Functions & Graph Transformations
Functions & Graph Transformations (Higher)
A function is a rule that takes an input and produces exactly one output. At GCSE, you need to use function notation and understand how graphs are transformed.
Function Notation
f(x) means "the function f applied to x".
If f(x) = 3x² − 2x + 5, then:
- f(1) = 3(1)² − 2(1) + 5 = 3 − 2 + 5 = 6
- f(−2) = 3(−2)² − 2(−2) + 5 = 12 + 4 + 5 = 21
- f(a) = 3a² − 2a + 5
Composite Functions
A composite function applies one function then another.
fg(x) means "apply g first, then f". Read right to left.
Worked Example: If f(x) = 2x + 1 and g(x) = x², find fg(3).
- g(3) = 3² = 9
- f(9) = 2(9) + 1 = 19
To find fg(x) as a formula: f(g(x)) = f(x²) = 2x² + 1.
Note: fg(x) ≠ gf(x) in general. gf(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
Inverse Functions
The inverse function f⁻¹(x) reverses what f does. To find it:
1. Write y = f(x)
2. Swap x and y
3. Rearrange to make y the subject
Worked Example: Find the inverse of f(x) = (3x − 5)/2.
- y = (3x − 5)/2
- Swap: x = (3y − 5)/2
- 2x = 3y − 5
- 2x + 5 = 3y
- y = (2x + 5)/3
- f⁻¹(x) = (2x + 5)/3
The graph of f⁻¹(x) is a reflection of f(x) in the line y = x.
Graph Transformations
These rules apply to any function y = f(x):
| Transformation | Effect on Graph |
|---|---|
| y = f(x) + a | Translate UP by a units |
| y = f(x) − a | Translate DOWN by a units |
| y = f(x + a) | Translate LEFT by a units |
| y = f(x − a) | Translate RIGHT by a units |
| y = −f(x) | Reflect in the x-axis |
| y = f(−x) | Reflect in the y-axis |
| y = af(x) | Stretch vertically by scale factor a |
| y = f(ax) | Stretch horizontally by scale factor 1/a |
The inside-outside rule: Changes inside the bracket (affecting x) are horizontal and do the opposite of what you expect. Changes outside are vertical and do what you expect.
Worked Example: The graph of y = f(x) passes through the point (4, 6). State the corresponding point on:
- y = f(x) + 3 → (4, 9) — y-coordinate increases by 3
- y = f(x − 2) → (6, 6) — x-coordinate increases by 2 (opposite!)
- y = −f(x) → (4, −6) — y-coordinate changes sign
- y = 2f(x) → (4, 12) — y-coordinate doubles
- y = f(3x) → (4/3, 6) — x-coordinate divides by 3
Using Vectors for Translations
A translation can be described using a column vector. y = f(x − a) + b is a translation by the vector (a, b) — a units right and b units up.
Worked Example: Describe the transformation from y = x² to y = (x − 3)² + 5.
- Translation by vector (3, 5) — 3 right, 5 up.
Exam Tips
- For f(x + a), the horizontal shift is the opposite direction to the sign — f(x + 3) moves LEFT
- When describing transformations, you must state the type (translation/reflection/stretch) AND the details (direction, scale factor, mirror line)
- Always check composite functions by substituting a simple value like x = 1
- The inverse of f only exists if f is one-to-one (each output comes from exactly one input)