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avatar+578 

\(y=x^2+8x+18\)

whats the vertex, focus and directrix

 Mar 12, 2018
 #1
avatar+17330 
+1

In vertex form:

y = x^2 +8x  +16     -16 +18     ('completing the square)

y = (x+4)^2 +2            Vertex  h, k   = -4,2    Upward opening parabola (coefficient of (x+4)^2 is positive: +1  )

 

Re-arranging

 

y-2 = 4p (x+4)^2          4p = 1   p= 1/4   distance from vertex to directrix and focus

 

Directrix =   y= 2- 1/4 = 1.75

Focus = y= 2+1/4 = 2 1/4       x = -4        so   -4, 2 1/4

 

Graph:

 Mar 12, 2018
 #2
avatar+98123 
+1

y = x^2 + 8x + 18       complete the square on x

 

y = x^2 + 8x + 16 + 18 - 16

 

y = (x + 4)^2 + 2

 

(y - 2) = (x + 4)^2    (1)

 

In the form 

 

4p (y - 2)  = ( x + 4)^2   .....it's clear from (1) that  p =1/4

 

This parbola turns upward

 

The vertex  is (-4, 2)

 

The  focus is given by :    ( -4 , 2+ p)  ⇒  (-4 , 2 + 1/4) ⇒ ( -4, 9/2)

 

The directrix is given by :

y   =  ( 2 - 1/4)  =  7/4

 

 

cool cool cool

 Mar 12, 2018

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