3-1 Symmetry3-1 Symmetry

Symmetry All Around UsSymmetry 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 SymmetryPoint Symmetry

Two distinct points P and P’ are symmetric Two distinct points P and P’ are symmetric with respect to a point, M, if and only if M with respect to a point, M, if and only if M is the midpoint of PP’. Point M is is the midpoint of PP’. Point M is symmetric with respect to itself.symmetric with respect to itself.

A figure with point symmetry can be A figure with point symmetry can be turned about a center point and, in less turned about a center point and, in less than a full turn, the image coincides with than a full turn, the image coincides with the original figure.the original figure.

ExamplesExamples

Symmetry with Respect to the Symmetry with Respect to the OriginOrigin

The graph of a relation S is The graph of a relation S is symmetric with respect to the symmetric with respect to the origin iff (a, b) origin iff (a, b) ЄЄ S implies that S implies that (-a, -b) (-a, -b) ЄЄ S. A function f(x) has S. A function f(x) has a graph that is symmetric with a graph that is symmetric with respect to the origin iffrespect to the origin iff

f(-x) = -f(x).f(-x) = -f(x).

Example of Symmetry with Example of Symmetry with Respect to the OriginRespect to the Origin

Example: Determine whether the graph of Example: Determine whether the graph of f(x) = -7xf(x) = -7x55 + 8x is symmetric with respect to the + 8x is symmetric with respect to the origin.origin.

Example: Determine whether the graph of Example: Determine whether the graph of f(x) = xf(x) = x22 - 2x - 1 is symmetric with respect to the - 2x - 1 is symmetric with respect to the origin.origin.

Line of SymmetryLine of Symmetry

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

Examples of Line SymmetryExamples of Line Symmetry

Symmetry with Respect to …Symmetry with Respect to …

the x-axisthe x-axis

(a, -b) (a, -b) ЄЄ S iff (a, b) S iff (a, b) ЄЄ S S

The graph of x = yThe graph of x = y22 – 4 – 4(0, 2) and (0, -2) are on the graph(0, 2) and (0, -2) are on the graph

ExampleExample

Symmetry with Respect to …Symmetry with Respect to …

the y-axisthe y-axis

(-a, b) (-a, b) ЄЄ S iff (a, b) S iff (a, b) ЄЄ S S

The graph of y = xThe graph of y = x22 - 4 - 4(2, 0) and (-2, 0) are on the graph(2, 0) and (-2, 0) are on the graph

ExampleExample

Symmetry with Respect to …Symmetry with Respect to …

the line y = xthe line y = x

(b, a) (b, a) ЄЄ S iff (a, b) S iff (a, b) ЄЄ S S

The graph of xy = 6The graph of xy = 6(2, 3) and (3, 2) are on the graph(2, 3) and (3, 2) are on the graph

ExampleExample

Symmetry with Respect to …Symmetry with Respect to …

the line y = -xthe line y = -x

(-b, -a) (-b, -a) ЄЄ S iff (a, b) S iff (a, b) ЄЄ S S

The graph of xy = 6The graph of xy = 6(3, 2) and (-2, -3) are on the graph(3, 2) and (-2, -3) are on the graph

ExampleExample

Even FunctionsEven Functions

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

All exponents are even.All exponents are even.

Odd FunctionsOdd Functions

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

All exponents are odd. All exponents are odd.

Determine whether the graph of x + yDetermine whether the graph of x + y2 2 = 1 is = 1 is symmetric with respect to the x-axis, y-axis, symmetric with respect to the x-axis, y-axis, the line y = x, the line y = -x, or none of these.the line y = x, the line y = -x, or none of these.