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INTERSECTIONS OF GRAPHS
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that is, 0.025x+2.5 is the price per pound at which buyers are willing to buy x million pounds of mutton. Find the intersection of the graphs of the supply and demand equations. This point (x, y) indicates the supply x at which the seller s price is equal to what the buyer is willing to pay. 5.8 Find the center and the radius of the circle passing through the points A(3, 0), B(0, 3), and C(6, 0). [Hint: The center is the intersection of the perpendicular bisectors of any two sides of ABC.] 5.9 Find the equations of the lines through the origin that are tangent to the circle with center at (3, 1) and radius 3. [Hint: A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the Pythagorean theorem may be used to give a second equation for the coordinates of the point of contact.] 5.10 Find the coordinates of the point on the line y = 2x + 1 that is equidistant from (0,0) and (5, 2). 5.11 Find the equation for a point (x, y) whose distance from the line x = 4 is twice as great as its distance from the line y = 6.
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Symmetry
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6.1 SYMMETRY ABOUT A LINE
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Two points P and Q are said to be symmetric with respect to a line L if P and Q are mirror images in L . More precisely, the segment PQ is perpendicular to L at a point A such that PA = QA (see Fig. 6-1).
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Fig. 6-1 (i) If Q(x, y) is symmetric to the point P with respect to the y-axis, then P is ( x, y) [see Fig. 6-2(a)]. (ii) If Q(x, y) is symmetric to the point P with respect to the x-axis, then P is (x, y) [see Fig. 6-2(b)].
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Fig. 6-2 A graph is said to be symmetric with respect to a line L if, for any point P on the graph, the point Q that is symmetric to P with respect to L is also on the graph, L is then called an axis of symmetry of the graph. See Fig. 6-3.
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Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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SYMMETRY
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Fig. 6-3 Consider the graph of an equation f (x, y) = 0. Then, by (i) above, the graph is symmetric with respect to the y-axis if and only if f (x, y) = 0 implies f ( x, y) = 0. And, by (ii) above, the graph is symmetric with respect to the x-axis if and only if f (x, y) = 0 implies f (x, y) = 0. EXAMPLES
(a) The y-axis is an axis of symmetry of the parabola y = x 2 [see Fig. 6-4(a)]. For if y = x 2 , then y = ( x)2 . The x-axis is not an axis of symmetry of this parabola. Although (1, 1) is on the parabola, (1, 1) is not on the parabola. (b) The ellipse x2 x2 + y2 = 1 [see Fig. 6-4(b)] has both the y-axis and the x-axis as axes of symmetry. For if + y2 = 1, then 4 4 ( x)2 + y2 = 1 4 and x2 + ( y)2 = 1 4
Fig. 6-4
SYMMETRY ABOUT A POINT
Two points P and Q are said to be symmetric with respect to a point A if A is the midpoint of the line segment PQ [see Fig. 6-5(a)]. The point Q symmetric to the point P(x, y) with respect to the origin has coordinates ( x, y). [In Fig. 6-5(b), POR is congruent to QOS. Hence, OR = OS and RP = SQ.] Symmetry of a graph about a point is de ned in the expected manner, in particular, a graph G is said to be symmetric with respect to the origin if, whenever a point P lies on G , the point Q symmetric to P with respect to the origin also lies on G . The graph of an equation f (x, y) = 0 is symmetric with respect to the origin if and only if f (x, y) = 0 implies f ( x, y) = 0.
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