Transformations & Graphs (AA HL)
Transformations describe how a graph moves, flips or stretches when its equation changes in a predictable way. This topic covers translating a graph vertically or horizontally, reflecting it in either axis, stretching it vertically or horizontally, and combining several of these transformations in the correct order to sketch or find the equation of a new graph.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus, examinable at both SL and HL.
| Code | Syllabus content |
|---|---|
| SL2.11 | Transformations of graphs: translations \(y=f(x)+b\) and \(y=f(x-a)\); reflections in both axes, \(y=-f(x)\) and \(y=f(-x)\); vertical stretch with scale factor \(p\), \(y=pf(x)\); horizontal stretch with scale factor \(\tfrac1q\), \(y=f(qx)\); composite transformations. Students should be aware of the relevance of the order in which transformations are performed. Link to composite functions (SL2.5). |
This is core AA syllabus content examinable at both SL and HL. Transformations of the form \(f(ax+b)\) are not required at SL, but are examinable at HL.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a translation?
A translation slides the graph without changing its shape. \(y=f(x)+b\) moves the graph \(b\) units vertically (up if \(b>0\)); \(y=f(x-a)\) moves it \(a\) units horizontally (right if \(a>0\)). Every point on the curve, and every feature like an intercept or asymptote, shifts by exactly the same amount.
e.g. \((2,5)\) on \(y=f(x)\); on \(y=f(x-3)+1\) it becomes \((5,6)\).
What is a reflection?
A reflection flips the graph across an axis. \(y=-f(x)\) reflects in the \(x\)-axis (flips vertically); \(y=f(-x)\) reflects in the \(y\)-axis (flips horizontally). Points on the axis of reflection don't move; everything else swaps sign in the relevant coordinate.
e.g. \((3,-4)\) on \(y=f(x)\); on \(y=-f(x)\) it becomes \((3,4)\).
What's the difference between a vertical and a horizontal stretch?
\(y=pf(x)\) stretches vertically by scale factor \(p\), multiplying every \(y\)-coordinate by \(p\). \(y=f(qx)\) stretches horizontally by scale factor \(\tfrac1q\), dividing every \(x\)-coordinate by \(q\) - the horizontal case is the one students most often get backwards.
e.g. \((4,10)\) on \(y=f(x)\); on \(y=f(2x)\) it becomes \((2,10)\).
Why does the order of transformations matter?
Each transformation acts on the graph as it currently stands, not on the original function - so applying a stretch then a translation is not the same as applying them the other way round. Read \(y=pf(x-a)+b\) from the inside out to get the order right.
e.g. \(f(x)=x\) at \(x=0\): \(2f(x)+3\) gives \(3\); \(2(f(x)+3)\) gives \(6\) - different results.
What happens to an asymptote under a transformation?
An asymptote transforms exactly like every other feature of the graph. A vertical translation shifts a horizontal asymptote by the same amount and leaves a vertical one unchanged; a horizontal translation does the reverse; a vertical stretch scales a horizontal asymptote about \(y=0\).
e.g. \(y=f(x)\) has asymptote \(y=-1\); on \(y=f(x)+4\) it becomes \(y=3\).
Key formulas
Six mappings cover every transformation on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
These are core syllabus definitions, not printed formulas - the formula booklet doesn't list transformation rules, so they need to be secure from memory.
| Mapping | Used for | Booklet? |
|---|---|---|
| \(y=f(x)+b\) | Vertical translation, \(b\) units up | Not in booklet |
| \(y=f(x-a)\) | Horizontal translation, \(a\) units right | Not in booklet |
| \(y=-f(x)\) | Reflection in the \(x\)-axis | Not in booklet |
| \(y=f(-x)\) | Reflection in the \(y\)-axis | Not in booklet |
| \(y=pf(x)\) | Vertical stretch, scale factor \(p\) | Not in booklet |
| \(y=f(qx)\) | Horizontal stretch, scale factor \(\tfrac1q\) | Not in the formula booklet - prior knowledge |
Vertical vs horizontal transformations
Every transformation has a vertical version (acting on \(y\)) and a horizontal version (acting on \(x\)) - and the horizontal versions behave in the opposite way to what you'd first guess.
| Feature | Vertical (acts on \(y\)) | Horizontal (acts on \(x\)) |
|---|---|---|
| Translation | \(y=f(x)+b\) - shifts up/down, sign matches direction | \(y=f(x-a)\) - shifts right/left, sign is reversed |
| Stretch | \(y=pf(x)\) - scale factor \(p\), direct | \(y=f(qx)\) - scale factor \(\tfrac1q\), reciprocal |
| Reflection | \(y=-f(x)\) - flips in the \(x\)-axis | \(y=f(-x)\) - flips in the \(y\)-axis |
| Point mapping | \((x,y)\to(x,\,py+b)\) | \((x,y)\to\left(\tfrac{x}{q}+a,\,y\right)\) |
Translations and reflections
These are the simplest transformations - they change position or orientation but never the graph's overall shape or scale.
Vertical translation
\[y=f(x)+b\]
Every \(y\)-value shifts by \(b\); the graph moves straight up (or down if \(b<0\)).
Not in the formula booklet - prior knowledgeHorizontal translation
\[y=f(x-a)\]
Set \(x-a=0\) to find where the shift lands: the graph moves \(a\) units right.
Not in the formula booklet - prior knowledgeReflections
\[y=-f(x)\quad\text{or}\quad y=f(-x)\]
\(-f(x)\) flips vertically (in the \(x\)-axis); \(f(-x)\) flips horizontally (in the \(y\)-axis).
Not in the formula booklet - prior knowledgeStretches and combined transformations
Stretches change the scale of the graph, and several transformations often appear together in one equation - always unpick them from the inside out.
Vertical stretch
\[y=pf(x)\]
Multiplies every \(y\)-value by \(p\) - the graph gets taller (\(p>1\)) or flatter (\(0
Horizontal stretch
\[y=f(qx)\]
Divides every \(x\)-value by \(q\) - so the scale factor is \(\tfrac1q\), not \(q\).
Not in the formula booklet - prior knowledgeOrder of operations
\[y=pf(x-a)+b\]
Work from the inside out: apply the transformation closest to \(f\) first, then move outward.
Not in the formula booklet - prior knowledgeWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
The point \((2,\,-5)\) lies on \(y=f(x)\). Find its image on each graph.
(a) \(y=f(x)+3\).
(b) \(y=f(x-4)\).
(c) \(y=-f(x)\).
Worked solution
(a) \((2,-2).\) A1
(b) \((6,-5).\) A1
(c) \((2,5).\) A1
The graph of \(y = x^2\) is transformed to \(y = -2(x+1)^2 - 3\).
(a) Describe the transformations.
(b) State the vertex.
(c) State the range.
Worked solution
(a) Transformations. Translation 1 left; M1
vertical stretch factor 2 and reflection in the \(x\)-axis; A1
translation 3 down. A1
(b) Vertex: \((-1, -3).\) A1
(c) Range: the parabola opens downward with maximum \(-3\), M1
so \(y \le -3.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Getting the sign backwards for a horizontal translation. \(y=f(x-a)\) shifts the graph right by \(a\) when \(a>0\), not left - it's the opposite of what the minus sign suggests.
- Using \(q\) instead of \(\tfrac1q\) as the horizontal stretch factor. \(y=f(qx)\) compresses the graph when \(q>1\); the scale factor is \(\tfrac1q\), not \(q\) itself.
- Applying combined transformations in the wrong order. Each transformation acts on the graph as it currently stands - work from the inside of the bracket outward, not in the order the words happen to be listed.
- Forgetting that labelled features move too. A vertex, intercept or asymptote transforms along with the rest of the curve - it's not just "the shape" that shifts, every coordinate does.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Compare an original graph and its transformed image visually - or check where a transformed graph crosses an axis or another curve - with no algebra.
- Enter both functions as Y1 and Y2 and graph them.
- 2nd → CALC → 5:intersect, move near the crossing, then press ENTER three times.TI-84
- menu → Analyze Graph → Intersection.Nspire
- Press G-Solve (SHIFT F5) → Intersection (ISCT).Casio
- Repeat for each crossing point.
Tip: Adjust the window/zoom so every intersection point is visible before using the tool.
A blank, flat or oddly-shaped graph after a transformation is almost always a window problem, not an equation problem - knowing the right zoom saves time and avoids missing features.
- Before graphing, set the x-range to match the domain in the question (e.g. \(0 \le t \le 20\)).
- ZOOM → 6:ZStandard resets to −10…10. ZOOM → 0:ZoomFit auto-scales y to your x-range. ZOOM → 4:ZDecimal gives clean decimal cursor steps.TI-84
- menu → Window/Zoom → Zoom-Standard, Zoom-Fit, or set the window manually with Window Settings.Nspire
- SHIFT → F3 (V-Window) to set ranges manually; Zoom (F2) → Auto fits y automatically.Casio
- If a function looks flat: widen the y-range (the variation is small). If nothing appears: check you're in the right x-range.
Tip: ZoomFit (Auto) is your best friend after a big vertical stretch or translation - it always shows the shape in your x-range.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Transformations & graphs questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
How do I remember which way y=f(x-a) shifts?
Set the inside equal to zero: \(x-a=0\) gives \(x=a\), which is where the point that used to be at \(x=0\) on the original graph now sits. So \(y=f(x-a)\) shifts right by \(a\) when \(a\) is positive - the sign inside the bracket is the opposite of the direction you might expect.
What's the difference between y=pf(x) and y=f(qx)?
\(y=pf(x)\) stretches vertically by scale factor \(p\) - it multiplies every \(y\)-value. \(y=f(qx)\) stretches horizontally by scale factor \(\tfrac1q\) - it divides every \(x\)-value. The horizontal case is the one most students get backwards.
Does the order of transformations matter?
Yes. Applying a stretch then a translation can give a different graph from applying them in the reverse order, because each transformation acts on the graph as it currently stands, not on the original function.
How do transformations affect a graph's asymptotes?
Exactly like any other feature - a vertical translation shifts a horizontal asymptote by the same amount, a horizontal translation shifts a vertical asymptote, and a vertical stretch scales a horizontal asymptote about \(y=0\).
Sub-topics
Transformations & Graphs broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA HL syllabus unit, in case you want to keep going.