Graph Transformations: f(x) + a, f(x + a), af(x) and f(ax)
Graph transformations let you take a graph of y = f(x) and translate, stretch or reflect it without plotting a single new point. On this page you will learn the four IGCSE Maths transformation forms, f(x) + a, f(x + a), af(x) and f(ax), and see exactly what each one does using an interactive explorer where you can drag a slider and watch the transformed graph move. Clear worked examples show you how to find image points, write translation vectors and work out the equation of a transformed graph, and the auto-marked practice rooms below generate unlimited randomised questions across five graded rooms so you can build real exam confidence.
What is a graph transformation?
A graph transformation takes the graph of \(y = f(x)\) and produces a new graph by translating it, stretching it, or reflecting it. The function \(f\) stays the same; what changes is what we do to the input \(x\) or the output \(f(x)\).
The single most useful idea on this page is this: anything done outside the bracket changes the \(y\)-coordinates and behaves exactly as you would expect, while anything done inside the bracket changes the \(x\)-coordinates and behaves the opposite way to what you would expect.
The IGCSE specification asks you to apply four forms to linear, quadratic, sine and cosine graphs, and to describe transformations and write transformed functions algebraically.
The four transformation forms
Worked Examples
💡 Example 1: Vertical translation
The graph of \(y = f(x)\), where \(f(x) = x^2 - 3x\), passes through the point \((4, 4)\).
The graph is translated by \(\left(\begin{array}{c} 0 \\ 2 \end{array}\right)\).
a) Find the image of the point \((4, 4)\).
b) Find the equation of the translated graph.
▬ solid original, ▬ dashed image
a) \((4, 4) \rightarrow (4, 4 + 2) = (4, 6)\)
b) \(y = f(x) + 2\)
\(y = x^2 - 3x + 2\)
What's happening?
A translation by \(\left(\begin{array}{c} 0 \\ 2 \end{array}\right)\) moves every point up 2. Only the \(y\)-coordinate changes, so we simply add 2 to the whole function.
💡 Example 2: Horizontal translation
\(f(x) = x^2 - 4x + 1\). Find the equation of \(y = f(x + 2)\) in its simplest form, and describe the transformation.
▬ solid original, ▬ dashed image
\(f(x+2) = (x+2)^2 - 4(x+2) + 1\)
\(= x^2 + 4x + 4 - 4x - 8 + 1\)
\(y = x^2 - 3\)
What's happening?
Replace every \(x\) with \(x + 2\), then expand and simplify. The transformation is a translation by \(\left(\begin{array}{c} -2 \\ 0 \end{array}\right)\): the graph moves left by 2 even though the bracket says \(+2\).
💡 Example 3: Stretches
The graph of \(y = f(x)\) has a maximum point at \((6, 8)\). Find the maximum point of
a) \(y = 2f(x)\) b) \(y = f(2x)\) c) \(y = \frac{1}{2}f(x)\)
▬ solid \(y = f(x)\), ▬ dashed \(y = 2f(x)\)
a) \((6, 2 \times 8) = (6, 16)\)
b) \((6 \times \frac{1}{2}, 8) = (3, 8)\)
c) \((6, \frac{1}{2} \times 8) = (6, 4)\)
What's happening?
\(2f(x)\) multiplies \(y\)-coordinates by 2. \(f(2x)\) multiplies \(x\)-coordinates by \(\frac{1}{2}\), squashing the graph towards the \(y\)-axis. \(\frac{1}{2}f(x)\) halves the \(y\)-coordinates, compressing the graph towards the \(x\)-axis.
💡 Example 4: Sine and cosine
The graph of \(y = \sin x\) has a maximum point at \((90, 1)\). Find the maximum point of
a) \(y = 3\sin x\) b) \(y = \sin 3x\) c) \(y = \sin x - 2\)
▬ solid \(y = \sin x\), ▬ dashed \(y = 3\sin x\)
a) \((90, 3 \times 1) = (90, 3)\)
b) \((90 \times \frac{1}{3}, 1) = (30, 1)\)
c) \((90, 1 - 2) = (90, -1)\)
What's happening?
Trigonometric graphs follow exactly the same rules. \(3\sin x\) stretches the wave taller, \(\sin 3x\) squashes it so three full waves fit where one did, and \(\sin x - 2\) slides the whole wave down 2.
💡 Example 5: Reflections
\(f(x) = x^2 + 2x\). The graph of \(y = f(x)\) passes through the point \((1, 3)\).
a) Find the image of \((1, 3)\) on \(y = -f(x)\) and on \(y = f(-x)\).
b) Find the equation of \(y = -f(x)\).
c) Find the equation of \(y = f(-x)\).
▬ solid \(y = f(x)\), ▬ dashed \(y = -f(x)\)
a) \(-f(x)\): \((1, 3) \rightarrow (1, -3)\)
\(f(-x)\): \((1, 3) \rightarrow (-1, 3)\)
b) \(y = -(x^2 + 2x) = -x^2 - 2x\)
c) \(y = (-x)^2 + 2(-x) = x^2 - 2x\)
What's happening?
\(-f(x)\) reflects in the \(x\)-axis, so the \(y\)-coordinate changes sign. \(f(-x)\) reflects in the \(y\)-axis, so the \(x\)-coordinate changes sign. For the equations, multiply the whole function by \(-1\), or replace every \(x\) with \(-x\).
How to describe each transformation in the exam
Examiners award marks for the type, the direction, and the value. Vague words like "moves" or "shifts" score nothing. Use this table.
| Form | Transformation | What to write |
|---|---|---|
| \(y = f(x) + a\) | Vertical translation | Translation by \(\left(\begin{array}{c} 0 \\ a \end{array}\right)\) |
| \(y = f(x + a)\) | Horizontal translation | Translation by \(\left(\begin{array}{c} -a \\ 0 \end{array}\right)\) |
| \(y = af(x)\) | Stretch parallel to the \(y\)-axis | Stretch, scale factor \(a\), in the \(y\) direction |
| \(y = f(ax)\) | Stretch parallel to the \(x\)-axis | Stretch, scale factor \(\frac{1}{a}\), in the \(x\) direction |
| \(y = -f(x)\) | Reflection | Reflection in the \(x\)-axis |
| \(y = f(-x)\) | Reflection | Reflection in the \(y\)-axis |
🔑 Key Points
- Outside the bracket acts on \(y\) and behaves as expected: \(f(x) + 3\) moves up 3, \(2f(x)\) doubles the heights.
- Inside the bracket acts on \(x\) and behaves the opposite way: \(f(x + 3)\) moves left 3, \(f(2x)\) squashes by \(\frac{1}{2}\).
- To find the equation of a transformed graph, substitute into \(f(x)\) and simplify.
- The sine and cosine graphs are translations of each other: \(\cos x = \sin(x + 90)\).
- Track one known point (usually a turning point) to check any sketch you draw.
⚠️ Common Pitfalls
- \(f(x + 3)\) moves the graph left, not right. The sign inside the bracket is the opposite of the direction.
- \(f(2x)\) has scale factor \(\frac{1}{2}\), not 2. Inside-the-bracket stretches use the reciprocal.
- Writing "the graph moves up" loses the mark. Write "translation by \(\left(\begin{array}{c} 0 \\ 3 \end{array}\right)\)".
- For combined transformations, apply them in the order given in the question.