IGCSE Graph Transformations: Translations & Reflections (H)
Once you can sketch basic graphs of functions, the next step is to move them around the grid. Graph Transformations involve taking a standard function, y = f(x), and applying mathematical operations to shift (translate) or flip (reflect) it.
This guide covers Topic 3.8 of the IGCSE Functions & Graphs Hub. In the Higher Tier exam, you must be able to sketch the result of a transformation and describe it using specific vector notation.
1. The Golden Rules of Transformations
To master this topic, you only need to remember two fundamental principles regarding where the change happens in the equation:
Rule 1: Outside vs. Inside
- Outside the bracket (Affects y): Operations applied to the whole function, e.g., f(x) + a or -f(x). These changes happen Vertically (up/down). They do exactly what you expect (Truthful).
- Inside the bracket (Affects x): Operations applied directly to x, e.g., f(x + a) or f(-x). These changes happen Horizontally (left/right). They do the Opposite of what you expect (Liars).
2. Translations (Shifting)
A translation moves the graph without changing its shape or orientation. We describe translations using column vectors.
| Notation | Effect | Vector | Logic |
|---|---|---|---|
| y = f(x) + a | Shift UP by a | 0a |
Outside = Vertical (Truthful) |
| y = f(x) - a | Shift DOWN by a | 0-a |
Outside = Vertical (Truthful) |
| y = f(x + a) | Shift LEFT by a | -a0 |
Inside = Horizontal (Liar) (+a goes left) |
| y = f(x - a) | Shift RIGHT by a | a0 |
Inside = Horizontal (Liar) (-a goes right) |
3. Reflections (Flipping)
A reflection creates a mirror image of the graph.
| Notation | Effect | Logic |
|---|---|---|
| y = -f(x) | Reflection in the x-axis (Upside down). | Outside changes y. Positive y becomes negative y. |
| y = f(-x) | Reflection in the y-axis (Left-Right flip). | Inside changes x. Positive x becomes negative x. |
Step-by-Step Worked Examples
Question: The graph of y = x² is transformed to y = x² + 3. Describe the transformation.
Methodology: Identify the change. Is it inside or outside?
Solution:
- The +3 is separate from the x². It is Outside.
- Outside means Vertical and Truthful.
- Answer: Translation by vector 03.
Question: The graph of y = f(x) has a turning point at (2, 5). Find the coordinates of the turning point on the graph of y = f(x - 4).
Methodology: Focus on the coordinates.
Solution:
- Identify Change: The -4 is inside the bracket.
- Logic: Inside is Horizontal and a Liar. "-4" sounds like left, so we go Right by 4.
- Apply to x: x-coordinate 2 becomes 2 + 4 = 6.
- Apply to y: y-coordinate stays the same (5).
Answer: (6, 5).
Question: Sketch the graph of y = -(x - 3)². Mark the turning point.
Methodology: This combines two transformations. Do them in order (BODMAS order: Brackets first).
Solution:
- Start: Standard y = x². Turning point (0,0).
- Inside Bracket (x - 3): Horizontal shift RIGHT by 3.
New Turning Point: (3, 0). Equation: y = (x - 3)². - Outside Negative -(...): Reflection in the x-axis.
The U-shape flips to an n-shape. The turning point (3, 0) stays on the axis.
Answer: An n-shaped parabola touching the x-axis at x=3.
Real-World Application (Global Context)
Transformations are the basis of animation and computer graphics.
Scenario: Video Game Physics
In a game like Angry Birds, the flight path of a bird is a parabola: y = -x².
• To launch the bird from a higher platform, the code applies a vertical translation: y = -x² + 5.
• To launch the bird from a different horizontal position, the code applies a horizontal translation: y = -(x - 10)².
• If the bird bounces off a wall, the trajectory might be reflected: y = f(-x).
Programmers use these exact transformation rules to render movement on screen.
Exam Technique and Common Pitfalls
1. The "Vector" Requirement
If an exam question asks you to "Describe the transformation," simply saying "it moves right" will lose marks. You MUST use the word Translation and write the Column Vector
2. The "Inside Liar" Trap
The most common error is thinking f(x + 2) moves to the right. Remember: Inside is a Liar. Plus means Negative direction (Left). Minus means Positive direction (Right).
3. Applying to the Wrong Coordinate
• f(x) + a changes the y-coordinate.
• f(x + a) changes the x-coordinate.
• -f(x) changes the sign of y.
• f(-x) changes the sign of x.
Mastering complex graph skills is vital, as these questions often appear at the end of the paper and require combining multiple logical steps.
Summary Checklist and Next Steps
Checklist:
- [ ] I can identify if a transformation is Vertical (outside) or Horizontal (inside).
- [ ] I can describe a translation using a column vector.
- [ ] I remember that horizontal transformations do the opposite of the sign.
- [ ] I can sketch the reflection of a graph in the x-axis and y-axis.
- [ ] I can track the coordinates of a turning point through a transformation.
Practice Resources
Mastering graph transformations requires practice to ensure you don't fall for the "Inside Liar" trap. Use our dedicated worksheet to test your skills on translating and reflecting functions.
Download Topic Worksheet: Graph Transformations (Translations & Reflections)
Looking for more practice? Access our complete library of IGCSE maths worksheets and answers:
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Next Steps:
Congratulations! You have completed Section 3: Functions and Graphs. You have mastered the visual language of mathematics. The next major section is Topic 4: Geometry and Trigonometry, where we apply logic to shapes and spaces.