Graphing Rational Functions: Unveiling Asymptotes and Holes

Algebra 2 Grades High School 18:36 Video
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Lesson Description

Master the art of graphing rational functions by understanding asymptotes, holes, intercepts, and strategic point plotting. Learn to analyze factored forms and apply division to identify key features.

Video Resource

Graph Rational Function Algebra 2

Kevinmathscience

Duration: 18:36
Watch on YouTube

Key Concepts

  • Factoring Rational Functions
  • Vertical, Horizontal, and Slant Asymptotes
  • Holes in Rational Functions
  • X and Y Intercepts
  • Graphing Techniques: Plotting Points

Learning Objectives

  • Factor rational functions to identify common factors that create holes.
  • Determine vertical, horizontal, and slant asymptotes of rational functions.
  • Find x and y intercepts of rational functions.
  • Graph rational functions using asymptotes, intercepts, and strategic point plotting.
  • Determine the domain of a rational function, accounting for vertical asymptotes and holes.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic concept of rational functions. Introduce the idea that these functions can have asymptotes and holes, which are critical to understand when graphing. Briefly preview the video.
  • Video Viewing and Note-Taking (20 mins)
    Play the Kevinmathscience video, pausing at key points to allow students to take notes. Encourage students to write down the steps for finding asymptotes, intercepts, and holes. Have students follow along with the example problem, solving it independently.
  • Asymptote Identification (15 mins)
    Discuss the three types of asymptotes: vertical, horizontal, and slant. Work through examples of each, emphasizing the rules for determining their presence based on the degrees of the numerator and denominator. Provide additional examples beyond what the video covers.
  • Hole Identification and Treatment (10 mins)
    Reinforce the concept of holes and how they arise from common factors in the numerator and denominator. Explain how to find the x-coordinate of a hole by setting the common factor to zero. And y-coordinate by plugging the x-coordinate into the reduced function.
  • Intercept Calculation (10 mins)
    Review the methods for finding x and y intercepts, emphasizing setting y=0 to find x-intercepts and x=0 to find y-intercepts.
  • Graphing Practice and Point Plotting (20 mins)
    Guide students through the process of graphing a rational function step-by-step. Emphasize the importance of plotting additional points, especially near asymptotes, to determine the shape of the graph. Encourage students to use graphing calculators to verify their results.
  • Wrap-up and Review (10 mins)
    Summarize the key steps involved in graphing rational functions. Answer any remaining questions. Preview the upcoming quiz to reinforce learning.

Interactive Exercises

  • Factoring Frenzy
    Provide students with a worksheet containing various rational functions. Have them factor the numerator and denominator of each function.
  • Asymptote Scavenger Hunt
    Give students a set of rational functions and ask them to identify all vertical, horizontal, and slant asymptotes.
  • Graphing Challenge
    Assign students to graph rational functions on their own using asymptotes, intercepts, and point plotting, and then compare with a partner.

Discussion Questions

  • Why is it important to factor rational functions before graphing them?
  • Explain the difference between a vertical asymptote and a hole.
  • How does the degree of the numerator and denominator affect the existence of horizontal and slant asymptotes?
  • Can a rational function cross a horizontal asymptote? If so, where?

Skills Developed

  • Factoring Polynomials
  • Analyzing Rational Functions
  • Graphing Functions
  • Problem-Solving
  • Critical Thinking

Multiple Choice Questions

Question 1:

Which of the following indicates the presence of a 'hole' in a rational function?

Correct Answer: A common factor in the numerator and denominator.

Question 2:

A rational function has a horizontal asymptote at y=0 when:

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

Question 3:

How do you find the x-intercept(s) of a rational function?

Correct Answer: Set y = 0 and solve for x.

Question 4:

What determines the location of a vertical asymptote?

Correct Answer: The roots of the denominator.

Question 5:

If the degree of the numerator is exactly one more than the degree of the denominator, the rational function has:

Correct Answer: A slant asymptote.

Question 6:

What is the first step in graphing a rational function?

Correct Answer: Factor the numerator and denominator.

Question 7:

Can a rational function cross a vertical asymptote?

Correct Answer: No, never.

Question 8:

What does it mean if a rational function has no horizontal asymptote?

Correct Answer: It may have a slant asymptote, or neither a slant nor a horizontal.

Question 9:

To find the y-coordinate of a hole, after finding the x-coordinate, you must...

Correct Answer: Plug the x-coordinate into the simplified equation (after cancellation).

Question 10:

Which of the following statements is true?

Correct Answer: Graphs can cross horizontal asymptotes in the middle of the graph.

Fill in the Blank Questions

Question 1:

If a factor cancels from both the numerator and denominator, there is a ______ at that point.

Correct Answer: hole

Question 2:

A ______ asymptote occurs when the degree of the numerator is one greater than the degree of the denominator.

Correct Answer: slant

Question 3:

Vertical asymptotes are found by setting the ______ equal to zero and solving for x.

Correct Answer: denominator

Question 4:

To find the y-intercept, substitute ______ for x and solve.

Correct Answer: 0

Question 5:

The values excluded from the domain of a rational function correspond to vertical asymptotes and ______.

Correct Answer: holes

Question 6:

A rational function has a horizontal asymptote at y=0 when the degree of the denominator is _______ than the degree of the numerator.

Correct Answer: greater

Question 7:

The line that a graph approaches but does not cross (except maybe in the middle of the graph) is called an ________.

Correct Answer: asymptote

Question 8:

Before graphing, it is important to ________ the rational function.

Correct Answer: factor

Question 9:

A graph ________ cross a vertical asymptote.

Correct Answer: cannot

Question 10:

If the degrees of the numerator and denominator are equal, the horizontal asymptote is y equals the leading coefficient of the numerator ________ by the leading coefficient of the denominator.

Correct Answer: divided

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.2.1:Factor polynomial expressions including, but not limited to, trinomials, differences of squares, sum and difference of cubes, and factoring by grouping, using a variety of tools and strategies. A2.A.2.3:Add, subtract, multiply, divide, and simplify rational expressions.

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