Curriculum  /  Math  /  11th Grade  /  Unit 4: Rational and Radical Functions  /  Lesson 1

Rational and Radical Functions

Lesson 1

Math

Unit 4

11th Grade

Lesson 1 of 18

Objective

Define rational functions. Identify domain restrictions of rational functions.

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.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.

Foundational Standards

  • F.IF.A.1
  • F.IF.C.8.A

Criteria for Success

  1. Describe how a rational function is like a numeric fraction and that you cannot have a denominator of zero because the function will be undefined. 
  2. Identify the shape formed by $${f(x)={1\over{x}}}$$ as a rational function.
  3. Find the domain of a rational function, including excluded values based on the denominator of the equation. 
  4. Define vertical and horizontal asymptotes, and compare to a domain restriction.
  5. Factor the numerator and denominator and use the zero product property to find excluded values from the domain. 
  6. Use [Y=] and [GRAPH] to graph rational functions.
  7. Use [WINDOW] to adjust the viewing window. Review how to adjust the scaling factor.

Tips for Teachers

A common error when students are using a calculator for rational functions is to forget to use parentheses to group the numerator together and the denominator together. Ensure that students remember this important step. 

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Anchor Problems

Problem 1

Below are two functions:

$${f(x)=1}$$

$${g(x)=x}$$

Graph the quotient $${{f(x)}\over{g(x)}}$$. Name this function as $${h(x).}$$

Guiding Questions

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Problem 2

From what you know about transformations, what value(s) of $${x }$$ cannot be part of the domain of the function: $${g(x)={1\over{x-6}} }$$? 

Guiding Questions

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Problem 3

What value(s) of $$x$$ cannot be part of the domain of the function? 
 

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

Guiding Questions

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Target Task

Problem 1

Find an equivalent rational expression in lowest terms, and identify the domain. 

$${{{x^2-7x+12}\over{6-5x+x^2}}}$$

References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 22 — Exit Ticket, Question #1

Algebra II > Module 1 > Topic C > Lesson 22 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Determine whether or not the rational expressions $${{x+4\over{(x+2)(x-3)}}}$$ and $${{x^2+5x+4\over{(x+1)(x+2)(x-3)}}}$$ are equivalent for $${x\neq -1}$$, $${x\neq -2}$$, and $${x\neq 3}$$. Explain how you know.

References

EngageNY Mathematics Algebra II > Module 1 > Topic C > Lesson 22 — Exit Ticket, Question #2

Algebra II > Module 1 > Topic C > Lesson 22 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

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 that require students to factor without then simplifying a rational expression and identifying domain. 
  • Include some problems where students need to name the parent functions of common functions. 
  • Include quadratic equations with real roots but not easily factorable. Identify the zeros. 
  • Include cubic and quartic functions as the denominator of the rational expression to practice finding zeros of other degrees other than quadratic. 
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Lesson 2

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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