Transformations of functions include reflections, stretches, compressions, and shifts.
Types of transformation
A function can be reflected across the x-axis by multiplying by -1 to give or .
A function can also be reflected across the y-axis by multiplying by -1, giving .
A function can be reflected across the line by swapping and in the equation, yielding (if can be isolated, this is equivalent to .
Stretches and compressions
Multiplying by any constant will stretch the graph vertically by a factor of the reciprocal of . Likewise, multiplying by any constant will do the same horizontally.
Subtracting any constant from (or adding it to ) will shift the graph up by units. Subtracting a constant from (giving will shift the graph units to the right.