Course 3 Unit 5 - Polynomial and Rational Functions

In units of the algebra and functions strand in Courses 1 and 2, students built a repertoire of function families that model linear, exponential, quadratic, and power (direct and inverse) relationships. They also developed skill in using algebraic expressions to represent those patterns of change, skill in manipulating the expressions to solve equations and inequalities and to gain insight into relationships, and skill in solving a wide variety of authentic quantitative problems. In this unit, polynomial and rational functions are added to students' toolkit of families of functions.

Unit Overview
Topics studied in this unit include the definition and properties of polynomials, operations on polynomials, completing the square, proof of the quadratic formula, solving quadratic equations (including complex number solutions), the vertex form of quadratic functions, definition and properties of rational functions, and operations on rational expressions.
     This unit is an introduction to polynomial and rational functions and expressions and should be accessible to most students. Further work toward developing proficiency with symbol manipulation occurs in subsequent Review tasks and in Course 4.

Objectives of the Unit
  • Recognize patterns in problem conditions and in data plots that can be described by polynomial and rational functions
  • Write polynomial and rational function rules to describe patterns in graphs, numerical data, and problem conditions
  • Use table, graph, or symbolic representations of polynomial and rational functions to answer questions about the situations they represent: (1) calculate y for a given x (i.e., evaluate functions); (2) find x for a given y (i.e., solve equations and inequalities); and (3) identify local max/min points and asymptotes
  • Rewrite polynomial and rational expressions in equivalent forms by expanding or factoring, by combining like terms, and by removing common factors in numerator and denominator of rational expressions
  • Add, subtract, and multiply polynomial and rational expressions and functions
  • Extend understanding and skill in work with quadratic functions to include completing the square, interpreting vertex form, and proving the quadratic formula
  • Recognize and calculate complex number solutions of quadratic equations

Sample Overview
The sample investigation is Investigation 2 of Lesson 1. In the first investigation, students learned how to model complex graphical patterns with polynomial functions of degree 3 and 4. They explored a variety of polynomial functions to discover the type of graph possibilities. They began to see the relationship between the degree of a polynomial function and other important features of its graph, especially the number of local maximum and local minimum points.
In the sample investigation provided below, students work with polynomial expressions and functions, learning how to combine them by addition and subtraction. They look for a pattern relating the degrees of components to the degree of the sum or difference of two polynomials and a pattern relating the degree of a polynomial expression to the number of zeroes of the corresponding function.
     This investigation makes use of the public-domain CPMP-Tools computer software. Select "Course 3" from the menu bar, then choose Algebra and CAS. Enter a polynomial equation with parameters a, b, and c (such as the one on Course 3 page 325) in the Y= tab. Then select the Graph tab to view.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Algebra and Functions Strand Continues
In Course 3, there is one more algebra unit, Unit 8, Inverse Functions. This unit develops student understanding of inverses of functions with a focus on logarithmic functions and their use in modeling and analyzing problem situations and data patterns. Topics studied in this unit include inverses of functions, logarithmic functions and their relation to exponential functions, properties of logarithms, equation solving with logarithms, and inverse trigonometric functions and their applications to solving trigonometric equations.
     Course 3 Unit 7, Recursion and Iteration, is technically a discrete mathematics unit, but working with sequences and series helps students strengthen their symbolic skills.
     Course 4: Preparation for Calculus extends student algebraic skills and understandings in equations and functions in algebra units but also in geometry units such as Unit 2, Vectors and Motion, and Unit 6, Surfaces and Cross Sections. (See the CPMP Courses 1-4 descriptions.)

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