3-1 Symmetry

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Point Symmetry Two distinct points P and P’ are symmetric with respect to a point, M, if and only if M is the midpoint of PP’. Point M is symmetric with respect to itself. A figure with point symmetry can be turned about a center point and, in less than a full turn, the image coincides with the original figure.

Examples

Symmetry with Respect to the Origin The graph of a relation S is symmetric with respect to the origin iff (a, b) Є S implies that (-a, -b) Є S. A function f(x) has a graph that is symmetric with respect to the origin iff f(-x) = -f(x).

Example of Symmetry with Respect to the Origin

Example: Determine whether the graph of f(x) = -7x 5 + 8x is symmetric with respect to the origin.

Example: Determine whether the graph of f(x) = x 2 - 2x - 1 is symmetric with respect to the origin.

Line of Symmetry Two distinct points P and P’ are symmetric with respect to a line ℓ iff ℓ is the perpendicular bisector of PP’. A point P is symmetric to itself with respect to the line ℓ iff P is on ℓ.

Examples of Line Symmetry

Symmetry with Respect to … the x-axis (a, -b) Є S iff (a, b) Є S The graph of x = y 2 – 4 (0, 2) and (0, -2) are on the graph

Example

Symmetry with Respect to … the y-axis (-a, b) Є S iff (a, b) Є S The graph of y = x (2, 0) and (-2, 0) are on the graph

Example

Symmetry with Respect to … the line y = x (b, a) Є S iff (a, b) Є S The graph of xy = 6 (2, 3) and (3, 2) are on the graph

Example

Symmetry with Respect to … the line y = -x (-b, -a) Є S iff (a, b) Є S The graph of xy = 6 (3, 2) and (-2, -3) are on the graph

Example

Even Functions Functions that are symmetric with respect to the y-axis are even. All exponents are even.

Odd Functions Functions that are symmetric with respect to the origin are odd functions. All exponents are odd.

Determine whether the graph of x + y 2 = 1 is symmetric with respect to the x-axis, y-axis, the line y = x, the line y = -x, or none of these.