Curriculum  /  Math  /  11th Grade  /  Unit 4: Rational and Radical Functions  /  Lesson 8

Rational and Radical Functions

Lesson 8

Math

Unit 4

11th Grade

Lesson 8 of 18

Objective

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

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

  • A.SSE.A.1

Criteria for Success

  1. Identify ways to manipulate algebraic expressions so that features can be shown. 
  2. Write the removable discontinuity as a coordinate point, $${(x, y).}$$
  3. Identify the features of the function, taking into account a removable discontinuity. 
  4. Use [TABLE] to show values of the function (and discontinuities). 
  5. Use [TBLSET] to change the table set-up to allow for manual entry (AUTO/ASK).
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Anchor Problems

25-30 minutes

Problem 1

Below is the graph and equation of the function $$f$$. 

 

$$f(x)=(x-2)(x-5)\over(x-2)(x+1)$$

 

And below is the graph and equation of the function $$g$$.

 

$$g(x)={(x-5)\over(x+1)}$$

 

How are the functions $$f$$ and $$g$$ similar Different? 

Guiding Questions

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Problem 2

How are the functions $${g(x)}$$ and $${h(x)}$$ different? Similar? 

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

 

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

Guiding Questions

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Target Task

5-10 minutes

Describe the asymptotes, intercepts, and coordinates of the removable discontinuity(s) in the function below. 

$${f(x)={x^2-6x+8\over{x-2}}}$$

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 write a coordinate point to fill in the removable discontinuity in a piecewise function. 
  • Include problems where students need to manipulate a rational function to identify the features, including any removable discontinuities.
  • Include problems where students need to identify the removable discontinuity graphically and algebraically. 
  • Include problems where students need to describe the difference between a removable discontinuity and a vertical asymptote. 

Next

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.

Lesson 9
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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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