IP EMaths Notes (Upper Sec, Year 3-4): 08) Functions and Transformations

Function language appears in calculator prompts, inverse-relationship questions, and graph sketching. Track how algebraic edits move or stretch the curve.

Key ideas

• \(f(x)\) gives the output when input \(x\) is substituted.
• Composite functions: \((f \circ g)(x) = f(g(x))\).
• Common transformations:
  • \(y = f(x) + k\): shift up by \(k\).
  • \(y = f(x - a)\): shift right by \(a\).
  • \(y = af(x)\): vertical stretch by factor \(a\).
  • \(y = f(ax)\): horizontal compression by factor \(a\).

Worked example - Transformation tracker

The graph of \(y = x^{2}\) is transformed to \(y = 3(x - 2)^{2} + 5\). Describe the sequence of transformations.

1. Start with \(y = x^{2}\).
2. \(x \mapsto x - 2\) translates the graph \(2\) units to the right.
3. Multiplying by \(3\) stretches it vertically by factor 3 (parabola becomes narrower).
4. Adding \(5\) shifts the graph up by \(5\) units.

So the sequence is: translate right by \(2\), stretch vertically by \(3\), then shift up by \(5\).

Try this

Given \(f(x) = 2x - 1\) and \(g(x) = x^{2} + 3\), find \((g \circ f)(x)\) and state the value when \(x = -2\).