Graphs & Transformations (AA SL)
A transformation moves, stretches, or reflects the graph of a function without changing its basic shape. This topic covers translations, reflections in the two axes, vertical and horizontal stretches, and how to combine several transformations to sketch or describe the resulting graph from a known parent function.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL2.11 | Transformations of graphs. Translations: \(y=f(x)+b\); \(y=f(x-a)\). Reflections (in both axes): \(y=-f(x)\); \(y=f(-x)\). Vertical stretch with scale factor \(p\): \(y=pf(x)\). Horizontal stretch with scale factor \(\tfrac1q\): \(y=f(qx)\). Composite transformations, e.g. using \(y=x^2\) to sketch \(y=3x^2+2\). The transformation \(y=f(ax+b)\) is not required at SL. |
Students are expected to be aware that the order in which transformations are performed can affect the resulting graph.
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 every point on a graph the same distance in the same direction, without changing its shape or orientation. \(y=f(x)+b\) translates vertically; \(y=f(x-a)\) translates horizontally.
e.g. if \((2,5)\) lies on \(y=f(x)\), then \((2,8)\) lies on \(y=f(x)+3\), since \(5+3=8\).
What is a reflection?
A reflection flips a graph across an axis, producing a mirror image. \(y=-f(x)\) reflects in the \(x\)-axis (flips vertically); \(y=f(-x)\) reflects in the \(y\)-axis (flips horizontally).
e.g. if \((2,5)\) lies on \(y=f(x)\), then \((-2,5)\) lies on \(y=f(-x)\).
What is a vertical stretch?
A vertical stretch with scale factor \(p\) multiplies every \(y\)-coordinate by \(p\), pulling the graph away from (or squashing it towards) the \(x\)-axis. Points on the \(x\)-axis are unaffected.
e.g. if \((2,5)\) lies on \(y=f(x)\), then \((2,10)\) lies on \(y=2f(x)\), since \(2\times5=10\).
What is a horizontal stretch?
A horizontal stretch with scale factor \(\tfrac1q\) comes from \(y=f(qx)\): every \(x\)-coordinate on the original graph is divided by \(q\). A value of \(q>1\) squashes the graph towards the \(y\)-axis.
e.g. if \((6,5)\) lies on \(y=f(x)\), then \((3,5)\) lies on \(y=f(2x)\), since \(f(2\times3)=f(6)=5\).
What is a parent function?
A parent function is the simplest version of a function family, before any transformation is applied - for example \(y=x^2\) is the parent of every quadratic, and \(y=\sin x\) is the parent of every sine curve. Transformations are described relative to this base graph.
e.g. \(y=3x^2+2\) is a vertical stretch (factor 3) then a translation (2 up) of the parent \(y=x^2\).
Key formulas
Four transformation rules cover every question 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 transformation rules are prior-knowledge algebraic facts rather than formula-booklet entries - none of them appear in the official booklet.
| Formula | Effect | 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 booklet |
Vertical vs horizontal transformations
Transformations applied "outside" the function act on \(y\)-values directly; transformations applied "inside" the function (to \(x\)) act in the opposite direction to what looks intuitive.
| Feature | Vertical (outside) | Horizontal (inside) |
|---|---|---|
| Translation | \(y=f(x)+b\): \(b\) up | \(y=f(x-a)\): \(a\) right |
| Reflection | \(y=-f(x)\): in \(x\)-axis | \(y=f(-x)\): in \(y\)-axis |
| Stretch | \(y=pf(x)\): factor \(p\) | \(y=f(qx)\): factor \(\tfrac1q\) |
| Direction is "as expected"? | Yes | No - it's reversed |
Translations and reflections
Each of these transformations acts on either the input or output of \(f\), moving or flipping the whole graph while keeping its shape.
Vertical translation
\[y=f(x)+b\]
Shifts the graph \(b\) units up (or \(|b|\) down if \(b<0\)).
Not in the formula booklet - prior knowledgeHorizontal translation
\[y=f(x-a)\]
Shifts the graph \(a\) units right (or left if \(a<0\)) - the sign flips inside the bracket.
Not in the formula booklet - prior knowledgeReflections
\[y=-f(x) \quad\text{and}\quad y=f(-x)\]
\(-f(x)\) flips vertically (in the \(x\)-axis); \(f(-x)\) flips horizontally (in the \(y\)-axis).
Not in the formula booklet - prior knowledgeStretches
Stretches scale distances from an axis rather than shifting the graph - one axis stays fixed while the other is scaled.
Vertical stretch
\[y=pf(x)\]
Multiplies every \(y\)-value by \(p\); points on the \(x\)-axis stay fixed.
Not in the formula booklet - prior knowledgeHorizontal stretch
\[y=f(qx)\]
Scale factor \(\tfrac1q\) - divides every \(x\)-value by \(q\); points on the \(y\)-axis stay fixed.
Not in the formula booklet - prior knowledgeComposite transformations
\[y=x^2 \;\longrightarrow\; y=3x^2+2\]
Apply transformations in the order the function is built: stretch by 3, then translate 2 up.
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 transformation maps \(y=f(x)\) to \(y=f(x)+3.\)
(a) Describe it.
(b) The point \((2,5)\) lies on \(y=f(x)\). Find its image.
Worked solution
(a) Vertical translation A1
\(3\) units up. A1
(b) \((2,8).\) A1
The graph of \(y = f(x)\) is transformed. Describe each resulting transformation.
(a) \(y = f(x) + 3\)
(b) \(y = f(x - 2)\)
(c) \(y = 2f(x)\)
(d) \(y = f(-x)\)
Worked solution
(a) \(f(x)+3\): adding outside shifts the output up - translation \(3\) units up. A1
(b) \(f(x-2)\): replacing \(x\) by \(x-2\) shifts the input - translation \(2\) units right. A1
(c) \(2f(x)\): multiplying the output doubles distances from the \(x\)-axis - vertical stretch, A1
scale factor \(2.\) A1
(d) \(f(-x)\): replacing \(x\) by \(-x\) - reflection A1
in the \(y\)-axis. 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 direction of a horizontal translation backwards. \(y=f(x-a)\) shifts the graph right by \(a\) when \(a>0\), not left - the sign inside the bracket is the opposite of the direction of movement.
- Applying transformations in the wrong order. \(y=3f(x)+2\) means "stretch by 3, then translate up 2" - doing the translation first gives a different (wrong) graph.
- Confusing \(y=pf(x)\) with \(y=f(px)\). The first stretches vertically by factor \(p\); the second stretches horizontally by factor \(\tfrac1p\) - multiplying \(x\) inside the brackets compresses the graph when \(p>1\), it doesn't stretch it.
- Forgetting that reflections and stretches fix a whole line, not just one point. \(y=-f(x)\) leaves every point on the \(x\)-axis unchanged (not just the origin); \(y=f(-x)\) leaves every point on the \(y\)-axis unchanged.
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.
Solves "line meets curve" and simultaneous problems visually, with no algebra - useful for checking where an original graph and its transformed image cross.
- 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 is almost always a window problem, and transformed graphs can easily drift off the default window. Knowing the right zoom saves time and avoids missing key 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. If nothing appears: check you're in the right x-range.
Tip: ZoomFit (Auto) is your best friend for unfamiliar or transformed functions - it always shows the shape in your x-range.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Graphs & transformations questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between y = f(x) + b and y = f(x + b)?
\(y=f(x)+b\) shifts the graph vertically by \(b\) units (up if \(b\) is positive). \(y=f(x+b)\) shifts it horizontally, but in the opposite direction to what you'd expect - it moves the graph \(b\) units to the left.
Does the order of transformations matter?
Yes. Applying a stretch then a translation can give a different graph to applying the translation then the stretch, unless the transformations are of different types acting independently (e.g. one vertical, one horizontal). Always apply them in the order the function is built up algebraically.
How do I sketch y = pf(x) from y = f(x)?
Every \(y\)-coordinate is multiplied by \(p\), so the graph is stretched vertically by scale factor \(p\). Points on the \(x\)-axis stay fixed, since \(0\) multiplied by anything is still \(0\).
Is this topic examined without a calculator?
Yes, most transformation questions are calculator-free - you're asked to describe or sketch the effect of a transformation, which is an algebraic skill. Your GDC becomes useful for checking a sketch or finding where two transformed graphs intersect. See the GDC guide for model-specific instructions.
Sub-topics
Graphs & Transformations broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA SL syllabus unit, in case you want to keep going.