Graphs can be transformed by translations, reflections and stretches. Understanding how changes to the equation affect the graph is a key Higher topic.
Put your revision into practice
Choose a topic and try a short quiz with instant explanations. Free, with no signup.
Revising transformation of graphs the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Transformation of graphs GCSE” on the site for the complete guide.
🔑
Key facts to remember
1f(x) + a: translation by (0, a) — shifts the graph up by a (or down if a < 0).
2f(x + a): translation by (−a, 0) — shifts the graph left by a (note the sign change).
3−f(x): reflection in the x-axis.
4f(−x): reflection in the y-axis.
5af(x): vertical stretch by scale factor a.
6f(ax): horizontal stretch by scale factor 1/a.
✍️
Worked examples
Example 1
The graph of y = f(x) passes through (2, 5). Write down the coordinates of this point after the transformation y = f(x + 3).
Working
f(x + 3) is a translation of (−3, 0)
x-coordinate: 2 − 3 = −1
y-coordinate: unchanged at 5
New point: (−1, 5)
Answer(−1, 5)
Example 2
Describe the transformation from y = x² to y = (x − 2)² + 3.
Working
f(x − 2): translation of (+2, 0) — shifts right by 2
+ 3: translation of (0, +3) — shifts up by 3
Overall: translation by vector (2, 3)
AnswerTranslation by vector (2, 3)
⚠️
Common mistakes
✗f(x + a) shifts left (not right) — the sign inside the bracket is opposite to the direction of movement.
✗Confusing vertical and horizontal stretches.
✗Applying transformations in the wrong order when combining them.
🎯
Exam tips
✓For translations inside the bracket f(x ± a), remember the shift is in the opposite direction to the sign.
✓Learn all six transformation rules thoroughly — they are regularly tested at Higher.
Ready to test yourself on Transformation of graphs?
Get AI-marked practice questions on exactly this subtopic.