Graph Transformations (AI HL)
Every graph you meet at HL - polynomial, exponential, trig, or a model built from real data - can be shifted, flipped, or stretched by changing its equation in predictable ways. The tricky part isn't any single transformation, it's applying several together in the right order. This page sets out each transformation with a worked example, and the order-of-operations trap that catches most students. It's part of the broader Function Concepts topic.
11 questions on this sub-topic.
Transformations of \(y=f(x)\)
Covered under IB syllabus reference AHL2.8. These apply to every function type in the SL and AHL syllabus, including ones built to model real-life situations. None of this is in the formula booklet - you're expected to know the effect of each rule on the graph.
Translations
\(y=f(x)+b\) \(y=f(x-a)\)
Translation by vector \(\binom{a}{b}\): \(a\) shifts horizontally (note the minus sign inside the brackets), \(b\) shifts vertically.
Reflections
\(y=-f(x)\) \(y=f(-x)\)
\(y=-f(x)\) reflects in the \(x\)-axis; \(y=f(-x)\) reflects in the \(y\)-axis.
Stretches
\(y=pf(x)\) \(y=f(qx)\)
Vertical stretch scale factor \(p\); horizontal stretch scale factor \(\dfrac{1}{q}\). The \(x\)-axis is invariant under a vertical stretch, the \(y\)-axis under a horizontal one.
Need the full syllabus wording and formula-booklet reference table? See Function Concepts.
Worked examples
The graph of \(y = f(x)\) is transformed to give \(y = 3f(x-2) + 1\).
(a) Describe the three transformations applied to \(y = f(x)\).
(b) The point \((4, 5)\) lies on \(y = f(x)\). Find the corresponding point on the transformed graph.
Worked solution
(a) Translation 2 right: \(x \to x - 2\) A1
Vertical stretch by scale factor 3: \(y \to 3y\) A1
Translation 1 up: \(y \to y + 1\) A1
(b) \(x: 4 \to 4+2 = 6\); \(y: 5 \to 3(5)+1 = 16\) → \((6,\, 16)\) A1
The pH of a solution is \(\text{pH}=-\log_{10}[H^+]\).
Find the pH when \([H^+]=3.2\times10^{-5}\), to 1 decimal places.
Worked solution
\(\text{pH}=-\log_{10}(3.2\times10^{-5}).\) M1
\(\log_{10}(3.2\times10^{-5})\approx-4.49\), so \(\text{pH}\) A1 \(\approx4.5\) (1 decimal places). A1
This isn't a transformation itself, but the \(-\log_{10}[H^+]\) rule is exactly a reflection-and-vertical-stretch of \(y=\log_{10}x\) applied to a real model - the same idea this topic covers, dressed as chemistry.
Common mistakes
- Getting the sign backwards on a horizontal translation. \(y=f(x-a)\) shifts the graph \(a\) units in the positive \(x\) direction - the minus sign inside the brackets produces a shift to the right, which feels the wrong way round on first meeting it.
- Applying transformations in the wrong order. \(y=3f(x-2)+1\) is built from \(y=f(x)\) by translating horizontally first, then stretching vertically, then translating vertically - reversing the order on a point-by-point mapping gives the wrong image point even though the final graph looks identical.
- Mixing up stretch scale factor with the value \(q\). In \(y=f(qx)\) the horizontal stretch scale factor is \(\dfrac{1}{q}\), not \(q\) itself - a large \(q\) compresses the graph towards the \(y\)-axis rather than stretching it out.
Ready to practise properly?
11 graph transformation questions, marked instantly like the real exam.
Quick answers
In what order do you apply multiple transformations?
Order matters and follows the order the operations are written in the function. For \(y = pf(x-a)+b\), apply the horizontal translation and any horizontal stretch first (they act on \(x\) before \(f\) is evaluated), then the vertical stretch, then the vertical translation. For GDC-based ways to check a transformed graph, see Function Concepts.
Does a horizontal stretch move points closer to or further from the y-axis?
A horizontal stretch with scale factor \(\dfrac{1}{q}\), given by \(y=f(qx)\), moves points closer to the \(y\)-axis when \(q>1\) and further away when \(0