Does the order of transformations always matter?

Not always, but often. Consider a vertical translation up by 2 then a vertical stretch by 3: that gives $3(f(x)+2)=3f(x)+6$. Reverse the order — stretch first, then translate — and you get $3f(x)+2$. Those are different graphs. Order does NOT matter only when the transformations act on different axes, such as a horizontal translation combined with a vertical stretch.

How do I remember whether $f(x+2)$ shifts left or right?

Ask which input makes the bracket behave like the original. For $f(x+2)$ the bracket equals zero when $x+2=0$, i.e. at $x=-2$. So the feature that used to sit at $x=0$ now sits at $x=-2$ — the graph has moved 2 units LEFT. Inside-bracket changes always go the opposite way to the sign.

What's the difference between $y=2f(x)$ and $y=f(2x)$?

$y=2f(x)$ is a vertical stretch, scale factor 2 — every $y$-coordinate is doubled. $y=f(2x)$ is a horizontal stretch, scale factor $\frac{1}{2}$ — every $x$-coordinate is halved (a horizontal compression). The '2' outside acts vertically the obvious way; the '2' inside acts horizontally the reciprocal way.

How much detail does a 'describe the transformation' question need?

Be precise and complete. For a translation, give the vector or the direction and number of units. For a reflection, name the line (the $x$-axis or the $y$-axis). For a stretch, give the scale factor AND the direction/axis. A description missing the scale factor or the axis will drop the accuracy mark even if the idea is right.

How do transformations affect an asymptote?

Treat the asymptote like any other feature and move it with the graph. A vertical asymptote is horizontal-only information, so it is shifted by horizontal translations and horizontal stretches but is untouched by vertical ones; a horizontal asymptote is moved by vertical translations, vertical stretches and reflection in the $x$-axis. For example, applying $y=f(x)+k$ to a curve with horizontal asymptote $y=0$ moves that asymptote to $y=k$.

Transformations of graphs — IB Maths AA HL

Revise Transformations of graphs for IB Mathematics: Analysis and Approaches (2.4)

Looking for free IB Mathematics: Analysis and Approaches notes on Transformations of graphs? This Higher Level lesson covers official syllabus point 2.4 on Maths AA HL. Use the visual steps, key terms, and quick checks below — then practise with IB criterion marking on Transformations of graphs. Built for students searching "IB Maths AA 2.4", "Transformations of graphs IB revision", or a free alternative to paid IB note sites.

Transformations of graphs

Understanding how to transform the graph of a function is a core skill in Analysis and Approaches. It lets you take a simple parent function, such as $y=x^2$ or $y=\sin x$ , and reshape it to model something more complex. In this lesson we work through the four transformation types on the syllabus — translations, reflections, and vertical and horizontal stretches — see how each one acts on key features like the vertex, intercepts and asymptotes, and then combine them in the right order. As in every calculation-and-description topic, the worked examples are laid out exactly the way our engine marks your own practice: the method and the description are on the page, mark by mark.

BY THE END, YOU CAN…

1Identify and describe translations, reflections and stretches of a function $y=f(x)$ with full, precise wording.
2Apply a single transformation, or a sequence of transformations, to a graph, to its equation, or to a point on the graph.
3Determine the effect of a transformation on key features such as the vertex, intercepts and asymptotes.
4Apply combined transformations in the correct order and explain why the order can change the result.
5Present descriptions and working so that method (M) and accuracy (A) marks, with follow-through, protect the final answer.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Shape-Shifting Functions

Learn how to move, flip and stretch the graph of a function using simple algebraic rules. Each rule changes a graph's position or shape in a completely predictable way, so from one parent function you can build a whole family of related graphs.

THINK OF IT LIKE…

Think of the graph as a photo on a screen. You can drag it up, down, left or right (a translation); flip it in a mirror (a reflection); or pull it taller or wider (a stretch). Each drag, flip and pull corresponds to one specific change in the equation — and the trick is knowing which change does what.

1
Start with the parent function $y=f(x)$ and note its key features: intercepts, vertex or turning points, and any asymptotes.
2
Look at changes INSIDE the brackets, like $f(x-h)$ or $f(bx)$ . These are horizontal, they act on the $x$ -coordinates, and they behave in the opposite way to what you expect.
3
Look at changes OUTSIDE the brackets, like $af(x)$ or $f(x)+k$ . These are vertical, they act on the $y$ -coordinates, and they behave the way you expect.
4
When several transformations are combined, apply them in a sensible order — usually stretches and reflections first, then translations — and track a general point $(x,y)$ so nothing is lost.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · Step 1

Loading simulation…

Fetching from GeoGebra…

GeoGebra Graph Transformations

Use “GeoGebra Graph Transformations” — shift curves or drag points to see equilibrium and slopes change.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step 1

Start with the parent function $y=f(x)$ and note its key features: intercepts, vertex or turning points, and any asymptotes.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

y = f(x) + k

y = af(x)

Full topic notes

Formal explanation with the rigour you need for the exam.

Translations: sliding the graph

A translation slides the whole graph to a new position without changing its shape, size or orientation. Vertical translations act outside the function and behave intuitively; horizontal translations act inside the brackets and behave in the opposite direction to the sign — the single most common source of error in this topic.

Vertical translation: $y = f(x) + k$ Shifts the graph by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ .

Horizontal translation: $y = f(x-h)$ Shifts the graph by the vector $\begin{pmatrix} h \\ 0 \end{pmatrix}$ .

For vertical shifts $y=f(x)+k$ : $k>0$ moves the graph UP, $k<0$ moves it DOWN.
For horizontal shifts the effect is reversed: $y=f(x-h)$ with $h>0$ (e.g. $f(x-3)$ ) moves the graph to the RIGHT.
Likewise $y=f(x+h)$ with $h>0$ (e.g. $f(x+3)$ ) moves the graph to the LEFT.
Translations move every key feature with the graph: the vertex, the intercepts and any asymptote all shift by the same vector.

Reflections: flipping the graph

A reflection flips the graph across a line to create a mirror image. On the SL syllabus we reflect in the $x$ -axis and the $y$ -axis. Reflecting in the $x$ -axis negates the output; reflecting in the $y$ -axis negates the input.

Reflection in the $x$ -axis: $y = -f(x)$ Every point $(x, y)$ maps to $(x, -y)$ .

Reflection in the $y$ -axis: $y = f(-x)$ Every point $(x, y)$ maps to $(-x, y)$ .

$y=-f(x)$ negates every $y$ -coordinate; the $x$ -intercepts stay fixed because their $y$ -value is already zero.
$y=f(-x)$ negates every $x$ -coordinate; the $y$ -intercept stays fixed because its $x$ -value is already zero.
Any point that lies on the mirror line — the axis of reflection — is invariant.

Stretches and compressions: scaling the graph

A stretch scales the graph away from an axis; a compression pushes it towards an axis. A change outside the function scales it vertically by exactly the factor shown; a change inside the brackets scales it horizontally by the RECIPROCAL of the factor shown — another reversal to watch for.

Vertical stretch: $y = af(x)$ Stretch parallel to the $y$ -axis, scale factor $a$ . The $x$ -axis is invariant.

Horizontal stretch: $y = f(bx)$ Stretch parallel to the $x$ -axis, scale factor $\frac{1}{b}$ . The $y$ -axis is invariant.

For a vertical stretch $y=af(x)$ : if $|a|>1$ it is a stretch, if $0<|a|<1$ it is a compression.
For a horizontal stretch $y=f(bx)$ : the factor is $\frac{1}{b}$ , so $|b|>1$ compresses (factor $<1$ ) and $0<|b|<1$ stretches (factor $>1$ ) — the counter-intuitive case.
Points on the invariant line do not move: for a vertical stretch the $x$ -intercepts are invariant; for a horizontal stretch the $y$ -intercept is invariant.
A horizontal asymptote is unaffected by a horizontal stretch; a vertical asymptote is unaffected by a vertical stretch.

Effect on key features: vertex, intercepts and asymptotes

You rarely need to redraw a whole curve — usually you just transform its key features and read off the result. Treat each feature (vertex, $x$ - and $y$ -intercepts, and asymptotes) as a point or line and move it by the same rule as the graph. The only subtlety is which features a given transformation actually touches.

Vertex / turning point: transform it like any point, applying horizontal changes to its $x$ -coordinate and vertical changes to its $y$ -coordinate.
$x$ -intercepts: moved by horizontal translations and horizontal stretches; unchanged by a vertical stretch or a reflection in the $x$ -axis (their $y$ -value is already 0).
$y$ -intercept: moved by vertical translations, vertical stretches and reflection in the $x$ -axis; unchanged by a horizontal stretch or reflection in the $y$ -axis.
Horizontal asymptote: moved by vertical translations, vertical stretches and reflection in the $x$ -axis. Vertical asymptote: moved by horizontal translations, horizontal stretches and reflection in the $y$ -axis.

Common mistakes examiners penalise

Shifting $f(x-h)$ the wrong way — $f(x-3)$ moves the graph 3 units to the RIGHT, not left. Inside-bracket changes act on $x$ in the opposite direction to their sign.
Getting the horizontal stretch factor wrong — $y=f(bx)$ has scale factor $\frac{1}{b}$ , not $b$ . So $y=f(2x)$ compresses horizontally by $\frac{1}{2}$ ; it does not stretch by 2.
Confusing $2f(x)$ with $f(2x)$ — the first is a vertical stretch (factor 2, acts on $y$ ); the second is a horizontal stretch (factor $\frac{1}{2}$ , acts on $x$ ).
Applying combined transformations in the wrong order — for $y=af(x)+k$ the stretch comes before the translation; reversing them changes the graph. State and follow a consistent order.
Giving an incomplete description — a stretch needs its scale factor AND its axis/direction; a translation needs its vector or direction-and-magnitude; a reflection needs the mirror line. Missing detail loses the accuracy mark.
Forgetting to move an asymptote — an asymptote is a feature of the graph and must be translated or stretched along with it; leaving it in place is a common slip in reciprocal and exponential questions.

Model answer — marked the way our engine marks it

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) for a correct approach or an answer mark (A) for a correct result — and follow-through (FT) means a wrong value early on need not cost you the marks that depend on it, provided your later working is correct on your own figures. The engine also accepts any equivalent correct form and applies 'ignore subsequent working' (ISW) once a correct answer is seen. But that protection only exists if your description and method are written down. Study how each mark below is earned by a specific line.

Where this leads

Transformations run through the whole functions strand. The same rules reshape the reciprocal, exponential and logarithmic graphs you meet next, where moving the asymptote correctly is the difference between a right and a wrong sketch. In trigonometry, amplitude, period and phase shift are simply the vertical stretch, horizontal stretch and horizontal translation of $\sin x$ and $\cos x$ . And composing transformations here is the graphical face of composing functions. Master the habit — identify inside versus outside, describe each transformation completely, fix an order, and track a general point — and every graph you meet later becomes a variation on a method you already own.

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

The graph of $y = f(x)$ passes through the point $A(-2, 6)$ . The graph is transformed by a reflection in the $x$ -axis followed by a translation of $\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ . Find the coordinates of the image of point $A$ . [4]

1
Track the single point $A(-2,6)$ through the two transformations in order.

EXAMPLE 2

Let $f(x) = x^2$ . The graph of $f$ is transformed to the graph of $g$ by a vertical stretch of scale factor 2, followed by a horizontal translation of 3 units to the right. (a) Find the equation of $g(x)$ . (b) The point $P(1,1)$ lies on $y=f(x)$ . Find the coordinates of its image on $y=g(x)$ . [5]

1
(a) Build the equation in the given order, starting from $f(x)=x^2$ .

EXAMPLE 3

The graph of $y=f(x)$ is transformed to $y=2f(x-3)$ . Describe the two transformations and find the image of the point $(1, 4)$ . [5]

1
Model answer — full working.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEATransformations of graphs

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What is the transformation represented by $y = f(x) + k$ ?

TAP OR SPACE TO FLIP

ANSWER

A vertical translation. The graph shifts up by $k$ units if $k>0$ and down by $|k|$ units if $k<0$ — a translation by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ .

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
For vertical shifts $y=f(x)+k$ : $k>0$ moves the graph UP, $k<0$ moves it DOWN.
✓
For horizontal shifts the effect is reversed: $y=f(x-h)$ with $h>0$ (e.g. $f(x-3)$ ) moves the graph to the RIGHT.
✓
Likewise $y=f(x+h)$ with $h>0$ (e.g. $f(x+3)$ ) moves the graph to the LEFT.
✓
Translations move every key feature with the graph: the vertex, the intercepts and any asymptote all shift by the same vector.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Get a Paper 1 transformations question marked: describe the transformations and find the image with full working

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

IB Mathematics: Analysis and Approaches guide (Topic 2 — Functions)↗

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Get a Paper 1 transformations question marked: describe the transformations and find the image with full working on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS2.3 Quadratic functions and equations NEXT →2.5 Rational functions and their graphs

Transformations of graphs

BY THE END, YOU CAN…

1Identify and describe translations, reflections and stretches of a function $y=f(x)$ with full, precise wording.
2Apply a single transformation, or a sequence of transformations, to a graph, to its equation, or to a point on the graph.
3Determine the effect of a transformation on key features such as the vertex, intercepts and asymptotes.
4Apply combined transformations in the correct order and explain why the order can change the result.
5Present descriptions and working so that method (M) and accuracy (A) marks, with follow-through, protect the final answer.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Shape-Shifting Functions

THINK OF IT LIKE…

1
Start with the parent function $y=f(x)$ and note its key features: intercepts, vertex or turning points, and any asymptotes.
2
Look at changes INSIDE the brackets, like $f(x-h)$ or $f(bx)$ . These are horizontal, they act on the $x$ -coordinates, and they behave in the opposite way to what you expect.
3
Look at changes OUTSIDE the brackets, like $af(x)$ or $f(x)+k$ . These are vertical, they act on the $y$ -coordinates, and they behave the way you expect.
4
When several transformations are combined, apply them in a sensible order — usually stretches and reflections first, then translations — and track a general point $(x,y)$ so nothing is lost.

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

GEOGEBRASTEP 1 / 4 · Step 1

Loading simulation…

Fetching from GeoGebra…

GeoGebra Graph Transformations

Use “GeoGebra Graph Transformations” — shift curves or drag points to see equilibrium and slopes change.

If the embed does not load, use Open on GeoGebra.

GeoGebra · GeoGebra Terms of Service

Step 1

Start with the parent function $y=f(x)$ and note its key features: intercepts, vertex or turning points, and any asymptotes.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

y = f(x) + k

y = af(x)

Full topic notes

Formal explanation with the rigour you need for the exam.

Translations: sliding the graph

Vertical translation: $y = f(x) + k$ Shifts the graph by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ .

Horizontal translation: $y = f(x-h)$ Shifts the graph by the vector $\begin{pmatrix} h \\ 0 \end{pmatrix}$ .

For vertical shifts $y=f(x)+k$ : $k>0$ moves the graph UP, $k<0$ moves it DOWN.
For horizontal shifts the effect is reversed: $y=f(x-h)$ with $h>0$ (e.g. $f(x-3)$ ) moves the graph to the RIGHT.
Likewise $y=f(x+h)$ with $h>0$ (e.g. $f(x+3)$ ) moves the graph to the LEFT.
Translations move every key feature with the graph: the vertex, the intercepts and any asymptote all shift by the same vector.

Reflections: flipping the graph

Reflection in the $x$ -axis: $y = -f(x)$ Every point $(x, y)$ maps to $(x, -y)$ .

Reflection in the $y$ -axis: $y = f(-x)$ Every point $(x, y)$ maps to $(-x, y)$ .

$y=-f(x)$ negates every $y$ -coordinate; the $x$ -intercepts stay fixed because their $y$ -value is already zero.
$y=f(-x)$ negates every $x$ -coordinate; the $y$ -intercept stays fixed because its $x$ -value is already zero.
Any point that lies on the mirror line — the axis of reflection — is invariant.

Stretches and compressions: scaling the graph

Vertical stretch: $y = af(x)$ Stretch parallel to the $y$ -axis, scale factor $a$ . The $x$ -axis is invariant.

Horizontal stretch: $y = f(bx)$ Stretch parallel to the $x$ -axis, scale factor $\frac{1}{b}$ . The $y$ -axis is invariant.

For a vertical stretch $y=af(x)$ : if $|a|>1$ it is a stretch, if $0<|a|<1$ it is a compression.
For a horizontal stretch $y=f(bx)$ : the factor is $\frac{1}{b}$ , so $|b|>1$ compresses (factor $<1$ ) and $0<|b|<1$ stretches (factor $>1$ ) — the counter-intuitive case.
Points on the invariant line do not move: for a vertical stretch the $x$ -intercepts are invariant; for a horizontal stretch the $y$ -intercept is invariant.
A horizontal asymptote is unaffected by a horizontal stretch; a vertical asymptote is unaffected by a vertical stretch.

Effect on key features: vertex, intercepts and asymptotes

Vertex / turning point: transform it like any point, applying horizontal changes to its $x$ -coordinate and vertical changes to its $y$ -coordinate.
$x$ -intercepts: moved by horizontal translations and horizontal stretches; unchanged by a vertical stretch or a reflection in the $x$ -axis (their $y$ -value is already 0).
$y$ -intercept: moved by vertical translations, vertical stretches and reflection in the $x$ -axis; unchanged by a horizontal stretch or reflection in the $y$ -axis.
Horizontal asymptote: moved by vertical translations, vertical stretches and reflection in the $x$ -axis. Vertical asymptote: moved by horizontal translations, horizontal stretches and reflection in the $y$ -axis.

Common mistakes examiners penalise

Shifting $f(x-h)$ the wrong way — $f(x-3)$ moves the graph 3 units to the RIGHT, not left. Inside-bracket changes act on $x$ in the opposite direction to their sign.
Getting the horizontal stretch factor wrong — $y=f(bx)$ has scale factor $\frac{1}{b}$ , not $b$ . So $y=f(2x)$ compresses horizontally by $\frac{1}{2}$ ; it does not stretch by 2.
Confusing $2f(x)$ with $f(2x)$ — the first is a vertical stretch (factor 2, acts on $y$ ); the second is a horizontal stretch (factor $\frac{1}{2}$ , acts on $x$ ).
Applying combined transformations in the wrong order — for $y=af(x)+k$ the stretch comes before the translation; reversing them changes the graph. State and follow a consistent order.
Giving an incomplete description — a stretch needs its scale factor AND its axis/direction; a translation needs its vector or direction-and-magnitude; a reflection needs the mirror line. Missing detail loses the accuracy mark.
Forgetting to move an asymptote — an asymptote is a feature of the graph and must be translated or stretched along with it; leaving it in place is a common slip in reciprocal and exponential questions.

Model answer — marked the way our engine marks it

Where this leads

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

1
Track the single point $A(-2,6)$ through the two transformations in order.

EXAMPLE 2

1
(a) Build the equation in the given order, starting from $f(x)=x^2$ .

EXAMPLE 3

The graph of $y=f(x)$ is transformed to $y=2f(x-3)$ . Describe the two transformations and find the image of the point $(1, 4)$ . [5]

1
Model answer — full working.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEATransformations of graphs

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What is the transformation represented by $y = f(x) + k$ ?

TAP OR SPACE TO FLIP

ANSWER

A vertical translation. The graph shifts up by $k$ units if $k>0$ and down by $|k|$ units if $k<0$ — a translation by the vector $\begin{pmatrix} 0 \\ k \end{pmatrix}$ .

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
For vertical shifts $y=f(x)+k$ : $k>0$ moves the graph UP, $k<0$ moves it DOWN.
✓
For horizontal shifts the effect is reversed: $y=f(x-h)$ with $h>0$ (e.g. $f(x-3)$ ) moves the graph to the RIGHT.
✓
Likewise $y=f(x+h)$ with $h>0$ (e.g. $f(x+3)$ ) moves the graph to the LEFT.
✓
Translations move every key feature with the graph: the vertex, the intercepts and any asymptote all shift by the same vector.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Get a Paper 1 transformations question marked: describe the transformations and find the image with full working

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

IB Mathematics: Analysis and Approaches guide (Topic 2 — Functions)↗

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS2.3 Quadratic functions and equations NEXT →2.5 Rational functions and their graphs

Revise Transformations of graphs for IB Mathematics: Analysis and Approaches (2.4)

In simple terms

Explore the concept

Key formulas

Full topic notes

Translations: sliding the graph

Reflections: flipping the graph

Stretches and compressions: scaling the graph

Effect on key features: vertex, intercepts and asymptotes

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Get a Paper 1 transformations question marked: describe the transformations and find the image with full working

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Translations: sliding the graph

Reflections: flipping the graph

Stretches and compressions: scaling the graph

Effect on key features: vertex, intercepts and asymptotes

Common mistakes examiners penalise

Model answer — marked the way our engine marks it

Where this leads

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Get a Paper 1 transformations question marked: describe the transformations and find the image with full working

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Transformations of graphs