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Find the focus and directrix for each parabola below:

 Feb 27, 2019
 #1
avatar+98091 
+2

a)  

y = -(1/4)(x - 2)^2 + 5

 

y - 5 = (-1/4)(x - 2)^2       multiply through by 4

 

4(y - 5) = -(x - 2)^2

 

The vertex  is ( 2, 5)

 

4p = 4

p = 1

 

This parabola turns downward

 

The focus is  (2, 5 - p)  = (2, 5 - 1)   =  (2, 4)

 

The directrix is  y = 5 + p =  5 + 1  = 6   ⇒   y = 6

 

 

cool cool cool

 Feb 27, 2019
edited by CPhill  Feb 27, 2019
 #2
avatar+98091 
+2

8y = (x + 1)^2 - 32

 

8y + 32 = (x + 1)^2

 

8(y + 4) = (x + 1)^2

 

The vertex is (-1, -4)

 

The parabola turns upward

 

8 = 4p

p = 2

 

The focus is  (-1, -4 + p)  =  (-1, -4 + 2)   = (-1, -2)

 

The directrix is  y = -4 - p = -4 - 2  =  -6  ⇒  y = -6

 

 

cool cool cool

 Feb 27, 2019

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