What are the basic types of transformations of graphs in Edexcel IGCSE Mathematics?

In Edexcel IGCSE Mathematics, the basic types of transformations of graphs include translations, reflections, stretches, and compressions. Translations involve shifting the graph horizontally or vertically. Reflections flip the graph over a specific axis. Stretches and compressions change the size of the graph either vertically or horizontally.

How does a vertical translation affect the graph of a function?

A vertical translation shifts the graph of a function up or down without altering its shape. This is achieved by adding or subtracting a constant to the function. For example, if you have a function f(x), the graph of f(x) + k will move k units up if k is positive, and k units down if k is negative.

What is the effect of reflecting a graph over the x-axis?

Reflecting a graph over the x-axis involves flipping it upside down. This transformation is achieved by multiplying the function by -1. For instance, if the original function is f(x), the reflected function will be -f(x). This changes the sign of all the y-values, effectively inverting the graph.

How do horizontal stretches and compressions transform a graph?

Horizontal stretches and compressions affect the width of the graph. A horizontal stretch occurs when the x-values are multiplied by a factor less than 1, making the graph wider. Conversely, a horizontal compression happens when the x-values are multiplied by a factor greater than 1, making the graph narrower. For a function f(x), the transformation f(ax) will compress the graph horizontally if a > 1 and stretch it if 0 < a < 1.

Can you explain how to perform a combination of transformations on a graph?

Performing a combination of transformations involves applying multiple transformations in sequence. For example, to apply a vertical stretch followed by a translation, you would first multiply the function by a constant to stretch it, and then add or subtract a constant to translate it. It's important to apply transformations in the correct order, as this can affect the final result. For instance, for the function f(x), applying a vertical stretch by a factor of 2 and then translating it up by 3 units would result in the function 2f(x) + 3.

Why is "$f(x + 2)$ shifts RIGHT instead of LEFT" a common mistake in Transformation of Graphs?

Why it happens: Sign appears positive but shift is negative. How to avoid it: f(x + a) shifts by -a (LEFT for positive a). Test: f(0) = value at x = 0 in original. f(0 + 2) = f(2) — so the graph at x = 0 in the new is what was at x = 2 in original — it's been shifted LEFT. Source: 4MA1/1H May/Jun 2024 — examiner report Q20

Why is "$f(2x)$ stretches by $2$ instead of by $1/2$" a common mistake in Transformation of Graphs?

Why it happens: Same logic as horizontal translation. How to avoid it: f(kx): horizontal scaling by 1/k. So f(2x) COMPRESSES by factor 2 (stretches by 1/2).

Why is "Confusing $-f(x)$ with $f(-x)$" a common mistake in Transformation of Graphs?

Why it happens: Both involve negatives. How to avoid it: -f(x): NEGATE the OUTPUT → reflect in x-axis. f(-x): NEGATE the INPUT → reflect in y-axis.

Sequences, Functions and GraphsEdexcel IGCSE Mathematics (4MA1)

Transformation of Graphs

Q: Can you explain how to perform a combination of transformations on a graph?

Performing a combination of transformations involves applying multiple transformations in sequence. For example, to apply a vertical stretch followed by a translation, you would first multiply the function by a constant to stretch it, and then add or subtract a constant to translate it. It's important to apply transformations in the correct order, as this can affect the final result. For instance, for the function f(x), applying a vertical stretch by a factor of 2 and then translating it up by 3 units would result in the function 2f(x) + 3.

Q: Why is "$f(x + 2)$ shifts RIGHT instead of LEFT" a common mistake in Transformation of Graphs?

Why it happens: Sign appears positive but shift is negative. How to avoid it: f(x + a) shifts by -a (LEFT for positive a). Test: f(0) = value at x = 0 in original. f(0 + 2) = f(2) — so the graph at x = 0 in the new is what was at x = 2 in original — it's been shifted LEFT. Source: 4MA1/1H May/Jun 2024 — examiner report Q20

Q: Why is "$f(2x)$ stretches by $2$ instead of by $1/2$" a common mistake in Transformation of Graphs?

Why it happens: Same logic as horizontal translation. How to avoid it: f(kx): horizontal scaling by 1/k. So f(2x) COMPRESSES by factor 2 (stretches by 1/2).

Q: Why is "Confusing $-f(x)$ with $f(-x)$" a common mistake in Transformation of Graphs?

Why it happens: Both involve negatives. How to avoid it: -f(x): NEGATE the OUTPUT → reflect in x-axis. f(-x): NEGATE the INPUT → reflect in y-axis.

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Transformation of Graphs, ready to print or save offline.

Transformation of Graphs Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Translate, stretch and reflect graphs using function notation. A* topic — counter-intuitive but rule-based.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

3.6 — Translate graphs vertically and horizontally.
3.6 — Reflect graphs in the $x$ -axis and $y$ -axis.
3.6 — Stretch graphs in horizontal and vertical directions.
3.6 — Combine multiple transformations.

Translations

$+a$ shifts UP. $f(x + a)$ shifts LEFT (sign flips for horizontal).

Vertical translation: $f(x) + a$ .

Adds $a$ to every $y$ -value.
Graph moves UP by $a$ (or DOWN if $a$ negative).

Example. $f(x) = x^{2}$ .

$f(x) + 3 = x^{2} + 3$ . Vertex moves from $(0,0)$ to $(0, 3)$ . Up by $3$ .
$f(x) - 2 = x^{2} - 2$ . Vertex at $(0, -2)$ . Down by $2$ .

Horizontal translation: $f(x + a)$ .

Replaces $x$ with $x + a$ INSIDE the function.
Graph moves LEFT by $a$ (counter-intuitive!).
$f(x - a)$ moves RIGHT by $a$ .

Example. $f(x) = x^{2}$ .

$f(x + 2) = (x + 2)^{2}$ . Vertex at $(-2, 0)$ . LEFT by $2$ .
$f(x - 3) = (x - 3)^{2}$ . Vertex at $(3, 0)$ . RIGHT by $3$ .

Why does horizontal translation work backwards?

Think: 'where is $f(0)$ in the new graph?' For $f(x + 2)$ : at $x + 2 = 0$ , i.e. $x = -2$ . So the value originally at $x = 0$ is now at $x = -2$ . Graph shifted LEFT.

Combined translation.

$f(x + a) + b$ : shift LEFT by $a$ AND UP by $b$ .

Vector notation: translation by $\binom{-a}{b}$ .

Worked qualitative. Sketch $y = (x - 2)^{2} + 3$ .

This is $f(x) = x^{2}$ translated by $\binom{2}{3}$ .
Vertex at $(2, 3)$ .
Up-opening parabola.

Edexcel tip. Always state translation as a VECTOR. Mark schemes credit the explicit vector.

$f(x) + a$ : UP by $a$ .
$f(x + a)$ : LEFT by $a$ (counter-intuitive).
$f(x - a)$ : RIGHT by $a$ .
Vector: $\binom{-a}{b}$ for $f(x + a) + b$ .

Reflections

$-f(x)$ : flip in $x$ -axis. $f(-x)$ : flip in $y$ -axis.

Reflection in $x$ -axis: $y = -f(x)$ .

Negate every $y$ -value.
Graph flips upside down.

Example. $f(x) = x^{2} + 1$ .

$-f(x) = -(x^{2} + 1) = -x^{2} - 1$ .
Down-opening parabola, vertex at $(0, -1)$ .

Reflection in $y$ -axis: $y = f(-x)$ .

Negate every $x$ -value.
Graph flips left-right.

Example. $f(x) = (x - 3)^{2}$ .

$f(-x) = (-x - 3)^{2} = (x + 3)^{2}$ .
Vertex moves from $(3, 0)$ to $(-3, 0)$ .

Symmetry test.

If $f(-x) = f(x)$ , the function is EVEN — symmetric about $y$ -axis. (E.g. $x^{2}, \cos x$ .)

If $f(-x) = -f(x)$ , the function is ODD — symmetric about origin. (E.g. $x, x^{3}, \sin x$ .)

Worked qualitative. Is $f(x) = x^{2} + 1$ symmetric about the $y$ -axis?

$f(-x) = (-x)^{2} + 1 = x^{2} + 1 = f(x)$ . Yes — even function.

Edexcel tip. Memorise: ' $-f$ ' negates OUTPUT (X-axis flip). ' $f(-x)$ ' negates INPUT (Y-axis flip).

$-f(x)$ : flip in $x$ -axis.
$f(-x)$ : flip in $y$ -axis.
Even functions: symmetric about $y$ -axis.
Odd functions: symmetric about origin.

Stretches

$kf(x)$ : vertical stretch by $k$ . $f(kx)$ : horizontal stretch by $1/k$ .

Vertical stretch: $y = kf(x)$ .

Multiplies every $y$ -value by $k$ .
Stretches graph VERTICALLY by factor $k$ .

Example. $f(x) = \sin x$ .

$2f(x) = 2\sin x$ . Amplitude doubles.

Horizontal stretch: $y = f(kx)$ .

Replaces $x$ with $kx$ INSIDE the function.
Stretches HORIZONTALLY by factor $1/k$ (inverse!).
$f(2x)$ : COMPRESSES by factor $2$ (stretches by $1/2$ ).

Example. $f(x) = \sin x$ has period $360°$ .

$f(2x) = \sin(2x)$ has period $180°$ . (Compressed.)
$f(x/2) = \sin(x/2)$ has period $720°$ . (Stretched.)

Why is horizontal stretch by $1/k$ ?

Think: 'where does the graph hit $y =$ value?' For $f(2x) = \sin(2x)$ : hit at $2x = 90°$ , i.e. $x = 45°$ . Originally hit at $x = 90°$ . So compressed (less $x$ to reach the same $y$ ).

Common stretches:

Function	Effect
$2f(x)$	Vertical stretch ×2
$\dfrac{1}{2}f(x)$	Vertical compression by 2 (or stretch ×1/2)
$f(2x)$	Horizontal compression by 2 (period halves)
$f(x/2)$	Horizontal stretch ×2 (period doubles)
$-f(x)$	Reflection (no stretch)
$f(-x)$	Reflection

Edexcel tip. When a question shows graph $y = f(x)$ and asks to draw $y = 2f(x)$ , multiply each $y$ -value by $2$ . Mark schemes credit accuracy at key points.

$kf(x)$ : vertical stretch by $k$ .
$f(kx)$ : horizontal stretch by $1/k$ .
Horizontal is the INVERSE.
Stretches change shape; reflections + translations don't.

How it’s examined

Graph transformations on Higher Tier (4-6 marks). Often given $y = f(x)$ as a graph; asked to draw various transformations. Examiner reports flag (1) wrong direction for horizontal translation, (2) confusing reflection axes, (3) wrong factor for horizontal stretch.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H May/Jun 2024. Last reviewed 2026-05-07.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Transformation of Graphs, ready to print or save as PDF.

Step-by-step worked examples — Transformation of Graphs

Step-by-step solutions to past-paper-style questions on transformation of graphs, written exactly the way a tutor would explain them at the board.

1Vertical translation $y = f(x) + a$
Higher• translation
Question
If $f(x) = x^{2}$ , sketch $y = f(x) + 3$ and describe the transformation.
Step-by-step solution
1. Step 1
  $y = f(x) + 3 = x^{2} + 3$ .
2. Step 2
  This is the graph of $y = x^{2}$ shifted UP by $3$ .
3. Step 3
  Vertex moves from $(0, 0)$ to $(0, 3)$ .
Answer
Translation by $\binom{0}{3}$ (up by $3$ ).
2Horizontal translation $y = f(x + a)$
Higher• Adapted from 4MA1/1H May/Jun 2024 Q20• translation
Question
If $f(x) = x^{2}$ , sketch $y = f(x + 2)$ .
Step-by-step solution
1. Step 1
  $y = (x + 2)^{2}$ .
2. Step 2
  Translation by $\binom{-2}{0}$ (LEFT by $2$ , NOT right).
3. Step 3
  Vertex moves from $(0, 0)$ to $(-2, 0)$ .
Answer
Translation by $\binom{-2}{0}$ (left by $2$ ).
Examiner tip
$f(x + a)$ shifts LEFT by $a$ (counter-intuitive!). $f(x - a)$ shifts RIGHT.
3Reflections
Higher• reflection
Question
$f(x) = x^{2} + 1$ . Describe (a) $y = -f(x)$ , (b) $y = f(-x)$ .
Step-by-step solution
1. Step 1
  (a) $-f(x) = -(x^{2} + 1) = -x^{2} - 1$ . Reflection in the $x$ -axis (flip upside down).
2. Step 2
  (b) $f(-x) = (-x)^{2} + 1 = x^{2} + 1 = f(x)$ . (Same — symmetric about $y$ -axis.)
Answer
(a) Reflection in $x$ -axis. (b) Same graph (symmetric).
Examiner tip
$-f(x)$ : reflect in X-AXIS. $f(-x)$ : reflect in Y-AXIS.
4Stretches
Higher• stretch
Question
$f(x) = \sin x$ . Describe (a) $y = 2f(x)$ , (b) $y = f(2x)$ .
Step-by-step solution
1. Step 1
  (a) $y = 2\sin x$ . Vertical stretch by factor $2$ . Amplitude doubles.
2. Step 2
  (b) $y = \sin(2x)$ . Horizontal stretch by factor $\dfrac{1}{2}$ (or compression). Period halves.
Answer
(a) Vertical stretch ×2. (b) Horizontal stretch by × $\dfrac{1}{2}$ .
Examiner tip
$kf(x)$ : VERTICAL stretch by $k$ . $f(kx)$ : HORIZONTAL stretch by $1/k$ (counter-intuitive — INVERSE for horizontal).

Key Definitions and Keywords — Transformation of Graphs

Definitions to memorise and the exact keywords mark schemes credit for transformation of graphs answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Translation
Examiner keyword
Sliding a graph without changing its shape. $f(x) + a$ : vertical. $f(x + a)$ : horizontal.
Reflection
Examiner keyword
$-f(x)$ : in $x$ -axis. $f(-x)$ : in $y$ -axis.
Stretch
Examiner keyword
$kf(x)$ : vertical stretch by $k$ . $f(kx)$ : horizontal stretch by $1/k$ .

Common Mistakes and Misconceptions — Transformation of Graphs

The traps other students keep falling into on transformation of graphs questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕ $f(x + 2)$ shifts RIGHT instead of LEFT
4MA1/1H May/Jun 2024 — examiner report Q20
Why it happens
Sign appears positive but shift is negative.
How to avoid it
$f(x + a)$ shifts by $-a$ (LEFT for positive $a$ ). Test: $f(0) =$ value at $x = 0$ in original. $f(0 + 2) = f(2)$ — so the graph at $x = 0$ in the new is what was at $x = 2$ in original — it's been shifted LEFT.
✕ $f(2x)$ stretches by $2$ instead of by $1/2$
Why it happens
Same logic as horizontal translation.
How to avoid it
$f(kx)$ : horizontal scaling by $1/k$ . So $f(2x)$ COMPRESSES by factor 2 (stretches by $1/2$ ).
✕Confusing $-f(x)$ with $f(-x)$
Why it happens
Both involve negatives.
How to avoid it
$-f(x)$ : NEGATE the OUTPUT → reflect in $x$ -axis. $f(-x)$ : NEGATE the INPUT → reflect in $y$ -axis.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Transformation of Graphs — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Transformation of Graphs

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Vertical translation y=f(x)+ay = f(x) + ay=f(x)+a

2Horizontal translation y=f(x+a)y = f(x + a)y=f(x+a)

3Reflections

4Stretches

Translation

Reflection

Stretch

✕f(x+2)f(x + 2)f(x+2) shifts RIGHT instead of LEFT

✕f(2x)f(2x)f(2x) stretches by 222 instead of by 1/21/21/2

✕Confusing −f(x)-f(x)−f(x) with f(−x)f(-x)f(−x)

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Transformation of Graphs — frequently asked questions

1Vertical translation $y = f(x) + a$

2Horizontal translation $y = f(x + a)$

✕ $f(x + 2)$ shifts RIGHT instead of LEFT

✕ $f(2x)$ stretches by $2$ instead of by $1/2$

✕Confusing $-f(x)$ with $f(-x)$