What is important to know about Transformation of graphs?

f(x) + a: translation by (0, a) — shifts the graph up by a (or down if a < 0).

What is important to know about Transformation of graphs?

f(x + a): translation by (−a, 0) — shifts the graph left by a (note the sign change).

What is important to know about Transformation of graphs?

−f(x): reflection in the x-axis.

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Algebra · Higher

Transformation of graphs

Q: What is important to know about Transformation of graphs?

f(x + a): translation by (−a, 0) — shifts the graph left by a (note the sign change).

Q: What is important to know about Transformation of graphs?

−f(x): reflection in the x-axis.

Q: What are common mistakes with Transformation of graphs?

f(x + a) shifts left (not right) — the sign inside the bracket is opposite to the direction of movement.

Q: What are common mistakes with Transformation of graphs?

Confusing vertical and horizontal stretches.

Q: What exam tips are there for Transformation of graphs?

For translations inside the bracket f(x ± a), remember the shift is in the opposite direction to the sign.

Q: What exam tips are there for Transformation of graphs?

Learn all six transformation rules thoroughly — they are regularly tested at Higher.

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

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)

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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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