中文（繁體）
中文（简体）
English
DSE Maths: Quadratic Functions and Graphs — Tutor Lesson Plan
Students solve equations but freeze at graphs. Three switches fix it: a opens the curve, the vertex gives the extreme value, the discriminant counts the intercepts.
Objectives — the student can complete the square, state the vertex and axis of symmetry, and use the discriminant to count x-intercepts.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Quadratic equations | Solve x² − 5x + 6 = 0 | x = 2 or 3 |
| Completing the square | Write x² − 4x as (x − 2)² − 4 | Correct |
| Function notation | What does f(2) mean? | Substitute x = 2 |
02 Lesson flow (60 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-8 min | Three switches | a, vertex, discriminant | Repeat them |
| 8-22 min | Completing the square | y = x² − 4x + 3 | Two alone |
| 22-36 min | Reading graphs | Vertex and axis from a graph | Write both |
| 36-50 min | Discriminant | Three cases of intercepts | Match five |
| 50-56 min | Applications | Maximum area problem | Set up and solve |
| 56-60 min | Exit check | Three questions marked | Finish alone |
03 Teaching script (read it out)
Scene 1: a decides the opening
In y = ax² + bx + c, a > 0 opens upwards with a minimum; a < 0 opens downwards with a maximum.
The larger |a|, the narrower the curve.
The larger |a|, the narrower the curve.
Student states the opening for three equations.
Scene 2: The vertex comes from completing the square
y = x² − 4x + 3 = (x − 2)² − 1, so the vertex is (2, −1) and the axis is x = 2.
The h is the value that makes the bracket zero; k is the extreme value.
The h is the value that makes the bracket zero; k is the extreme value.
Student does y = x² + 6x + 5: vertex (−3, −4).
Scene 3: The discriminant counts intercepts
Δ = b² − 4ac. Positive: two intercepts (two real roots). Zero: one touching point. Negative: none.
Equations and graphs are two views of the same thing.
Equations and graphs are two views of the same thing.
Note: no real roots does not mean no solution at all.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Completing the square | y = x² − 4x + 3 | (x − 2)² − 1 | Vertex (2, −1), axis x = 2 |
| Opening and extreme value | y = −2x² + 4x | a < 0, maximum | Maximum 2 at (1, 2) |
| Discriminant | y = x² + 2x + 5 | Δ = 4 − 20 = −16 | No x-intercepts |
| Application | 20 m of fence, largest rectangle | Area x(10 − x) | 25 m² at x = 5 |
Sanity check (teach this): The vertex y-value is the extreme value: a minimum when a > 0, a maximum when a < 0.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Vertex of y = x² + 6x + 5 | (−3, −4) | Completing the square |
| Axis of y = 2x² − 8x + 1 | x = 2 | Use −b/2a |
| Intercepts of y = x² − 2x + 3 | None, Δ = −8 | Sign of Δ |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| Constant dropped | x² − 4x + 3 written as (x − 2)² + 3 | Subtract the 4 back: −1. |
| Vertex sign flipped | y = (x − 2)² − 1 gives (−2, −1) | Minus two inside means h = +2. |
| Minimum claimed when a < 0 | y = −x² said to have minimum 0 | Downward opening gives a maximum. |
| Two roots claimed when Δ < 0 | negative discriminant, two roots | No real roots. |
07 Homework (paper, section and question numbers)
- Secondary maths chapter index — Use the three questions on this page
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| Vertex of y = x² − 4x + 3 | (2, −1) | All right: move to logarithms |
| Extreme value when a < 0 | Maximum | One wrong: re-teach the three switches |
| What Δ = 0 means | One repeated root | Two wrong: repeat scene 3 |
09 Tutor FAQ
Memorise the vertex formula?
Learn x = −b/2a to check, but completing the square is needed for maximum and minimum problems.
How accurate must the sketch be?
Mark the vertex, axis and intercepts; no need to plot points.
How is this different from quadratic equations?
That chapter finds roots; this one reads shape and extreme values.
A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.
