Maura M.

asked • 05/07/17

What are the focus and tricks of the parabola represented by the equation : 8(y-6)=(x+2)^2

what is the focus and directrix of this equation 

David R.

The standard form for a parabola is (x-h)2=4p(y-k). 
 
(where h is the horizontal shift, k is vertical shift, and 4p is the vertical stretch)
 
In this form the focus is located at (h, k+p)
 
and the directrix is a y=k-p.
 
For the unit parabola, (x-0)2=1(y-0) the focus is at (0, 1/4) and directrix at y=-1/4
 
The focus is the center of the parabola, so when you vertically or horizontally shift the unit parabola, the focus will move by an equal distance and direction. Only vertical shifts will affect the directrix because it is written in the form of a horizontal line (y=k-p).
 
Stretching your parabola will affect both the focus and directrix equally, but in opposite directions. The effect on the focus and directrix is proportional to the coefficient before the y term. By doubling the unit parabola to (x-0)2=2(y-0), the focus moves from (0, 1/4) to (0, 1/2), and directrix to y=-1/2. Just remember that p is 1/4 the coefficient of the y term. p is also the distance from the vertex(h,k) or base of a parabola, to the focus and directrix, which is why the vertical terms are written, k+p and k-p respectively.
 
Given your equation, 8(y-6)=(x+2)2, the focus would be located at (-2, 8) and directrix at y=4
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05/07/17

1 Expert Answer

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Kemal G. answered • 05/07/17

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Hi Maura,
 
The parabola has a vertical axis of symmetry. A vertical axis has a focus at (h,k+p) and the equation (x-h)^2=4p(y-k).
Let's write 8(y-6) = (x+2)^2 in the form above.
4*2(y-6) = (x - (-2))^2     h = -2, k = 6, p = 2
the focus is at (-2, 8) and directrix is a straight line equal to y = k - p 
directrix is at y = 4
 
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