GCSE Maths • Foundation & Higher
GCSE Graph Transformations Revision Guide
Learn how to translate, reflect and stretch graphs, then combine transformations in the correct order using clear function notation and coordinate rules.
Foundation
Exam practice
Worked examples
What You'll Learn
Translate graphs
Move a graph horizontally or vertically and write the transformed equation.
Reflect graphs
Recognise reflections in the x-axis and y-axis from equations and coordinates.
Stretch graphs
Apply vertical and horizontal scale factors without confusing inside and outside changes.
Combine transformations
Carry out two or more transformations in the stated order and record intermediate results.
Transform coordinates and shapes
Apply vectors, mirror lines and invariant-line stretch rules accurately.
Key Rules and Formulas
Graph Transformation Lessons
Translations
Move graphs and coordinate shapes using horizontal and vertical shifts or column vectors.
Reflections
Reflect graphs in coordinate axes and reflect points or shapes in specified mirror lines.
Stretches
Apply vertical and horizontal stretches and distinguish them from enlargements.
Combined Transformations
Apply multiple transformations one at a time, in the stated order.
CORE SUBTOPICS
Translations
For a coordinate translation by vector (p, q), every point follows (x, y) → (x + p, y + q). The top component is horizontal; the bottom component is vertical.
Reflections
Every point and its image must be the same perpendicular distance from the mirror line. Points on the mirror line stay fixed.
Stretches
A stretch changes distances in one direction only. It is not the same as an enlargement, which scales all distances from a centre.
Combined Transformations
Apply transformations one at a time and use the output of one step as the input to the next. In general, changing the order changes the final image.
Worked Example
Translate a graph
QUESTION
The graph y = x² is translated 2 units right and 3 units up. Write its new equation.
- 1. Start with f(x) = x².
- 2. Right by 2: replace x with x − 2, giving y = (x − 2)².
- 3. Up by 3: add 3 outside, giving y = (x − 2)² + 3.
- Answer y = (x − 2)² + 3
- Common GCSE Mistakes
- Using f(x − a) for a move left instead of right.
- Using f(x − a) for a move left instead of right.
- Changing both coordinates during a one-direction stretch.
- Describing a geometric stretch without its invariant line.
- Performing combined transformations in the wrong order.
- Treating a stretch and an enlargement as the same transformation.
- Transforming only one vertex of a shape or failing to join corresponding vertices correctly.
- Exam Tips
- Mark several recognisable points on the original graph before transforming it.
- For horizontal transformations, test one known x-value to confirm the direction.
- Write an intermediate equation or set of coordinates in combined questions.
- Use tracing paper only where the exam permits it; still provide the required mathematical description.
- Include vector notation for translations and all defining information for rotations, enlargements and stretches.
Ready to Practice?
Printable Worksheet
Practise simplifying, unit conversions, sharing totals, recipe scaling, combined ratios and best-buy comparisons.
Interactive Quiz
Check ratio order, equivalent ratios, total parts, unit prices and common misconceptions.
Exam-Style Questions
Apply ratio methods to money, measures, recipes, maps and multi-stage GCSE contexts.
Watch the Graph Transformations Tutorial
A clear walkthrough of translations, reflections, vertical and horizontal stretches, and combined transformations using key points and function notation.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
The equation changes the input needed to obtain the same output. To move f(x) right by a units, the original input x is reached when the new input is x − a, so the equation is y = f(x − a).
The first changes every y-value and reflects the graph in the x-axis. The second changes every input and reflects the graph in the y-axis.
For a shape, give a column vector with horizontal movement on top and vertical movement below. For a function graph, you may also describe the horizontal and vertical displacement in words.
For a geometric stretch, state the scale factor, direction and invariant line. For a graph defined from y = f(x), state whether the stretch is horizontal or vertical and give the scale factor.