Standard Form of Parabolas

Algebra 2 · Axiom Academy

LESSON Standard Form of Parabolas Read the vertex, focus, directrix, and which way it opens straight out of (x-h)^2 = 4p(y-k) . 1. Equal Distances Define the Parabola Fix a point F (the focus ) and a line (the directrix ). The parabola is every point P whose distance to the focus equals its perpendicular distance to the directrix. Neither is squared, neither is halved — they are simply equal . Watch the blue segment (to the focus) and the red drop (to the directrix): as P slides, both change length, but they match at every instant — that tie is the parabola. All points a fixed distance from one center. Constant sum of distances to two foci. Constant difference of distances to two foci. Equal distances to one focus and one directrix. When the axis of symmetry is vertical, the parabola opens up or down and the equation puts the x term squared : vertex (h,k) , focus , directrix y = k-p Here |p| is the distance from the vertex to the focus, which is exactly the distance from the vertex to the directrix — the vertex is their midpoint. Take the example (x-2)^2 = 8(y-1) . Match it to the form: h=2 , k=1 , and 4p = 8 , so p = 2 . y = k-p = -1 — down p=2 from the vertex. 3. Which Way It Opens: Four Orientations Two decisions set the orientation. Which variable is squared fixes the axis, and the sign of p fixes the direction along that axis. 4. Writing the Equation From a Focus and Directrix

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