Horizontal Asymptotes: Mastering Rational Functions

Algebra 2 Grades High School 2:14 Video
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Lesson Description

Learn to identify and determine horizontal asymptotes of rational functions by comparing the degrees of the numerator and denominator.

Video Resource

Find Horizontal Asymptote of a Rational Function

Mario's Math Tutoring

Duration: 2:14
Watch on YouTube

Key Concepts

  • Rational Functions
  • Horizontal Asymptotes
  • Degree of a Polynomial
  • Ratio of Leading Coefficients
  • Slant Asymptotes

Learning Objectives

  • Students will be able to determine the horizontal asymptote of a rational function by comparing the degrees of the numerator and denominator.
  • Students will be able to identify when a rational function has a horizontal asymptote at y=0.
  • Students will be able to recognize when a rational function has a slant asymptote instead of a horizontal asymptote.
  • Students will be able to calculate the horizontal asymptote given the leading coefficients when the degree of the numerator and denominator are the same.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a rational function and the concept of asymptotes. Briefly discuss the importance of understanding asymptotes for graphing rational functions. Introduce the video 'Find Horizontal Asymptote of a Rational Function' by Mario's Math Tutoring.
  • Video Viewing (7 mins)
    Play the video 'Find Horizontal Asymptote of a Rational Function.' Instruct students to take notes on the three cases discussed in the video: numerator degree equals denominator degree, numerator degree is less than denominator degree, and numerator degree is greater than denominator degree.
  • Discussion and Examples (10 mins)
    After watching the video, facilitate a class discussion on the three cases for determining horizontal asymptotes. Work through additional examples on the board, ensuring students understand how to identify the highest-powered terms and compare their degrees. Include examples where the function needs to be simplified first.
  • Practice Problems (10 mins)
    Provide students with a worksheet containing various rational functions. Have them determine the horizontal asymptote for each function. Circulate the classroom to provide assistance and answer questions.
  • Wrap-up (3 mins)
    Review the main points of the lesson. Emphasize the importance of comparing the degrees of the numerator and denominator. Briefly mention the concept of slant asymptotes and refer students to the linked video for further exploration.

Interactive Exercises

  • Degree Comparison Game
    Present students with pairs of polynomials (numerator and denominator). Have them compete in teams to quickly identify the degree of each polynomial and predict the horizontal asymptote (or lack thereof).
  • Whiteboard Practice
    Divide the class into small groups. Assign each group a rational function and have them work together on a whiteboard to determine the horizontal asymptote. Groups then present their solutions to the class.

Discussion Questions

  • What is the relationship between the degrees of the numerator and denominator and the existence of a horizontal asymptote?
  • When does a rational function have a horizontal asymptote at y=0? Explain why this occurs.
  • How do you determine the equation of a horizontal asymptote when the degrees of the numerator and denominator are the same?
  • What is a slant asymptote, and when does it occur in a rational function?

Skills Developed

  • Analytical Skills
  • Problem-Solving
  • Critical Thinking
  • Mathematical Reasoning

Multiple Choice Questions

Question 1:

What is the horizontal asymptote of the function f(x) = (3x^2 + 2x + 1) / (x^2 - 4)?

Correct Answer: y = 3

Question 2:

If the degree of the numerator of a rational function is less than the degree of the denominator, what is the horizontal asymptote?

Correct Answer: y = 0

Question 3:

Which of the following functions has a slant asymptote?

Correct Answer: f(x) = (x^2 + 1) / (x + 2)

Question 4:

What is the horizontal asymptote of the function f(x) = (5x) / (x^3 + 1)?

Correct Answer: y = 0

Question 5:

When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is:

Correct Answer: The ratio of the leading coefficients

Question 6:

For which function would you use polynomial long division to find the equation of an asymptote?

Correct Answer: f(x) = (x^2 + 1) / (x + 2)

Question 7:

What type of asymptote exists for the function f(x) = (x^3+2)/(x+4)?

Correct Answer: Slant

Question 8:

What is the horizontal asymptote of f(x) = (7x - 3) / (2x + 5)?

Correct Answer: y = 7/2

Question 9:

The horizontal asymptote of f(x) = 4 / (x^2 + 3) is:

Correct Answer: y = 0

Question 10:

If a rational function has a horizontal asymptote at y=2, what does this tell you about the end behavior of the function?

Correct Answer: The function approaches 2 as x approaches infinity or negative infinity

Fill in the Blank Questions

Question 1:

If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = _______.

Correct Answer: 0

Question 2:

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

Correct Answer: slant

Question 3:

To find the equation of a slant asymptote, you perform _______ _______.

Correct Answer: polynomial long division

Question 4:

When the degrees of the numerator and denominator are the same, the horizontal asymptote is the _______ of the leading coefficients.

Correct Answer: ratio

Question 5:

The horizontal asymptote describes the end _______ of a rational function.

Correct Answer: behavior

Question 6:

For f(x) = (4x^3 + 2x) / (x^3 - 5), the horizontal asymptote is y = _______.

Correct Answer: 4

Question 7:

If a rational function has no horizontal asymptote, and the degree of the numerator is more than the denominator, it may have a _______ asymptote.

Correct Answer: slant

Question 8:

The line y=0 is also known as the _______ axis.

Correct Answer: x

Question 9:

A rational function is a fraction where both the numerator and the denominator are _______.

Correct Answer: polynomials

Question 10:

The highest power of x in a polynomial is called the _______.

Correct Answer: degree

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.

Teaching Materials

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