Hi there! Thank you for the opportunity to help you with your question!

The first part for solving this is rewriting this equation in vertex form y = a(x-h)^2 + k

8y = -x^2 +6x + 7

y = -(x^2-6x-7)/8

Then we have to complete the square:

y=-[(x-3)^2 - 9 - 7]/8 = -[(x-3)^2]/8 + 2

Going back to standard form, this says a = -1/8, h = 3 and k =2.

1. When the equation is in vertex form, the focus is at the point (h, k + 1/(4a) ). Replacing h, k, and a we get that the focus is at (3, 2-1/(4/8) ) = (3, 0)

Step 2. The directrix of the parabola is given by the line y = k -1/(4a) = 2 + 2 = 4

y = 4 is the equation of the directrix

Step 3

The vertex is located at (h,k), or (3, 2)

The points which lie on the line through the focus and parallel to the directrix are the points that have y = 0

0 = -(x^2-6x-7)/8

(x-7)(x+1) = 0

x = 7, -1

So the points are (7,0), (-1,0)

Please let me know if you need any clarification. Always glad to help!

Oct 6th, 2015

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