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Parabolas Date: ____________ Parabolas Parabola—Set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix. Focus Directrix Vertex Parabolas with Vertex (0,0) that Open Up/Down y = ax2 +a → opens up, -a → opens down 1 vertex: (0, 0) focus: (0, ) 4a 1 directrix: y = – 4a 1 c= 4a c is the distance from the vertex Parabolas with Vertex (0,0) that Open Left/Right x = ay2 +a → opens right, -a → opens left 1 focus: ( , 0) 4a 1 directrix: x = – 4a 1 c= 4a vertex: (0, 0) c is the distance from the vertex Parabola Opens Up/Down y – k = a(x – h)2 +a → opens up, -a → opens down vertex: (h, k) 1 focus: (h, k + ) 4a 1 directrix: y = k – 4a Parabola Opens Left/Right x – h = a(y – k)2 +a → opens right, -a → opens left 1 vertex: (h, k) focus: (h + , k) 4a 1 directrix: x = h – 4a Write an equation of a parabola with a vertex at the origin and the given focus. focus at (6,0) 1 6= · 4a 4a opens to the right Solve for a 24a = 1 24 24 1 a= 24 x = ay2 x= 1 24 y² Write an equation of a parabola with a vertex at the origin and the given directrix. directrix x = 9 x = ay2 opens to the left 1 9=· -4a Solve for a 4a -36a = 1 -36 -36 1 a= -36 x =- 1 36 y² Write an equation of a parabola with a vertex at the origin and the given focus. focus at (0,7) 1 7= · 4a 4a opens up y = ax2 Solve for a 28a = 1 28 28 1 a= 28 y= 1 28 x² 2 x = ⅛y Graph opens right vertex: (0, 0) 1 = 2 4(⅛) y focus: (2,0) directrix: x = –2 x 1 2 y = x Graph 12 opens down vertex: (0, 0) 1 = -3 1 4(- 12 ) y focus: (0,-3) directrix: y = 3 x 2 y = ¼x Graph opens up vertex: (0, 0) 1 =1 1 4( 4 ) y focus: (0,1) directrix: y = -1 x 1 2 x = y Graph 16 opens left vertex: (0, 0) 1 = -4 4(-1/16) y focus: (-4,0) directrix: x = 4 x 1 2 y – 3 = − (x + 3) Graph 16 opens down vertex: (-3, 3) 1 = -4 1 4(- 16 ) y focus: (-3,-1) directrix: y = 7 x Write the equation in standard form and graph the parabola. y2 – 8y + 8x + 8 = 0 y2 – 8y = -8x – 8 16 = -8x – 8 +16 y2 – 8y + ___ (y – 4)2 = -8x + 8 (y – 4)2 = -8(x – 1) -8 -⅛(y – 4)2 = x – 1 x – 1 = -⅛(y – 4)2 x – 1 = -⅛(y – 4)2 vertex: (1, 4) y opens left 1 = -2 1 4(- 8 ) focus: (-1,4) directrix: x = 3 x Write the equation in standard form and graph the parabola. x2 – 6x + 6y + 18 = 0 x2 – 6x = -6y – 18 9 = -6y − 18+9 x2 – 6x + ___ (x – 3)2 = -6y – 9 (x – 3)2 = -6(y + 1.5 ) -6 - 6 (x – 3)2 = y +1.5 1 y + 1.5 = - 6 (x − 3)2 1 y + 1.5 = - 6 (x − 3)2 1 vertex: (3,-1.5) y opens down 1 = -1.5 1 4(- 6 ) focus: (3,-3) directrix: y = 0 x