IB Maths AA SL
Topic 2 — Functions
Paper 1 & 2
~11 min read
Composite Transformations of Graphs
When more than one transformation acts on a graph, you need to apply them in the right order — otherwise you’ll end up with the wrong equation. Here’s the full rulebook for combining translations, reflections, and stretches without losing marks.
📘 What you need to know
- A composite transformation is two or more single transformations applied to the same graph.
- Order matters — but only between transformations that act on the same axis. A vertical and a horizontal transformation can swap freely.
- For y = af(x) + b, the order is: stretch first (×a), then translate (+b) — the same order as evaluating the expression.
- Big trap: af(x) + b is not the same as a[f(x) + b] — the second one becomes af(x) + ab.
- If a is negative, you’ve also got a reflection in the x-axis wrapped into the stretch.
Quick recap — the seven single transformations
Before tackling composites, make sure these are second nature. Every composite is just two or three of these stacked together:
→
y = f(x − a)
Translation right by a · vector a0
←
y = f(x + a)
Translation left by a · vector −a0
↑↓
y = f(x) + b
Translation up by b (down if negative)
⇄
y = f(−x)
Reflection in the y-axis
⇅
y = −f(x)
Reflection in the x-axis
↕
y = af(x)
Vertical stretch, scale factor a
↔
y = f(ax)
Horizontal stretch, scale factor 1a
Does the order of transformations matter?
This is the question that decides whether you get full marks or zero. The short answer is: sometimes. The rule is much simpler than it looks once you split transformations into “vertical” and “horizontal” groups.
⚠ Same axis: keep them in order
Two vertical transformations (or two horizontal ones) must be applied in the order they’re written.
stretch ×3→+2
Different from “+2 first, then ×3”, which gives a different equation.
✓ Different axes: swap freely
A vertical transformation and a horizontal one don’t talk to each other. Do them in any order.
stretch ×2 (vertical)⇄translate left 3
Both orders give the same final graph.
🤔 Why does the same-axis order matter?
Think about it like the order of operations in arithmetic. The expression 3 × x + 2 is not the same as 3 × (x + 2). When we transform a graph, we’re literally doing operations on y-values (or x-values). Stretch-then-translate gives a different result from translate-then-stretch.
But a vertical transformation only changes y‘s, and a horizontal one only changes x‘s. They’re operating on totally different things, so they can’t interfere with each other — the order doesn’t matter.
Composite vertical transformations: y = af(x) + b
This is the most common composite you’ll meet in IB exams. The equation tells you exactly what to do, in the right order — just read it like a calculation:
Start
y = f(x)
original graph
×a→
Step 1
y = af(x)
vertical stretch
+b→
Final
y = af(x) + b
+ translation
Use the BIDMAS / order-of-operations idea: the equation af(x) + b is built by first multiplying f(x) by a, then adding b. Apply the transformations in the same order.
⚠ Stretch-then-translate ≠ translate-then-stretch
If you do the translation first and then the stretch, you don’t end up with af(x) + b — you end up with something different. Watch what happens:
Stretch first, then translate
f(x) → af(x) → af(x) + b
Final: y = af(x) + b ✓
Translate first, then stretch
f(x) → f(x) + b → a[f(x) + b]
Final: y = af(x) + ab ✗
The + b on the right has accidentally been multiplied too — that’s why the stretch must always come first when you read off af(x) + b.
What if a is negative?
A negative a hides a reflection in the x-axis inside the stretch. For example, y = −2f(x) + 3 actually contains three operations: stretch by 2, reflect in the x-axis, then translate up by 3. Since reflection and stretch are both vertical, the order between them doesn’t matter — but they both come before the translation.
Composite horizontal transformations
The same logic applies on the x-axis side, but with the usual horizontal twist (everything is opposite). For an equation like y = f(2x − 6), factor inside the bracket first to see what’s really going on:
Now read this in the standard order:
- Translation: x − 3 inside the bracket → translate right by 3
- Stretch: 2x means stretch horizontally by scale factor 12
For horizontal composites, the convention is: translate first, then stretch. Because the stretch is a multiplication on the x-coordinate, and it would scale any earlier translation by the same factor — the same trap as the vertical case, just mirrored.
For SL exams, horizontal composites are rare — most questions focus on the af(x) + b form. But knowing the factoring trick gives you a clean way to handle f(ax + c) if it does appear.
Worked examples
WE 1Sketch a composite vertical transformation
The graph of y = f(x) has a maximum at A(−1, 5) and a minimum at B(3, −3). Find the new coordinates of A and B on the graph of y = 3f(x) − 2.
Step 1: Read the equation in order
First ×3 (stretch), then −2 (translate down).
Step 2: Apply the stretch first
y = 3f(x) ⇒ y-coords × 3
A(−1, 5) → (−1, 15)
B(3, −3) → (3, −9)
Step 3: Now translate by 0−2
y-coords − 2
(−1, 15) → (−1, 13)
(3, −9) → (3, −11)
A′(−1, 13), B′(3, −11)
always stretch first, then translate!
WE 2Show that order matters
Starting with f(x) = x2, find the equation after:
(a) a vertical stretch by 2, then a translation up by 5
(b) a translation up by 5, then a vertical stretch by 2
(a) Stretch first
f(x) → 2f(x) = 2x2
Translate up 5: 2x2 + 5
(a): y = 2x2 + 5
(b) Translate first
f(x) → f(x) + 5 = x2 + 5
Stretch ×2: 2(x2 + 5) = 2x2 + 10
(b): y = 2x2 + 10
different equations — same-axis order MATTERS
WE 3Mixed composite (vertical + horizontal)
The point P(2, 6) lies on y = f(x). Find the image of P after the transformations y = f(x − 4) + 1.
Step 1: Spot the two transformations
f(x − 4) ⇒ translate right by 4
+ 1 ⇒ translate up by 1
Step 2: Different axes — order doesn’t matter
Apply both shifts to P(2, 6):
x: 2 + 4 = 6
y: 6 + 1 = 7
P′(6, 7)
a “translate by 41” handles both at once
WE 4Composite with a reflection
Describe, in order, the transformations that map y = f(x) onto y = −2f(x) + 4.
Step 1: Spot the building blocks
Coefficient −2 ⇒ stretch ×2 + reflection in x-axis
+ 4 ⇒ translate up by 4
Step 2: List in the correct order
All vertical, so order matters between stretch/reflection and the translation.
1. Vertical stretch, factor 2
2. Reflection in the x-axis
3. Translation by 04
stretch & reflection can swap (both vertical, same step) — but BOTH come before the +4
WE 5Build an equation from a word description
The graph of y = f(x) is transformed by, in order: a vertical stretch by scale factor 12, a reflection in the x-axis, and finally a translation by 0−3. Write the equation of the new graph.
Step 1: Apply the stretch
y = 12f(x)
Step 2: Apply the reflection
Multiply by −1: y = −12f(x)
Step 3: Apply the translation (down 3)
Subtract 3 outside: y = −12f(x) − 3
always do same-axis transformations in the order given!
💡 Top tips
- Read the equation like a calculation. For af(x) + b, multiplication comes first, then addition — and the transformations follow the same order.
- Stretch / reflection always before translation. When the same axis has multiple transformations, the multiplicative ones (stretch, reflection) come before the additive ones (translation).
- Vertical vs horizontal don’t interact. A vertical transformation and a horizontal one can be done in any order. They never interfere with each other.
- Update named points step-by-step. Don’t try to do two transformations in one go for a sketch — apply each one and write the new coordinates after every step.
- Negative coefficients always hide a reflection. Spot it explicitly, name it, and list it as a separate transformation.
- Always state the scale factor + axis for stretches, the vector for translations, and the axis for reflections in your written answer.
- If you’re asked to describe a sequence of transformations, write them as a numbered list — examiners can mark each step independently.
⚠ Common mistakes
- Doing translation before stretch on the same axis. This gives af(x) + ab instead of af(x) + b — the constant term gets accidentally scaled.
- Missing the hidden reflection. For y = −3f(x), students often write only “stretch by 3” and forget the reflection in the x-axis.
- Applying horizontal and vertical operations in the wrong “axis” group. A common slip is treating f(2x) + 3 as if the +3 affects the input — it doesn’t. + 3 is a vertical translation, separate from the horizontal stretch.
- Forgetting to factor for horizontal composites. f(2x − 6) needs to be factored as f(2(x − 3)) before you can read off the translation correctly.
- Listing transformations in the wrong order in written answers. Examiners want them in the order they’re applied — write them as a numbered list and double-check.
- Forgetting brackets in algebra. When applying a stretch after a translation in working, missing brackets gives the wrong distributive expansion.
- Saying “two translations” when one is a stretch. Always check whether each operation is multiplicative (stretch/reflection) or additive (translation).
That’s the full Transformations of Graphs section in the bag — translations, reflections, stretches, and now composites. The “inside vs outside” rule has carried us all the way through. Practise a handful of past-paper sketches and you’ll be exam-ready on this topic.
Need help with Composite Transformations?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
Book Free Session →