Rational and Radical Functions

Students will extend their understanding of inverse functions to functions with a degree higher than 1, and factor and simplify rational expressions to reveal domain restrictions and asymptotes.

Unit Summary

In Unit 4, Rational and Radical Functions, students will extend their understanding of inverse functions to functions with a degree higher than 1. Alongside this concept, students will factor and simplify rational expressions and functions to reveal domain restrictions and asymptotes. Students will become fluent in operating with rational and radical expressions and use the structure to model contextual situations. In this unit, students will also revisit the concept of an extraneous solution, first introduced in Unit 1, through the solution of radical and rational equations. 

The unit begins with Topic A, where there is a focus on understanding the graphical and algebraic connections between rational and radical expressions, as well as fluently writing these expressions in different forms. In Topic B, students delve deeper into rational equations and functions and identify characteristics such as the $$x$$- and $$y$$-intercepts, asymptotes, and removable discontinuities based on the relationship between the degree of the numerator and denominator of the rational expression. Students will also connect these features with the transformation of the parent function of a rational function. In Topic C, students solve rational and radical equations, identifying extraneous solutions, then modeling and solving equations in situations where rational and radical functions are necessary. Students will connect the domain algebraically with the context and interpret solutions.

Assessment

This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Essential Understandings

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  • A rational function is a ratio of polynomial functions. If a rational function does not have a constant in the denominator, the graph of the rational function features asymptotic behavior and can have other features of discontinuity. 
  • Rational and radical equations that have algebraic numerators or denominators operate within the same rules as fractions. 
  • Extraneous solutions may result due to domain restrictions in rational or radical functions. 
  • Rational functions can be used to model situations in which two polynomials or root functions are divided. 

Vocabulary

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Vertical and horizontal asymptote Invertible functions
Rational function Zero product property 
Rational expression Asymptotic discontinuities (infinite)
Domain restriction Removable discontinuities
Square root / cube root End behavior
Extraneous solutions Sign chart

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate “Unit Summary."
  • Notice the progression of concepts through the unit using “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions 

Lesson Map

Topic A: Introduction to Rational and Radical Functions and Expressions

Topic B: Features of Rational Functions and Graphing Rational Functions

Topic C: Solve Rational and Radical Equations and Model with Rational Functions

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.D.6 — Rewrite simple rational expressions in different forms; write a(x /b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

  • A.APR.D.7 — Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Building Functions
  • F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

  • F.BF.B.4 — Find inverse functions.

Creating Equations
  • A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

High School — Number and Quantity
  • N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

  • N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Interpreting Functions
  • F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

  • F.IF.C.7.D — Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Reasoning with Equations and Inequalities
  • A.REI.A.2 — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

  • A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.A.1

Building Functions
  • F.BF.B.3

  • F.BF.B.4.A

Creating Equations
  • A.CED.A.4

Expressions and Equations
  • 8.EE.A.1

Interpreting Functions
  • F.IF.A.1

  • F.IF.B.4

  • F.IF.C.8

  • F.IF.C.8.A

Reasoning with Equations and Inequalities
  • A.REI.A.1

Seeing Structure in Expressions
  • A.SSE.A.1

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.