Transformations of Graphs
Transformations of Graphs is a key algebra topic at GCSE Maths. This Higher worksheet gives you exam-style questions on applying transformations to graphs, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Changes inside the bracket move the graph the opposite way you'd expect.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
A transformation of a graph shifts, stretches or reflects it. The original curve keeps its shape in a recognisable way, but its position or size changes according to how the equation is altered.
Two translations are needed at GCSE. Writing \(f(x) + a\) moves the graph up by \(a\), which behaves as you would expect. Writing \(f(x + a)\) moves it left by \(a\), which is the opposite of what most students expect.
That counter-intuitive horizontal rule is the whole difficulty. A change inside the bracket affects the \(x\) direction and works backwards; a change outside affects the \(y\) direction and works normally.
Revision notes
Vertical translations
\(f(x) + a\) moves the graph up by \(a\), and \(f(x) - a\) moves it down.
This is intuitive because you are adding to the output. The vector is \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).
Horizontal translations
\(f(x + a)\) moves the graph left by \(a\), and \(f(x - a)\) moves it right.
The direction is opposite to the sign, because you are changing the input before the function acts. The vector is \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).
Reflections
\(-f(x)\) reflects the graph in the \(x\)-axis, flipping it vertically.
\(f(-x)\) reflects it in the \(y\)-axis, flipping it horizontally. The same inside-outside logic applies: outside affects \(y\), inside affects \(x\).
Key points
- \(f(x) + a\) moves the graph up by \(a\).
- \(f(x) - a\) moves it down.
- \(f(x + a)\) moves it LEFT by \(a\).
- \(f(x - a)\) moves it right.
- \(-f(x)\) reflects in the x-axis.
- \(f(-x)\) reflects in the y-axis.
Worked examples
Example 1
Describe the transformation from \(y = f(x)\) to \(y = f(x) + 3\).
Working
Example 2
Describe the transformation from \(y = f(x)\) to \(y = f(x - 4)\).
Working
Example 3
The point \((2, 5)\) is on \(y = f(x)\). Find its image on \(y = f(x) + 2\).
Working
Common mistakes
Moving the graph right for \(f(x + a)\).
A change inside the bracket moves the graph the opposite way, so f(x + 3) moves left.
Confusing inside and outside changes.
Outside the bracket affects y; inside affects x.
Changing both coordinates for a single translation.
A vertical shift alters only the y coordinate.
Mixing up the two reflections.
−f(x) flips vertically; f(−x) flips horizontally.
Exam tips
- Ask first whether the change is inside or outside the bracket.
- Remember horizontal shifts go the opposite way to the sign.
- Describe transformations fully, giving direction and distance.
- Track a single point through the transformation to check.
Key terms
- Translation
- A shift of a graph without changing its shape.
- Reflection
- A flip of a graph in an axis.
- Image
- The result after a transformation.
- Vector
- A description of a translation as horizontal and vertical movement.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.