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Algebra · Higher

Transformation of graphs

Graphs can be transformed by translations, reflections and stretches. Understanding how changes to the equation affect the graph is a key Higher topic.

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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.
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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

  1. f(x + 3) is a translation of (−3, 0)
  2. x-coordinate: 2 − 3 = −1
  3. y-coordinate: unchanged at 5
  4. New point: (−1, 5)
Answer(−1, 5)
Example 2

Describe the transformation from y = x² to y = (x − 2)² + 3.

Working

  1. f(x − 2): translation of (+2, 0) — shifts right by 2
  2. + 3: translation of (0, +3) — shifts up by 3
  3. Overall: translation by vector (2, 3)
AnswerTranslation by vector (2, 3)
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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.
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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.

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