Derive the equation of a parabola from a focus and directrix.
Write the standard form of a parabola from its vertex, orientation, and p-value
Problem
Write the standard focus-directrix form of a parabola with vertex \((0,~0)\), signed focal parameter \(p~=~2\), and upward opening. Identify the vertical template and substitute \(4p\) with the correct sign.
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What this problem is really about
Orientation decides which coordinate expression is squared, while the vertex supplies both shifts. We’ll use the vertical template indicated by the opening, preserve the sign of p, substitute the parameters before simplifying the four-p coefficient, and recover the focus and directrix as a geometric check.
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