Quadratic Function Graphs

Overview

In this lesson (Lesson 3 of 9), students investigate how changing a quadratic in standard form and vertex form affects its graph. They will sketch graphs, identify turning points, and determine maximum/minimum values by linking algebraic features to graph features.

Learning intentions

• Students will recognise and compare quadratic functions in standard form and vertex form.
• Students will determine key graph features (parabolic shape, intercepts, axis of symmetry, and turning point) from the equation.
• Students will sketch quadratic graphs accurately with or without technology.
• Students will determine zeros and turning points (and hence maximum/minimum values) for selected quadratics.

Success criteria

• I can rewrite or interpret a quadratic in a way that reveals the vertex (turning point).
• I can identify the axis of symmetry and turning point directly from vertex form.
• I can sketch a parabola using enough points to match its equation and check intercepts.
• I can find the maximum or minimum value and relate it to the turning point on the graph.

Curriculum links

• QCAA Mathematics Methods — Unit 1 / Topic 1: Surds and quadratic functions: Recognise and determine features of quadratic graphs including parabolic nature, turning points, axis of symmetry and intercepts.
• QCAA Mathematics Methods — Unit 1 / Topic 1: Surds and quadratic functions: Sketch quadratic function graphs with or without technology.
• QCAA Mathematics Methods — Unit 1 / Topic 1: Surds and quadratic functions: Determine turning points and zeros of quadratic functions, with and without technology.
• QCAA Mathematics Methods — Unit 1 / Topic 1: Surds and quadratic functions: Model and solve problems that involve quadratic functions.

Lesson structure (60 minutes)

1. 0–5 min · Quick activation. Teacher displays two quadratic equations—one in vertex form and one in standard form—and asks students to predict which has a maximum and which has a minimum based on the coefficient and “opening” direction; students make brief predictions and justify orally.
2. 5–15 min · Direct teach: interpreting forms. Teacher explains how vertex form (y=a(x-h)^2+k) reveals the turning point ((h,k)), axis of symmetry (x=h), and how standard form (y=ax^2+bx+c) supports finding intercepts and turning points using technology or structured methods; students fill a two-column “equation feature → graph feature” table.
3. 15–28 min · Guided sketching (teacher + student). Teacher works through one example equation in each form: first identify turning point and axis, then plot symmetrical points to sketch; students copy the steps, then choose one additional equation (provided) and sketch it using the same checklist: turning point, axis of symmetry, intercepts, then at least 3 points.
4. 28–40 min · Find zeros and maximum/minimum values. Teacher models determining zeros for a chosen quadratic (e.g., by factorisation when possible, or using technology/suitable algebra method when not) and shows how the turning point value (k) (vertex form) or computed turning point value (standard form) gives the maximum/minimum; students complete a short worked set for two quadratics and record answers as: zeros, turning point, maximum/minimum value.
5. 40–53 min · Independent practice: form matching and graph checks. Teacher provides 4 cards (two in vertex form, two in standard form) where students must match each equation to a labelled sketch region (opens up/down, vertex location, and approximate intercepts), then refine by drawing a quick graph and checking intercepts; students work on paper, then verify using a graphing tool only if available/appropriate.
6. 53–60 min · Exit ticket (single-page). Teacher gives one final quadratic in each form and asks: (1) state turning point and axis of symmetry, (2) sketch a quick parabola, and (3) give maximum/minimum value; students submit within the final minutes.

Resources

• Printed equation cards (2–4 vertex form + 2–4 standard form)
• Graph paper or digital grid (graphing software optional)
• Checklist for quadratic graph features (turning point, axis symmetry, intercepts, points)
• Calculator and/or graphing technology (optional, teacher-controlled)
• Student response sheets for the table and exit ticket
• Coloured pens/pencils for marking points (optional)

Assessment

• Formative check during Step 2: completed “equation feature → graph feature” table for accuracy of turning point/axis/intercepts.
• Formative check during Step 4: teacher circulates/observes and reviews zeros and maximum/minimum value reasoning (even if brief, must show link to graph features).
• Exit ticket at Step 6: turning point, axis of symmetry, sketch quality (enough points with correct orientation), and correct maximum/minimum value.

Differentiation

• Support: Provide a structured sentence starter for interpreting vertex form (e.g., “Since the equation is (y=a(x-h)^2+k), the turning point is…”), and a point-plotting template (start with vertex, then reflect across the axis).
• Support for standard form: Supply a “find turning point” scaffold (either by technology prompts or a guided method your class has used previously) and require at least correct axis symmetry and turning point value before full intercept checking.
• Extension: Ask students to compare two quadratics with the same turning point but different (a) values, predicting how the graph width changes while keeping the vertex fixed, then confirm from their sketches.
• SEN/EAL considerations: Allow working through steps in a clear order; provide colour-coding conventions (vertex in one colour, axis in another) and accept oral explanations if writing is difficult, as long as graph features are correctly identified.