Chapter 2 More on Functions

Section 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry 2.5 Transformations 2.6 Variation and Applications

2.4 Symmetry 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 y = x2 + 2 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 not equivalent to the original so the graph is not symmetric with respect to the x-axis.

Example continued Test y = x 2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin. y-axis: We replace x with x: The resulting equation is equivalent to the original equation, so the graph is symmetric with respect to the y-axis.

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).

Example Determine whether the function is even, odd, or neither. 1. We see that h(x) = h(x). Thus, h is even. y = x4  4x2

Example Determine whether the function is even, odd, or neither. 2. y = x4  4x2 We see that h(x)  h(x). Thus, h is not odd.