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Asymptote Adventures: Graphing Rational Functions

Asymptote Adventures: Graphing Rational Functions

PreAlgebra Grades High School 4:12 Video

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Lesson Description

Master the art of finding horizontal, vertical, and slant asymptotes to accurately graph rational functions. Learn to identify holes and discontinuities.

Video Resource

Finding Horizontal and Vertical Asymptotes Graphing Rational Functions

Mario's Math Tutoring

Duration: 4:12

Watch on YouTube

Key Concepts

Horizontal Asymptotes
Vertical Asymptotes
Slant Asymptotes
Removable Discontinuities (Holes)
Degree of Polynomials

Learning Objectives

Students will be able to determine horizontal asymptotes by comparing the degrees of the numerator and denominator.
Students will be able to identify vertical asymptotes by finding the zeros of the denominator after simplifying the rational function.
Students will be able to find the equation of a slant asymptote using polynomial long division.
Students will be able to recognize and identify removable discontinuities (holes) in rational functions.
Students will be able to graph rational functions, including asymptotes and holes.

Educator Instructions

Introduction (5 mins)
Begin by reviewing the definition of a rational function and the concept of asymptotes. Briefly discuss why asymptotes are important for understanding the behavior of functions. Introduce the video by Mario's Math Tutoring as a helpful resource for learning about finding asymptotes.
Horizontal Asymptotes (15 mins)
Watch the section of the video from 0:10 to 2:03. Discuss the three cases for determining horizontal asymptotes: 1. Degree of denominator > degree of numerator: y = 0 2. Degree of denominator = degree of numerator: y = ratio of leading coefficients 3. Degree of numerator > degree of denominator: No horizontal asymptote (may have a slant asymptote)
Slant Asymptotes (10 mins)
Watch the section of the video from 2:03 to 2:14. Explain how to find the equation of a slant asymptote using polynomial long division. Emphasize that a slant asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator.
Vertical Asymptotes and Holes (15 mins)
Watch the section of the video from 3:27 to the end. Explain the process of factoring the numerator and denominator to identify vertical asymptotes and holes. Emphasize that a vertical asymptote occurs at a zero of the denominator that does *not* cancel with a factor in the numerator. A hole occurs at a value where a factor cancels from both the numerator and denominator.
Practice Problems (15 mins)
Work through several example problems as a class, guiding students through the steps of finding horizontal, vertical, and slant asymptotes, and identifying holes. Provide additional practice problems for students to work on individually or in small groups.

Interactive Exercises

Asymptote Scavenger Hunt
Provide students with a set of rational functions. Students work individually or in groups to identify all asymptotes (horizontal, vertical, and slant) and any holes. The first student/group to correctly identify all asymptotes and holes wins.
Graphing Challenge
Assign students a rational function to graph. Students must show all work for finding asymptotes and holes and create an accurate graph of the function, including all key features.

Discussion Questions

Why is it important to factor the numerator and denominator before identifying vertical asymptotes?
How does the degree of the numerator and denominator affect the end behavior of a rational function?
Can a rational function have both a horizontal and a slant asymptote? Why or why not?
Why do holes occur in the graph of a rational function?

Skills Developed

Algebraic Manipulation
Problem-Solving
Analytical Thinking
Graphing Techniques

Multiple Choice Questions

Question 1:

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

y = 0

y = 3

y = 1

There is no horizontal asymptote

Correct Answer: y = 3

Question 2:

Where does a vertical asymptote occur in a rational function?

At zeros of the numerator

At zeros of the denominator after simplification

At zeros of both the numerator and denominator

Where the function crosses the x-axis

Correct Answer: At zeros of the denominator after simplification

Question 3:

When does a rational function have a slant asymptote?

When the degree of the numerator is less than the degree of the denominator

When the degree of the numerator is equal to the degree of the denominator

When the degree of the numerator is exactly one greater than the degree of the denominator

When the degree of the denominator is exactly one greater than the degree of the numerator

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

Question 4:

How do you find the equation of a slant asymptote?

By finding the y-intercept of the function

By using the quadratic formula

By performing polynomial long division

By finding the derivative of the function

Correct Answer: By performing polynomial long division

Question 5:

What is a 'hole' in the graph of a rational function?

A vertical asymptote

A point where the function is undefined but continuous

A point where the function crosses the x-axis

A horizontal asymptote

Correct Answer: A point where the function is undefined but continuous

Question 6:

The function f(x) = (x-2)/(x^2-4) has:

A vertical asymptote at x=2

A hole at x=2

Both a hole and a vertical asymptote at x=2

Neither a hole nor a vertical asymptote at x=2

Correct Answer: A hole at x=2

Question 7:

What is the horizontal asymptote of f(x) = (5x)/(x^2 + 1)?

y = 0

y = 5

y = 1

There is no horizontal asymptote

Correct Answer: y = 0

Question 8:

Which of the following functions would have a slant asymptote?

f(x) = (x+1)/(x^2-1)

f(x) = (x^2-1)/(x+1)

f(x) = (x^2+1)/(x^2-1)

f(x) = (x+1)/(x+1)

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

Question 9:

How can you identify a removable discontinuity (hole) in a rational function?

By looking for x-intercepts

By finding factors that cancel in the numerator and denominator

By determining the domain of the function

By graphing the function and observing gaps

Correct Answer: By finding factors that cancel in the numerator and denominator

Question 10:

What is the vertical asymptote of the function f(x) = 1/(x-3)?

x = 0

x = 3

x = -3

There is no vertical asymptote

Correct Answer: x = 3

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 more than the degree of the denominator.

Correct Answer: slant

Question 3:

Before identifying vertical asymptotes, it is important to _______ the numerator and denominator.

Correct Answer: factor

Question 4:

A _______ occurs when a factor cancels out from both the numerator and the denominator.

Correct Answer: hole

Question 5:

To find the equation of a slant asymptote, use polynomial _______.

Correct Answer: division

Question 6:

The horizontal asymptote of f(x) = (4x^3 + x)/(2x^3 - 5) is y = _______.

Correct Answer: 2

Question 7:

The vertical asymptote of f(x) = 1/(x + 5) is x = _______.

Correct Answer: -5

Question 8:

If a rational function has a hole at x = a, then the function is ________ at x = a.

Correct Answer: undefined

Question 9:

As x approaches positive or negative infinity, the graph of a function approaches its ________ asymptote.

Correct Answer: horizontal

Question 10:

The function f(x) = (x^2 - 9)/(x - 3) has a hole at x = _______.

Correct Answer: 3

Educational Standards

PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.3.2:Graphically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation.

Teaching Materials

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