Combining transformations

Transformations applied in a different order can produce different functions. Follow a point through each step.

Example

Start with \(y=\mathrm{f}(x)\). Stretch vertically by \(3\), then translate up by \(2\): \(y=3\mathrm{f}(x)+2\), with \((u,v)\mapsto(u,3v+2)\).

Translate up by \(2\), then stretch vertically by \(3\): \(y=3(\mathrm{f}(x)+2)=3\mathrm{f}(x)+6\).

For \(\mathrm{f}(x)=x^2\), the resulting vertices are \((0,2)\) and \((0,6)\), respectively.