1. 3-1 Symmetry

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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) = -7x 5 + 8x is symmetric with respect to the origin.

8. Example: Determine whether the graph of f(x) = x 2 - 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 = y 2 – 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 = x 2 - 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 + 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.