transformation of functions

A transformation changes the appearance or position of a graph. The general form for transforming a function is [latex]y=\pm af(b(x-h))+k[/latex].
The most common transformations are:

• Vertical shifts: move the graph up or down.
  [latex]f(x) + k[/latex] shifts up by [latex]k[/latex]; [latex]f(x) - k[/latex] shifts down.

• Horizontal shifts: move the graph left or right.
  [latex]f(x - h)[/latex] shifts right by [latex]h[/latex]; [latex]f(x + h)[/latex] shifts left.

• Reflections: flip the graph.
  [latex]-f(x)[/latex] reflects across the x-axis; [latex]f(-x)[/latex] reflects across the y-axis.

• Stretches and compressions: make the graph taller or wider.
  [latex]a f(x)[/latex] vertically stretches if [latex]|a| > 1[/latex], compresses if [latex]|a| < 1[/latex].