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Unformatted text preview: SECTION 21 83 CHAPTER 2 Section 21 1. The set of all points for which the x coordinate is 0 is the y axis. 3. The set of all points for which the x and y coordinates are negative is quadrant III. 5. The set of all points for which the x coordinate is positive and the y coordinate is negative is quadrant IV. 7. The set of all points for which x is positive, excluding those points for which y = 0 (positive x axis), includes quadrants I and IV. 9. The set of all points for which xy < 0 includes those points for which the x coordinate is positive and the y coordinate is negative (quadrant IV) and also those points for which the x coordinate is negative and the y coordinate is positive (quadrant II). 11. 13. 15. Point A has coordinates (2, 4). Its reflection through the y axis is A’ (–2, 4). Point B has coordinates (3, –1). Its reflection through the y axis is B’ (–3, –1). Point C has coordinates (–4, 0). Its reflection through the y axis is C’ (4, 0). Point D has coordinates (–5, 2). Its reflection through the y axis is D’ (5, 2). 17. Point A has coordinates (–3, –3). Its reflection through the origin is A’ (3, 3). Point B has coordinates (0, 4). Its reflection through the origin is B’ (0, –4). Point C has coordinates (–3, 2). Its reflection through the origin is C ’(3, –2). Point D has coordinates (5, –1). Its reflection through the origin is D’ (–5, 1). 19. y = 2 x – 4 Test y axis Test x axis Test origin Replace x with – x : Replace y with – y : Replace x with – x and y with – y : y = 2(– x ) – 4 – y = 2 x – 4 – y = 2(– x ) – 4 y = –2 x – 4 y = –2 x + 4 y = 2 x + 4 The graph has none of these symmetries. x y4 2 4 4 84 CHAPTER 2 GRAPHS AND FUNCTIONS 21. y = 1 2 x Test y axis Test x axis Test origin Replace x with – x : Replace y with – y : Replace x with – x and y with – y : y = 1 2 (– x ) – y = 1 2 x – y = 1 2 (– x ) y = – 1 2 x y = – 1 2 x y = 1 2 x x y 4 2 The graph has symmetry with respect to the origin. We reflect the portion of the graph in quadrant I through the origin, using the origin symmetry. 23.  y  = x Test y axis Test x axis Test origin Replace x with – x : Replace y with – y : Replace x with – x and y with – y :  y  = – x – y  = x – y  = – x  y  = x  y  = – x x y 4 4 The graph has symmetry with respect to the x axis. We reflect the portion of the graph where y ¡ 0 through the x axis, using the x axis symmetry. 25.  x  =  y  Test y axis Test x axis Origin symmetry Replace x with – x : Replace y with – y : follows automatically. – x  =  y   x  = – y  x =  y   x  =  y  x y 4 4 The graph has all three symmetries....
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 Spring '08
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