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Parabolas Date: ____________

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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 Vertex Directrix

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**Parabolas with Vertex (0,0) that Open Up/Down**

y = ax2 +a → opens up, -a → opens down focus: (0, ) 1 4a vertex: (0, 0) directrix: y = – 1 4a c = 1 4a c is the distance from the vertex

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**Parabolas with Vertex (0,0) that Open Left/Right**

x = ay2 +a → opens right, -a → opens left focus: ( , 0) 1 4a vertex: (0, 0) directrix: x = – 1 4a c = 1 4a c is the distance from the vertex

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**Parabola Opens Up/Down**

y – k = a(x – h)2 +a → opens up, -a → opens down focus: (h, k ) 1 4a vertex: (h, k) directrix: y = k – 1 4a

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**Parabola Opens Left/Right**

x – h = a(y – k)2 +a → opens right, -a → opens left focus: (h , k) 1 4a vertex: (h, k) directrix: x = h – 1 4a

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**Write an equation of a parabola with a vertex at the origin and the given focus.**

focus at (6,0) opens to the right x = ay2 6 = 1 4a 4a · · 4a Solve for a 24a = 1 24 24 x = y² 24 1 a = 24 1

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**Write an equation of a parabola with a vertex at the origin and the given directrix.**

directrix x = 9 opens to the left x = ay2 9 = - 1 4a -4a · · -4a Solve for a -36a = 1 -36 -36 x =- y² 36 1 a = -36 1

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**Write an equation of a parabola with a vertex at the origin and the given focus.**

focus at (0,7) opens up y = ax2 7 = 1 4a 4a · · 4a Solve for a 28a = 1 28 28 y = x² 28 1 a = 28 1

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**x = ⅛y2 Graph opens right vertex: (0, 0) 1 = 2 4(⅛) focus: (2,0)**

directrix: x = –2

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**Graph y = - x2 opens down vertex: (0, 0) 1 = -3 4(- ) focus: (0,-3)**

12 y = - x2 opens down vertex: (0, 0) x y 1 = -3 1 4(- ) 12 focus: (0,-3) directrix: y = 3

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**y = ¼x2 Graph opens up vertex: (0, 0) 1 = 1 4( ) focus: (0,1)**

4( ) 4 focus: (0,1) directrix: y = -1

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**x = - y2 Graph opens left vertex: (0, 0) 1 = -4 4(-1/16) focus: (-4,0)**

directrix: x = 4

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**Graph y – 3 = − (x + 3)2 opens down vertex: (-3, 3) 1 = -4 4(- )**

16 opens down vertex: (-3, 3) x y 1 = -4 1 4(- ) 16 focus: (-3,-1) directrix: y = 7

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**Write the equation in standard form and graph the parabola**

Write the equation in standard form and graph the parabola. y2 – 8y + 8x + 8 = 0 y2 – 8y = -8x – 8 y2 – 8y + ___ = -8x – 8 16 +16 (y – 4)2 = -8x + 8 (y – 4)2 = -8(x – 1) -8 -⅛(y – 4)2 = x – 1 x – 1 = -⅛(y – 4)2

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**x – 1 = -⅛(y – 4)2 vertex: (1, 4) opens left 1 = -2 4(- )**

4(- ) 8 focus: (-1,4) directrix: x = 3

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**Write the equation in standard form and graph the parabola**

Write the equation in standard form and graph the parabola. x2 – 6x + 6y + 18 = 0 x2 – 6x = -6y – 18 x2 – 6x + ___ = -6y − 18 9 +9 (x – 3)2 = -6y – 9 (x – 3)2 = -6(y ) -6 - (x – 3)2 = y +1.5 1 6 y = - (x − 3)2 1 6

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**y + 1.5 = - (x − 3)2 vertex: (3,-1.5) opens down 1 = -1.5 4(- )**

6 vertex: (3,-1.5) x y opens down 1 = -1.5 1 4(- ) 6 focus: (3,-3) directrix: y = 0

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