Subject:
Mathematics
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
• Algebra
• College Algebra
• Polynomial
• Polynomial Functions
• Rational Functions
• Tmm0012
Language:
English

# Polynomial and Rational Functions ## Overview

This material covers Chapter 5: Polynomial and Rational Funcitons chapter of the OpenStax College Algebra Text. This module contains an overview of learning objectives mapped to the OTM state standards, worksheets that correspond to chapter sections, interactive Desmos Activities that pair with the chapter, and a list of supplemental videos that correspond to the chapter content.

Photo by Jimmy Chang on Unsplash

# Polynomial and Rational Functions

Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating images is now governed by mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.

(From OpenStax College Algebra)

Chapter Sections

• 5.2 Power Functions and Polynomial Functions
• 5.3 Graphs of Polynomial Functions
• 5.4 Dividing Polynomials
• 5.5 Zeros of Polynomial Functions
• 5.6 Rational Functions
• 5.7 Inverses and Radical Functions
• 5.8 Modeling Using Variation

OTM Outcomes

1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.). In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

1b. Convert between different representations of a function.*

1c. Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.*

2a. Recognize function families as they appear in equations and inequalities and choose an appropriate solution methodology for a particular equation or inequality and can communicate reasons for that choice.*

2c. Distinguish between exact and approximate solutions and which solution methodologies result in which kind of solutions.*

2d. Demonstrate an understanding of the correspondence between the solution to an equation, the zero of a function, and the point of intersection of two curves.*

3a. Purposefully create equivalences and indicate where they are valid.*

3b. Recognize opportunities to create equivalencies in order to simplify workflow.*

4a. Interpret the function correspondence and behavior of a given model in terms of the context of the model.*

4c. Determine parameters of a model given the form of the model and data.*

4d. Determine a reasonable applied domain for the model as well as articulate the limitations of the model.*

5a. Anticipate the output from a graphing utility and make adjustments, as needed, in order to efficiently use the technology to solve a problem.*

5d. Use technology and algebra in concert to locate and identify exact solutions.*

6a. Recognize when a result (theorem) is applicable and use the result to make sound logical conclusions and provide counter-examples to conjectures.*

Chapter Section Objectives and related OTM standards:

• Recognize characteristics of parabolas.(1a)
• Understand how the graph of a parabola is related to its quadratic function (1a, 1b, 4c)
• Determine a quadratic function’s minimum or maximum value (1a, 3b)
• Solve problems involving a quadratic function’s minimum or maximum value (2c, 4a, 4d)

5.2 Power Functions and Polynomial Functions

• Identify power functions (1a)
• Identify end behavior of power functions (1a)
• Identify polynomial functions (1a)
• Identify the degree and leading coefficient of polynomial functions (1a, 2a, 2d)

5.3 Graphs of Polynomial Functions

• Recognize characteristics of graphs of polynomial functions (1a)
• Use factoring to ﬁnd zeros of polynomial functions (2a)
• Identify zeros and their multiplicities (1a, 2d)
• Determine end behavior (1a)
• Understand the relationship between degree and turning points (1a)
• Graph polynomial functions (1a, 5a)
• Use the Intermediate Value Theorem (6a)

5.4 Dividing Polynomials

• Use long division to divide polynomials.
• Use synthetic division to divide polynomials.

5.5 Zeros of Polynomial Functions

• Evaluate a polynomial using the Remainder Theorem (3b)
• Use the Factor Theorem to solve a polynomial equation (2a, 5d, 6a)
• Use the Rational Zero Theorem to find rational zeros (2a, 5d)
• Find zeros of a polynomial function (2a, 5d)
• Use the Linear Factorization Theorem to find polynomials with given zeros (3a, 6a)
• Use Descartes’ Rule of Signs (6a)
• Solve real-world applications of polynomial equations (2a, 4a)

5.6 Rational Functions

• Use arrow notation (1a)
• Solve applied problems involving rational functions (1b, 4a)
• Find the domains of rational functions (1a)
• Identify vertical asymptotes (1a)
• Identify horizontal asymptotes (1a, 2a)
• Graph rational functions (1a, 2a, 5a)

• Find the inverse of an invertible polynomial function (1a, 1c)
• Restrict the domain to find the inverse of a polynomial function (1a, 1c)

5.8 Modeling Using Variation

• Solve direct variation problems (1b, 4c, 5d)
• Solve inverse variation problems (1b, 4c, 5d)
• Solve problems involving joint variation (1b, 4c, 5d)

# Chapter 5 Worksheets

The files below are worksheets created to support practice and learning for chapter 5 of the OpenStax College text.

# Chapter 5 Desmos Activites

The links below will direct to pre-built Desmos activites. These can be easily copied, remixed, or compiled into activites for teachers to customize to their class and objectives. For more on how educators can integrate the desmos activites into their class please click here to link to the desmos teacher support site:

Below is a short explanation of the Desmos activities linked.

The quadratic bundle contains a collection of activities that are appropriate for this section. The activities that apply directly to this section are listed below:

Will it hit the hoop
In this activity, students predict whether various basketball shots will go through the hoop, and then model these shots with parabolas to check their predictions. Students use draggable points to model in this activity, and do not need to be familiar with symbolic forms of quadratic functions in advance.

Polygraph: Parabolas
Students play a game in pairs where one member of the pair picks a single parabola from a collection and the other asks yes/no questions in an attempt to guess which parabola their partner chose. The game provides students with a reason for noticing important features of parabolas—how else will you tell them apart?—and this in turn provides a need for words to name these features. “Does your graph cross the x-axis twice?” is a common question in Polygraph that leads to naming roots, for example. Some of this vocabulary development is built into the structure of the activity; classroom discussion following the activity can provide more of it.

Polygraph: Parabolas, part 2
This activity follows up on Polygraph: Parabolas, using the discussions (and students' informal language) in that activity to develop academic vocabulary related to the graphs of quadratic functions.For this activity, we recommend that students using a screen reader pair up with a sighted partner.

Match my Parabola
In this activity, students work through a series of scaffolded quadratic graphing challenges to develop their proficiency with standard, vertex, factored, and other quadratic function forms.

Marbleslides: Parabolas
In this delightful and challenging activity, students will transform parabolas so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

Card sort: Parabolas
There are many strategies for determining the shape of a graph given its equation. In this activity, students will find the shape of a parabola by using its form to reveal its characteristics. The activity begins with a review of both the characteristics and forms of a parabola. Later, students will determine characteristics of the graph of a parabola given either in standard form, vertex form, or intercept form.

Build a bigger field
Students will use quadratic models to optimize the area of a field for a given perimeter. This is the Desmos treatment of a task that's as old as fields themselves. We emphasize estimation, construction, and formulation, in addition to the graphing and solution you find in traditional treatments.

Penny Circle
This task bounces students from a table to a graph to symbols in five steps—without words describing why one abstraction is more useful than another. Students need to understand those differences. A table is great because it lets us forget about the physical pennies. A graph is great because it shows us the shape of the model. And the algebraic function is great because it lets us compute and predict. Penny Circle highlights those strengths.

Two truths and a lie: Parabolas
Students will practice their understanding of the features and vocabulary of a parabola by creating a parabola, writing two true and one false statement about it, and inviting their peers to separate truth from lies.

Constructing polynomials
In this activity, students will consider properties of polynomial functions such as end behavior, leading terms, and properties of roots. They will explore connections between those properties and the factored forms of the equations of the polynomials.

The Intermediate Value Theorem
This activity asks students to speculate on the existence of roots based on graphs of functions. This is intended as a first exposure to the ideas of the Intermediate Value Theorem, and to provide fodder for classroom conversation about important ideas behind the theorem, including:
(1) The "continuous" condition is essential (though so very often overlooked by students), and
(2) The Intermediate Value Theorem is silent on where there are NOT roots, only on where there ARE roots (given the right conditions).

Polynomial Equation Challenges
In this activity, students will create polynomial equations (of degree 2, 3, and 4) to match given zeros and points. Students will explore how the factored form of the equations relates to the zeros and the order of those zeros.

Polygraph: Rational Functions
Students play a game in pairs where one member of the pair picks a single graph of a rational funciton from a collection and the other asks yes/no questions in an attempt to guess which graph their partner chose. The game provides students with a reason for noticing important features of rational functions—how else will you tell them apart?—and this in turn provides a need for words to name these features.

Marbleslides: Rationals
In this delightful and challenging activity, students will transform rational functions so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.

# Chapter 5 Supplemental Videos

Below is a document that links to supplemental videos for this chapter. Note that these videos are not created by the publisher of the text, so some verbiage or problem solving strategies may vary from what is presented in the text.