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Sec 2.6

Families of Functions

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Parent FunctionParent Function: the simpliest form in a set of functions that form a family.Transformation: different functions in the family.

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Graph y = xGraph the following equations on the graph above.y = x + 2y = x + 4y = x – 2y = x – 4

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Graph y = x2

Graph the following equations on the graph above.y = x2 + 1y = x2 + 2y = x2 – 2y = x2 – 4

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Graph y = x2

Graph the following equations on the graph above.y = (x ­ 1)2y = (x ­ 4)2y = (x + 2)2y = (x + 3)2

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Transformations of f(x)Vertical Translations:Translation up k units, k > 0; y = f(x) + kTranslation down k units, k < 0; y = f(x) – k

Horizontal Translations:Translation right h units, h > 0; y = f(x – h)Translation left h units, h < 0; y = f(x + h)

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Transformations of f(x) cont.Vertical stretches and compressions:Vertical stretch, a > 1; y = af(x)Vertical compression, 0 < a < 1; y = af(x)Reflection:In the x­axis; y = ­f(x)In the y­axis; y = f(­x)

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General Equation

y=a f(x ­ h) + k

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Piecewise FunctionPiecewise function: a function that has different rules for different parts of the domain.

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Graph

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Graph

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Graph