# MAT 523 Theory of Functions for Teachers

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### Course Description

Topics from Function Theory including: mathematical models, linear functions, nonlinear functions, transformations, limits, continuity, functions of several variables.

MAT 523 meets for three hours of lecture per week.

### Prerequisites

MAT 543, graduate standing and one year of full time secondary teaching.

### Objectives

After completing MAT 523 the student should be able to

• represent functions numerically, graphically, and algebraically
• use a graphing calculator effectively in exploring properties of functions
• state the relevant mathematical definitions and results about functions and apply them to solving problems
• demonstrate understanding of the abstract concepts of functions and be able to identify these concepts within models of particular functions in context
• identify function concepts within the secondary curriculum
• demonstrate knowledge of current research on teaching and learning about functions

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 523.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of topics covered in MAT 523 and a project to be written and presented to the class.

### Text

Functions, Modeling Change, by the Shell Centre.

#### Outline of contents

• representing functions and other relations numerically, graphically, algebraically
• examples of relations which are 1-1, many-1, 1-many, and many-many
• identifying basic properties of functions given in each of the different representations, for example:
• domain and range
• x- and y-intercepts
• intervals on which the function is increasing/decreasing
• comparison of slope for linear functions and variable rates of change for nonlinear functions
• intervals on which the function is concave up/down
• whether the function is one-to-one or many-to-one
• given x, find f(x)
• given b, solve for x such that f(x) = b, f(x) < b, f(x) > b.
• identify horizontal and vertical asymptotes
• intervals on which the function is continuous/smooth
• existence of and properties of inverse functions
• introduction to the concepts, language and notation of calculus
• the study of representatives of classic families of functions:
• linear functions
• other polynomial functions
• exponential functions
• logarithmic functions
• rational functions
• trigonometric functions
• modeling given situations with functions and identifying basic concepts associated with functions in particular contexts
• the relationship between functions of a discrete variable, sequences, and functions of a continuous variable, for example:
• connection with patterns
• when to connect the dots on a graph
• progression of exponents from whole numbers to integers to rationals to reals
• extension of the domain of trigonometric functions from basic angles in a right triangle to their full domain within the real numbers
• using line/curve of best fit to analyze data
• identifying functions and relations throughout the secondary mathematics curriculum, for example:
• the need for inverse functions in solving equations
• the expression on each side of an equation or an inequality as a function (particularly when an expression = or < or > a constant)
• the number of elements as a function of the subsets of a given set
• the probability of an event as a function of the subsets of a sample space
• the number of vertices, the measure of the largest angle, the area, the perimeter each as a function of the set of polygons
• the number of factors as a function of the counting numbers
• the correspondence between a fraction representation and a decimal representation of rational numbers
• selected topics regarding current research on both teachers and students understanding of functions

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 523. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.