Let $f(x) = \frac{1}{x}$ . The function $g(x)$ is obtained by translating $f(x)$ right by $3$ units and up by $2$ units. Write down an expression for $g(x)$ . Show algebraically that $g(x)$ can be written in the form $h(x) = \frac{2x-5}{x-3}$ . State the domain and range of $h(x)$ .

Translate right 3: $\frac{1}{x-3}$ . A1 Translate up 2: $g(x) = \frac{1}{x-3} + 2$ . A1 Write over a common denominator: $g(x) = \frac{1 + 2(x-3)}{x-3}$ . M1 $= \frac{1 + 2x - 6}{x-3}$ . A1 $= \frac{2x-5}{x-3} = h(x)$ . AG Domain: $x \in \mathbb{R}, \ x \ne 3$ . A1 Range: $y \in \mathbb{R}, \ y \ne 2$ . A1

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IB Maths AA SL · Unit 2: Functions

Transformations of Graphs (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

Changes outside the function move the graph vertically and do what they say: $f (x) + b$ moves it up $b$ , and $p f (x)$ stretches it vertically by $p$ . Changes inside act horizontally and do the opposite: $f (x - a)$ moves it right $a$ , and $f (q x)$ stretches it by $\frac{1}{q}$ . Name each transformation fully, with its type and value.

49 questions
Paper 1: 41
Paper 2: 8
3 worked examples

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Transformations of Graphs notes

Translations

$y = f (x) + b$ is a translation up by $b$ ; $y = f (x - a)$ is a translation right by $a$ . Together, $y = f (x - a) + b$ is the translation by the vector $(\binom{a}{b})$ .

Stretches

$y = p f (x)$ is a vertical stretch with scale factor $p$ : every $y$ -value is multiplied by $p$ . $y = f (q x)$ is a horizontal stretch with scale factor $\frac{1}{q}$ : every $x$ -value is divided by $q$ .

Reflections

$y = - f (x)$ reflects the graph in the $x$ -axis; $y = f (- x)$ reflects it in the $y$ -axis.

Combined transformations and mapping points

Follow a point through each change. On $y = 3 f (x - 2) + 1$ , the point $(x, y)$ on $y = f (x)$ moves to $(x + 2, 3 y + 1)$ , so $(1, 4)$ moves to $(3, 13)$ . Order matters when a stretch and a translation act in the same direction: $3 f (x) + 1$ (stretch, then up 1) is not the same as $3 (f (x) + 1)$ (up 1, then stretch).

Describing a transformation in words

Examiners want the type and the value: "vertical stretch, scale factor 3", "translation by the vector $(\binom{2}{0})$ ", "reflection in the $x$ -axis". A description with only one of these, or a translation in the wrong direction, loses the mark.

Key formulas

Translations: $f (x) + b$ up $b$ ; $f (x - a)$ right $a$ Learn it
Stretches: $p f (x)$ vertical × $p$ ; $f (q x)$ horizontal × $\frac{1}{q}$ Learn it
Reflections: $- f (x)$ in the $x$ -axis; $f (- x)$ in the $y$ -axis Learn it

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Transformations of Graphs worked examples

Worked example 1: Mapping coordinates · easy

The point $P (4, - 2)$ lies on $y = f (x)$ . Find the coordinates of the image of $P$ under $y = f (x - 2) + 3$ .

Solution

4 steps · 1 to fill in

Examiner tip: Transformations inside

f (\dots)

affect

x

counter-intuitively (opposite direction to the sign). Transformations outside affect

y

normally.

Worked example 2: Algebraic composition of transformations · medium

Given $f (x) = 3 x^{2} - 2 x$ , find $g (x)$ obtained by translating $f$ by the vector $(\binom{2}{0})$ then applying a vertical stretch of scale factor $4$ .

Solution

4 steps

Examiner tip: Always place

(x - c)

inside parentheses before expanding; forgetting these brackets is the most common cause of transformation-expansion errors.

Worked example 3: Describing complex sequences · hard

Describe a sequence of three transformations mapping $f (x) = \sqrt{x}$ onto $g (x) = 3 \sqrt{- x - 2}$ .

Solution

4 steps

Examiner tip: When horizontal reflections and translations combine, always factor the negative sign INSIDE (i.e.

f (- (x + c))

) to read the correct translation direction.

Play with it Translate a parabola: drag the sliders

The dashed curve is y = f(x), where f(x) = x². Change h and k so that y = f(x − h) + k passes through all the points. Write the translation as a vector.

Want more? Open the full Grapher.

Transformations of Graphs exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 5 marks · Paper 2

Let $f (x) = x^{2} - 4$ . The graph of $g (x)$ is obtained by translating the graph of $f (x)$ by the vector $(\begin{matrix} - 2 \\ 3 \end{matrix})$ .

Find an expression for $g (x)$ , fully expanded.
Use your graphic display calculator to sketch both $f (x)$ and $g (x)$ on the same axes.
Find the exact coordinates of the vertex of the graph of $g (x)$ .

Check your answer in practice →

Question 2 · medium · 6 marks · Paper 1

Let $f (x) = \frac{1}{x}$ . The function $g (x)$ is obtained by translating $f (x)$ right by $3$ units and up by $2$ units.

Write down an expression for $g (x)$ .
Show algebraically that $g (x)$ can be written in the form $h (x) = \frac{2 x - 5}{x - 3}$ .
State the domain and range of $h (x)$ .

Full worked solution Free, no sign-in

Translate right 3: $\frac{1}{x - 3}$ . A1
Translate up 2: $g (x) = \frac{1}{x - 3} + 2$ . A1
Write over a common denominator: $g (x) = \frac{1 + 2 (x - 3)}{x - 3}$ . M1
$= \frac{1 + 2 x - 6}{x - 3}$ . A1
$= \frac{2 x - 5}{x - 3} = h (x)$ . AG
Domain: $x \in R, x \neq 3$ . A1
Range: $y \in R, y \neq 2$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

Consider the cubic function $f (x) = - x^{3} + 2 x^{2} + 5$ . Let $g (x) = f (x - 2)$ .

Use your GDC to find the coordinates of the local maximum and local minimum points on the graph of $f (x)$ .
Hence, write down the coordinates of the local maximum and local minimum points on the graph of $g (x)$ .
Find the coordinates of the two points of intersection between the graphs of $f (x)$ and $g (x)$ .

Attempt it and see the mark scheme →

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Where students lose marks on transformations of graphs

Examiner's noteTransformations and inverses: incomplete descriptions and f⁻¹ notation

Where marks are lost: Candidates described only one of two transformations, gave a translation in the wrong direction, mishandled a vertical stretch (multiplying a function by a constant), misapplied log laws before transforming, and did not recognise f⁻¹ notation or that the domain of f⁻¹ is the range of f.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Transformations of Graphs on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Graphs: intersections, zeros, max and min on your calculator

Transformations of Graphs FAQ

Are graph transformations in the IB formula booklet?

No. None of the transformation rules are printed in the AA SL booklet, so learn them: inside the bracket acts horizontally and does the opposite of what it looks like; outside acts vertically and does what it says.

Why does f(x − 2) move the graph right, not left?

The new graph reaches each y-value when x − 2 equals the old x-value, so every point happens 2 units later, which is to the right.

How many IB Maths AA SL transformations of graphs questions are there?

There are 49 exam-style transformations of graphs questions in the AA SL question bank (Paper 1: 41 · Paper 2: 8), graded 16 easy, 19 medium, 10 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is transformations of graphs on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 41 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 2: Equations of Straight Lines · Exponential and Logarithmic Functions · Functions: Domain, Range & Inverses · Graphing Functions · Quadratic Functions · Rational and Reciprocal Functions

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Combined transformations.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL functions knowledge organiser

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Transformations of Graphs (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Transformations of Graphs notes

Translations

Stretches

Reflections

Combined transformations and mapping points

Describing a transformation in words

Key formulas

Transformations of Graphs worked examples

Worked example 1: Mapping coordinates · easy

Solution

Worked example 2: Algebraic composition of transformations · medium

Solution

Worked example 3: Describing complex sequences · hard

Solution

Transformations of Graphs exam-style questions

Question 1 · easy · 5 marks · Paper 2

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Where students lose marks on transformations of graphs

Transformations of Graphs on your calculator

Transformations of Graphs FAQ

Next steps

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