Analysing Quadratic Graphs

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Curriculum Alignment

New Zealand Curriculum Area:
Mathematics and Statistics – Level 6
Strand: Algebra
Achievement Objective:
Students will form and solve quadratic equations and interpret graphical representations of relationships.
Key Progress Outcome: Students will explore and interpret parabolic graphs derived from quadratic equations. They will develop an understanding of how the parameters of the equation affect the graph and solve contextual problems using graphical methods.

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Lesson Overview

Lesson Duration: 60 minutes
Class Level: Year 11 (NCEA Level 1, approx. 15–16 years old)
Class Size: 30 students
Topic Focus: Analysing and interpreting quadratic graphs — identification of key features including vertex, axis of symmetry, direction of opening, intercepts, and transformations.
Big Idea: Graphs tell stories — understanding the shape and features of a quadratic graph helps us understand patterns and solve real world problems.

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Learning Intentions

By the end of this lesson, students will be able to:

• Identify and describe the key characteristics of a quadratic graph: vertex, roots, axis of symmetry, and direction of opening.
• Connect the form of a quadratic equation (standard and vertex form) to its corresponding graph.
• Use transformations (translations, reflections, vertical stretches) to sketch and describe the changes in quadratic graphs.
• Apply their knowledge to solve contextual problems involving quadratic graphs.

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Success Criteria

Students will demonstrate success when they can:

• Accurately sketch a given quadratic graph with correct annotations.
• Verbally explain how changes to the equation (e.g., ( y = a(x - h)^2 + k )) affect the graph's shape and position.
• Solve a real-world quadratic problem using graphical interpretation.
• Collaborate effectively in group activities and communicate mathematical thinking clearly.

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Lesson Structure (60 minutes)

1. Ignite Curiosity (5 mins)

Activity: Image Spark
Display a real-life parabolic image: e.g., a fountain arc, a skateboard ramp, or the path of a kicked rugby ball.
Prompt Discussion Questions:

• “What shape is this?”
• “What do you think this picture has to do with algebra?”
• “Can we use mathematics to predict how high the ball goes or where it lands?”

Connect student observations to the idea of parabolic motion and quadratic graphs.

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2. Explicit Teaching (10 mins)

Use a dynamic graphing software (e.g., Desmos on interactive board, or GeoGebra visual projection) to show the equation:

( y = a(x - h)^2 + k )

Teacher-Led Demonstration:

• Begin with the standard ( y = x^2 )
• Vary a, h, and k values and have students observe what changes.
• Use guided questioning:
  “What happens when ‘a’ is negative?”
  “What does the ‘h’ shift do to the graph?”

Highlight and annotate key features on the projected graph:

• Vertex (turning point)
• Axis of symmetry
• Direction (up/down)
• Intercepts (x and y)

Ensure clear distinction between standard and vertex form. Emphasise interpretation over memorisation.

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3. Interactive Activity – Graph Match Challenge (15 mins)

Group Work: 2–3 students per group

Give each group a set of quadratic equations and a matching set of graphed functions (on card). Students must match each equation to its correct graph using reasoning.
Example Set:

• Equations:

  • ( y = -2(x + 1)^2 + 3 )
  • ( y = (x - 4)^2 )
  • ( y = x^2 - 9 )

• Graphs printed on laminated cards

Bonus challenge: Given a graph with hidden labels, write the equation back in vertex form.

Teacher Role: Move between groups, listen for misconceptions, and prompt deeper thinking. Elicit language and reasoning from quieter students.

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4. Real-Life Application Task (15 mins)

Contextual Task: "Kick for Gold!" – Analysing a Rugby Kick

Provide a context:
A kicker launches the rugby ball with a parabolic path described by:

[ h(t) = -5(t - 3)^2 + 45 ]

Where h(t) is height in metres, and t is time in seconds after the ball is kicked.

Student Tasks:

• Identify vertex: What is the maximum height? When does this occur?
• When does the ball hit the ground? (Find x-intercepts)
• Sketch the graph, labelling key features.
• Use the graph to explain whether the ball clears a 3m crossbar located 5 seconds after the kick.

This task blends mathematical analysis with meaningful sport context—appealing to student interests, especially in NZ.

Students work in pairs and submit written working and annotated sketch. Extension for fast finishers: Model a different kick by adjusting the equation and analysing the effects.

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5. Reflection & Plenary (10 mins)

Strategy: 3–2–1 Exit Cards

Hand students small slips to complete:

• 3 things I learned
• 2 transformations I can describe
• 1 question I still have

Quickly collect and skim for misconceptions to address in the next lesson.

Bonus Wrap-Up Game (time permitting):
“Graph It!” – Call out simple transformations (e.g., “Reflect it!”, “Shift 2 up!”), and students sketch mini graphs quickly on whiteboards.

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Teacher Notes

• Differentiation:

  • Students needing support can receive graph templates and guiding questions.
  • Extension students can explore completing the square to connect standard and vertex forms.

• Formative Assessment Opportunities:

  • During group work, listen for mathematical language used.
  • Use exit slips to spot areas for reteaching or extension.

• Connection to Next Lesson:

  • Introduce solving quadratic equations graphically or algebraically (e.g., using factorisation or the quadratic formula).

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Materials Needed

• Graphing projector tool (e.g., Desmos/GeoGebra)
• Laminated graph and equation cards
• Exit slip cards
• Student books or graph paper
• Mini whiteboards and markers
• Printed copies of the Rugby Kick task

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Reflective Teacher Prompt

“How many of my students were confidently articulating graph features today—both visually and algebraically? Who needs more concrete examples, and who’s ready for abstract generalisations in their own modelling?”

By linking mathematical structure to real-world visuals and combining digital tools with tactile resources, this lesson encourages conceptual depth without sacrificing engagement.

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Māori and Pasifika Integration Opportunity

• Use local sporting examples (e.g., Māori and Pasifika stars in rugby league or sevens) to increase relevance.
• Introduce te reo Māori terms for mathematical concepts.
• Allow students to collaboratively name their graph cards with phrases in both English and te reo (e.g., tīmatanga = starting point, rahi = height).

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Final Thought

This lesson reframes quadratic graphs as storytelling tools – mathematical models rich with narrative possibility. By giving students agency to explore, match, model, and question, we move beyond compliance learning to true mathematical thinking.