Graphs of Rational Functions

Curriculum Connections

BC Mathematics 10: Functions and Relations

• Big Idea: "Relations and functions are used to describe, analyse, and represent mathematical situations from real life."
• Content Focus: Understanding the behaviour of rational functions, including asymptotes and graph characteristics (BC Ministry of Education, 2023)
• Specific Learning Standards:
  • Explain characteristics of rational functions in various forms, including asymptotes, intercepts, and behaviour near asymptotes.
  • Sketch graphs of rational functions using transformations and asymptotic behaviour. (Mathematics 10 - “Relations and Functions” – Learning Standards 10.1 & 10.2)

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Learning Objectives

By the end of the 60-minute lesson, students will be able to:

1. Define rational functions and recognise their general form as ( f(x) = \frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials and ( q(x) \neq 0 ).
2. Identify and classify vertical, horizontal, and oblique (slant) asymptotes algebraically from the function equation.
3. Analyse the behaviour of rational functions near their asymptotes, predicting where the graph approaches but does not touch these lines.
4. Sketch basic graphs of rational functions using knowledge of intercepts, asymptotes, and end behaviour.
5. Apply reasoning to solve problems involving rational function graphs, enhancing their interpretation skills.

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Materials and Resources

• Whiteboard and markers
• Graphing calculators or tablets with graph plotting app (optional)
• Chart paper and markers for group work
• Printed worksheets with practice problems (rational functions with various asymptotes)
• Graph paper for individual sketching
• Visual aids: large printed or digital examples of rational function graphs showing asymptotes

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Lesson Structure

1. Introduction (10 minutes)

• Engagement: Begin by showing a graph of a simple rational function such as ( f(x) = \frac{1}{x} ). Ask students what they notice about the graph and what values ( x ) cannot take (x ≠ 0).
• Brief Review: Recap polynomials briefly to highlight numerator and denominator in rational functions.
• Definition: Present the formal definition of a rational function. Write ( f(x) = \frac{p(x)}{q(x)} ) on the board and emphasise ( q(x) \neq 0 ).
• Learning Goals: Share the learning objectives for the lesson clearly with the class.

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2. Direct Instruction & Modelling (15 minutes)

• Vertical Asymptotes:

  • Explain that vertical asymptotes occur where ( q(x) = 0 ) (denominator zero), provided numerator ( p(x) ) does not also equal zero at the same point (discuss holes briefly as extension).
  • Example: ( f(x) = \frac{2x+3}{x-1} ) → vertical asymptote at ( x = 1 ).
  • Show how the function behaves as ( x \rightarrow 1^{\pm} ) (use limits approach, explain graph ‘blows up’ near vertical asymptotes).

• Horizontal Asymptotes:

  • Discuss degrees of numerator and denominator polynomials:
    • Degree numerator < degree denominator → horizontal asymptote at ( y=0 )
    • Degrees equal → horizontal asymptote at ratio of leading coefficients
    • Degree numerator > degree denominator → no horizontal asymptote (possible oblique/slant asymptote)

• Oblique (Slant) Asymptotes:

  • Occur when degree numerator = degree denominator + 1.
  • Use polynomial long division to find the slant asymptote. Show an example.

• Graph Behaviour near Asymptotes:

  • Sketch a quick graph showing the function approaching vertical and horizontal or oblique asymptotes without crossing them (unless at holes).

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3. Guided Practice (15 minutes)

• Divide students into 5 groups of 5. Provide each group with a different rational function. Tasks:
  1. Identify vertical asymptotes.
  2. Determine horizontal or oblique asymptotes, showing the calculations.
  3. Sketch a rough graph indicating the asymptotes and general behaviour near these lines.
• Circulate and prompt students with questions such as:
  • What happens to ( f(x) ) as ( x \to ) vertical asymptote values from the left and right?
  • What is the end behaviour of the function?
• Each group records their findings on chart paper.

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4. Independent Practice (15 minutes)

• Distribute worksheets with 5 problems of varying difficulty: from finding asymptotes algebraically, to sketching graphs and interpreting graphs to answer questions about behaviour.
• Students work individually to solve these, focusing on applying the concepts learned.
• Teacher assists and offers targeted feedback to individuals as needed.

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5. Wrap-Up and Assessment (5 minutes)

• Quickly review key points: definition of rational functions, types of asymptotes, and sketching strategies.
• Formative Assessment:
  • Exit Ticket: Each student writes on a sticky note one vertical asymptote and one horizontal or oblique asymptote for a given rational function ( f(x) = \frac{x^2+1}{x-2} ).
  • Collect these for a quick check on understanding.
• Preview next lesson on transformations of rational functions or more advanced problem solving with holes in graphs (removable discontinuities).

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Assessment Ideas

• Formative:

  • Exit Tickets assessing identification of asymptotes.
  • Observation during group work for gauging understanding of graph behaviour near asymptotes.

• Summative:

  • Create a short quiz at the end of the unit requiring students to:
    • Identify vertical, horizontal, and oblique asymptotes from given functions.
    • Sketch a graph using asymptotes and intercepts.
    • Describe the behaviour near asymptotes in writing.

• Use rubric based on accuracy, clarity of sketches, and explanation quality aligned with BC Curriculum Core Competencies: Communication and Thinking.

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Cross-Curricular Connections

• Science: Discuss how rational functions model real-world phenomena such as rates, chemical reactions, or population models where values approach limits but never reach them.
• Technology: Explore using graphing software or calculators to explore rational functions beyond hand-sketching.

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Inclusive and Differentiation Strategies

• Provide printed guides with step-by-step examples.
• Allow use of graphing calculators/tablets for visual learners or students who need technological support.
• Pair stronger students with peers during group tasks for peer scaffolding.
• Challenge interested students with complex rational functions involving holes and higher-degree polynomials as extension.

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This lesson plan is carefully designed to engage Grade 10 learners with concrete and visual approaches aligned with British Columbia's mathematics curriculum. The lesson interweaves conceptual understanding with practical skills for graphing rational functions, ensuring students build solid foundations for later studies in mathematics.