Lesson 10 | Rational and Radical Functions | 11th Grade Mathematics

Objective

Identify features of rational functions with a larger degree in the denominator than in the numerator. Describe how to calculate these features algebraically.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

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.

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

Interpreting Functions

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

Foundational Standards

The foundational standards covered in this lesson

F.IF.B.4

Interpreting Functions

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. 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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe horizontal and vertical asymptotes and identify their location on a graph of a rational function.
Describe how to calculate the vertical and horizontal asymptotes of rational functions algebraically.
Describe how you know that the end behavior of functions with a larger degree in the denominator than in the numerator will approach zero.
Write a rational function with features of asymptotes, $${x- }$$and $${y-}$$intercepts, and end behavior described.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

What do all of these functions have in common?

$${j(x)={2x^2-2\over{x^3}}}$$	$${h(x)={x^2-2\over{x^4-81}}}$$	$${l(x)={3x+3\over{x^3-2x^2-3x}} }$$

Guiding Questions

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Problem 2

What are the $${x-}$$ and $${y-}$$intercepts and end behavior of the function $${h(x)}$$?

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

A rational function has a horizontal asymptote of zero and a vertical asymptote of zero.
What information can you determine about the equation of the rational function from this information?

Problem 2

Below is a function. What are the vertical and horizontal asymptotes of the function? What is the end behavior of the function?

$${g(x)={x+2\over{x^2}}}$$

$${h(x)={x^3+2\over{x^2}}}$$

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students need to write a function that contains particular properties pertaining to the asymptotes, intercepts, and end behavior.
Include problems that ask students to compare features and properties of different types of rational functions.

Mathematics Vision Project: Secondary Mathematics Three Module 4: Rational Expressions and Functions — Lesson 4.1

Lesson 9

Lesson 11

Lesson Map

Topic A: Introduction to Rational and Radical Functions and Expressions

Define rational functions. Identify domain restrictions of rational functions.

A.APR.D.6 F.IF.B.5

Identify domain restrictions algebraically for non-invertible functions.

F.BF.B.4 F.IF.C.7.B

Graph and transform square root and cubic root functions.

F.BF.B.3 F.IF.C.7.B

Write rational functions in equivalent radical form and identify domain restrictions of rational and radical functions.

F.IF.B.5 N.RN.A.2

Write radical and rational exponent expressions in equivalent forms.

N.RN.A.2

Multiply and divide rational expressions and simplify using equivalent expressions.

A.APR.D.6 A.APR.D.7

Add and subtract rational expressions.

A.APR.D.6 A.APR.D.7

Topic B: Features of Rational Functions and Graphing Rational Functions

Identify asymptotic discontinuities (also known as infinite discontinuities) and removable discontinuities in a rational function and describe why these discontinuities exist.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with equal degrees in the numerator and the denominator. Describe how to calculate features of these types of rational functions algebraically.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with a larger degree in the denominator than in the numerator. Describe how to calculate these features algebraically.

A.APR.D.6 F.IF.C.7.D

Identify features of rational functions with a larger degree in the numerator than in the denominator. Describe how to calculate these features algebraically.

A.APR.D.6 F.IF.C.7.D

Analyze the graph and equations of rational functions and identify features. Use features of a rational function to identify and construct appropriate equations and graphs.

A.APR.D.6 F.IF.C.7.D

Describe transformations of rational functions.

F.BF.B.3 F.IF.C.7.D

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

Solve simple radical equations.

A.REI.A.2

Solve radical equations and identify extraneous solutions.

A.REI.A.2

Solve rational equations.

A.APR.D.6 A.REI.A.2 A.REI.D.11

Write and solve rational functions for contextual situations.

A.APR.D.6 A.CED.A.2 N.Q.A.1

Analyze rational and radical functions in context and write rational functions for percent applications.

A.APR.D.6 A.CED.A.2 A.REI.A.2

Rational and Radical Functions

Objective

Common Core Standards

Core Standards

Arithmetic with Polynomials and Rational Expressions

Interpreting Functions

Foundational Standards

Interpreting Functions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Problem 1

Problem 2

Additional Practice

Lesson Map

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