Mathematics · 11th Grade · Rational and Radical Relationships · Weeks 1-9

Graphing Rational Functions: Vertical Asymptotes

Students will identify and graph vertical asymptotes of rational functions based on their denominators.

• Common Core State Standards
• CCSS.Math.Content.HSF.IF.C.7d

About This Topic

Asymptotes and discontinuity are fundamental concepts in the study of rational functions. Students learn to identify where a function is undefined and how it behaves near those points. Vertical asymptotes represent values that the function can never reach, while horizontal asymptotes describe the long term behavior of the function. This topic is essential for understanding limits and the behavior of complex systems where certain inputs are restricted.

In the Common Core framework, analyzing these features helps students build a comprehensive understanding of function families. They learn to distinguish between 'holes' (removable discontinuities) and 'breaks' (non-removable discontinuities). This distinction is vital for modeling real world phenomena like electrical circuits or population limits. Students grasp this concept faster through structured discussion and peer explanation, where they can compare different rational functions and identify their unique boundaries.

Key Questions

1. Explain what causes a vertical asymptote in a rational function.
2. Predict the behavior of a rational function as it approaches a vertical asymptote.
3. Compare the graphical representation of a vertical asymptote to a hole in the graph.

Learning Objectives

• Identify the values of x that make the denominator of a rational function equal to zero.
• Calculate the equations of vertical asymptotes for given rational functions.
• Compare the graphical behavior of a rational function near a vertical asymptote versus a hole.
• Explain the algebraic condition that leads to a vertical asymptote in a rational function.
• Graph rational functions, accurately plotting vertical asymptotes.

Before You Start

Factoring Polynomials

Why: Students need to factor quadratic and higher-order polynomials to find the roots of the denominator.

Solving Equations

Why: Students must be able to solve equations, specifically setting polynomial denominators equal to zero and finding their roots.

Understanding Functions

Why: Students should have a foundational understanding of what a function is, including its domain and range, and how to evaluate functions for given inputs.

Key Vocabulary

Rational Function	A function that can be written as the ratio of two polynomial functions, p(x)/q(x), where q(x) is not the zero polynomial.
Vertical Asymptote	A vertical line, x = a, that the graph of a function approaches but never touches. It occurs where the denominator of a simplified rational function is zero.
Denominator	The part of a fraction that is below the line, indicating the number of equal parts into which the whole is divided.
Undefined	A mathematical expression that does not have a meaning or cannot be evaluated, such as division by zero.

Watch Out for These Misconceptions

Common Misconception

Students often think a graph can never cross any asymptote.

What to Teach Instead

Show examples of functions that cross their horizontal asymptotes. Use a collaborative graphing activity to demonstrate that horizontal asymptotes only describe behavior at the far ends of the graph, not the middle.

Common Misconception

Students may confuse vertical asymptotes with holes.

What to Teach Instead

Use a hands-on factoring activity. Show that if a factor cancels out, it creates a hole; if it remains in the denominator, it creates a vertical asymptote. Peer discussion helps clarify this 'canceling' logic.

Active Learning Ideas

Gallery Walk: Function Features

Post several rational function graphs around the room. Students move in groups to identify the vertical asymptotes, horizontal asymptotes, and any holes, recording their findings on a shared chart.

30 min·Small Groups

Inquiry Circle: The Mystery of the Hole

Pairs are given two similar looking rational functions, one with a hole and one with a vertical asymptote. They must simplify the expressions and use a graphing tool to discover why one factor cancels out while the other creates a break.

25 min·Pairs

Think-Pair-Share: Asymptote Predictions

Students are given a set of rational equations and must predict the horizontal asymptote based on the degrees of the numerator and denominator. They then share their logic with a partner before verifying with a graph.

15 min·Pairs

Real-World Connections

• In electrical engineering, the impedance of certain circuits can be modeled by rational functions. Vertical asymptotes can represent frequencies where the circuit's response becomes infinitely large, indicating resonance or instability.
• Chemical engineers use rational functions to model reaction rates or concentrations that approach limits. Vertical asymptotes might signify conditions where a reaction becomes impossible or infinitely fast, guiding process design.

Assessment Ideas

Quick Check

Provide students with the rational function f(x) = (x+2)/(x^2 - 9). Ask them to: 1. Factor the denominator. 2. Identify the x-values that make the denominator zero. 3. State the equations of the vertical asymptotes.

Exit Ticket

On one side of an index card, write a rational function with a vertical asymptote. On the other side, write a rational function with a hole. Students must identify the vertical asymptote for the first function and the location of the hole for the second function.

Discussion Prompt

Pose the question: 'How is the behavior of a rational function as x approaches a vertical asymptote different from its behavior as x approaches a hole?' Guide students to discuss limits and the concept of a function approaching infinity versus approaching a specific y-value.

Frequently Asked Questions

What causes a vertical asymptote in a rational function?

A vertical asymptote occurs at any x-value that makes the denominator zero, provided that the factor in the denominator does not cancel out with a factor in the numerator.

How does active learning help students understand asymptotes?

Asymptotes are abstract boundaries that can be hard to visualize. Active learning strategies like gallery walks and collaborative investigations allow students to see many different examples and discuss the patterns they observe. By comparing functions with their peers, students can better understand the algebraic reasons for different types of discontinuity, moving beyond simple memorization of rules.

Can a function have more than one horizontal asymptote?

Standard rational functions have at most one horizontal asymptote. However, more complex functions, like those involving square roots or absolute values, can have two different horizontal asymptotes.

What is a removable discontinuity?

A removable discontinuity, or a 'hole,' occurs when a factor in the denominator cancels with a factor in the numerator. The function is undefined at that point, but the graph looks continuous otherwise.