by scalars. Students develop an understanding of how multiplication of a matrix results in a transformation of the points in the coordinate plane. Students examine 3x3 matrices in order to find the determinant. Students will see that a system of linear equations can be represented and solved algebraically as a matrix equation and the results may be verified with technology.
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 3: Vectors and Parametrics
In this unit, students are introduced to algebraic vectors and parametric equations. Students interpret algebraic vectors as an ordered pair of real numbers in the coordinate plane and use operations to identify the components of a vector. Students recognize vector quantities as having both magnitude and direction, and calculate for a vector and resultant vector. Students perform operations on vectors to model and solve work applications, as well as find the dot product, unit vector, and angle between vectors. They represent vectors as a one column matrix and perform scalar and matrix multiplication as transformations of vectors on the coordinate plane.
For parametric equations, students define a curve using functions with respect to a common variable. Students then model a path of an object using parametric equations and define the curve by eliminating the parameter.
Unit 4: Polynomial Functions
In this unit, students build on their understanding of quadratic functions to explore polynomial functions. Students build new polynomial functions using the arithmetic operations of addition, subtraction, multiplication, and composition. Students analyze key features of graphs of polynomial functions including domain and range, zeros, local extrema, intervals of increasing and decreasing, and concavity. Students make connections between end behavior, the leading coefficient, and the degree, and then graph polynomial functions based on these key features. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal description of a real-world context. They solve polynomial equations in context using similar representations. Students find all roots of a polynomial function using algebraic methods and use them to sketch the graph.
Unit 5: Rational Functions
In this unit, students apply their understanding of transformations to build rational functions. Students identify key features of functions, including domain, range, end behavior, local (relative) and/or absolute extrema, continuity, zeros, intercepts, symmetry, holes, asymptotes, and whether the function is even/odd. Students solve rational equations by hand, checking for extraneous solutions. Students solve rational inequalities by hand, checking for extraneous solutions. Students calculate limits of rational functions algebraically and estimate limits from graphs and tables of values. Students decompose a fraction with a factorable quadratic denominator and a linear or constant numerator.
Unit 6: Radical and Special Functions
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 7: Logarithmic and Logistic Functions
In this unit, students build on their prior work with exponential functions to explore the logarithmic and logistic function families. Students revisit the concept of an inverse as they are introduced to logarithmic functions. Students apply properties of logarithms and exponents to solve logarithmic and logistic equations.
Students describe key features of the graphs of logarithmic and logistic functions, showing intercepts and end behavior, of logarithmic and logistic functions. They use transformations to sketch and analyze the graphs, focusing on domain and range restrictions. As the unit progresses, students build fluency with rewriting and/or identifying equivalent logarithmic expressions. Students solve real-world problems involving transformed logarithmic models, verify appropriateness of models, and make predictions based on the model.
Unit 8: Limits and Series
In this unit, students extend understanding of functions to explore sequences and series. First, students determine whether a sequence converges or diverges. They find the partial sum of an arithmetic or geometric sequence, as well as the sum of a converging geometric sequence. Students develop an understanding that an infinite series is found by taking the limit of partial sums.
Students apply their understanding of functions and use a problem-solving approach in order to calculate limits algebraically as well as sketch functions without the use of a graphing calculator and identify key elements of the function including intercepts, end behavior, turning points, and intervals for which the function is increasing/decreasing/or remains constant using limits. They use limits to describe continuity of functions and be able to discuss one-sided limits and determine if a limit fails to exist at a value. Students calculate average rate of change of a functions and use limits to find the instantaneous rate of change of a function at a point and instantaneous rates of change as defined by limits and discuss the use of limits to develop the concept of the derivative of a function. Students use functions to model real-world situations and make predictions about future behavior.
Unit 1: Graphical Analysis of Functions
In this unit, students analyze the key features of graphs of functions and make connections between domain, range, relative extrema, intervals increasing/decreasing, end behavior, continuity, zeros, and symmetry. Students create an equation, graph, table, or verbal description given key properties including increasing or decreasing behavior, the sign of the leading coefficient, degree, and end behavior.
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 2: Matrices
In this unit, students are introduced to matrices. They use a matrix to represent and manipulate real world data using matrix addition and subtraction, matrices products, and multiplication of matrices by scalars. Students develop an understanding of how multiplication of a matrix results in a transformation of the points in the coordinate plane. Students examine 3x3 matrices in order to find the determinant. Students will see that a system of linear equations can be represented and solved algebraically as a matrix equation and the results may be verified with technology.
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 3: Vectors and Parametrics
In this unit, students are introduced to algebraic vectors and parametric equations. Students interpret algebraic vectors as an ordered pair of real numbers in the coordinate plane and use operations to identify the components of a vector. Students recognize vector quantities as having both magnitude and direction, and calculate for a vector and resultant vector. Students perform operations on vectors to model and solve work applications, as well as find the dot product, unit vector, and angle between vectors. They represent vectors as a one column matrix and perform scalar and matrix multiplication as transformations of vectors on the coordinate plane.
For parametric equations, students define a curve using functions with respect to a common variable. Students then model a path of an object using parametric equations and define the curve by eliminating the parameter.
Unit 4: Polynomial Functions
In this unit, students build on their understanding of quadratic functions to explore polynomial functions. Students build new polynomial functions using the arithmetic operations of addition, subtraction, multiplication, and composition. Students analyze key features of graphs of polynomial functions including domain and range, zeros, local extrema, intervals of increasing and decreasing, and concavity. Students make connections between end behavior, the leading coefficient, and the degree, and then graph polynomial functions based on these key features. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal description of a real-world context. They solve polynomial equations in context using similar representations. Students find all roots of a polynomial function using algebraic methods and use them to sketch the graph.
Unit 5: Rational Functions
In this unit, students apply their understanding of transformations to build rational functions. Students identify key features of functions, including domain, range, end behavior, local (relative) and/or absolute extrema, continuity, zeros, intercepts, symmetry, holes, asymptotes, and whether the function is even/odd. Students solve rational equations by hand, checking for extraneous solutions. Students solve rational inequalities by hand, checking for extraneous solutions. Students calculate limits of rational functions algebraically and estimate limits from graphs and tables of values. Students decompose a fraction with a factorable quadratic denominator and a linear or constant numerator.
Unit 6: Radical and Special Functions
Students build on their work with quadratic and absolute value functions and extend their repertoire of functions to begin to explore other function families. Students discover the connections between changes in algebraic equations and transformations in the corresponding graphs. Students compare properties of two functions using multiple representations, such as, a graph, table of values, and verbal descriptions. Students begin to fit real-world data and scenarios to potential function families. This unit serves as an introduction to function families, transformations, and modeling and lays the groundwork for deeper study of individual function families.
Unit 7: Logarithmic and Logistic Functions
In this unit, students build on their prior work with exponential functions to explore the logarithmic and logistic function families. Students revisit the concept of an inverse as they are introduced to logarithmic functions. Students apply properties of logarithms and exponents to solve logarithmic and logistic equations.
Students describe key features of the graphs of logarithmic and logistic functions, showing intercepts and end behavior, of logarithmic and logistic functions. They use transformations to sketch and analyze the graphs, focusing on domain and range restrictions. As the unit progresses, students build fluency with rewriting and/or identifying equivalent logarithmic expressions. Students solve real-world problems involving transformed logarithmic models, verify appropriateness of models, and make predictions based on the model.
Unit 8: Limits and Series
In this unit, students extend understanding of functions to explore sequences and series. First, students determine whether a sequence converges or diverges. They find the partial sum of an arithmetic or geometric sequence, as well as the sum of a converging geometric sequence. Students develop an understanding that an infinite series is found by taking the limit of partial sums.
Students apply their understanding of functions and use a problem-solving approach in order to calculate limits algebraically as well as sketch functions without the use of a graphing calculator and identify key elements of the function including intercepts, end behavior, turning points, and intervals for which the function is increasing/decreasing/or remains constant using limits. They use limits to describe continuity of functions and be able to discuss one-sided limits and determine if a limit fails to exist at a value. Students calculate average rate of change of a functions and use limits to find the instantaneous rate of change of a function at a point and instantaneous rates of change as defined by limits and discuss the use of limits to develop the concept of the derivative of a function. Students use functions to model real-world situations and make predictions about future behavior.
