Report

• CIRCLES General Equation: (x – h)2 + (y – k)2 = r2 where (h, k) is the center and r is the radius Graph the following equation: (x + 2)2 + y2 = 25 Note: Addition and Same Coefficients center (-2,0) • ELLIPSES General Equation: (x – h)2 + (y – k)2 = 1 a2 b2 where (h, k) is the center. The vertices are a units right and left from (h, k) and b units up and down from (h, k) Center: (h, k) Note: Addition and Different Coefficients b a Major axis: 2b Minor axis: 2a a b since 0 < a < b (ellipse is vertical) Graph the following equation: (x – 7)2 + (y + 2)2 = 1 25 16 Center: (7, -2) Since a2 = 25, then a = 5 Major axis Since b2 = 16, then b = 4 Minor axis Major: (7 + 5, -2) (12, -2) (7 5, -2) (2, -2) Minor: (7, -2 + 4) (7, 2) (7, -2 4) (7, -6) Graph the following equation: 4x2 + 25y2 = 400 400 400 400 x2 + y2 100 16 = 1 Center: (0, 0) Since a2 = 100, then a = 10 Major axis Since b2 = 16, then b = 4 Minor axis Major: (0 + 10, 0) (10, 0) (0 10, 0) (-10, 0) Minor: (0, 0 + 4) (0, 4) (0, 0 4) (0, -4) • HYPERBOLAS I. General Equation: (x – h)2 – (y – k)2 = 1 a2 b2 where (h, k) is the center. The vertices are (h + a, k) and (h – a, k) and opens left and right (transverse axis is horizontal). Note: Subtraction and (+) xcoefficient (h – a, k) (h + a, k) m of asymptotes = b/a • HYPERBOLAS II. General Equation: (y – k)2 – (x – h)2 = 1 b2 a2 where (h, k) is the center. The vertices are (h, k + b) and (h, k – b) and opens up and down (transverse axis is vertical). Note: Subtraction and (+) y coefficient m of asymptotes = b/a Graph the following equation: 25x2 – 4y2 = 400 400 400 400 x2 16 – y2 = 100 Center: (0, 0) Since a2 = 16, then a = 4 Since b2 = 100, then b = 10 Vertices: (0 + 4, 0) (4, 0) (0 4, 0) (-4, 0) 1 m = 10/4 = 5/2 Vertices (since it opens left and right) • Putting Equations in Standard Form by Completing the Square Graph the following equation: 4y2 + 9x2 – 24y – 72x + 144 = 0 Group the x terms and y terms 9x2 – 72x + 4y2 – 24y = - 144 Factor and leave blanks to complete the square: 9(x2 – 8x + 16 ) + 4(y2 – 6y + 9 ) = - 144 + 144 + 36 Complete the square and fill in blanks 9(x – 4)2 + 4(y – 3)2 = 36 Ellipse Graph the following equation: 4y2 + 9x2 – 24y – 72x + 144 = 0 9(x – 4)2 + 4(y – 3)2 = 36 36 36 36 (x – 4)2 + (y – 3)2 = 1 Center: (4, 3) 4 9 Since a2 = 4, then a = 2 Minor axis Since b2 = 9, then b = 3 Major axis Minor: (4 + 2, 3) (6, 3) (4 2, 3) (2, 3) Major: (4, 3 + 3) (4, 6) (4, 3 3) (4, 0) Graph the following equation: 12y2 – 4x2 + 72y + 16x + 44 = 0 12y2 + 72y – 4x2 + 16x = - 44 12(y2 + 6y + 9 ) – 4(x2 – 4x + 4 ) = - 44 + 108 + - 16 ) 12(y + 3)2 – 4(x – 2)2 = 48 (y + 3)2 – (x – 2)2 = 1 Hyperbola – opens up & down Center: (2, -3) 4 12 Since b2 = 4, then b = 2 Since a2 = 12, then a = 2 3 3.5 Vertices: (2, -3 + 2) (2, -1) (2, -3 – 2) (2, -5) Vertices (since it opens up & down)