Transformations of Graphs

How algebraic changes move and reshape a graph

A graph transformation changes the position or shape of a familiar curve. GCSE Higher and IGCSE students often work from y = f(x), then interpret a related graph such as y = f(x) + 3 or y = f(x - 2). This topic connects algebra with coordinates and the broader idea of a transformation

Key transformation rules

For y = f(x) + a, translate the graph up by a. For y = f(x) - a, translate it down. For y = f(x - a), translate right by a, while y = f(x + a) translates left. The graph y = -f(x) is reflected in the x-axis, and y = f(-x) is reflected in the y-axis. Changes outside the bracket act vertically; changes inside act horizontally and in the opposite direction.

Step-by-step method

Identify the parent graph, match the equation to a rule, transform important points such as intercepts and turning points, then sketch the same basic shape in its new position. Use the language of translations and reflections

Worked examples

Example 1: If (1, 4) lies on y = f(x), then (1, 7) lies on y = f(x) + 3. Example 2: Under y = f(x - 5), the point (2, 6) moves to (7, 6). Example 3: Under y = -f(x), the point (-3, 2) becomes (-3, -2). These skills support work on quadratic equations and graphs

Common mistakes and exam advice

The most common mistake is moving a graph in the wrong horizontal direction. Students may also transform only one point or change the shape during a translation. Mark several key points and check that all have moved consistently. Understanding gradient and intercepts helps when lines are transformed.

Practice exercises and answers

Exercise 1: Describe y = f(x) + 4. Answer: translate 4 units upwards. Exercise 2: A point (3, -1) lies on y = f(x). Find its image on y = f(x - 2). Answer: (5, -1). Exercise 3: A point (-4, 7) lies on y = f(x). Find its image on y = f(-x). Answer: (4, 7), a reflection in the y-axis.

Related learning across the website

Develop this topic through straight-line graph lessons, Algebra Lessons, GCSE algebra help, equations, functions and formulae, enlargements, rotations and Geometry Lessons

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can target confusion between horizontal and vertical changes, develop accurate sketches and provide immediate feedback on function notation.

Contact and lesson enquiry

Students and parents can enquire about one-to-one maths lessons for personalised graph support.