Transformations of functions: translations and reflections [Higher tier]
Given a base function y = f(x) you can shift, stretch or reflect it predictably. The four GCSE transformations are:
| Transformation | Effect on graph |
|---|---|
y = f(x) + a | Translate up by a (parallel to y) |
y = f(x + a) | Translate left by a (parallel to x) — note: opposite sign to intuition! |
y = -f(x) | Reflect in the x-axis |
y = f(-x) | Reflect in the y-axis |
A shorthand: changes outside f act on y (vertical), changes inside f act on x (horizontal and inverted).
Vertical translation y = f(x) + a
Adds a to every y-value. y = x² shifted up 3 becomes y = x² + 3. The graph moves up 3 units; its shape is unchanged.
Horizontal translation y = f(x + a)
Adds a inside the bracket. y = (x + 3)² is y = x² shifted left by 3. (Yes — left, even though the sign is +. Intuition: at x = -3 we get the same y as we had at x = 0 originally.)
Reflection in x-axis: y = -f(x)
Multiplies every y by -1. y = x² becomes y = -x², an upside-down parabola.
Reflection in y-axis: y = f(-x)
Replaces x with -x. The graph is mirrored left-right. For y = x³, y = (-x)³ = -x³ — happens to coincide with x-axis reflection because x³ is an odd function.
Combinations
Order matters. y = -f(x) + 2 first reflects, then translates.
y = f(x − 4) + 1: shift right 4, up 1. Vertex of y = (x − 4)² + 1 is at (4, 1).
✦Worked example
Given y = f(x) has a turning point at (2, 5), find the turning point of:
y = f(x) − 3→ (2, 2) (down 3)y = f(x + 4)→ (-2, 5) (left 4)y = -f(x)→ (2, -5) (reflect in x-axis)y = f(-x)→ (-2, 5) (reflect in y-axis)
✦Worked example— Worked example — sketching
Sketch y = (x − 2)² − 3 from y = x².
Translate y = x² right 2 (vertex (2, 0)), then down 3 (vertex (2, -3)). Same U-shape.
⚠Common mistakes— Common mistakes (examiner traps)
- Wrong direction for horizontal shift.
y = f(x − 5)shifts right by 5, not left. The minus inside means "x is replaced by x − 5, which becomes 0 when x = 5", i.e. the new origin is at x = 5. - Confusing reflections.
-f(x)flips top-bottom;f(-x)flips left-right. Memorise: outside → outside (y-axis flip means inside; got it? Check again.) Outside the bracket → vertical effect; inside the bracket → horizontal effect. - Forgetting symmetry of even/odd functions. For
y = x²(even),f(-x) = f(x)— y-axis reflection looks unchanged. - Translating both ways at once incorrectly.
y = f(x + 1) + 2shifts left 1, up 2. - Misreading the original turning point. Always start from the given key feature, then apply the shift to it.
➜Try this— Quick check
Curve y = f(x) has a minimum at (1, -2). Find the minimum of y = f(x − 3) + 5.
Shift right 3, up 5: minimum at (4, 3).
AI-generated · claude-opus-4-7 · v3-deep-algebra