Symmetry

1. Symmetry • Two points, P and P₁, are symmetric with respect to linel when they are the same distance from l, measured along a perpendicular line to l. Line l is the axis of symmetry.

2. Reflections • Two points symmetric with respect to a line are called reflections of each other across the line. The line is a line of symmetry.

3. Symmetry a.) A graph with b.) A graph with c.) A graph with x-axis symmetryy-axis symmetry origin symmetry for every (x,y) the for every (x,y) the for every (x,y) the point (x,-y) is also point (-x,y) is also point (-x,-y) is also on the graph. on the graph. on the graph.

4. Testing for symmetry • y = x² + 2 • To test for symmetry replace x with –x and y with –y . • Check to see if the equation is still equivalent to the original equation. • If it is there is symmetry to that axis. • Try x² + y² = 2

5. Point Symmetry • Two points, P and P₁, are symmetric with respect to a point Q when they are the same distance from Q. P₁ is said to be the image of P.

6. Symmetric with Respect to Origin • Two points are symmetric with respect to the origin if and only if both their x- and y-coordinates are additive inverses of each other. • Example: The point symmetric (3, -5) with respect to the origin is (-3, 5) • What would it be for point (4, -9)?