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The equation of a parabola is given.

 

y=−1/6x2+7x−80

 

What is the equation of the directrix of the parabola?

 

Enter your answer in the box.

i wrote the other one wrong sorry.

 May 23, 2018
 #1
avatar+250 
+1

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

 May 24, 2018
 #2
avatar+250 
+1

hope this helps! :)

 May 24, 2018
 #3
avatar+8205 
+1

First you need to complete the square.

 \(y=-\dfrac{1}{6}x^2+7x-80\)

\(y = \dfrac{-1}{6}(x^2-42x+480)\\ y = \dfrac{-1}{6}((x-21)^2+39)\\ y = \dfrac{-1}{6}(x-21)^2-\dfrac{13}{2}\\ (y-(-\dfrac{13}{2})) = -\dfrac{1}{6}(x-21)^2\\ -6(y-(-\dfrac{13}{2})) = (x-21)^2\\ 4(\dfrac{-3}{2})(y-(-\dfrac{13}{2})) = (x-21)^2\\\)

We can see from the equation that vertex of parabola = (21,-13/2) and focus is 3/2 units below the vertex.

Therefore the equation of directrix is y = (-13/2 + 3/2), i.e., y = -5.

 May 24, 2018
 #4
avatar+250 
+2

whoops, sorry forgot to put that in my answer thx MW

 May 24, 2018

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