Find the focus, directrix, axis of symmetry, and latus rectum of any parabola from its equation. Converts vertex form to conic standard form x² = 4py.
x² = 4py (vertical) or y² = 4px (horizontal)
A parabola is the set of all points equidistant from the focus and the directrix. For any point P on the parabola, the distance from P to the focus equals the distance from P to the directrix.
In standard conic form x² = 4py (vertex at origin), the focus is at (0, p) and the directrix is y = −p. The vertex is always the midpoint between focus and directrix — exactly p units from each.
When the vertex is at (h, k), the full standard form is (x−h)² = 4p(y−k), which is exactly what you get by expanding the vertex form y = a(x−h)² + k and noting that a = 1/(4p).
p = 1/(4a) → focus = vertex ± p → directrix = vertex ∓ p
p is the focal distance — the distance from the vertex to the focus, and equally, from the vertex to the directrix.
Large |p| (small |a|) → wider, flatter parabola. Small |p| (large |a|) → narrower, steeper parabola. The latus rectum — the chord through the focus perpendicular to the axis — has length |4p| = |1/a|, which gives an easy way to sketch the width.
Sign of a (and p): a > 0 → parabola opens up or right, focus is above/right of vertex. a < 0 → opens down or left, focus is below/left of vertex.
One-on-one Algebra 2 tutoring connects the geometric definition to the algebra — we work through focus, directrix, and all four conics until the pattern clicks for your actual tests.