Transformations of functions

Overview

Welcome to the definitive guide for Transformations of Functions, section 3.4 of the OCR GCSE Further Mathematics specification. This topic is a cornerstone of advanced mathematical thinking, moving beyond simple plotting to manipulating entire graphs with precision. In this guide, we will deconstruct the four key transformations: translations, reflections, and both vertical and horizontal stretches. A solid grasp of this topic is crucial, as it not only accounts for a significant portion of exam marks (AO1: 50%) but also provides a foundation for understanding more complex functions in A-Level Mathematics. Candidates are expected to handle function notation like f(x+a) and f(kx) with fluency, often in multi-step, composite transformation questions. This guide will equip you with the terminology, techniques, and exam strategies to turn this challenging topic into a reliable source of marks.

Key Concepts

Concept 1: Translations (Shifting the Graph)

A translation slides a graph to a new position without rotating, reflecting, or changing its size. The key is to understand how the function's equation dictates the movement.

• Vertical Translation: y = f(x) + a
  This transformation moves the graph vertically. If 'a' is positive, the graph shifts up. If 'a' is negative, it shifts down. This is an intuitive transformation; it happens outside the function, directly affecting the y-values.
  Example: f(x) + 2 moves the graph of f(x) up by 2 units.

• Horizontal Translation: y = f(x + a)
  This transformation moves the graph horizontally. This is where candidates often make mistakes. If 'a' is positive, the graph shifts left. If 'a' is negative, it shifts right. This is counter-intuitive. It happens inside the function, affecting the x-values. Think of it as what you need to do to x to bring it back to its original value.
  Example: f(x - 3) moves the graph of f(x) right by 3 units.

Concept 2: Reflections (Flipping the Graph)

Reflections create a mirror image of the graph across a specific line.

• Reflection in the x-axis: y = -f(x)
  This transformation flips the graph across the x-axis. All the y-coordinates are multiplied by -1. What was positive becomes negative, and vice-versa.

• Reflection in the y-axis: y = f(-x)
  This transformation flips the graph across the y-axis. All the x-coordinates are multiplied by -1. The left side of the graph becomes the right side, and vice-versa.

Concept 3: Stretches (Scaling the Graph)

Stretches expand or compress the graph either vertically or horizontally.

• Vertical Stretch: y = af(x)
  This transformation stretches the graph vertically by a scale factor of 'a'. It is a stretch parallel to the y-axis. If a > 1, the graph gets taller. If 0 < a < 1, the graph gets shorter.

• Horizontal Stretch: y = f(kx)
  This is the other counter-intuitive transformation. It stretches the graph horizontally by a scale factor of 1/k. It is a stretch parallel to the x-axis. If k > 1, the graph gets narrower. If 0 < k < 1, the graph gets wider.

Mathematical Relationships

Practical Applications

While abstract, function transformations have real-world applications in fields like signal processing (stretching and shifting sound waves), computer graphics (moving and resizing objects on a screen), and even in economics to model shifts in supply and demand curves. Understanding these transformations provides a powerful toolkit for modeling and interpreting real-world phenomena.