L4 · Linear functions clinic

Learning goal

“I can use everything in this unit on unfamiliar problems: generate an arithmetic sequence by restricting a function's domain to the natural numbers, predict values by interpolation and extrapolation (and say which I am doing), and move freely among situations, tables, graphs and function notation.”

Interpolation — predicting a value BETWEEN points you already have.

Extrapolation — predicting a value BEYOND the points you have. It assumes the pattern continues - sometimes it doesn't, so extrapolated values carry risk.

Worked example

Take the function f(x) = 4x + 3 and restrict its domain to the natural numbers n = 1, 2, 3, 4, ... Step 1: Evaluate at consecutive natural numbers: f(1) = 7; f(2) = 11; f(3) = 15; f(4) = 19. Step 2: The outputs form a SEQUENCE: 7, 11, 15, 19, ... Each term grows by the same amount: 11 - 7 = 4, 15 - 11 = 4, 19 - 15 = 4 - an arithmetic sequence with common difference 4. Step 3: Notice WHERE the 4 lives: it is the coefficient of x - the slope. A linear function's slope becomes the sequence's common difference, and the constant (3) is the value "before the first term" (f(0)), which is why the first term is 3 + 4 = 7, not 3.

Check yourself — explain it

Why is the domain of a sequence function the natural numbers rather than the reals?

Paired problem — example still visible

Generate the first four terms of the sequence from a(n) = 2n + 9, and state the common difference. -> a(1) = 11; a(2) = 13; a(3) = 15; a(4) = 17 - 11, 13, 15, 17; difference 2.

Faded completion — last step is yours

"a(n) = 7n - 1: a(1) = ___, a(2) = ___, a(3) = ___. Common difference: ___" -> 6; 13; 20; 7

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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