3 1 Symmetry Symmetry All Around Us Symmetry

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3 -1 Symmetry

3 -1 Symmetry

Symmetry All Around Us Symmetry at the Beach Line Symmetry & Rotational Symmetry -

Symmetry All Around Us Symmetry at the Beach Line Symmetry & Rotational Symmetry - All you need to Know + Symmetry in the World, Symmetry Games, symmetry of the face, Symmetry Quiz and Worksheets

Point Symmetry Two distinct points P and P’ are symmetric with respect to a

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

Examples

Symmetry with Respect to the Origin The graph of a relation S is symmetric

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 of Symmetry with Respect to the Origin

Example: Determine whether the graph of f(x) = -7 x 5 + 8 x

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

Example: Determine whether the graph of f(x) = x 2 - 2 x -

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

Line of Symmetry Two distinct points P and P’ are symmetric with respect to

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

Examples of Line Symmetry

Symmetry with Respect to … the x-axis (a, -b) Є S iff (a, b)

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

Example

Symmetry with Respect to … the y-axis (-a, b) Є S iff (a, b)

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

Example

Example

Symmetry with Respect to … the line y = x (b, a) Є S

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

Example

Symmetry with Respect to … the line y = -x (-b, -a) Є S

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

Example

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

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

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

Odd 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

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.

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