Symmetry of graphs. Circles Symmetry of graphs. Circles 1 / 10 Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Symmetry of graphs. Circles 2 / 10 Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Symmetry of graphs. Circles 2 / 10 Sends (x,y) to (-x,-y) Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Symmetry of graphs. Circles 2 / 10 Examples with Symmetry What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Symmetry of graphs. Circles 2 / 10 What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry Symmetry of graphs. Circles 2 / 10 What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry Symmetry of graphs. Circles 2 / 10 What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry Symmetry of graphs. Circles 2 / 10 What is Symmetry? Take some geometrical object. It is called symmetric if some geometric move preserves it Today we will be interested in reflection across the x-axis, reflection across the y-axis and reflection across the origin. Reflection across y reflection across x reflection across (0; 0) Sends (x,y) to (-x,y) Sends (x,y) to (x,-y) Sends (x,y) to (-x,-y) Examples with Symmetry Symmetry of graphs. Circles 2 / 10 (4,2) (-4,-2) (-4,2) Reflect a point: Reflection across the x−axis is given by (x; y) 7! (x; −y) Reflection across the y−axis is given by (x; y) 7! (−x; y) reflection across the origin is given by (x; y) 7! (−x; −y) Example: For the point P = (4; −2), write down the coordinates of and draw The reflection of P across the x-axis. The reflection of P across the y-axis. The reflection of P across the origin. Symmetry of graphs. Circles 3 / 10 (-4,-2) (-4,2) Reflect a point: Reflection across the x−axis is given by (x; y) 7! (x; −y) Reflection across the y−axis is given by (x; y) 7! (−x; y) reflection across the origin is given by (x; y) 7! (−x; −y) Example: For the point P = (4; −2), write down the coordinates of and draw The reflection of P across the x-axis. (4,2) The reflection of P across the y-axis. The reflection of P across the origin. Symmetry of graphs. Circles 3 / 10 (-4,2) Reflect a point: Reflection across the x−axis is given by (x; y) 7! (x; −y) Reflection across the y−axis is given by (x; y) 7! (−x; y) reflection across the origin is given by (x; y) 7! (−x; −y) Example: For the point P = (4; −2), write down the coordinates of and draw The reflection of P across the x-axis. (4,2) The reflection of P across the y-axis. (-4,-2) The reflection of P across the origin. Symmetry of graphs. Circles 3 / 10 Reflect a point: Reflection across the x−axis is given by (x; y) 7! (x; −y) Reflection across the y−axis is given by (x; y) 7! (−x; y) reflection across the origin is given by (x; y) 7! (−x; −y) Example: For the point P = (4; −2), write down the coordinates of and draw The reflection of P across the x-axis. (4,2) The reflection of P across the y-axis. (-4,-2) The reflection of P across the origin.(-4,2) Symmetry of graphs. Circles 3 / 10 (b) (c) (a) (a, b, c) Analytic symmetry: The graph of an equation is symmetric about the y-axis if you can replace x by −x and get an equivalent equation The graph of an equation is symmetric about the x-axis if you can replace y by −y and get an equivalent equation The graph of an equation is symmetric about the origin if you can replace x by −x and y by −y and get an equivalent equation Check these examples. Testing for symmetry for graphs of equations A graph is called Symmetric with respect to a reflection if that reflection does not change the graph. The following graphs are symmetric about (a) reflection across the x-axis (b) reflection across the y-axis (c) reflection about the origin. y = x 2 − 4 y = x 3 − 3x x = y 2 + 1 x 2 + y 2 = 4 Symmetry of graphs. Circles 4 / 10 (c) (a) (a, b, c) Analytic symmetry: The graph of an equation is symmetric about the y-axis if you can replace x by −x and get an equivalent equation The graph of an equation is symmetric about the x-axis if you can replace y by −y and get an equivalent equation The graph of an equation is symmetric about the origin if you can replace x by −x and y by −y and get an equivalent equation Check these examples. Testing for symmetry for graphs of equations A graph is called Symmetric with respect to a reflection if that reflection does not change the graph.