SECTION 1.3 Intercepts; Symmetry; Graphing Key Equations 21 x = − y x 4 2 ( ) x, y 3− ( ) − − = 3 4 5 2 3, 5 ( ) − 1− 3− 1, 3 ( ) − − 1 3− 1, 3 ( ) − 3 5 3, 5 ( ) Table 4 Figure 31 y x 4 2 = − x y (3, 5) 5 (2, 0) (–1, –3) (1, –3) (0, –4) (–3, 5) (–2, 0) 5 –5 –5 Credit: Alex Staroseltsev/Shutterstock DEFINITION Symmetry with Respect to the x -Axis A graph is symmetric with respect to the x -axis if, for every point ( ) x y , on the graph, the point ( ) − x y , is also on the graph. Since ≥ x 0 2 for all x, we deduce from the equation = − y x 4 2 that ≥ − y 4 for all x. This information, the intercepts, and the points from Table 4 enable us to graph = − y x 4. 2 See Figure 31. Now Work PROBLEM 17 2 Test an Equation for Symmetry with Respect to the x -Axis, the y -Axis, and the Origin Another helpful tool for graphing equations by hand involves symmetry , particularly symmetry with respect to the x -axis, the y -axis, and the origin. Symmetry often occurs in nature. Consider the picture of the butterfly. Do you see the symmetry? Figure 32 illustrates the definition. Note that when a graph is symmetric with respect to the x -axis, the part of the graph above the x -axis is a reflection (or mirror image) of the part below it, and vice versa. Figure 33 illustrates the definition. When a graph is symmetric with respect to the y -axis, the part of the graph to the right of the y -axis is a reflection of the part to the left of it, and vice versa. DEFINITION Symmetry with Respect to the y -Axis A graph is symmetric with respect to the y-axis if, for every point ( ) x y , on the graph, the point ( ) −x y , is also on the graph. Points Symmetric with Respect to the x -Axis If a graph is symmetric with respect to the x -axis, and the point ( ) 3, 2 is on the graph, then the point ( ) − 3, 2 is also on the graph. EXAMPLE 2 Points Symmetric with Respect to the y -Axis If a graph is symmetric with respect to the y -axis and the point ( ) 5, 8 is on the graph, then the point ( ) −5, 8 is also on the graph. EXAMPLE 3 Figure 32 Symmetry with respect to the x -axis (x, –y) (x, y) (x, –y) (x, y) (x, y) (x, –y) x y Figure 33 Symmetry with respect to the y -axis (–x, y) (–x, y) (x, y) (x, y) x y
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