Section 2.4 Symmetry Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives  Determine whether a graph is symmetric with respect to the x-axis, the y-axis, and the origin.  Determine whether a function is even, odd, or neither even nor odd.

Symmetry Algebraic Tests of Symmetry x-axis: If replacing y with  y produces an equivalent equation, then the graph is symmetric with respect to the x-axis. y-axis: If replacing x with  x produces an equivalent equation, then the graph is symmetric with respect to the y-axis. Origin: If replacing x with  x and y with  y produces an equivalent equation, then the graph is symmetric with respect to the origin.

Example Test x = y for symmetry with respect to the x-axis, the y-axis, and the origin. x-axis: We replace y with  y: The resulting equation is equivalent to the original so the graph is symmetric with respect to the x-axis.

y-axis: We replace x with  x: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the y-axis. Example continued Test x = y for symmetry with respect to the x-axis, the y-axis, and the origin.

Example continued Origin: We replace x with  x and y with  y: The resulting equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.

Even and Odd Functions If the graph of a function f is symmetric with respect to the y-axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f(  x). If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f(  x) =  f(x).

Determine whether the function is even, odd, or neither. 1. We see that h(x) = h(  x). Thus, h is even. Example y = x 4  4x 2

Determine whether the function is even, odd, or neither. 2. Example We see that h(  x)   h(x). Thus, h is not odd. y = x 4  4x 2