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Transformations of functions include reflections, stretches, compressions, and shifts.

Types of transformation

Reflections

A function y=f(x) can be reflected across the x-axis by multiplying y by -1 to give -y=f(x) or y=-f(x) .

A function can also be reflected across the y-axis by multiplying x by -1, giving y=f(-x) .

A function can be reflected across the line y=x by swapping x and y in the equation, yielding x=f(y) (if y can be isolated, this is equivalent to y=f^{-1}(x) .

Stretches and compressions

Multiplying y by any constant a will stretch the graph vertically by a factor of the reciprocal of a . Likewise, multiplying x by any constant will do the same horizontally.

Shifts

Subtracting any constant k from y (or adding it to f(x)) will shift the graph up by k units. Subtracting a constant from x (giving y=f(x-h) will shift the graph h units to the right.

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