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Special Pythagorean Triple r2 = 16 a2 b2 c2 r=4 Center @ (0, 0) Center @ (-2, 1) 32 4 2 5 2 5 4 3 5 4 3 4 4 r2 = 25 r=5 34 4 5 5 4 5 (___, 0) Substitute in 0 for y and solve for x. x 22 0 12 16 x 22 1 16 x 22 15 x 2 15 x 2 15 2 2 15,0 15 ,0; 2 15 ,0 (0, ___) Substitute in 0 for x and solve for y. 0 22 y 12 16 2 4 y 1 16 y 12 12 y 1 12 y 1 2 3 0,1 2 3 0,1 2 3 ; 0,1 2 3 ( h, k ) x h2 y k 2 r 2 x 12 y 22 r 2 x 12 y 22 r 2 4 12 2 22 r 2 32 42 r 2 9 16 25 r 2 Substitute in h and k as 1 and -2. Substitute in x and y as 4 and 2. Solve for r2. x 12 y 22 25 Write the standard equation of a circle with the endpoints of the diameter as ( -2, 7) and ( -14, -9). ( x, y ) GENERAL FORM OF THE EQUATION OF A CIRCLE: Graph x y 4 x 6 y 9 0 Convert to Group x terms and y terms together and move the constant to the other side. 2 2 x h2 y k 2 r 2 by completing the square. x 2 4 x y 2 6 y 9 Complete the square of the x’s and y’s. (+2)2 y 2 6 y ___ (-3)2 9 ___ 4 ___ 9 x 2 4 x ___ x y x y x 22 y 32 4 Graph Center @ (-2, 3) 2 x 2 2 y 2 12 x 8 y 24 0 r2 = 4 r=2 Divide everything by 2. Why? x 2 y 2 6 x 4 y 12 0 (-3)2 y 2 4 y ___ (-2)2 12 ___ 9 ___ 4 x 2 6 x ___ x x 32 y 22 25 y Center @ (3, 2) x x 2 y 2 ax by c 0 y r2 = 25 r=5 x 0 y a x 2 y (-a) x x y a 0 y a 2 2 2 2 2 Square both sides to remove radical. x2 y a y a FOIL the binomials. x 2 y 2 2ay a 2 y 2 2ay a 2 2 Focus Cancel like terms on each side. a a 2a x 2 2ay 2ay Solve for x2. 4a a -a -a Directrix 2 x 2 4ay Graph the following equations. y 12 x 2 x=-3 The y is squared and the coefficient on the x is positive, the parabola opens to the right. 4a = 12, a = 3 and the vertex is at (0, 0). 6 V 3 F 6 Graph the following equations. x 2 16 y The x is squared and the coefficient on the y is negative, the parabola y=4 opens down. 4a = -16, a = -4 and the vertex is at (0, 0). V 4 8 F 8 Graph the following equations. y 2 8 x x=2 The y is squared and the coefficient on the x is negative, the parabola opens to the left. 4a = -8, a = -2 and the vertex is at (0, 0). 4 F V 2 4 Graph the following equations. x 2 2 8 y 1 The x is squared and the coefficient on the y is positive, the parabola opens up. 4a = 8, a = 2 and the vertex is at (2, -1). 4 F 2 y=-3 V 4 Graph the following equations. y 2 2 y 4 x 17 0 x=3 We need to complete the square of the y-terms to put in graphing form. Isolate the y-terms. 1 y 2 2 y (-1) ___2 4 x 17 ___ y 1 2 4 x 16 Factor out the 4 as the GCF. y 1 2 4x 4 The y is squared and the coefficient on the x is positive, the parabola opens to the right. 4a = 4, a = 1 and the vertex is at (4, 1). V 2 F 2 Graph the following equations. x2 6x 4 y 1 0 We need to complete the square of the x-terms to put in graphing form. Isolate the x-terms. 9 x 2 6 x (+3) ___2 4 y 1 ___ x 3 2 4y 8 Factor out the 4 as the GCF. x 3 2 4 y 2 2 F y=-3 The x is squared and the coefficient on the y is positive, the parabola opens up. 4a = 4, a = 1 and the vertex is at (-3, -2). V 2 Equation format is ... …plug in x & y to solve for 4a. Draw a rough graph. (2,3) Draw a rough graph. V V(-4, ?) 9 4a 2 9 4a 2 Equation format is ... …distance from V to F is 1, a = 1, 2 y k 4a x h and plug in the vertex values. F Draw a rough graph. F(-4, 4) 3 y = -2 32 4a2 y 2 4ax 9 2 y x 2 Equation format is ... x h 2 4a y k 4–3=1 V(-4, 1) y 22 41x 1 y 22 4x 1 …distance from F to the directrix line is 6, V is halfway, so a = 3. Plug in a and the vertex values. x 42 43 y 1 x 42 12 y 1