Function transformations

WJEC Higher candidates need four standard transformations. Memorise both the algebraic form and the geometric effect.

Notation

Original function: y = f(x).

Transformed function notation appears as:

• f(x) + a
• f(x + a)
• −f(x)
• f(−x)

y = f(x) + a — vertical translation

Adds a to every output. Translation by vector (0, a). Up if a > 0, down if a < 0.

y = f(x + a) — horizontal translation

Replaces x with (x + a). Translation by vector (−a, 0). The sign FLIPS — adding inside the bracket moves LEFT.

Examples:

• f(x − 3) → translation 3 right.
• f(x + 2) → translation 2 left.

y = −f(x) — reflection in x-axis

Negate the output. Every (x, y) maps to (x, −y). The graph flips upside-down.

y = f(−x) — reflection in y-axis

Negate the input. Every (x, y) maps to (−x, y). The graph flips left-right.

Combining transformations

f(x − 2) + 3: shift right 2, then up 3. Vector (2, 3).

The shifts can be done in any order because they're independent.

Effect on key features

If the original turning point is (p, q):

• f(x) + a → (p, q + a).
• f(x + a) → (p − a, q).
• −f(x) → (p, −q).
• f(−x) → (−p, q).

WJEC exam tip

When asked to sketch on the same axes as a printed graph, copy the shape using a faint pencil line first, then ink the labelled key points. Show the new turning point, intercepts, and any asymptotes — these earn the A1 marks.