Graphing Rational Functions: A Step-by-Step Guide

Algebra 2 Grades High School 22:27 Video
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Lesson Description

Master the art of graphing rational functions by identifying key features such as asymptotes, intercepts, and holes. Learn through examples and sign analysis.

Video Resource

Graphing Rational Functions Step-by-Step (Complete Guide 3 Examples)

Mario's Math Tutoring

Duration: 22:27
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Key Concepts

  • Rational Functions
  • Asymptotes (Vertical, Horizontal, Slant)
  • Intercepts (x and y)
  • Holes (Removable Discontinuities)
  • Sign Analysis

Learning Objectives

  • Students will be able to factor rational functions to identify holes and asymptotes.
  • Students will be able to determine vertical, horizontal, and slant asymptotes of a rational function.
  • Students will be able to find x and y intercepts of a rational function.
  • Students will be able to use sign analysis to determine the behavior of the graph near vertical asymptotes.
  • Students will be able to graph rational functions accurately by identifying key features.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a rational function and its general form. Briefly discuss the importance of understanding rational functions in various mathematical contexts.
  • Factoring and Identifying Holes (10 mins)
    Explain the importance of factoring the numerator and denominator of a rational function. Demonstrate how to identify holes (removable discontinuities) by canceling common factors. Provide examples.
  • Finding Asymptotes (15 mins)
    Explain how to determine vertical asymptotes by setting the non-canceled factors in the denominator equal to zero. Discuss the rules for finding horizontal asymptotes based on the degree of the numerator and denominator. Introduce the concept of slant asymptotes and how to find them using polynomial long division. Show examples of each type of asymptote.
  • Finding Intercepts (10 mins)
    Explain how to find the y-intercept by setting x=0 and solving for y. Explain how to find the x-intercept by setting y=0 and solving for x (which is equivalent to setting the numerator equal to zero). Provide examples.
  • Sign Analysis (15 mins)
    Explain how to use sign analysis to determine the behavior of the graph near vertical asymptotes. Demonstrate how to choose test points on either side of the vertical asymptotes and evaluate the sign of the function at those points. Relate the sign to whether the graph approaches positive or negative infinity.
  • Graphing and Examples (20 mins)
    Work through multiple examples of graphing rational functions step-by-step. Emphasize the importance of accurately plotting asymptotes and intercepts. Use sign analysis to sketch the graph's behavior near vertical asymptotes. Discuss how the graph approaches horizontal or slant asymptotes as x approaches infinity or negative infinity.
  • Conclusion (5 mins)
    Summarize the key steps in graphing rational functions. Review the concepts of holes, asymptotes, intercepts, and sign analysis. Encourage students to practice graphing rational functions on their own.

Interactive Exercises

  • Rational Function Matching
    Provide students with a set of rational functions and a set of graphs. Ask them to match each function to its corresponding graph.
  • Asymptote Identification
    Give students various rational functions and ask them to identify the vertical, horizontal, and slant asymptotes (if any).
  • Intercept Calculation
    Provide students with rational functions and ask them to calculate the x and y intercepts.
  • Graph Sketching Practice
    Have students practice sketching the graphs of rational functions on graph paper, using the techniques learned in the lesson.

Discussion Questions

  • Why is it important to factor the numerator and denominator of a rational function before graphing?
  • How do you determine whether a rational function has a horizontal, vertical, or slant asymptote?
  • Explain the relationship between the sign of a rational function and the behavior of its graph near a vertical asymptote.
  • Can a rational function cross a horizontal or slant asymptote? Why or why not?
  • What are the key features to look for when graphing a rational function?

Skills Developed

  • Factoring Polynomials
  • Solving Equations
  • Graphing Functions
  • Analytical Thinking
  • Problem-Solving

Multiple Choice Questions

Question 1:

Which of the following is the first step in graphing a rational function?

Correct Answer: Factor the numerator and denominator

Question 2:

A hole in a rational function occurs when:

Correct Answer: A factor in the numerator and denominator cancels out

Question 3:

Vertical asymptotes are found by setting:

Correct Answer: The denominator equal to zero

Question 4:

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:

Correct Answer: y = 0

Question 5:

A slant asymptote occurs when:

Correct Answer: The degree of the numerator is one greater than the degree of the denominator

Question 6:

To find the y-intercept, you set:

Correct Answer: x = 0

Question 7:

To find the x-intercept, you set:

Correct Answer: y = 0

Question 8:

What does sign analysis help determine about the graph?

Correct Answer: The behavior of the graph near vertical asymptotes

Question 9:

Can a rational function cross a vertical asymptote?

Correct Answer: No, never

Question 10:

Which method is used to find the equation of a slant asymptote?

Correct Answer: Polynomial Long Division

Fill in the Blank Questions

Question 1:

A rational function is a polynomial divided by another ___________.

Correct Answer: polynomial

Question 2:

A ___________ discontinuity, or hole, occurs when a factor cancels out in the numerator and denominator.

Correct Answer: removable

Question 3:

To find the vertical asymptotes, set the ___________ equal to zero and solve for x.

Correct Answer: denominator

Question 4:

If the degree of the denominator is higher than the degree of the numerator, the horizontal asymptote is y = ___________.

Correct Answer: 0

Question 5:

A slant asymptote exists when the degree of the numerator is exactly ___________ than the degree of the denominator.

Correct Answer: one greater

Question 6:

To find the y-intercept, substitute x = ___________ into the rational function.

Correct Answer: 0

Question 7:

To find the x-intercept(s), set y = ___________, which means setting the numerator equal to zero.

Correct Answer: 0

Question 8:

___________ analysis helps determine whether the graph approaches positive or negative infinity near vertical asymptotes.

Correct Answer: Sign

Question 9:

Polynomial ___________ is used to determine the equation of a slant asymptote.

Correct Answer: long division

Question 10:

It is okay for a rational function to ___________ a horizontal or slant asymptote.

Correct Answer: cross

Educational Standards

A2.F.1.6:Graph a rational function and identify the domain (including holes), range, x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology (excluding slant or oblique asymptotes). A2.A.2.3:Add, subtract, multiply, divide, and simplify rational expressions. A2.A.1.3:Solve one-variable rational equations and check for extraneous solutions.

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