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]