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

## Types of transformation

### Reflections

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.

### Shifts

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.

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