Graphing Rational Functions: Unveiling Holes and Asymptotes

PreAlgebra Grades High School 4:49 Video
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Lesson Description

Master the art of graphing rational functions by identifying holes (removable discontinuities), vertical and horizontal asymptotes, and intercepts. Learn to analyze function behavior using factoring and sign analysis.

Video Resource

Holes, Removable Discontinuities, Graphing Rational Functions

Mario's Math Tutoring

Duration: 4:49
Watch on YouTube

Key Concepts

  • Factoring rational functions
  • Identifying and locating removable discontinuities (holes)
  • Finding horizontal and vertical asymptotes
  • Determining x and y intercepts
  • Sign analysis for function behavior near asymptotes

Learning Objectives

  • Students will be able to factor rational functions to identify removable discontinuities.
  • Students will be able to determine the coordinates of holes in the graph of a rational function.
  • Students will be able to find horizontal and vertical asymptotes of rational functions.
  • Students will be able to determine the x and y intercepts of rational functions.
  • Students will be able to use sign analysis to determine the behavior of a rational function near its asymptotes and intercepts.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of a rational function and the concept of discontinuities. Briefly discuss vertical asymptotes as familiar discontinuities. Introduce the idea of 'removable discontinuities' or 'holes' as a new type of discontinuity that occurs when factors cancel in the numerator and denominator.
  • Factoring and Identifying Holes (10 mins)
    Explain the first crucial step: factoring both the numerator and denominator of the rational function. Emphasize that if a factor appears in both the numerator and denominator, it indicates a potential hole. Show how canceling this common factor simplifies the function, but the original factor still imposes a restriction on the domain.
  • Locating the Coordinates of the Hole (10 mins)
    Demonstrate how to find the x-coordinate of the hole by setting the canceled factor equal to zero and solving for x. Explain that this x-value is where the function is undefined due to the original form. Then, show how to substitute this x-value into the simplified function (after canceling the factor) to find the corresponding y-coordinate of the hole.
  • Finding Asymptotes and Intercepts (15 mins)
    Review how to find horizontal asymptotes by comparing the degrees of the numerator and denominator. Explain the rules for horizontal asymptotes based on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator. Review how to find vertical asymptotes by setting the remaining factors in the denominator (after canceling) equal to zero. Show how to find the x-intercepts by setting the numerator equal to zero and solving for x, and how to find the y-intercept by setting x equal to zero and solving for y.
  • Sign Analysis and Graphing (15 mins)
    Introduce the concept of sign analysis to determine the function's behavior near the vertical asymptotes. Explain how to choose test values slightly to the left and right of each vertical asymptote and substitute them into the factored form of the function (after canceling the common factor). Analyze the sign of the result (positive or negative) to determine whether the function approaches positive or negative infinity near the asymptote. Use the information gathered (holes, asymptotes, intercepts, and sign analysis) to sketch the graph of the rational function. Emphasize that the graph behaves like the simplified function everywhere except at the hole.
  • Practice Problems (10 mins)
    Work through several practice problems, having students identify the holes, asymptotes, and intercepts, and then sketch the graph. Encourage students to ask questions and work together.

Interactive Exercises

  • Graphing Challenge
    Divide students into small groups and give each group a different rational function to analyze and graph. Have each group present their findings to the class.
  • Error Analysis
    Present students with graphs of rational functions that contain errors (e.g., incorrectly placed holes or asymptotes). Have students identify and correct the errors.

Discussion Questions

  • Why is it important to factor the numerator and denominator when graphing rational functions?
  • How does the presence of a common factor in the numerator and denominator affect the graph of a rational function?
  • Explain the difference between a vertical asymptote and a hole in the graph of a rational function.
  • How does sign analysis help you determine the behavior of a rational function near its vertical asymptotes?

Skills Developed

  • Factoring polynomials
  • Analyzing rational functions
  • Graphing functions
  • Problem-solving
  • Critical thinking

Multiple Choice Questions

Question 1:

What is the first step in graphing a rational function to identify removable discontinuities?

Correct Answer: Factor the numerator and denominator

Question 2:

A removable discontinuity, or hole, occurs in a rational function when:

Correct Answer: A factor cancels in the numerator and denominator

Question 3:

How do you find the x-coordinate of a hole in a rational function?

Correct Answer: Set the canceled factor equal to zero

Question 4:

Where do vertical asymptotes occur on a rational function?

Correct Answer: Where the simplified denominator equals zero

Question 5:

How do you locate the y-coordinate of a hole?

Correct Answer: Substitute the x-coordinate of the hole into the simplified function after canceling factors

Question 6:

What is the purpose of sign analysis when graphing rational functions?

Correct Answer: To determine the behavior of the graph near vertical asymptotes

Question 7:

When is there a horizontal asymptote at y=0?

Correct Answer: Degree of numerator is less than degree of denominator

Question 8:

How do you find the x-intercept?

Correct Answer: Set y=0 and solve for x

Question 9:

What do you call the discontinuity when a factor in the numerator and denominator cancel?

Correct Answer: Removable Discontinuity

Question 10:

When can you say there is no horizontal asymptote?

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

Fill in the Blank Questions

Question 1:

A removable discontinuity is also known as a __________.

Correct Answer: hole

Question 2:

Vertical asymptotes occur where the __________ of the simplified rational function equals zero.

Correct Answer: denominator

Question 3:

To locate a hole's y-coordinate, substitute the x-coordinate into the __________ function.

Correct Answer: simplified

Question 4:

The x-intercepts are found by setting the __________ equal to zero.

Correct Answer: numerator

Question 5:

__________ analysis helps determine the function's behavior near vertical asymptotes.

Correct Answer: Sign

Question 6:

Before graphing, the first step is to _________ both the numerator and denominator.

Correct Answer: factor

Question 7:

A hole occurs when a factor appears in both the numerator and __________.

Correct Answer: denominator

Question 8:

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

Correct Answer: 0

Question 9:

Horizontal asymptotes describe the __________ behavior of the function.

Correct Answer: end

Question 10:

To find the y-intercept, set x equal to _________ and solve for y.

Correct Answer: 0

Educational Standards

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

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