IB Maths AA: Functions and Graphs — Tutor Lesson Plan (vertex, discriminant, transformations)

2026-10-28

An IB functions question does not ask you to draw a graph, it asks you to read the graph out of the algebra. One quadratic carries the whole lesson.

Objectives — the student can complete the square to read a vertex, use the discriminant to classify roots, and derive a transformed graph from f of x.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Substitution	For f of x = 2x squared – 3x + 1, what is f of 2?	3
Completing the square	Vertex of x squared – 6x + 5	(3, -4)
Discriminant	How many real roots has x squared + 2x + 5?	None

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Three forms	Standard, vertex and factorised form	State how to convert
8-24 min	Vertex by completing the square	2x squared – 3x + 1	Get (0.75, -0.125)
24-36 min	The discriminant	b squared – 4ac and the root type	Get 1, two distinct roots
36-48 min	Transformations	Translation, stretch, reflection	Write 2(x-1) squared + 4
48-56 min	Answer format	IB wants method, not only the answer	Rewrite one answer
56-60 min	Exit check	Three questions marked	Finish alone

03 Teaching script (read it out)

Scene 1: Completing the square: factor out a first

2x squared – 3x + 1 equals 2(x squared – 1.5x) + 1, which is 2[(x – 0.75) squared – 0.5625] + 1, so 2(x – 0.75) squared – 0.125.
The vertex is (0.75, -0.125) and the axis of symmetry is x = 0.75. IB awards method marks, so show the squaring step.

Student explains why the vertex value is not 1.

Scene 2: The discriminant in three cases

Delta equals b squared minus 4ac, so 9 – 8 = 1, positive, giving two distinct roots at (3 plus or minus 1) over 4, that is 1 and 0.5.
Delta zero means a repeated root where the curve touches the axis; a negative value means no real roots.

Note: write Delta before judging, never quote the rule without the line of working.

Scene 3: Transformations: order changes the result

For g of x = 2(x – 1) squared + 4 the graph moves right 1, stretches vertically by 2, then moves up 4.
Stretching before translating scales the horizontal shift, so the two orders give different graphs.

Student calculates g of 0 (6).

04 Question types and worked steps

Type	Example	Steps	Answer
Completing the square	2x squared – 3x + 1	Factor out, complete, read the vertex	(0.75, -0.125)
Discriminant	b squared – 4ac	9 – 8 = 1	Two distinct roots, 1 and 0.5
Transformation	2(x – 1) squared + 4	Shift, stretch, shift	Vertex (1, 4)
Inverse	solve y = f of x for x	Swap and solve, state the domain	Give the domain

Sanity check (teach this): The vertex value must equal f evaluated at the axis. If the completed form disagrees with direct substitution, a inside the bracket was not scaled.

05 Independent practice (3 questions)

Question	Answer	What to watch
f of 2 for 2x squared – 3x + 1	3	Multiply before subtracting
Discriminant of 2x squared – 3x + 1	1	Signs in b squared minus 4ac
The two roots	1 and 0.5	Written as (3 plus or minus 1) over 4

06 Common mistakes and what to say

Mistake	Student writes	Say this
Forgot to scale by a	vertex written (1.5, -1.25)	After factoring out 2, halve 1.5 again inside the bracket.
Sign error in the discriminant	Delta written -1	b squared is added, 4ac subtracted.
Transformation order reversed	stretch before shift	Shift first, otherwise the horizontal move is scaled.
Answer only	IB method marks lost	IB needs the working; a right answer alone earns half.

07 Homework (paper, section and question numbers)

IB maths chapter index — Use the three questions on this page

08 Exit check (last 5 minutes)

Question	Answer	Decision
Value of f of 2	3	All three right: move to trigonometry
Discriminant and root type	1, two distinct roots	One wrong: re-teach completing the square
The two roots	1 and 0.5	Two wrong: repeat scene 2

09 Tutor FAQ

How does IB differ from DSE?

IB supplies a formula booklet, so method and explanation carry the marks.

Is a calculator allowed?

Paper 1 is usually non-calculator, so completing the square must be fluent.

How many marks for transformations?

Usually four to six, and the order of steps must be shown.

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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.