Derive the equation of a parabola from a focus and directrix.
Convert a parabola equation to standard focus-directrix form
Problem
Convert \(y~=~(1/8)x^{2}\) to standard focus-directrix form by isolating the squared coordinate. Identify the resulting coefficient as \(4p\) and preserve the vertex shifts.
Big Picture
What this problem is really about
Converting to focus-directrix form is an equivalent rearrangement, so both sides must be scaled together. We’ll clear the fractional coefficient, put the squared coordinate first, preserve the zero vertex shifts, interpret the resulting factor as four times p, and reverse the algebra to confirm the original equation returns.
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