Curriculum  /  Math  /  11th Grade  /  Unit 4: Rational and Radical Functions  /  Lesson 9

Rational and Radical Functions

Lesson 9

Math

Unit 4

11th Grade

Lesson 9 of 18

Objective

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.

Common Core Standards

Core Standards

  • 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.

Foundational Standards

  • F.IF.B.4

Criteria for Success

  1. Describe horizontal and vertical asymptotes, and identify their location on a graph of a rational function. 
  2. Describe how to calculate the vertical and horizontal asymptotes of rational functions algebraically. 
  3. Identify the end behavior of rational functions with the same degree in the numerator and denominator. 
  4. Identify the $${x-}$$ and $${y-}$$intercepts of rational functions with the same degree in the numerator and denominator. 
  5. Use [Y=] and [GRAPH] to graph rational functions and horizontal asymptotes. 
  6. "Turn on" and "turn off" functions in [Y=].

Tips for Teachers

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Anchor Problems

25-30 minutes

Problem 1

Write a table of values for the function from $${-5<x<5}$$: 

$${f(x)={x^2-2\over{x^2}}}$$

Guiding Questions

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Problem 2

What do all of these functions have in common? 

$${f(x)={3x^3 + 2x\over{x^3-8}}}$$

$${g(x)={x^4-2x^2\over{x^4}}}$$

$${h(x)={x^2-4x\over{-2x^2-8}}}$$

           

             

            

 

Guiding Questions

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Problem 3

What is the end behavior, $$y$$-intercepts, and $$x$$-intercepts of $$f(x)$$? 

 

$$f(x)={3x^3+2x\over{x^3-8}}$$

 

Guiding Questions

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Target Task

5-10 minutes

A great target task for this lesson is question #21 on the 2012 Public Practice Exam: AP Calculus AB released by the College Board. We are unable to reproduce the content online here due to the policies of the College Board; however, teachers can find the exam and question at apcentral.collegeboard.org.

Ask these questions with the Target Task:

  • What is the vertical asymptote of the function you chose? 
  • What is the end behavior? 
  • What are the $${x-}$$ and $${y-}$$ intercepts of the function you chose? 

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.

Next

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

Lesson 10
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

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

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