Graphing Rational Functions: Vertical Asymptotes

Active learning works because graphing rational functions relies on students visualizing abstract concepts. When students manipulate functions and discuss their features, they build mental models of asymptotes and discontinuities that static notes cannot provide.

• 11th Grade
• Mathematics
• 3 activities
• 15 min–30 min

Learning Objectives

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

Gallery Walk

⏱ 30 min·Small Groups

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.

Prepare & details

Explain what causes a vertical asymptote in a rational function.

Facilitation Tip: During the Gallery Walk, arrange the stations so students move clockwise to avoid crowding around one function at a time.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Inquiry Circle

⏱ 25 min·Pairs

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.

Prepare & details

Predict the behavior of a rational function as it approaches a vertical asymptote.

Facilitation Tip: For The Mystery of the Hole, provide colored pencils so students can trace canceled factors and highlight holes in their graphs.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 15 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.

Prepare & details

Compare the graphical representation of a vertical asymptote to a hole in the graph.

Facilitation Tip: In Asymptote Predictions, assign partners who have different approaches to graphing so they must reconcile their methods during discussion.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach graphing by connecting algebra to visual behavior. Start with simple examples where students factor and cancel terms, then gradually introduce more complex functions. Avoid teaching rules like 'set denominator to zero' in isolation; instead, emphasize why those values matter. Research shows that students grasp asymptotes better when they explain the function’s behavior in their own words rather than labeling parts of a graph.

What to Expect

By the end of these activities, students should confidently identify vertical asymptotes and holes, explain their causes through factoring, and describe a function’s behavior near these points without relying on memorized rules.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share: Asymptote Predictions, watch for students who assume graphs cannot cross any asymptote.

What to Teach Instead

During Think-Pair-Share, ask partners to graph f(x) = (x^2 - 1)/(x - 1) and f(x) = (2x)/(x^2 + 1) side by side. Have them explain why one crosses its horizontal asymptote while the other does not.

Common Misconception

During Collaborative Investigation: The Mystery of the Hole, watch for students who confuse canceled factors with vertical asymptotes.

What to Teach Instead

During Collaborative Investigation, give groups a whiteboard to write both the original and simplified functions. Ask them to mark canceled terms in red and discuss how those terms affect the graph’s continuity.

Assessment Ideas

Quick Check

After Gallery Walk: Function Features, ask students to complete a quick-check with f(x) = (x+2)/(x^2 - 9). Have them factor the denominator, identify the x-values that make it zero, and state the vertical asymptotes.

Exit Ticket

During Collaborative Investigation: The Mystery of the Hole, collect each group’s whiteboard work. Ask for one rational function with a vertical asymptote and one with a hole, and have students identify the features on the board before leaving.

Discussion Prompt

After Think-Pair-Share: Asymptote Predictions, 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?' Use student responses to assess their understanding of limits and continuity.

Extensions & Scaffolding

• Challenge students to create their own rational function with a vertical asymptote at x = 4 and a hole at x = -1, then trade with a partner to identify each other’s features.
• Scaffolding: Provide partially completed graphs with missing factors or asymptotes for students to fill in before sketching the full function.
• Deeper exploration: Ask students to research real-world phenomena modeled by rational functions (e.g., drug concentration in the bloodstream) and present how vertical asymptotes appear in those contexts.

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.