Graph Transformations (AA HL)
Every transformation of \(y=f(x)\) is really just a rule for relabelling points on the original graph - stretch, reflect or shift, and the underlying shape carries over unchanged. The tricky part is combining several transformations at once and getting both the order and the direction right, which is exactly where IB exam questions like to test you. This page is part of the broader Transformations & Graphs topic.
37 questions on this sub-topic.
The transformation rules
Covered under IB syllabus reference SL2.11: translations \(y=f(x)+b\) and \(y=f(x-a)\); reflections in both axes, \(y=-f(x)\) and \(y=f(-x)\); vertical stretch \(y=pf(x)\); horizontal stretch \(y=f(qx)\); and composite transformations, where the order applied matters.
Translations and reflections
\(y=f(x)+b,\ \ y=f(x-a),\ \ y=-f(x),\ \ y=f(-x)\)
\(b\) shifts up/down, \(a\) shifts right/left (note the minus sign inside the bracket), \(-f(x)\) reflects in the \(x\)-axis, \(f(-x)\) reflects in the \(y\)-axis.
Stretches
\(y=pf(x)\) (vertical, factor \(p\)); \(y=f(qx)\) (horizontal, factor \(\tfrac1q\))
A vertical stretch multiplies every \(y\)-value by \(p\). A horizontal stretch has scale factor \(\tfrac1q\), not \(q\) itself - a common source of errors.
For combined transformations, build up the function step by step from the inside out. For the full syllabus wording and formula-booklet reference table, see Transformations & Graphs.
Worked examples
The graph of \(y=f(x)\) is translated by \(\begin{pmatrix}3\\-2\end{pmatrix}.\) Write the new function in terms of \(f.\)
Worked solution
Translation by \(\begin{pmatrix}3\\-2\end{pmatrix}\) means right 3 (\(x \to x-3\)) M1
and down 2, so \(y = f(x-3) - 2.\) A1
\(y=x^2\) is transformed to \(y=2(x-3)^2+1.\) Describe a sequence of transformations and state the vertex.
Worked solution
factor 2 (\(y\) M1 \(= 2x^2\)). A1
right 3 M1 and up 1. A1 Vertex \((3, 1).\) A1
The point \((2,5)\) lies on \(y=f(x).\)
(a) Find the corresponding point on \(y=f(2x)\).
(b) Find the corresponding point on \(y=-f(x)+3.\)
Worked solution
(a) \(y = f(2x)\): \(2x = 2 \Rightarrow x = 1.\) M1
Point \((1, 5).\) A1
(b) \(y = -f(x) + 3\): M1
\(-5 + 3 = -2.\) A1
Point \((2, -2).\) A1
Common mistakes
- 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.
- 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.
Ready to practise properly?
38 graph transformation questions, marked instantly like the real exam.
Quick answers
Which way does y = f(x - a) shift the graph?
It shifts the graph in the positive \(x\)-direction (to the right) by \(a\) when \(a\) is positive - the opposite of what the minus sign inside the bracket suggests.
In what order should combined transformations be applied?
Work from the inside of the function bracket outward: apply whatever is done to \(x\) first, then whatever is applied to the whole function afterward. This does not always match the order the transformations are described in words.