A function is a rule that sends each input to exactly one output; 'one-to-one' means different inputs give different outputs. Transformations are tested constantly: know exactly which way \(f(x + a)\) and \(f(ax)\) move the graph and what happens when transformations are combined.
Contents: Mappings and common functions (MM1.7)Transformations (MM8.2-MM8.4)Must-knowTrapsReal questions
Mappings and common functions (MM1.7)
- Many-to-one: \(x^2\), \(|x|\), \(\sin x\) (a horizontal line can cut the graph more than once). One-to-one: \(x^3\), \(2^x\), \(\log x\), \(x^3 + x\).
- \(f(x) = \sqrt x\) is defined for \(x \geq 0\) and always non-negative; \(f(x) = |x|\) is \(x\) for \(x \geq 0\) and \(-x\) for \(x < 0\).
- Composition: \(f(g(x))\) means apply \(g\) first. \(f(g(x))\) and \(g(f(x))\) are usually different.
- \(\sqrt{x^2} = |x|\), so \(\sqrt{g(x)}\) with \(g(x) = x^2 + 3\) ... care: \(\sqrt{x^2} = x\) only when \(x \geq 0\).
Transformations (MM8.2-MM8.4)
- \(y = f(x) + a\): translate \(a\) up. \(y = f(x + a)\): translate \(a\) to the left. \(y = af(x)\): stretch vertically by \(a\). \(y = f(ax)\): stretch horizontally by factor \(\frac1a\) (squash if \(a > 1\)).
- \(y = -f(x)\) reflects in the \(x\)-axis; \(y = f(-x)\) reflects in the \(y\)-axis.
- Order matters for combinations: \(y = 2f(x - 3)\) is 'translate then stretch'; \(y = f(2x - 2) = f(2(x - 1))\) is 'stretch by \(\frac12\) then translate \(1\) right'.
- Track a point: if \((4, 6)\) is on \(y = f(x)\) then \((2, 17)\) is on \(y = 3f(2x) - 1\) (halve \(x\), then \(3y - 1\)).
- For \(y = a(x + b)^2 + c\) the vertex is \((-b, c)\); for \(y = mx + c\), \(m\) is the gradient and \(c\) the intercept.
- An exponential translation is a stretch in disguise: \(2^{x-3} = \frac18\cdot 2^x\); a stretch of \(\log\) is a translation: \(\log(kx) = \log k + \log x\).
Must-know (no formula booklet)
- \(f(x + a)\): left by \(a\)
- \(f(ax)\): horizontal factor \(1/a\)
- \(af(x)\): vertical factor \(a\)
- \(f(-x)\): reflect in \(y\)-axis
- \(\sqrt{x^2} = |x|\)
Classic traps
- \(f(x + 3)\) moves the graph to the left, not the right.
- For \(y = f(2x + 6)\) factorise first: \(f(2(x + 3))\) - translate 3 left, then squash by \(\frac12\) (or squash then translate 3 left if you translate by 6 first - do not mix the orders).
- Two transformations that look different can give the same graph (e.g. reflecting \(x^2\) in the \(y\)-axis does nothing).
Worked example
Q. The maximum of \(y = f(x)\) is at \((3, 8)\). Where is the maximum of \(y = f(x - a) + b\)?
A. At \((3 + a, 8 + b)\): the translation moves every point by \(a\) right and \(b\) up.