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R.2 Circles The graph of an equation the graph of the function y = x2 is the set of all points (x, y) that satisfy y = x2 e.g. (2, 4), (-3, 9), etc. we can take any equation and graph it by placing all the points that satisfy it on a coordinate system Watch while we graph x2 + y2 = 16. It’s a circle! (but it’s not a function. Why?) If we graph (x - 2)2 + (y - 1)2 = 4, we get the graph: y (2, 1) x Notice that the center is at (2, 1), and the radius is 2. If we expand our equation, we get it in another form: x2 + y2 - 4x - 2y + 1 = 0 (same circle!) R.2-1 Circles: Four views and twelve skills General form: 2 2 Ax + Ay + Bx + Cy + D = 0 (A>0) complete the square expand Standard form: ( x - h )2 + ( y - k )2 = r2 read off std. form substitute in std form Parameters: Center: (h, k) Radius: r draw the picture read off picture Graph: R.2-2 Example: Find the standard form of a circle with center (5, -1) and radius 2. Example: Find the standard form of the circle with general form x2 + y2 - 4x - 2y + 1 = 0, and graph it. Example: Find the equation of the circle with the graph: y (-1, 3) x Example: Find a circle with center at (0, 5) and the point (2,5) on the circle. R.2-3