Build an exact graph specification for the parabola with focus \((3,~1)\) and directrix \(x~=~-1\). Derive the vertex, signed \(p\), axis, opening, standard equation, latus-rectum line, \(\text{both}~\text{endpoints}\), and total latus-rectum width.

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(y−1)²=8(x−1); endpoints (3,−1) and (3,3).

(x−1)²=8(y−1); endpoints (3,−3) and (3,5).

(y−1)²=16(x−1); endpoints (5,−7) and (5,9).

Vertex (1,1); p=2; axis y=1; opens right; (y−1)²=8(x−1); latus-rectum line x=3; endpoints (3,−3) and (3,5); total width 8.

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Curriculum context

Course: Math II
Standard: G-GPE.2
Category: Geometry
Domain: Expressing Geometric Properties with Equations
Objective: Derive the equation of a parabola from a focus and directrix.
Problem type: Build an exact parabola graph specification from focus and directrix