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The equation of a parabola is given.

y=−1/6x^2+7x−80

What is the equation of the directrix of the parabola?

Acceptfully  May 29, 2017
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y= (−1/6)x^2+7x−80      multiply both sides by -6

-6y  =    x^2   - 42x  +  480       subtract 480 from both sides

-6y  - 480   =  x^2  -  42x    take (1/2)  of 42  = 21.....square this  = 441  and add to both sides

-6y - 480 + 441  =   x^2  - 42x  +  441       simplify the left, factor the right

-6y - 39   =  (x  - 21)^2      factor the left side as

-6 (y  +  39/6)   =  ( x - 21)^2       (1)

Usiing the form

4p (y - k)  =  ( x  - h)      we  can   write (1)  as

4 (-3/2)(y - (-39/6) )   =  ( x  - 21)^2

The vertex  = ( x, k)  =  ( 21, -39/6)   and    p  = -3/2

And the directrix   is given by    y  = k - p  →   y  = -39/6 - (-3/2)  =  -39/6 + 3/2  = -39/6 + 9/6  =

-30/6  = - 5

See the graph, here :  https://www.desmos.com/calculator/hqmz4lhcvk

CPhill  May 29, 2017
edited by CPhill  May 29, 2017

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