Graphing Rational Functions: Asymptotes, Intercepts, and Behavior

PreAlgebra Grades High School 13:50 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to graph rational functions by identifying vertical, horizontal, and slant asymptotes, finding x and y-intercepts, and analyzing the function's behavior near asymptotes. Explore examples with varying degrees of numerator and denominator.

Video Resource

Rational Functions Graphing

Mario's Math Tutoring

Duration: 13:50
Watch on YouTube

Key Concepts

  • Vertical Asymptotes
  • Horizontal Asymptotes
  • Slant Asymptotes
  • X and Y Intercepts
  • Holes (Removable Discontinuities)
  • Sign Analysis

Learning Objectives

  • Identify and determine vertical, horizontal, and slant asymptotes of rational functions.
  • Calculate x and y-intercepts of rational functions.
  • Determine the location and y-coordinate of holes in rational functions.
  • Use sign analysis to determine the behavior of a rational function near vertical asymptotes.
  • Graph rational functions based on their asymptotes, intercepts, and behavior.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a rational function and its general form. Introduce the concept of asymptotes and their significance in graphing rational functions. Briefly mention the types of asymptotes (vertical, horizontal, slant) and holes.
  • Vertical Asymptotes (10 mins)
    Explain how to find vertical asymptotes by setting the denominator of the rational function equal to zero and solving for x. Emphasize that vertical asymptotes represent values where the function is undefined. Show examples from the video (1:19).
  • Horizontal Asymptotes (10 mins)
    Explain the rules for determining horizontal asymptotes based on the degrees of the numerator and denominator: degree(numerator) < degree(denominator) => y = 0 (0:12), degree(numerator) = degree(denominator) => y = ratio of leading coefficients (6:00), degree(numerator) > degree(denominator) => no horizontal asymptote (slant asymptote exists). Show examples from the video.
  • Slant Asymptotes (10 mins)
    Explain how to find slant asymptotes when the degree of the numerator is exactly one greater than the degree of the denominator. Demonstrate the use of polynomial long division to find the equation of the slant asymptote (11:30). Show examples from the video.
  • X and Y Intercepts (5 mins)
    Explain how to find x-intercepts by setting the numerator equal to zero and solving for x. Explain how to find y-intercepts by setting x equal to zero and solving for y. Show examples from the video (1:58 & 8:30).
  • Holes (Removable Discontinuities) (5 mins)
    Explain how to identify holes when a factor cancels from both the numerator and the denominator. Explain how to find the y-coordinate of the hole by substituting the x-value into the simplified function (7:12). Show examples from the video.
  • Sign Analysis (10 mins)
    Demonstrate how to use sign analysis to determine the behavior of the function near vertical asymptotes. Choose test points to the left and right of each vertical asymptote and determine whether the function approaches positive or negative infinity (2:49 & 9:30). Show examples from the video.
  • Graphing Rational Functions (10 mins)
    Summarize the steps for graphing rational functions: find asymptotes, find intercepts, find holes, perform sign analysis, and sketch the graph. Work through additional examples.

Interactive Exercises

  • Asymptote Identification
    Provide students with a set of rational functions and ask them to identify the vertical, horizontal, and slant asymptotes.
  • Intercept Calculation
    Provide students with a set of rational functions and ask them to calculate the x and y-intercepts.
  • Graphing Practice
    Provide students with a set of rational functions and ask them to graph them, identifying all key features (asymptotes, intercepts, holes, behavior near asymptotes).

Discussion Questions

  • How does the relationship between the degrees of the numerator and denominator affect the existence and location of horizontal and slant asymptotes?
  • What is the significance of vertical asymptotes and holes in the graph of a rational function?
  • How does sign analysis help in understanding the behavior of a rational function near its vertical asymptotes?
  • Can a rational function cross a horizontal asymptote? Why or why not?
  • How do you determine the y-coordinate of a hole in a rational function?

Skills Developed

  • Analytical Thinking
  • Problem Solving
  • Graphing
  • Algebraic Manipulation
  • Function Analysis

Multiple Choice Questions

Question 1:

A rational function has a vertical asymptote at x = a if:

Correct Answer: The denominator is zero at x = a

Question 2:

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

Correct Answer: y = 0

Question 3:

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

Correct Answer: y = ratio of leading coefficients

Question 4:

A rational function has a slant asymptote if:

Correct Answer: Degree of numerator = degree of denominator + 1

Question 5:

The x-intercepts of a rational function are found by:

Correct Answer: Setting the numerator equal to zero

Question 6:

A 'hole' in the graph of a rational function occurs when:

Correct Answer: A factor cancels from both numerator and denominator

Question 7:

To find the y-coordinate of a hole:

Correct Answer: Substitute the x-value of the hole into the simplified function

Question 8:

Sign analysis helps determine:

Correct Answer: The horizontal asymptote

Question 9:

Which of the following statements is true about crossing horizontal asymptotes?

Correct Answer: A rational function can cross a horizontal asymptote

Question 10:

Which of these functions would have a slant asymptote?

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

Fill in the Blank Questions

Question 1:

A vertical asymptote occurs where the __________ of the rational function equals zero.

Correct Answer: denominator

Question 2:

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = __________.

Correct Answer: 0

Question 3:

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

Correct Answer: slant

Question 4:

To find the x-intercept(s), set the __________ of the rational function equal to zero and solve.

Correct Answer: numerator

Question 5:

A hole occurs in the graph when a factor __________ from both the numerator and the denominator.

Correct Answer: cancels

Question 6:

Sign analysis involves testing points on either side of the __________ asymptote.

Correct Answer: vertical

Question 7:

The equation of a slant asymptote is found using polynomial __________.

Correct Answer: division

Question 8:

The y-intercept is found by setting x = __________ in the rational function.

Correct Answer: 0

Question 9:

As X approaches infinity, the graph of a rational function will approach the __________ asymptote.

Correct Answer: horizontal

Question 10:

A removable discontinuity is also known as a __________.

Correct Answer: hole

Educational Standards

PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.3.2:Graphically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans