Graphing Quadratics

Overview

This 60-minute lesson is aimed at Year 10 students focusing on graphing quadratic functions. The lesson fits into the “Mastering Algebraic Concepts” unit (lesson 12 of 20) following the National Curriculum for England guidelines for Key Stage 4 Mathematics. Students will learn to graph quadratic functions, identify key features such as the vertex and axis of symmetry, and understand their real-world applications.

National Curriculum Reference

Mathematics Programme of Study: Key Stage 4 - KS4 (Years 10-11)

Algebra - Graphs:
- Plot and interpret graphs of quadratic functions, recognising connections between algebraic and graphical representations.
- Identify key features of parabolas: turning point (vertex), axis of symmetry.
- Solve quadratic equations by graphing.

Learning Objectives

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

Plot at least two quadratic functions accurately on Cartesian axes.
Identify the vertex (turning point) and axis of symmetry from the graph and equation.
Understand how changes in the quadratic equation affect the graph’s shape and position.

Success Criteria

I can plot quadratic functions using a table of values correctly.
I can identify and label the vertex and axis of symmetry on the graph.
I can explain how the coefficients in the quadratic equation affect the graph’s shape and position.
I can sketch two different quadratic graphs and describe their differences.

Resources

Graph paper (squared)
Scientific calculator or CAS tool
Whiteboard and markers
Printed worksheets with quadratic equations and plotting tables
Interactive graphing software (optional)
Ruler and pencil

Lesson Structure

Starter (10 minutes)

Objective: Activate prior knowledge of quadratic functions and coordinate graphs.

Quick recap: What is a quadratic function? Write general form: ( y = ax^2 + bx + c ).
Elicit responses on how to find y-values for given x-values.
Briefly remind how to plot points on Cartesian plane.
Mini-quiz: Students complete 5 quick questions filling in y-values for quadratic equations from given x-inputs.

Introduction (10 minutes)

Objective: Demonstrate the graphing process and key features of a parabola.

Teacher models creating a table of values for ( y = x^2 - 4x + 3 ) for x = 0 to 4, plotting the points on the graph.
Draw the parabola through plotted points.
Identify the vertex and axis of symmetry visually from the graph. (Vertex form can be left for a later lesson.)
Describe axis of symmetry as a vertical line through the vertex, formula ( x = -\frac{b}{2a} ) introduced verbally (deeper algebraic understanding comes in later lessons).
Discuss shape changes if a > 0 (opens upwards) and a < 0 (opens downwards).

Main Activity (25 minutes)

Objective: Students practise graphing quadratic functions and identifying features.

Provide students with two quadratic functions:
1. ( y = x^2 - 4x + 3 )
2. ( y = -2x^2 + 8x - 6 )
Students:
- Construct tables of values for x from 0 to 4 (and negative x if they feel confident).
- Plot the points carefully on graph paper.
- Draw the parabola smoothly.
- Identify and label the vertex (highest or lowest point).
- Draw and label the axis of symmetry line.
Teacher circulates to scaffold learning, ask probing questions like:
- “What can you say about the vertex coordinates?”
- “How does the leading coefficient affect the graph’s shape?”
- “Can you find the axis of symmetry without calculating it explicitly?”

Plenary (10 minutes)

Objective: Review and consolidate learning, check understanding.

Class discussion: Students share their graphs and findings.
Teacher leads discussion on how each graph differed and why.
Generate a quick mind-map on the board about graph characteristics: vertex, axis of symmetry, direction of opening.
Exit ticket: Students write one sentence on what the vertex means and one on why the axis of symmetry helps graphing.

Assessment and Feedback

Formative: Observation during plotting and questioning.
Mark students’ graphs for accuracy of points, smooth curve, labelled key features.
Use exit ticket responses to gauge conceptual understanding.

Differentiation

Support: Provide scaffolding sheets with partially filled tables and steps for plotting.
Challenge: Extend by asking students to predict vertex coordinates using ( x = -\frac{b}{2a} ) formula and verify by graphing.

Homework

Sketch graphs of two more quadratic functions with given equations, label vertex and axis of symmetry.
Write a brief paragraph explaining how changing the coefficient (a) changes the shape.

Cross-Curricular Links

Physics: Projectile motion and parabolic trajectories (contextualise graphing quadratics).
Art: Symmetry and curve drawing.

This lesson plan not only fulfils the National Curriculum requirements on algebraic graphing but also integrates active learning, real-world context, and personalised challenge. It supports building confidence in graphical skills pivotal for GCSE success and practical maths applications.

Teaching Instructions