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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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