Conic Sections
GradesAlgebra II & TrigonometryConic SectionsParabolas — Standard Form

Parabolas — Standard Form

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Parabolas — Standard Form

Write the standard form of a parabola and identify vertex, focus, and directrix.

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Its standard form depends on the direction of opening.

Vertical Parabola

(x − h)² = 4p(y − k)Vertex: (h, k)Opens up if p > 0, down if p < 0Focus: (h, k+p); Directrix: y = k−p

Horizontal Parabola

(y − k)² = 4p(x − h)Vertex: (h, k)Opens right if p > 0, left if p < 0Focus: (h+p, k); Directrix: x = h−p
(x−h)2=4p(y−k)focus: (h,k+p),directrix: y=k−p(x - h)^2 = 4p(y - k) \qquad \text{focus: }(h, k+p),\quad \text{directrix: }y = k-p
✏️Identify Vertex, Focus, and Directrix
Parabola: (x − 2)² = 8(y + 1). Here h=2, k=−1, 4p=8 so p=2. Vertex: (2, −1). Focus: (2, −1+2) = (2, 1). Directrix: y = −1−2 = −3.
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Remember This!

The latus rectum (chord through the focus, perpendicular to axis of symmetry) has length |4p|. Useful for sketching accurate parabolas.

✏️ Try It!

For the parabola y² = 12x, what is the location of the focus?

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