Schaum's Outline of Theory and Problems of Beginning Calculus

42 SYMMETRY[CHAP. 6Consider the graph of an equation f(x, y) = 0. Then, by (i) above, the graph is symmetric withrespect to the y-axis if and only if f(x, y) = 0 implies f( - x, y) = 0. And, by (ii) above, the graph issymmetric with respect to the x-axis if and only iff(x, y) = 0 impliesf(x, -y) = 0.EXAMPLES(a) The y-axis is an axis of symmetry of the parabola y = x2 [see Fig. 6-4(u)]. For if y = x2, then y = (- x)~. Thex-axis is not an axis of symmetry of this parabola. Although (1, 1) is on the parabola, (1, - 1) is not on theparabola.XL(b) The ellipse - + y2 = 1 [see Fig. 6-4(b)] has both the y-axis and the x-axis as axes of symmetry. For if4X2- + y2 = 1, then4(_x)2+y2 = 1 and -+(-y)’= X2 14 46.2 SYMMETRY ABOUT A POINTTwo points P and Q are said to be symmetric with respect to U point A if A is the midpoint of theline segment PQ [see Fig. 6-5(u)].The point Q symmetric to the point P(x, y) with respect to the origin has coordinates (-x, -y). [InFig. 6-5(b), APOR is congruent to AQOS. Hence, = 0s and = a.]Symmetry of a graph about a point is defined in the expected manner. In particular, a graph Y issaid to be symmetric with respect to the origin if, whenever a point P lies on Y, the point Q symmetric toP with respect to the origin also lies on 9. The graph of an equation f(x, y) = 0 is symmetric withrespect to the origin if and only iff(x, y) = 0 impliesf( -x, -y) = 0.t’0