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.

Standards Addressed (Common Core):
- CCSS.Math.Content.HSA-SSE.A.1 - CCSS.Math.Content.HSA-APR.D.6 - CCSS.Math.Content.HSA-APR.D.7 - CCSS.Math.Content.HSA-REI.D.11 - CCSS.Math.Content.HSF-BF.B.4 - CCSS.ELA-Literacy.RST.11-12.2 - CCSS.ELA-Literacy.RST.11-12.5 - CCSS.ELA-Literacy.WHST.11-12.1 - CCSS.ELA-Literacy.WHST.11-12.2 - CCSS.ELA-Literacy.WHST.11-12.4

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. 

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.

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.

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.