22 CHAPTER 1 Graphs Now Work PROBLEMS 25 AND 35(b) Figure 34 illustrates the definition. Symmetry with respect to the origin may be viewed in three ways: • As a reflection about the y -axis, followed by a reflection about the x -axis • As a projection along a line through the origin so that the distances from the origin are equal • As half of a complete revolution about the origin Figure 34 Symmetry with respect to the origin (x, y) (x, y) (–x, –y) (–x, –y) x y Tests for Symmetry To test the graph of an equation for symmetry with respect to the • x-Axis Replace y by −y in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the x -axis. • y-Axis Replace x by −x in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the y -axis. • Origin Replace x by −x and y by −y in the equation and simplify. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin. DEFINITION Symmetry with Respect to the Origin A graph is symmetric with respect to the origin if, for every point ( ) x y , on the graph, the point ( ) − − x y , is also on the graph. Points Symmetric with Respect to the Origin If a graph is symmetric with respect to the origin, and the point ( ) − 4, 2 is on the graph, then the point ( ) −4, 2 is also on the graph. EXAMPLE 4 When the graph of an equation is symmetric with respect to a coordinate axis or the origin, the number of points that you need to plot in order to see the pattern is reduced. For example, if the graph of an equation is symmetric with respect to the y -axis, then once points to the right of the y -axis are plotted, an equal number of points on the graph can be obtained by reflecting them about the y -axis. Because of this, before we graph an equation, we should first determine whether it has any symmetry. The following tests are used for this purpose. Testing an Equation for Symmetry Test + = x y 4 9 36 2 2 for symmetry. Solution EXAMPLE 5 x-Axis: To test for symmetry with respect to the x -axis, replace y by −y. Since ( ) + − = x y 4 9 36 2 2 is equivalent to + = x y 4 9 36 2 2 because ( ) ( )( ) − = − − = y y y y , 2 2 the graph of the equation is symmetric with respect to the x -axis. y-Axis: To test for symmetry with respect to the y -axis, replace x by −x. Since ( ) − + = x y 4 9 36 2 2 is equivalent to + = x y 4 9 36 2 2 because ( ) ( )( ) − = − − = x x x x , 2 2 the graph of the equation is symmetric with respect to the y -axis. Origin: To test for symmetry with respect to the origin, replace x by −x and y by −y. Since ( ) ( ) − + − = x y 4 9 36 2 2 is equivalent to + = x y 4 9 36, 2 2 the graph of the equation is symmetric with respect to the origin. The results of Example 5 suggest that an equation may have more than one type of symmetry.
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