What are the basic types of transformations in graphs covered in Edexcel International A Levels Pure Mathematics 1?

In Edexcel International A Levels Pure Mathematics 1, the basic types of transformations include translations, reflections, stretches, and compressions. Translations move the graph horizontally or vertically, reflections flip the graph over a line, and stretches/compressions change the graph's size either vertically or horizontally.

How do you perform a vertical translation on a graph?

To perform a vertical translation on a graph, you add or subtract a constant from 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 multiplying the function by -1. For a function f(x), the reflection is represented by -f(x). This transformation inverts the graph, flipping it upside down across the x-axis.

How can you identify a horizontal stretch in a graph transformation?

A horizontal stretch occurs when you multiply the x-variable by a factor less than 1 in the function. For instance, in the function f(ax) where 0 < a < 1, the graph is stretched horizontally by a factor of 1/a. This makes the graph wider.

What is the significance of understanding graph transformations in Pure Mathematics 1?

Understanding graph transformations is crucial in Pure Mathematics 1 as it helps students visualize and analyze the behavior of functions. This knowledge is essential for solving complex problems and is a foundational skill for further studies in mathematics and related fields.

Why is "Treating $y = f(x + 3)$ as a shift to the RIGHT by 3" a common mistake in Graphs and transformations?

Why it happens: Plus inside 'feels' like positive direction. How to avoid it: y = f(x + a) moves LEFT by a (counter-intuitive). y = f(x - a) moves RIGHT by a. Test with y = (x - 2)^2: vertex shifts to x = 2 — RIGHT, not left. The sign inside is OPPOSITE to the actual shift direction. Source: WMA11/01 examiner reports — most-flagged transformation error

Why is "Saying $y = f(2x)$ is a horizontal stretch by factor 2" a common mistake in Graphs and transformations?

Why it happens: Mistakenly applying the multiplier directly. How to avoid it: y = f(ax) stretches horizontally by factor 1/a. So y = f(2x) is a COMPRESSION (stretch factor 1/2). y = f(x/3) is a stretch by factor 3.

Why is "Sketching $y = k/x$ as just a single branch" a common mistake in Graphs and transformations?

Why it happens: Forgetting the negative values of x. How to avoid it: y = k/x has TWO branches. For k > 0: Q1 and Q3. For k < 0: Q2 and Q4. ALWAYS draw both branches and both asymptotes (x = 0 and y = 0).

Why is "Drawing a cubic with positive leading coefficient as $\cap$-shape" a common mistake in Graphs and transformations?

Why it happens: Confusing cubic with quadratic. How to avoid it: Positive cubic: bottom-left to top-right (S-shape going UP). Negative cubic: top-left to bottom-right. A cubic always goes from - to + (sign depends on leading coefficient) — it's NOT bounded like a parabola.

Why is "Sketching without labelling intercepts" a common mistake in Graphs and transformations?

Why it happens: Rushing the diagram. How to avoid it: 'Sketch, showing intercepts' is the standard wording — you LOSE B marks if intercepts aren't labelled. Always include x-intercepts AND the y-intercept (and stationary points if requested). Source: Persistent issue in WMA11/01 examiner reports

Edexcel International A Levels Mathematics (XMA01-YMA01)

Graphs and transformations

Q: What is the significance of understanding graph transformations in Pure Mathematics 1?

Understanding graph transformations is crucial in Pure Mathematics 1 as it helps students visualize and analyze the behavior of functions. This knowledge is essential for solving complex problems and is a foundational skill for further studies in mathematics and related fields.

Q: Why is "Treating $y = f(x + 3)$ as a shift to the RIGHT by 3" a common mistake in Graphs and transformations?

Why it happens: Plus inside 'feels' like positive direction. How to avoid it: y = f(x + a) moves LEFT by a (counter-intuitive). y = f(x - a) moves RIGHT by a. Test with y = (x - 2)^2: vertex shifts to x = 2 — RIGHT, not left. The sign inside is OPPOSITE to the actual shift direction. Source: WMA11/01 examiner reports — most-flagged transformation error

Q: Why is "Saying $y = f(2x)$ is a horizontal stretch by factor 2" a common mistake in Graphs and transformations?

Why it happens: Mistakenly applying the multiplier directly. How to avoid it: y = f(ax) stretches horizontally by factor 1/a. So y = f(2x) is a COMPRESSION (stretch factor 1/2). y = f(x/3) is a stretch by factor 3.

Q: Why is "Sketching $y = k/x$ as just a single branch" a common mistake in Graphs and transformations?

Why it happens: Forgetting the negative values of x. How to avoid it: y = k/x has TWO branches. For k > 0: Q1 and Q3. For k < 0: Q2 and Q4. ALWAYS draw both branches and both asymptotes (x = 0 and y = 0).

Q: Why is "Drawing a cubic with positive leading coefficient as $\cap$-shape" a common mistake in Graphs and transformations?

Why it happens: Confusing cubic with quadratic. How to avoid it: Positive cubic: bottom-left to top-right (S-shape going UP). Negative cubic: top-left to bottom-right. A cubic always goes from - to + (sign depends on leading coefficient) — it's NOT bounded like a parabola.

Q: Why is "Sketching without labelling intercepts" a common mistake in Graphs and transformations?

Why it happens: Rushing the diagram. How to avoid it: 'Sketch, showing intercepts' is the standard wording — you LOSE B marks if intercepts aren't labelled. Always include x-intercepts AND the y-intercept (and stationary points if requested). Source: Persistent issue in WMA11/01 examiner reports

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Short Notes - Graphs and transformations

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Detailed Study Notes

Detailed notes on Pure Mathematics 1 for Edexcel International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Graphs and Transformations Study Notes — Edexcel IAL Mathematics P1 (WMA11/01, 2018 spec — 2026 onwards)

Sketching cubics and reciprocals, finding intersections, and the four transformations of $y = f(x)$ . Inside vs outside is the make-or-break rule.

At a glance

Cubic $y = (x - a)(x - b)(x - c)$ : three $x$ -intercepts; smooth S-curve.
Reciprocal $y = k/x$ : two branches; asymptotes $x = 0$ , $y = 0$ .
Translation: $f(x + a)$ shifts LEFT by $a$ ; $f(x) + a$ shifts UP by $a$ .
Stretch: $a\,f(x)$ vertical $\times a$ ; $f(ax)$ horizontal $\times 1/a$ .
Reflection: $-f(x)$ in $x$ -axis; $f(-x)$ in $y$ -axis.
Intersection: solve curves simultaneously.
Always label intercepts on sketches.

What you’ll learn

Mapped to the Pearson Edexcel International A Levels XMA01-YMA01 syllabus (2026 onwards).

4.1 — Understand and use the graphs of functions; sketch curves defined by simple equations including polynomials and $y = a/x$ , $y = a/x^{2}$ .
4.2 — Interpret algebraic solutions of equations graphically; use intersection points of graphs to solve equations.
4.3 — Understand the effect of single transformations on the graph of $y = f(x)$ , including $y = af(x)$ , $y = f(x) + a$ , $y = f(x + a)$ , $y = f(ax)$ .
4.4 — Understand combinations of these transformations.

Sketching cubic curves

Three intercepts (when factored) + $y$ -intercept + shape = full sketch.

Factored cubic. $y = (x - a)(x - b)(x - c)$ .

$x$ -intercepts: $x = a, b, c$ .
$y$ -intercept: set $x = 0$ : $y = (-a)(-b)(-c) = -abc$ .
End behaviour: positive leading coefficient → comes from bottom-left, goes to top-right. Negative → top-left to bottom-right.

Example. Sketch $y = (x - 2)(x + 1)(x + 3)$ .

$x$ -intercepts: $2, -1, -3$ .
$y$ -intercept: $(-2)(1)(3) = -6$ .
Leading coefficient $+1$ (positive), so comes from $-\infty$ , goes to $+\infty$ .
Between $-3$ and $-1$ : local max. Between $-1$ and $2$ : local min.

Repeated roots. $y = (x - 2)^{2}(x + 1)$ .

$x = 2$ is a DOUBLE root: curve TOUCHES the axis at $x = 2$ but doesn't cross.
$x = -1$ is a single root: curve CROSSES.

Why? A double factor $(x - 2)^{2}$ is always non-negative, so the sign of $y$ doesn't change as $x$ passes through 2. A single factor changes sign.

Triple root. $y = (x - 1)^{3}$ has a point of inflection at $x = 1$ — the curve passes through with horizontal tangent.

Common graph transformations: translation, stretch, and reflection — INSIDE the bracket changes x; OUTSIDE changes y.

Quick reference.

Roots	Behaviour at the root
Single $(x - a)$	Curve crosses the axis
Double $(x - a)^{2}$	Curve touches the axis (tangent)
Triple $(x - a)^{3}$	Inflection on the axis

$x$ -intercepts from factors.
$y$ -intercept by setting $x = 0$ .
Positive cubic: bottom-left to top-right.
Double root: touches; triple: inflection.

Reciprocal curves — $y = k/x$ and $y = k/x^{2}$

Hyperbolas with axis asymptotes. Number of branches and quadrants depend on the form.

$y = k/x$ (the rectangular hyperbola).

Asymptotes: vertical $x = 0$ , horizontal $y = 0$ .
For $k > 0$ : branches in Q1 (upper right) and Q3 (lower left).
For $k < 0$ : branches in Q2 (upper left) and Q4 (lower right).
Symmetry: $y = x$ and $y = -x$ are axes of symmetry.

Example. Sketch $y = 6/x$ .

Asymptotes: $x = 0$ , $y = 0$ .
Sample points in Q1: $(1, 6), (2, 3), (3, 2), (6, 1)$ .
Reflect through origin to get Q3 points: $(-1, -6), (-2, -3), \ldots$ .

$y = k/x^{2}$ .

Asymptotes: same as above.
ALWAYS positive when $k > 0$ (since $x^{2} > 0$ ). So branches in Q1 and Q2 (upper half plane).
For $k < 0$ : branches in Q3 and Q4 (lower half plane).
Symmetry: $y$ -axis ( $f(-x) = f(x)$ ).

Comparison.

Form	Sign behaviour	Quadrants ( $k > 0$ )
$y = k/x$	Odd: $f(-x) = -f(x)$	Q1 and Q3
$y = k/x^{2}$	Even: $f(-x) = f(x)$	Q1 and Q2

Edexcel tip. Always draw asymptotes as DASHED lines, label them with their equation ( $x = 0$ , $y = 0$ ), and include at least one sample point on each branch.

$y = k/x$ — two opposite quadrants, odd symmetry.
$y = k/x^{2}$ — two adjacent quadrants, even symmetry.
Asymptotes: $x = 0$ , $y = 0$ , drawn dashed.
Label both asymptotes and at least one sample point.

Transformations — the inside/outside rule

OUTSIDE affects $y$ (vertical). INSIDE affects $x$ (horizontal). Inside is BACKWARDS.

Four transformations of $y = f(x)$ .

Transformation	Effect on graph
$y = f(x) + a$	Translation up by $a$
$y = f(x + a)$	Translation left by $a$
$y = a f(x)$	Vertical stretch by factor $a$
$y = f(ax)$	Horizontal stretch by factor $1/a$

Rule: OUTSIDE the function = vertical (intuitive). INSIDE = horizontal (counter-intuitive — sign and factor are inverted).

Translations.

$y = f(x) + 5$ : every point moves UP 5 units.
$y = f(x - 2)$ : every point moves RIGHT 2 (because $f(x - 2)$ has its 'old $x = 0$ ' value at the new $x = 2$ ).
$y = f(x + 3)$ : every point moves LEFT 3.

Stretches.

$y = 3f(x)$ : every $y$ -coordinate $\times 3$ . (Vertical stretch by 3.)
$y = f(2x)$ : every $x$ -coordinate $\times 1/2$ . (Horizontal compression — the curve is squashed.)
$y = f(x/3)$ : every $x$ -coordinate $\times 3$ . (Horizontal stretch by 3.)

Reflections.

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

Combined transformations. Apply in this order: stretch/reflection first, then translation.

Example. $y = f(2x + 4) = f(2(x + 2))$ — horizontal stretch by $1/2$ , then translation left by 2 (NOT 4!).

Quick test. If $y = f(x)$ passes through $(4, 6)$ , where does $y = f(2x - 8) + 3$ pass?

$f(2x - 8) = f(2(x - 4))$ : horizontal compression by $1/2$ , then translation right by 4. Point $(4, 6)$ becomes $(2 + 4, 6) = (6, 6)$ .
Then $+ 3$ : vertical translation up by 3. Becomes $(6, 9)$ .

Edexcel checkpoint. Always write $f(ax + b)$ as $f(a(x + b/a))$ to see the order clearly. Mark scheme treats stretch-then-translate as the canonical order.

Outside = vertical. Inside = horizontal.
Inside is BACKWARDS: $+a$ inside moves LEFT; $\times a$ inside compresses by $1/a$ .
Reflections: $-f(x)$ in $x$ -axis; $f(-x)$ in $y$ -axis.
Combined: factorise inside, apply stretch then translation.

Points of intersection — solve simultaneously

Set the two equations equal; solve the resulting equation.

Method. To find where $y = f(x)$ and $y = g(x)$ meet, set $f(x) = g(x)$ and solve for $x$ . Then substitute each $x$ back into one equation (the simpler one) to find $y$ .

Example 1: line meets parabola.

$y = x + 2$ and $y = x^{2} + 4x - 4$ .

Set equal: $x + 2 = x^{2} + 4x - 4 \Rightarrow x^{2} + 3x - 6 = 0$ .
Quadratic formula: $x = \dfrac{-3 \pm \sqrt{33}}{2}$ .
Find $y$ from line: $y = x + 2$ gives $y = \dfrac{1 \pm \sqrt{33}}{2}$ .
Two intersection points.

Example 2: line meets cubic.

$y = x^{3} - 4x$ and $y = 3x$ .

Set equal: $x^{3} - 4x = 3x \Rightarrow x^{3} - 7x = 0$ .
Factorise: $x(x^{2} - 7) = 0$ → $x = 0, \pm\sqrt{7}$ .
Three intersection points: $(0, 0)$ , $(\sqrt{7}, 3\sqrt{7})$ , $(-\sqrt{7}, -3\sqrt{7})$ .

Number of intersections.

Curve A	Curve B	Max number of intersections
Line	Line	1 (unless parallel)
Line	Parabola	2
Line	Cubic	3
Parabola	Parabola	4
Two cubics	—	9

In general: the maximum number of intersections is the product of the degrees.

Tangent. If the line is tangent to the curve, the resulting equation has a REPEATED root → discriminant zero (or factor appears twice).

Edexcel tip. When intersections are asked, the question often specifies "the coordinates of the points of intersection" — both $x$ and $y$ for each, paired correctly. Marks are usually awarded as: M for setting equal, M for solving, A for $x$ -values, A for $y$ -values, A for paired coordinates.

Set the curves equal.
Solve for $x$ .
Substitute back to find $y$ .
Pair coordinates carefully.
Tangency ⇔ repeated root.

Quick recap

Cubics: three intercepts when factored; smooth S-curve.
Reciprocal $k/x$ : two opposite-quadrant branches with axis asymptotes.
Translation: $f(x + a)$ left; $f(x) + a$ up.
Stretch: $af(x)$ vertical $\times a$ ; $f(ax)$ horizontal $\times 1/a$ .
Reflection: $-f(x)$ in $x$ -axis; $f(-x)$ in $y$ -axis.
Intersection: solve $f(x) = g(x)$ .
Sketches must show intercepts and asymptotes.

Memorise this

Verbatim phrases and definitions Edexcel mark schemes credit.

Inside = horizontal, backwards. $f(x + a)$ moves LEFT.
Outside = vertical, forwards. $f(x) + a$ moves UP.
Stretch inside is reciprocal. $f(2x)$ compresses by $1/2$ .
Reflection: $-f(x)$ in $x$ -axis, $f(-x)$ in $y$ -axis.
Hyperbola $y = k/x$ : asymptotes $x = 0$ , $y = 0$ .
Cubic ends: positive coef → up-and-up.

How it’s examined

Graphs and transformations is a 6-10 mark presence in WMA11/01 P1, typically split into a sketching question (cubic or reciprocal) and a transformation question (often combined with a specific point being tracked). Mark scheme awards B-marks for each labelled feature — intercepts, asymptotes, shape, vertex. A* candidates label EVERY intercept as $(x, y)$ and use dashed lines for asymptotes. The most-missed marks: failing to draw both branches of a reciprocal, treating $f(x + 3)$ as a rightward shift, or forgetting the reciprocal in horizontal stretches.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WMA11/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Graphs and transformations

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

1Sketch a factorised cubic (4 marks, P1)
Core• Style: WMA11/01 January 2024 Q7• cubic, sketch
Question
Sketch the curve with equation $y = (x - 2)(x + 1)(x + 3)$ , showing clearly the coordinates of any points where the curve meets the coordinate axes.
Step-by-step solution
1. Step 1
  $x$ -intercepts (set $y = 0$ ): $x = 2$ , $x = -1$ , $x = -3$ .
2. Step 2
  $y$ -intercept (set $x = 0$ ): $y = (-2)(1)(3) = -6$ .
3. Step 3
  Shape: coefficient of $x^{3}$ is $+1$ (positive), so the cubic goes from bottom-left to top-right. As $x \to -\infty$ , $y \to -\infty$ . As $x \to \infty$ , $y \to \infty$ .
4. Step 4
  Sketch: smooth cubic curve crossing $x$ -axis at $-3, -1, 2$ and $y$ -axis at $-6$ . Local max between $-3$ and $-1$ ; local min between $-1$ and $2$ .
Answer
Cubic through $(-3, 0)$ , $(-1, 0)$ , $(2, 0)$ with $y$ -intercept $(0, -6)$ .
Examiner tip
B1 for each correct intercept ( $x$ -intercepts: 1 mark for all three; $y$ -intercept: 1 mark). B1 for correct shape (smooth cubic, positive leading coefficient). B1 for the curve going through all the marked points. A* sketch labels ALL FOUR intercepts as $(x, y)$ pairs.
2Sketch $y = k/x$ (3 marks, P1)
Core• Style: WMA11/01 June 2023 Q6• reciprocal, sketch, asymptote
Question
Sketch the graph of $y = \dfrac{6}{x}$ . State the equations of any asymptotes.
Step-by-step solution
1. Step 1
  The graph is a hyperbola. Since $k = 6 > 0$ , the curve sits in the FIRST and THIRD quadrants.
2. Step 2
  Asymptotes: $x = 0$ (vertical, $y$ -axis) and $y = 0$ (horizontal, $x$ -axis). The curve approaches but never touches them.
3. Step 3
  Sample points: $(1, 6)$ , $(2, 3)$ , $(6, 1)$ in Q1; $(-1, -6)$ , $(-2, -3)$ , $(-6, -1)$ in Q3.
Answer
Hyperbola with branches in Q1 and Q3. Asymptotes: $x = 0$ and $y = 0$ .
Examiner tip
B1 for correct shape (hyperbola in Q1 + Q3, not just one branch). B1 for both asymptotes stated. B1 for the asymptotes drawn as dashed lines (or for at least one labelled point on a branch). Common error: only drawing one branch.
3Translation transformation (3 marks, P1)
Core• Style: WMA11/01 June 2024 Q7• transformation, translation
Question
The curve $y = f(x)$ has a turning point at $(2, -5)$ . State the coordinates of the turning point of the curve with equation $y = f(x + 3) + 4$ .
Step-by-step solution
1. Step 1
  $y = f(x + 3)$ is a horizontal translation by $-3$ (LEFT by 3 units). New $x$ : $2 - 3 = -1$ .
2. Step 2
  $y = f(x) + 4$ is a vertical translation by $+4$ (UP by 4 units). New $y$ : $-5 + 4 = -1$ .
3. Step 3
  Combined: turning point moves from $(2, -5)$ to $(-1, -1)$ .
Answer
$(-1, -1)$
Examiner tip
B1 for the horizontal shift (note: $+3$ INSIDE means LEFT 3, not right). B1 for the vertical shift. B1 for both coordinates correct. The 'inside vs outside' rule is the most-failed concept: $f(x + a)$ shifts LEFT by $a$ ; $f(x) + a$ shifts UP by $a$ .
4Vertical and horizontal stretches (4 marks, P1)
Extended• Style: WMA11/01 January 2023 Q7• transformation, stretch
Question
The curve $y = f(x)$ passes through the point $(4, 6)$ . State the coordinates of the corresponding point on (a) $y = 2f(x)$ , (b) $y = f(2x)$ , (c) $y = -f(x)$ , (d) $y = f(-x)$ .
Step-by-step solution
1. Step 1
  (a) $y = 2f(x)$ — vertical stretch by factor 2. $y$ doubles, $x$ unchanged: $(4, 12)$ .
2. Step 2
  (b) $y = f(2x)$ — horizontal stretch by factor $1/2$ (i.e. compression). $x$ halves, $y$ unchanged: $(2, 6)$ .
3. Step 3
  (c) $y = -f(x)$ — reflection in the $x$ -axis. $y$ flips sign: $(4, -6)$ .
4. Step 4
  (d) $y = f(-x)$ — reflection in the $y$ -axis. $x$ flips sign: $(-4, 6)$ .
Answer
(a) $(4, 12)$ . (b) $(2, 6)$ . (c) $(4, -6)$ . (d) $(-4, 6)$ .
Examiner tip
B1 for each correct coordinate. The trickiest is (b): $f(2x)$ HALVES the $x$ -coordinate (stretch factor $1/2$ horizontally, not 2). Outside vs inside: outside affects $y$ ; inside affects $x$ — and INVERSELY.
5Intersection of cubic and line (6 marks, P1)
Extended• Style: WMA11/01 June 2024 Q8• intersection
Question
Find the coordinates of the points where the curve $y = x^{3} - 4x$ meets the line $y = 3x$ .
Step-by-step solution
1. Step 1
  Set equations equal: $x^{3} - 4x = 3x$ .
2. Step 2
  Rearrange:
  $x^{3} - 7x = 0$
3. Step 3
  Factorise: $x(x^{2} - 7) = 0$ . So $x = 0$ or $x = \pm \sqrt{7}$ .
4. Step 4
  Find $y$ from the LINE $y = 3x$ : $x = 0 \Rightarrow y = 0$ ; $x = \sqrt{7} \Rightarrow y = 3\sqrt{7}$ ; $x = -\sqrt{7} \Rightarrow y = -3\sqrt{7}$ .
Answer
$(0, 0)$ , $(\sqrt{7}, 3\sqrt{7})$ , $(-\sqrt{7}, -3\sqrt{7})$ .
Examiner tip
M1 for setting equations equal; M1 for rearranging to one side; A1 for factorising correctly; A1 for the three $x$ -values; A1 for the three $y$ -values; A1 for paired coordinates. ECF applies. Note: a cubic and a line can meet at 1, 2, or 3 points — don't be surprised by three solutions.

Key Formulae — Graphs and transformations

The formulae you need to memorise for graphs and transformations on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Horizontal translation
$y = f(x + a) \quad \text{shifts } y = f(x) \text{ LEFT by } a \text{ units}$
$a$
translation amount (positive a = leftward)
When to use
When a constant is added INSIDE the function. Counter-intuitive: '+a inside' moves LEFT.
Example
$y = (x + 3)^{2}$ is $y = x^{2}$ shifted LEFT by 3, so vertex at $(-3, 0)$ .
Vertical translation
$y = f(x) + a \quad \text{shifts } y = f(x) \text{ UP by } a \text{ units}$
$a$
translation amount
When to use
When a constant is added OUTSIDE the function.
Example
$y = x^{2} + 5$ is $y = x^{2}$ shifted up by 5.
Vertical stretch
$y = a f(x) \quad \text{stretches } y = f(x) \text{ vertically by factor } a$
$a$
stretch factor (a > 0)
When to use
Multiply outside. Each $y$ -coordinate gets multiplied by $a$ .
Example
$y = 3\sin x$ has amplitude 3 (compared to $\sin x$ 's amplitude 1).
Horizontal stretch
$y = f(ax) \quad \text{stretches } y = f(x) \text{ horizontally by factor } 1/a$
$a$
factor inside the function (a > 0)
When to use
Multiply inside. Each $x$ -coordinate gets DIVIDED by $a$ (counter-intuitive).
Example
$y = \sin(2x)$ has period $\pi$ (compared to $\sin x$ 's period $2\pi$ ).
Reflections
$y = -f(x) \text{ reflects in } x\text{-axis}; \quad y = f(-x) \text{ reflects in } y\text{-axis}$
$—$
negative sign placement
When to use
Outside negative → flip vertically. Inside negative → flip horizontally.
Example
$y = -e^{x}$ is the reflection of $y = e^{x}$ in the $x$ -axis; $y = e^{-x}$ is the reflection in the $y$ -axis.

Key Definitions and Keywords — Graphs and transformations

Definitions to memorise and the exact keywords mark schemes credit for graphs and transformations answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Asymptote
Examiner keyword
A line that a curve approaches but never touches. For $y = k/x$ : vertical asymptote $x = 0$ , horizontal asymptote $y = 0$ .
Transformation
Examiner keyword
A geometric change to a curve. Four standard transformations: translation (shift), stretch (scale), reflection (flip), and combinations thereof.
Stretch factor
$y = a f(x)$ has vertical stretch factor $a$ . $y = f(ax)$ has horizontal stretch factor $1/a$ (note the reciprocal).
Point(s) of intersection
$(x, y)$ values where two curves cross. Found by solving the two equations simultaneously.
Turning point
A local maximum or minimum of a curve. Where the gradient changes sign.
Related: calculus, completing the square

Common Mistakes and Misconceptions — Graphs and transformations

The traps other students keep falling into on graphs and transformations questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Treating $y = f(x + 3)$ as a shift to the RIGHT by 3
WMA11/01 examiner reports — most-flagged transformation error
Why it happens
Plus inside 'feels' like positive direction.
How to avoid it
$y = f(x + a)$ moves LEFT by $a$ (counter-intuitive). $y = f(x - a)$ moves RIGHT by $a$ . Test with $y = (x - 2)^{2}$ : vertex shifts to $x = 2$ — RIGHT, not left. The sign inside is OPPOSITE to the actual shift direction.
✕Saying $y = f(2x)$ is a horizontal stretch by factor 2
Why it happens
Mistakenly applying the multiplier directly.
How to avoid it
$y = f(ax)$ stretches horizontally by factor $1/a$ . So $y = f(2x)$ is a COMPRESSION (stretch factor $1/2$ ). $y = f(x/3)$ is a stretch by factor 3.
✕Sketching $y = k/x$ as just a single branch
Why it happens
Forgetting the negative values of $x$ .
How to avoid it
$y = k/x$ has TWO branches. For $k > 0$ : Q1 and Q3. For $k < 0$ : Q2 and Q4. ALWAYS draw both branches and both asymptotes ( $x = 0$ and $y = 0$ ).
✕Drawing a cubic with positive leading coefficient as $\cap$ -shape
Why it happens
Confusing cubic with quadratic.
How to avoid it
Positive cubic: bottom-left to top-right (S-shape going UP). Negative cubic: top-left to bottom-right. A cubic always goes from $-\infty$ to $+\infty$ (sign depends on leading coefficient) — it's NOT bounded like a parabola.
✕Sketching without labelling intercepts
Persistent issue in WMA11/01 examiner reports
Why it happens
Rushing the diagram.
How to avoid it
'Sketch, showing intercepts' is the standard wording — you LOSE B marks if intercepts aren't labelled. Always include $x$ -intercepts AND the $y$ -intercept (and stationary points if requested).

Graphs and transformations — frequently asked questions

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

Graphs and transformations

Short Notes - Graphs and transformations

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Sketch a factorised cubic (4 marks, P1)

2Sketch y=k/xy = k/xy=k/x (3 marks, P1)

3Translation transformation (3 marks, P1)

4Vertical and horizontal stretches (4 marks, P1)

5Intersection of cubic and line (6 marks, P1)

Horizontal translation

Vertical translation

Vertical stretch

Horizontal stretch

Reflections

Asymptote

Transformation

Stretch factor

Point(s) of intersection

Turning point

✕Treating y=f(x+3)y = f(x + 3)y=f(x+3) as a shift to the RIGHT by 3

✕Saying y=f(2x)y = f(2x)y=f(2x) is a horizontal stretch by factor 2

✕Sketching y=k/xy = k/xy=k/x as just a single branch

✕Drawing a cubic with positive leading coefficient as ∩\cap∩-shape

✕Sketching without labelling intercepts

Graphs and transformations — frequently asked questions

2Sketch $y = k/x$ (3 marks, P1)

✕Treating $y = f(x + 3)$ as a shift to the RIGHT by 3

✕Saying $y = f(2x)$ is a horizontal stretch by factor 2

✕Sketching $y = k/x$ as just a single branch

✕Drawing a cubic with positive leading coefficient as $\cap$ -shape