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# math help

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May 6, 2019

#1
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1.  y  = x^2 - 8x + 12

y - 12  =  x^2 - 8x           complete the square on x

y - 12  + 16  =   x^2 - 8x + 16      simplify

(y + 4)  = (x - 4)^2

We have the form  (4p)(y - k) = (x - h)^2

The vertex  =   (h, k) =   ( 4, - 4)

Axis of symmetry    x  = 4

p  = (1/4)

The focus  =  ( 4 , - 4 + p)  =  ( 4, - 4 + 1/4)  =  ( 4, -15/4)

The directrix  =  y = (k - p)  =   - 4 - 1/4   =   -17/4

Here's a graph : https://www.desmos.com/calculator/am3svgafgw   May 6, 2019
#2
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x = 2y^2 -8y + 3

x - 3  = 2(y^2 - 4y)         complete the square on y

x - 3 + 8  = 2(y^2 - 4y + 4)       simplify

(x + 5)  = 2(y - 2)^2

(1/2)(x + 5)  = (y - 2)^2

Vertex  =  ( -5, 2)

We have the form   (4p)(x + 5) = (y - 2)^2

4p = 1/2

p = 1/8

This parabola opens to the right

The axis of symmetry is y = 2

The focus  =  ( -5 + p , 2)  = (-5 + 1/8, 2)  = ( -39/8, 2)

The directrix is    x = -5 - p  =  -5 - 1/8  =  -41/8   Here's the graph :

May 6, 2019