Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 61 SYMMETRY 43EXAMPLES(a) The ellipse graphed in Fig. 6-4(b) is symmetric with respect to the origin becauseX2( XI2- + y2 = 1 implies - + (-Y)~ = 14 4(b) The hyperbola xy = 1 (see Fig. 6-6) is symmetric with respect to the origin, for if xy = 1, then (-x)(- y) = 1.(c) If y = ax, then -y = a(-x). Hence, any straight line through the origin is symmetric with respect to theorigin.Fig. 6-6Solved Problems6.1 Determine whether the line y = --x (see Fig. 6-7) is symmetric with respect to: (a) the x-axis;(b) the y-axis; (c) the origin.(U) The line is not symmetric with respect to the x-axis, since (-1, 1) is on the line, but (-1, -l), thereflection of (- 1, 1) in the x-axis, is not on the line.(b) The line is not symmetric with respect to the y-axis, since (- 1, 1) is on the line, but (1, l), the reflectionof (- 1, 1) in the y-axis, is not on the line.t',UI 2 3Fig. 6-7 Fig. 6 8