214 CHAPTER 2 Graphs and Functions (b) The graph with center 10, 02 and radius 3 is shown in Figure 15. S Now Try Exercises 11(b) and 17(b). EXAMPLE 2 Graphing Circles Graph each circle discussed in Example 1. (a) 1x + 322 + 1y - 422 = 36 (b) x2 + y2 = 9 SOLUTION (a) Writing the given equation in center-radius form 3x - 1-3242 + 1y - 422 = 62 gives 1-3, 42 as the center and 6 as the radius. See Figure 14. 6 (–3, 4) (x + 3)2 + (y – 4)2 = 36 (x, y) –9 –2 3 10 x y Figure 14 (0, 0) x2 + y2 = 9 (x, y) –3 3 3 –3 3 x y Figure 15 The circles graphed in Figures 14 and 15 of Example 2 can be generated on a graphing calculator by first solving for y and then entering two expressions for y1 and y2. See Figures 16 and 17. In both cases, the graph of y1 yields the top half of the circle, and that of y2 yields the bottom half. It is necessary to use a square viewing window to avoid distortion when graphing circles. y1 = 4+!36−(x+3)2 −8.4 −21.8 16.4 15.8 y2 = 4−!36−(x+3)2 Figure 16 The graph of this circle has equation 1x + 322 + 1y - 422 = 36. y1 = !9−x2 −4.1 4.1 6.6 −6.6 y2 = −!9−x2 Figure 17 7 The graph of this circle has equation x2 + y2 = 9. General Form Consider the center-radius form of the equation of a circle, and rewrite it so that the binomials are expanded and the right side equals 0. 1x - h22 + 1y - k22 = r2 Center-radius form x2 - 2xh + h2 + y2 - 2yk + k2 - r2 = 0 x2 + y2 + 1-2h x + 1-2k2y + 1h2 + k2 - r2 = 0 Properties of real numbers Don’t forget the middle term when squaring each binomial. (1)1* (1)1* (1++)++1* D E F Square each binomial, and subtract r2. If r 70, then the graph of this equation is a circle with center 1h, k2 and radius r.
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