Algebra: Sequences, Functions and Graphs

This lesson extends your knowledge of sequences to quadratics, introduces function notation and inverse/composite functions, and explores the full range of graphs you meet at GCSE including graph transformations.

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Sequences

Arithmetic Sequences (Review)

nth term = dn + c, where d = common difference.

Quadratic Sequences

A sequence where the second differences are constant.

Example: 3, 7, 13, 21, 31, … First differences: 4, 6, 8, 10, … (not constant) Second differences: 2, 2, 2, … → constant, so quadratic.

The nth term is of the form an² + bn + c, where a = second difference / 2.

Finding the quadratic nth term: Second difference = 2 → a = 1 (so n² term) Subtract n² from the sequence: 3−1, 7−4, 13−9, 21−16, 31−25 = 2, 3, 4, 5, 6 → linear, nth term = n + 1. Full nth term: n² + n + 1

Check: n=1: 1+1+1=3 ✓; n=3: 9+3+1=13 ✓

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Function Notation

f(x) = 3x − 2 means "apply the rule 3x − 2 to the input x."

Evaluating: f(4) = 3(4) − 2 = 10

Composite Functions [H]

fg(x) means apply g first, then apply f to the result.

If f(x) = 2x + 1 and g(x) = x², then: fg(x) = f(g(x)) = f(x²) = 2x² + 1 gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)²

Note: fg(x) ≠ gf(x) in general.

Inverse Functions [H]

f⁻¹(x) reverses the function f. If f maps x to y, then f⁻¹ maps y back to x.

Finding the inverse: write y = f(x), rearrange to make x the subject, then replace x with f⁻¹(x) and y with x.

Example: f(x) = (2x + 3)/5 y = (2x + 3)/5 → 5y = 2x + 3 → x = (5y − 3)/2 f⁻¹(x) = (5x − 3)/2

The graph of f⁻¹(x) is a reflection of f(x) in the line y = x.

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Types of Graphs

Graph type	General form	Shape
Linear	y = mx + c	Straight line
Quadratic	y = ax² + bx + c	Parabola (U or ∩)
Cubic	y = ax³ + bx² + cx + d	S-shaped curve
Reciprocal	y = k/x	Hyperbola (two branches)
Exponential growth	y = aˣ (a > 1)	Rapidly increasing curve
Exponential decay	y = aˣ (0 < a < 1)	Rapidly decreasing curve

Sketching: mark intercepts, turning points, and general shape. No need for a detailed table of values when sketching.

Key Features of a Quadratic y = ax² + bx + c