Answer:
To convert the given conic equation from standard form to graphing form (also known as the vertex form), we need to complete the square for both x and y terms.
Given equation: x^2 + 2x + u^2 - 8y + 1 = 0
First, let's complete the square for the x terms:
x^2 + 2x can be rewritten as (x + 1)^2 - 1, since (x + 1)^2 expands to x^2 + 2x + 1.
Now we need to complete the square for the y terms:
Since we only have -8y, let's rewrite it as -8(y - 1), so that we have a perfect square in y.
Substitute the transformed x and y terms back into the equation:
(x + 1)^2 - 1 + u^2 - 8(y - 1) + 1 = 0
Simplify the equation:
(x + 1)^2 + u^2 - 8(y - 1) = 0
Now, we have the conic equation in graphing (vertex) form:
(x + 1)^2 + u^2 - 8(y - 1) = 0