S4 Maths | Graph transformations prep guide (for tutors) | translation, stretch, reflection, vertex

2026-11-01

Tutor lesson-prep guides · Secondary Mathematics

Students read y = f(x – 3) as a shift left. This lesson uses a machine that reacts three seconds late so the direction finally sticks.

Objectives — Students can: (1) describe how f(x) + k, f(x – h), a f(x), f(bx) and -f(x) change a graph; (2) track the vertex and the roots from the original curve to the image; (3) tell an outside multiplier from an inside one and explain why a horizontal shift runs against intuition.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Quadratic functions	complete the square for the vertex	vertex (1, -1)
Function notation	f(x) is an input-output machine	f(2) means x = 2
Solving equations	factorising for roots	roots 0 and 2

02 Lesson flow (50 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Reference curve	y = x squared – 2x, vertex (1, -1)	roots 0 and 2
8-18 min	Adding outside	f(x) + 3	vertex (1, 2)
18-30 min	Adding inside	f(x – 3)	vertex (4, -1)
30-40 min	Outside vs inside multiplier	2f(x) and f(2x)	vertex (0.5, -1)
40-46 min	Reflection	-f(x) + 3	vertex (1, 4)
46-50 min	Exit check	three questions collected	independent work

03 Teaching script (read it out)

Scene 1: The reference curve: fix three points first

y = f(x) = x squared – 2x = (x – 1) squared – 1, vertex (1, -1), opening upward, cutting the x-axis at 0 and 2, with y-intercept 0.
Before any transformation, students should write those three references down: vertex and two roots. Then every transformation is tracked on those points rather than redrawing the whole curve, which is faster and safer. Point out that shape-preserving moves only need the vertex, while stretches also change the width.

Student task: complete the square, then write the vertex, both roots and the y-intercept.

Scene 2: Translations: outside the bracket and inside it

Outside: y = f(x) + k adds k to every output, so the whole graph rises by k. For f(x) + 3 the vertex becomes (1, 2). Inside: y = f(x – 3) needs x to be 3 larger to feed the same input, so the graph moves right by 3 and the vertex goes to (4, -1); expanded, that is x squared – 8x + 15, with roots 3 and 5.
Rule: minus inside shifts right, plus inside shifts left, the opposite of intuition. A machine that reacts three seconds late delivers its output three seconds late. Outside changes y, inside changes x in the reverse direction. The usual mistake is shifting left and then mixing in the old left-plus rule.

Student task: write the vertex and roots of f(x + 2) and contrast its direction with f(x – 3).

Scene 3: Stretches and reflections: outside changes height, inside changes width

Vertical: y = 2f(x) doubles every y, so the vertex y goes from -1 to -2 while the roots stay put, since 0 doubled is still 0. Horizontal: y = f(2x) halves x to produce the same output, so the graph is squeezed by half and the vertex x drops from 1 to 0.5; expanded it is 4x squared – 4x with roots 0 and 1.
Reflections: -f(x) flips vertically, f(-x) flips horizontally, and both together rotate the graph 180 degrees about the origin. Combined example: -f(x) + 3 flips then rises 3, vertex (1, 4) with roots -1 and 3. Order matters: inside first, then outside multiply, then outside add. Make students number the steps.

Student task: step from f(x) to -f(x) + 3, writing the vertex and roots at each stage.

04 Question types and worked steps

Type	Example	Steps	Answer
Vertical shift	f(x) + 3	add 3 to y, shape unchanged	vertex (1, 2)
Horizontal shift	f(x – 3)	minus inside shifts right	vertex (4, -1)
Vertical stretch	2f(x)	double y, roots unchanged	vertex (1, -2)
Horizontal stretch	f(2x)	halve x	vertex (0.5, -1)

Sanity check (teach this): The vertex starts at (1, -1); after a rise of 3 the y must be 2, so a student who answers x = 4 has swapped the axes. A horizontal squeeze makes the vertex x smaller (0.5); a larger value means it was treated as a stretch. After both a flip and a rise the vertex y is 4, higher than the original, because the flip happens first. Always check whether the roots still sit on the x-axis.

05 Independent practice (3 questions)

Question	Answer	What to watch
f(x) has vertex (2, -4); vertex of f(x) + 5?	(2, 1)	only y changes
y = f(x + 2) shifts which way?	left by 2	plus inside shifts left
f(x) has roots 1 and 4; roots of 3f(x)?	still 1 and 4	0 times 3 is still 0

06 Common mistakes and what to say

Mistake	Student writes	Say this
Reading f(x – 3) as a left shift	direction reversed	Minus 3 inside means the input is late, so the graph moves right.
Confusing 2f(x) with f(2x)	both called a stretch by two	Outside changes height, inside changes width.
Stretch changes the roots	saying 2f(x) doubles the roots	Roots are where y = 0, and zero doubled is still zero.
Wrong order of operations	adding before multiplying	Inside first, then outside multiply, then outside add.

07 Homework (paper, section and question numbers)

Secondary Mathematics chapter index (quadratics, logarithms, trigonometry, probability) — do quadratics before transformations
Secondary prep guides overview — all subjects

08 Exit check (last 5 minutes)

Question	Answer	Decision
Which way does y = f(x – 4) move?	right by 4	wrong means redo act two
What does f(3x) do to the graph?	squeezes it horizontally by one third	wrong means redo act three
Do the roots of 2f(x) change?	no	two wrong means back to act one

09 Tutor FAQ

Which year is this for?

S4, the functions and graphs unit of the HKDSE syllabus, and useful revision in S5.

How many rules to remember?

Five: add outside moves up or down, add inside moves sideways, multiply outside stretches vertically, multiply inside stretches horizontally, minus reflects.

Is there paired practice?

The homework links to the maths index with quadratics, logarithms and trigonometry.

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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.