2-6 Families of Functions

Objective

To analyze transformations of functions

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Different nonvertical lines have different slopes, or y-intercepts, or both. They are graphs of different linear functions. For two such lines, you can think of one as a transformation of the other.

Essential Understanding There are sets of functions, called families, in which each function is a transformation of a special function called the parent.

The linear functions form a family of functions. Each linear function is a transformation of the function y = x . y equals x .  The function y = x y equals x  is the parent linear function.

A parent function is the simplest form in a set of functions that form a family. Each function in the family is a transformation of the parent function.

One type of transformation is a translation. A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation. For a positive constant k and a parent function f(x), f ( x ) ± k f open x close plus minus k  is a vertical translation. For a positive constant h , f ( x ± h ) h comma f open x plus minus h close  is a horizontal translation.

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