Unit 3: Polynomial and Rational Functions
Linear functions are used to measure many situations, but curved
situations such as the path of a baseball hit into the air or
concentrations and dilutions of solutions will need polynomial and
rational functions investigated in this unit. Local extrema, intercepts,
and asymptotes will help to interpret the behavior of these situations
and make extrapolations.
Unit 3 begins with the study of polynomials and their behaviors and
then concentrates on quadratic functions, equations, and applications.
Your function toolkit will be expanded to include rational and root
functions that include new challenges when graphing and solving for
domains, inverses, and long-term behavior. These new techniques will be
important in higher level courses in mathematics such as calculus.
Unit 3 Time Advisory
This unit should take approximately 18.25 hours to complete.
☐ Subunit 3.1: 2.75 hours
☐ Subunit 3.2: 3.75 hours
☐ Subunit 3.3: 3.75 hours
☐ Subunit 3.4: 3.25 hours
☐ Subunit 3.5: 2.75 hours
☐ Review and Exam: 2 hours
Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- graph polynomial and rational functions analytically and using
technology;
- write the equation of a polynomial and rational function from a
graph;
- solve a quadratic equation using the quadratic formula;
- compute extrema of polynomial functions analytically or using
technology;
- identify the properties of polynomial and rational functions such as
degree, maximum number of zeros, maximum number of turns,
multiplicity of zeros, vertical asymptotes, horizontal asymptotes,
and long-run behavior;
- compute inverse functions of polynomials and rational functions; and
- determine an appropriate approach, solve polynomial and rational
application problems, and then interpret the solution.
3.1 Power Functions and Polynomial Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational
Functions” (PDF)
Instructions: Read pages 155–161 of Chapter 3 to learn about power
and polynomial functions, their terminology, long-run behavior, and
how to solve for their intercepts. Note that this reading covers the
material you need to know for subunits 3.1.1 through 3.1.3.
Reading this section should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
3.1.1 Define Power and Polynomial Functions Note: This subunit is covered in the reading assigned beneath subunit 3.1. To learn about power functions, their long-term behavior, and polynomial functions, read the information on pages 155 through 157.
- Web Media: Brightstorm: “Precalculus – Power Functions”
Link: Brightstorm: “Precalculus – Power
Functions” (YouTube)
Instructions: Watch this video to learn the basics about power functions and view them graphically. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 5 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.2 Terminology of Polynomial Functions Note: This subunit is covered in the reading assigned beneath subunit 3.1. To learn the terminology of polynomial functions, read the information on pages 158 and 159.
- Web Media: YouTube: Brightstorm: “Precalculus – Polynomial
Functions”
Link: YouTube: Brightstorm: “Precalculus – Polynomial
Functions” (YouTube)
Instructions: Watch this video to learn the definition of a polynomial function and the terminology related to polynomial functions. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.1.3 Behaviors of Polynomials Note: This subunit is covered in the reading assigned beneath subunit 3.1. To learn how to determine the long- and short-run behavior of a polynomial and how to compute the intercepts, read the information on pages 159 and 160.
Web Media: Mathispower4u: “Summary of End Behavior or Long Run Behavior of Polynomial Functions” Link: Mathispower4u: “Summary of End Behavior or Long Run Behavior of Polynomial Functions” (YouTube)
Instructions: Watch this video to discover how to determine the intercepts and end behavior of a polynomial function. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 15 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.1: Power and Polynomial Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.1: Power and Polynomial Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.1, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.
Completing this assessment should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.2 Quadratic Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational
Functions” (PDF)
Instructions: Read pages 163–171 of Chapter 3 to learn about
functions, one-to-one functions, and function notation. Note that
this reading covers the material you need to know for subunits 3.2.1
through 3.2.4.
Reading this section should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
3.2.1 Define Quadratic Function Note: This subunit is covered in the reading assigned beneath subunit 3.2. To learn about the second-degree polynomial and how to use it to model the area of a garden, read the information on pages 163 and 164.
- Web Media: YouTube: Khan Academy’s “Introduction to the Quadratic
Equation”
Link: YouTube: Khan Academy’s “Introduction to the Quadratic
Equation” (YouTube)
Instructions: This video will introduce you to the quadratic equation, show you how to graph it, and help you learn common terms related to quadratic functions. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 20 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.2 Forms of Quadratic Functions Note: This subunit is covered in the reading assigned beneath subunit 3.2. To learn the three different forms of a quadratic function, read the information on page 165.
- Web Media: YouTube: gpTeacher’s “Converting Vertex Form and
Factored Form to Standard Form”
Link: YouTube: gpTeacher’s “Converting Vertex Form and Factored
Form to Standard
Form” (YouTube)
Instructions: Watch this video to learn how to convert vertex and factored form into standard form. Notice the step-by-step process explained for each. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.3 The Vertex Note: This subunit is covered in the reading assigned beneath subunit 3.2. To learn how to compute the vertex of a quadratic function and how to use it to answer real-world questions about maximizing or minimizing, read the information on pages 166 through 168.
- Web Media: YouTube: Dr. Phil Clark’s “Quadratic Functions”
Link: YouTube: Dr. Phil Clark’s “Quadratic
Functions” (YouTube)
Instructions: Watch this video to learn how to compute the vertex, axis of symmetry, domain and range, x-intercepts and y-intercepts for quadratic functions. Take detailed notes and watch the video as many times as you need. Note that this web media also covers the material you need to know for subunit 3.2.4.
Watching this video should take approximately 25 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.2.4 Intercepts and the Quadratic Formula Note: This subunit is covered in the reading assigned beneath subunit 3.2. To learn how to find the horizontal and vertical intercepts of a quadratic by using the vertex form and by using the Quadratic Formula, read the information on pages 168 through 171. This subunit is also covered in the web media assigned beneath subunit 3.2.3. Note the computations for x- and y-intercepts.
Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.2: Quadratic Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.2: Quadratic Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.2, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.
This assessment should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 7a: Sections 3.1–3.2” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 7a: Sections 3.1–3.2” (PDF)
Instructions: Once you have completed the readings and viewed the material for all of subunits 3.1 and 3.2, complete this quiz.Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 7a: Sections 3.1-3.2 Answer Key” (PDF).
Completing this assessment should take approximately 30 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.3 Graphs of Polynomial Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational
Functions” (PDF)
Instructions: Read pages 176–184 of Chapter 3 to learn more about
polynomials, their intercepts, polynomial inequalities, and modeling
using polynomials and technology. Note that this reading covers the
material you need to know for subunits 3.3.1 through 3.3.5.
Reading this section should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
3.3.1 Intercepts Note: This subunit is covered in the reading assigned beneath subunit 3.3. To learn how to find the horizontal and vertical intercepts using factoring and technology, read the information on pages 176 and 177.
- Web Media: YouTube: Mathispower4u: “Find the Intercepts of a
Polynomial Function (Factorable)”
Link: YouTube: Mathispower4u: “Find the Intercepts of a Polynomial
Function
(Factorable)” (YouTube)
Instructions: Watch this video for an example of how to find the vertical and horizontal intercepts of a polynomial function. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.2 Graphical Behavior at Intercepts Note: This subunit is covered in the reading assigned beneath subunit 3.3. To learn how to determine whether a horizontal intercept “passes through” or “bounces off” the horizontal axis, read the information on pages 177 through 179.
- Web Media: YouTube: Mathispower4u: “Turning Points and X
Intercepts of a Polynomial Function”
Link: YouTube: Mathispower4u: “Turning Points and X Intercepts of a
Polynomial
Function” (YouTube)
Instructions: Watch this video to determine how the x-intercepts and degree of a polynomial affect the graphical behavior. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 15 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.3 Solving Polynomial Inequalities Note: This subunit is covered in the reading assigned beneath subunit 3.3. To solve for the intervals where a polynomial is greater than or less than zero, read the information on pages 179 through 181.
- Web Media: YouTube: Mathispower4u: “Solve a Polynomial Inequality
in Factored Form”
Link: YouTube: Mathispower4u: “Solve a Polynomial Inequality in
Factored
Form” (YouTube)
Instructions: Watch this video to see an example by hand and graphically solving a polynomial inequality in factored form. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 15 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.4 Writing Equations using the Intercepts Note: This subunit is covered in the reading assigned beneath subunit 3.3. To learn how to use known horizontal intercepts and write the equation of a polynomial, read the information on pages 181 and 182.
- Web Media: YouTube: Mathispower4u: “Find an Equation of a Degree 4
Polynomial Function from the Graph of the Function”
Link: YouTube: Mathispower4u: “Find an Equation of a Degree 4
Polynomial Function from the Graph of the
Function” (YouTube)
Instructions: Watch this video for an example of how to find an equation of a polynomial function using the graph only. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.3.5 Estimating Extrema Note: This subunit is covered in the reading assigned beneath subunit 3.3. To learn how to use technology to find the maximum or minimum values of a polynomial, read the information on pages 182 and 183.
Web Media: YouTube: Mathispower4u: “How to Determine Relative Extrema on the Graphing Calculator” Link: YouTube: Mathispower4u: “How to Determine Relative Extrema on the Graphing Calculator” (YouTube)
Instructions: Watch this video to see how to use a graphing calculator to estimate the extrema of a polynomial function. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.3: Polynomial Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.3: Polynomial Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.3, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.
Completing this assessment should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 8: Sections 3.2–3.3” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 8: Sections 3.2–3.3” (PDF)
Instructions: Once you have completed the readings and viewed the material for all of subunits 3.2 and 3.3, complete this quiz.
Once you have completed this assessment, you may check your answers against the Saylor Foundation’s “Quiz 8: Sections 3.2-3.3 Answer Key” (PDF).
Completing this assessment should take approximately 30 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.4 Rational Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational
Functions” (PDF)
Instructions: Read pages 188–200 of Chapter 3 to learn about
rational functions, asymptotes, holes, and rational function graphs.
Note that this reading covers the material you need to know for
subunits 3.4.1 through 3.4.3.
Reading this section should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
3.4.1 Defining Rational Functions and Properties Note: This subunit is covered in the reading assigned beneath subunit 3.4. To read some examples of rational functions and learn notation for short-run and long-run behavior, read the information on pages 188 through 191.
- Web Media: Tony Baker’s “Definition and Evaluation of Rational
Functions”
Link: Tony Baker’s “Definition and Evaluation of Rational
Functions” (YouTube)
Instructions: Watch this video to learn what a rational function is and how to interpret the equation of a rational function. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.4.2 Finding Asymptotes, Intercepts, and Holes Note: This subunit is covered in the reading assigned beneath subunit 3.4. To learn how to use algebra to find horizontal and vertical asymptotes, horizontal and vertical intercepts, and when a function has a hole, read the information and many examples on pages 191 through 197.
- Web Media: YouTube: Mathispower4u: “Find the Intercepts and
Asymptotes of a Rational Function”
Link: YouTube: Mathispower4u: “Find the Intercepts and Asymptotes
of a Rational
Function” (YouTube)
Instructions: Watch this video to determine the intercepts and asymptotes of a rational function. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 15 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.4.3 Writing a Rational Function from Intercepts and Asymptotes Note: This subunit is covered in the reading assigned beneath subunit 3.4. To learn how to work in reverse and use the horizontal asymptotes, vertical asymptotes, and intercepts to write the equation of a rational function, read the information on pages 198 through 200.
Web Media: YouTube: Mathispower4u: “Find a Rational Function Given the Vertical Asymptotes and Intercepts” Link: YouTube: Mathispower4u: “Find a Rational Function Given the Vertical Asymptotes and Intercepts” (YouTube)
Instructions: Watch this video to determine the equation of a rational function if you are given the intercepts and asymptotes. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.4: Rational Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.4: Rational Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.4, complete this activity. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.
Completing this assessment should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 9: Section 3.4” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Quiz 9: Section 3.4” (PDF)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.4, complete this quiz.
When you have finished, you may check your answers against the Saylor Foundation’s “Quiz 9: Section 3.4 Answer Key” (PDF).
Completing this assessment should take approximately 30 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.5 Inverses and Radical Functions
- Reading: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of
Functions: “Chapter 3: Polynomial and Rational
Functions” (PDF)
Instructions: Read pages 206–210 of Chapter 3 to learn about
constructing and using inverses of polynomial and rational
functions. Note that this reading covers the material you need to
know for subunits 3.5.1 through 3.5.2.
Reading this section should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons
Attribution-ShareAlike 3.0 United States
License. It is
attributed to David Lippman and Melonie Rasmussen.
3.5.1 Inverses of Polynomials: Radical Functions Note: This subunit is covered in the reading assigned beneath subunit 3.5. To find out how to take the inverse of second- and third-order polynomials and to see which has a domain restriction, read the information on pages 206 through 209.
- Web Media: YouTube: Mathispower4u: “Restrict the Domain to Make a
Function 1 to 1, Then Find the Inverse”
Link: YouTube: Mathispower4u: “Restrict the Domain to Make a
Function 1 to 1, Then Find the
Inverse” (YouTube)
Instructions: Watch this video to learn how to restrict the domain of a second-order polynomial and then find the inverse. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
3.5.2 Inverses of Rational Functions Note: This subunit is covered in the reading assigned beneath subunit 3.5. To learn how to take the inverse of a rational function that is the ratio of two linear functions, read the information on pages 209 and 210.
Web Media: YouTube: Mathispower4u: “Find the Inverse of a Rational Function” Link: YouTube: Mathispower4u: “Find the Inverse of a Rational Function” (YouTube)
Instructions: Watch this video, which explains how to find the inverse of a rational function when x is in the numerator and the denominator. Take detailed notes and watch the video as many times as you need.
Watching this video should take approximately 10 minutes.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.5: Inverses and Radical Functions – Graded Homework” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Section 3.5: Inverses and Radical Functions – Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 3.5, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the activity. Your answers will be scored upon submission.
Completing this assessment should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use displayed on the webpage above.
Unit 3 Review - Assessment: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 3 Review Problems” Link: Washington State Board for Community & Technical Colleges: MyOpenMath: “Chapter 3 Review Problems” (HTML)
Instructions: Once you have completed the readings and viewed the
material for all of unit 3, complete this assessment. You will need
to create a free account in order to access the material. If you
have already created an account, log in to access the assessment.
Your answers will be scored upon submission. This will provide a
review for your unit assessment.
Completing this review should take approximately 1 hour.
Terms of use: Please respect the copyright and terms of use
displayed on the webpage above.
Unit 3 Exam - Assessment: Washington State Board for Community & Technical Colleges: “Chapter 3: Exam A”
Link: Washington State Board for Community & Technical Colleges:
[“Chapter 3: Exam
A”](http://school.saylor.org/mod/quiz/view.php?id=1459) (HTML)
Instructions: Without the use of a calculator, complete this
assessment to gauge your understanding of the topics covered in this
unit. The correct answers will be displayed when you click
the “Submit” button.
Completing this exam should take approximately 1 hour.