 # K12MATH011: Algebra II

Unit 5: Rational Functions   The first variation you will experience from polynomial functions is the set of rational functions, which involves expressions that contain variables in the denominator (bottom) of fractions. Such expressions cannot accept values for denominators that cause a zero in the bottom of any fraction. For instance, in the function f(x) = 3/x - 4 , x cannot be equal to 4 or the expression will have a zero in the bottom. You will learn about operations on rational functions, how to simplify rational expressions, how to graph rational functions, and how to find solutions of rational equations and inequalities. Finding solutions follows from what you learn from polynomials and often requires checking the answer(s), as some potential solutions may cause a zero in a denominator, which cannot be allowed.

Unit 5 Time Advisory
Completing this unit should take approximately 10 hours and 30 minutes.

☐    Subunit 5.1: 2 hours and 15 minutes

☐    Subunit 5.2: 2 hours and 15 minutes

☐    Subunit 5.3: 45 minutes

☐    Subunit 5.4: 1 hour and 30 minutes

☐    Subunit 5.5: 3 hours and 45 minutes

Unit5 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Find the results of operations on rational expressions. - Graph a rational function. - Determine the inverse of a rational function. - Find solutions for rational equations and inequalities.

5.1 Inverse, Joint, and Combined Variation   Rational expressions fall out of the original study of inverse, joint, and combined variation. This is a good historical starting point for this unit. Rational expressions have variables in the denominator of fractions. Rate of change problems in calculus often involve rational expressions, and these types of problems are used in many different disciplines. For instance, in business there are rate of production problems involving numbers of laborers that form inverse variations. In physics, almost any problem involving the distance and gravity between objects in space results in an inverse variation. And in chemistry, energy equations for chemical reactions are often written as direct variation.

• Explanation: CK-12: “Direct Variation” Link: CK-12: “Direct Variation” (HTML and YouTube)

Instructions: This is an easy-to-read, yet detailed discussion of direct variation. It is especially useful for using direct variation in real-world applications. It also contains videos with further details.

Reading the material and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Explanation: CK-12: “Inverse Variation Models” Link: CK-12: “Inverse Variation Models” (HTML and YouTube)

Instructions: This is an easy-to-read, yet detailed discussion of inverse variation. It is especially useful for using inverse variation in real-world applications. It also contains a video with further details on proportionality.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Direct and Inverse Variation” Link: Khan Academy’s “Direct and Inverse Variation” (HTML)

Instructions: This is a short self-assessment on direct and inverse variation. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

5.2 Graphing Rational Functions   One of the first things we learn about fractions is that zero can’t be in the denominator. For this reason, rational functions can be tricky, as they may have several values that can cause zeros in denominators. These values are called “asymptotes” and the graphs of functions at these asymptotes may be wild. For that reason, follow the content below and read or watch as many times as necessary, until the ideas make sense.

• Explanation: CK-12: “Graphs of Rational Functions” Link: CK-12: “Graphs of Rational Functions” (HTML and Vimeo)

Instructions: This video discusses graphing rational functions, and then proceeds into a detailed outline on the process. Note: There is a second video on the page as a follow-up to the topic.

Watching the video and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Asymptotes of Rational Functions” Link: Khan Academy’s “Asymptotes of Rational Functions” (YouTube)

Instructions: This video provides a detailed discussion and step-by-step approaches to graphing rational functions. Graphs of rational functions flow around their asymptotes (the x-values where the function is undefined; that is, the value for x causes a zero in the denominator), so asymptotes are a great place to begin the discussion.

Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (Common Core):

• Web Media: GeoGebra: “Intro Graphing Rational Functions” Link: GeoGebra: “Intro Graphing Rational Functions” (HTML)

Instructions: This simple interactive graph allows you to change values in the basic transformation rational form (f(x) = a/(x - h) + k) to view how the graph itself changes. Note carefully what each slider controls and which values control horizontal and vertical asymptotes. After trying different variations, answer the five sets of questions at the bottom of the page. Write at least one full paragraph for each question.

Practicing with this application and writing notes should take approximately 1 hour.

Standards Addressed (Common Core):

5.3 Multiplying and Dividing Rational Expressions   We multiply top to top and bottom to bottom of a fraction, then reduce. It’s helpful, therefore, to factor everywhere first. Division merely requires flipping the divisor into its reciprocal and then multiplying.

5.4 Adding and Subtracting Rational Expressions   Rational expressions may only be added if the terms are written with common denominators. We see this is true for any set of fractions that are added. To find common denominators for rational expressions, we must first factor each denominator and then use the factors to find a smallest common denominator.

• Explanation: CK-12: “Applications of Adding and Subtracting Rational Expressions” Link: CK-12: “Applications of Adding and Subtracting Rational Expressions” (HTML and Vimeo)

Instructions: This page begins with a video on adding and subtracting rational expressions. It then contains some pretty detailed discussion of the process of adding and subtracting algebraic fractions, including methods for finding common denominators. The page ends with a video that shows the examples on the page as they are worked out.

Reading the material, watching the videos, and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

Instructions: This video discusses adding and subtracting rational expressions, with a good explanation on finding common denominators. Note: This is one of three videos on the topic.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Adding and Subtracting Rational Expressions 2” Link: Khan Academy’s “Adding and Subtracting Rational Expressions 2” (YouTube)

Instructions: This video discusses adding and subtracting rational expressions. Note: This is the second of three videos on the topic.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Adding and Subtracting Rational Expressions 3” Link: Khan Academy’s “Adding and Subtracting Rational Expressions 3” (YouTube)

Instructions: This video discusses adding and subtracting rational expressions. Note: This is the last of three videos on the topic.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

5.5 Solving Rational Equations and Inequalities   Solving rational equations is fairly easy if the lowest common denominator is used to eliminate fractions on both sides of the equation first. Polynomial methods may be used to solve the equation once that is done. Caution needs to be taken that the solution found does not cause a zero in any denominator of any of the original expressions of the equation.

• Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 4: Rational Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 4: Rational Functions” (PDF)

Instructions: Read pages 301–351, and then try the problems at the end of the chapter. Answers are on the pages that follow. The text is a bit technical, but coupled with the videos and other content in this unit, it should become clearer. Write, in your own words, a clear but brief study guide on graphing rational functions. In this study guide, discuss how to find solutions, how to find all asymptotes, and then how to determine where the curves are going (that is, on each end, is the curve going up or down?).

Completing this activity should take approximately 2 hours.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Solving Rational Equations 1” Link: Khan Academy’s “Solving Rational Equations 1” (YouTube)

Instructions: This video discusses the basics of solving rational equations and emphasizes the use of the lowest common denominator to simplify the equation before solving. Note: Always check the final answer for any nonpolynomial function, as the potential answer may cause the equation to be undefined. Note also that this is the first of three videos.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Solving Rational Equations 2” Link: Khan Academy’s “Solving Rational Equations 2” (YouTube)

Instructions: This video discusses the basics of solving rational equations and emphasizes the use of the lowest common denominator to simplify the equation before solving. Note: Always check the final solution for any nonpolynomial function, as the potential answer may cause the equation to be undefined. Note also that this is the second of three videos.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Solving Rational Equations 3” Link: Khan Academy’s “Solving Rational Equations 3” (YouTube)

Instructions: This video discusses the basics of solving rational equations. This emphasizes the use of the lowest common denominator to simplify the equation before solving. Note: Always check the final solution for any nonpolynomial function, as the potential answer may cause the equation to be undefined. Note also that this is the last of three videos.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):