Problem

Build an exact graph specification for the parabola with focus \((0,~2)\) and directrix \(y~=~-2\). Derive the vertex, signed \(p\), axis, opening, standard equation, latus-rectum line, \(\text{both}~\text{endpoints}\), and total latus-rectum width.

Unworked coordinate plane showing focus (0,2) and horizontal directrix y=−2; the vertex, p-value, parabola, equation, and latus rectum are not shown.

Big Picture

What this problem is really about

An exact graph specification needs reproducible landmarks, not just a curved sketch. We’ll derive the vertex, signed p, axis, opening, and equation from the focus-directrix geometry, place the latus rectum through the focus, locate its symmetric endpoints using p, and verify those anchors against both distance and equation conditions.

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Preview only

x²=8y; endpoints (−2,2) and (2,2).

x²=16y; endpoints (−8,4) and (8,4).

x²=8y; endpoints (−4,2) and (4,2).

x²=−8y; endpoints (−4,−2) and (4,−2).

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Build an exact graph specification for the parabola with focus \((0,~2)\) and directrix \(y~=~-2\). Derive the vertex, signed \(p\), axis, opening, standard equation, latus-rectum line, \(\text{both}~\text{endpoints}\), and total latus-rectum width.

What this problem is really about

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