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3-1 Symmetry

3-1 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.

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3-1 Symmetry

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  1. 3-1 Symmetry

  2. 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

  3. 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.

  4. Examples

  5. 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).

  6. Example of Symmetry with Respect to the Origin

  7. Example: Determine whether the graph of f(x) = -7x5 + 8x is symmetric with respect to the origin.

  8. Example: Determine whether the graph of f(x) = x2 - 2x - 1 is symmetric with respect to the origin.

  9. 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.

  10. Examples of Line Symmetry

  11. Symmetry with Respect to … the x-axis (a, -b) Є S iff (a, b) Є S The graph of x = y2 – 4 (0, 2) and (0, -2) are on the graph

  12. Example

  13. Symmetry with Respect to … the y-axis (-a, b) Є S iff (a, b) Є S The graph of y = x2 - 4 (2, 0) and (-2, 0) are on the graph

  14. Example

  15. 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

  16. Example

  17. 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

  18. Example

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

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

  21. Determine whether the graph of x + y2 = 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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