Question 48599
<pre><font size = 4><b>Find an equation for the parabola with focus at (-8,-4) and vertex at (6,-4)
i do not know how to figure this one out.  will you please help me.  
i dont know what to do.  thank you a lot.
aloha!

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The equation of a parabola with vertex (h,k) is either

(x - h)² = 4p(y - k)

where the vertex is (h, k), 
|p| = distance from vertex to focus,
p positive if focus is above vertex (opens upward), and
p is negative if focus is below vertex (parabola opens downward).
focus (h, k+p), and 
directrix is the horizontal line whose equation is y = k-p

or

(y - k)² = 4p(x - h)

where the vertex is (h, k),
|p| = distance from vertex to focus,
p is positive if focus is right of vertex, (parabola opens to the right)
and p is negative if focus is left of vertex, (parabola opens to the left)
focus (h+p, k), and directrix is the vertical line whose equation is
x = h-p

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Since the vertex is (6, -4), h = 6 and k = -4
Since the vertex (6, -4) and focus (-8, -4) have the same y-coordinate,
it is of the second type.
The distance from vertex to focus is 14 units. Since the focus is left
of the vertex, p is negative, and so p = -14. (Parabola opens to the left)

So the equation is

(y - k)² = 4p(x - h) or

(y - (-4))² = 4(-14)(x - 6)

(y + 4)² = -56(x - 6)

If you were asked for the equation of the directrix,
it would be x = k-p or x = -4-(-14) = -4+14 or x = 10


{{{ graph( 300, 300, -60, 40, -50, 50, sqrt(-56(x-6))-4, -sqrt(-56(x-6))-4) }}}
Edwin<pre>