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Solve each problem below:

 Feb 20, 2019
 #1
avatar+100570 
+3

a)   y = (1/8) ( x - 6)^2 + 4

 

We are working backwards, here

 

(y - 4) = (1/8) ( x - 6)^2

 

8(y - 4) = ( x - 6)^2           the vertex is ( 6, 4)

 

4p = 8   ⇒    p = 2

 

The parabola turns upward  so  the focus is  ( 6 , 4 + p)  =  ( 6 , 4 + 2) = (6, 6)

 

And the directrix is   y = 4 - p  =  4 - 2   = 2   ⇒  y = 2

 

 

cool cool cool

 Feb 20, 2019
 #2
avatar+100570 
+3

(b)   y = (-1/4) ( x + 2)^2 + 1

 

y - 1  =  -(1/4)(x + 2)^2

 

4 ( y - 1) = - ( x + 2)^2        the parabola turns downward

 

The vertex is ( -2, 1)

 

4p = 4      so   p = 1

 

The focus is ( -2, 1 - p) = (-2, 1 - 1 ) = (-2, 0)

 

The directrix is    y = 1 + p  =  1 + 1   = 2   ⇒    y = 2

 

cool cool cool

 Feb 20, 2019
 #3
avatar+100570 
+3

(c)   y = (1/16) ( x - 3)^2       easier!!!

 

16 y = (x - 3)^2

 

16 ( y - 0) = (x - 3)^2

 

The vertex is (3, 0)       and this turns upward

 

4p = 16

p = 4

 

The focus is  ( 3, 0 + p)  = ( 3, 0 + 4) = ( 3, 4)

 

The directrix is    y = 0 - p  = 0 - 4  =  -4  ⇒    y = -4

 

 

cool cool cool

 Feb 20, 2019
 #4
avatar+100570 
+3

(d)   y = 2 ( x + 1)^2 - 5             add 5 to both sides

 

y + 5 =  2 ( x + 1)^2             multiply both sides by 1/2

 

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

 

The vertex is ( -1, -5)       and this opens upward

 

4p = 1/2

p = 1/8

 

The focus is ( -1, -5 + 1/8) = (-1, -39/8 )

 

The directrix is y = -5 - p =  - 5 - 1/8 =   -41/8  ⇒  y = -41/8

 

 

cool  cool  cool

 Feb 20, 2019

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