Section 2.4 Symmetry

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 Consider the points (4, 2) and (4, −2) that appear on the graph of x = y2 shown on the next slide. Points like these have the same x-value but opposite y-values are reflections of each other across the x-axis. If, for any point (x, y) on a graph, the point (x, −y) is also on the graph, then the graph is said to be symmetric with respect to the x-axis.

Symmetry If we fold the graph of the x-axis, the parts above and below the x-axis will coincide.

Symmetry Consider the points (3, 4) and (-3, 4) that appear on the graph of y = x2 – 5 on the following slide. Points like this have the same y-value but opposite x-values and are reflections of each other across the x-axis. If, for any point (x, y) on a graph, the point (−x, y) is also on the graph, then the graph is said to be symmetric with respect to the y-axis.

Symmetry If we fold the graph on the y-axis, the parts to the left and to the right of the y-axis will coincide.

Symmetry Consider the points that appear on the graph of x2 = y2 + 2 on the following slide. Note that if we take the opposites of the coordinates of one pair, we get the other pair. If, for any point (x, y) on a graph, the point (−x, −y) is also on the graph, then the graph is said to be symmetric with respect to the origin.

Symmetry Visually, if we rotate the graph 180° about the origin, the resulting figure coincides with the original.

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 equation, so the graph is not symmetric with respect to the x-axis..

Example continued Test y = x2 + 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).

Determining Even and Odd Functions Given the function f(x): Find f(−x) and simplify. If f(x) = f(−x), then f is even. Find –f(x), simplify, and compare with f(−x) from step (1). If f(−x) = −f(x), the f is odd. Except for the function f(x) = 0, a function cannot be both even and odd. Thus if f(x) ≠ 0 and we see in step (1) that f(x) = f(−x) (that is, f is even), we need not continue.

Example Determine whether the function is even, odd, or neither. 1. We see that f(x) ≠ f(x). Thus, f is not even.

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

Example a) We see that the graph appears to be symmetric with respect to the origin. The function is odd.