Graph Transformations Quizzes

Introduction

Graph transformations are an important part of GCSE Maths, helping students understand how the graph of a function changes when modifications are made to its equation. Mastering graph transformations allows students to interpret shifts, stretches, compressions, and reflections of functions. This skill is essential for higher-level topics like quadratics, cubics, and trigonometric functions, and it often appears in both foundation and higher-tier exams.

Core Concepts

What are Graph Transformations?

A graph transformation describes how the graph of a base function, such as \(y = f(x)\), changes when its equation is modified. Transformations include translations, reflections, stretches, and compressions.

Types of Transformations

• Translations: Moving the graph horizontally or vertically.
• Reflections: Flipping the graph across an axis.
• Stretches and Compressions: Scaling the graph vertically or horizontally.

Rules & Steps

1. Translation (Shifts)

Translations move the graph without changing its shape.

• Vertical shift: \(y = f(x) + k\) → shift graph up by \(k\) units if \(k>0\), down if \(k<0\).
• Horizontal shift: \(y = f(x - h)\) → shift graph right by \(h\) units if \(h>0\), left if \(h<0\).

Example: \(y = (x-3)^2 + 2\)

• Base graph: \(y = x^2\)
• Horizontal shift: right 3 units
• Vertical shift: up 2 units

2. Reflection

Reflections flip the graph across an axis.

• Across x-axis: \(y = -f(x)\)
• Across y-axis: \(y = f(-x)\)

Example: \(y = -(x-2)^2 + 3\)

• Base graph: \(y = x^2\)
• Reflect across x-axis
• Shift right 2 units and up 3 units

3. Stretch and Compression

Scaling changes the size of the graph either vertically or horizontally.

• Vertical stretch/compression: \(y = af(x)\)
• \(|a|>1\): vertical stretch
• \(0<|a|<1\): vertical compression
• Horizontal stretch/compression: \(y = f(bx)\)
• \(|b|>1\): horizontal compression
• \(0<|b|<1\): horizontal stretch

Example: \(y = 2(x-1)^2\)

• Vertical stretch by factor 2
• Shift right 1 unit

4. Combining Transformations

Multiple transformations can be applied in one equation. The general order is:

1. Horizontal shifts/compressions/stretches inside the function
2. Reflections inside the function
3. Vertical stretches/compressions/reflections outside the function
4. Vertical shifts outside the function

Example: \(y = -2(x+3)^2 + 5\)

• Horizontal shift: left 3 units
• Reflection across x-axis
• Vertical stretch by factor 2
• Vertical shift: up 5 units

Worked Examples

1. Translate \(y = x^2\) right 4 units, up 3 units: $$ y = (x-4)^2 + 3 $$
2. Reflect \(y = x^3\) across x-axis: $$ y = -x^3 $$
3. Vertical stretch of \(y = \sqrt{x}\) by factor 3: $$ y = 3\sqrt{x} $$
4. Horizontal compression of \(y = \sin(x)\) by factor 2: $$ y = \sin(2x) $$
5. Combination: \(y = -\frac{1}{2}(x-1)^2 + 4\)
   • Base: \(y = x^2\)
   • Horizontal shift: right 1 unit
   • Reflection across x-axis
   • Vertical compression by 1/2
   • Vertical shift: up 4 units

Common Mistakes

• Applying transformations in the wrong order.
• Confusing horizontal and vertical shifts; remember “inside” affects x, “outside” affects y.
• For reflections, forgetting to apply the negative to the correct part of the function.
• Incorrectly interpreting vertical or horizontal stretches/compressions.
• Combining multiple transformations without step-by-step checking.

Applications

• Physics: displacement-time or velocity-time graphs
• Economics: cost, revenue, and profit curves
• Engineering: designing curves and trajectories
• Computer graphics: scaling and reflection of objects
• Problem-solving: modelling shifts and changes in functions

Strategies & Tips

• Always identify the base function first (e.g., \(y = x^2\), \(y = x^3\), \(y = \sin x\)).
• Analyze transformations step by step: horizontal → reflection/stretch → vertical.
• Sketch rough graphs to visualize changes before plotting accurately.
• Check points such as intercepts and vertices to ensure transformations are correct.
• Practice multiple combinations to become confident with transformations in exams.

Summary

Graph transformations allow students to understand how mathematical functions behave under shifts, reflections, and scaling. Mastery of these transformations is essential for GCSE Maths exams, problem-solving, and preparation for advanced topics. Regular practice with base functions and transformations, step-by-step analysis, and careful graphing will improve accuracy and confidence. Attempt the quizzes and exercises to consolidate your understanding of graph transformations and enhance your skills in interpreting and manipulating functions.