2.2.4 Absolute-Value Functions
Key Objectives • Graph and transform absolute-value functions. • Write rules for transformed absolute-value functions. Key Terms • An absolute-value function is a function whose rule contains an absolute-value expression. The parent absolute-value function is f ( x ) = | x |. The graph of this function has a V shape, its vertex is at (0, 0), its domain is all real numbers, and its range is all nonnegative real numbers. Because graphs of functions are transformations of the graph of their parent function, the graph of an absolute-value function is a transformation of the graph of f ( x ) = | x |. Example 1 Translating Absolute-Value Functions Translating the absolute-value parent function f ( x ) = | x | vertically k units gives the image g ( x ) = | x | + k , where f is translated up when k > 0 and down when k < 0. Generally, if any function f ( x ) is vertically translated k units, then the equation of the image g is g ( x ) = f ( x ) + k . Translating the absolute-value parent function f ( x ) = | x | horizontally h units gives the image g ( x ) = | x − h |, where f is translated right when h > 0 and left when h < 0. Generally, if any function f ( x ) is horizontally translated k units, then the equation of the image g is g ( x ) = f ( x − h ).
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