Conic Sections: Parabolas

Graphing a Parabola with a Vertex (h, k)

If we move the vertex of a parabola away from the origin then we can say the vertex is defined as (h, k). Again, the equation of a parabola with a  vertex of (h, k) falls into two categories.

Horizontal Directrix

y minus k equals 1 divided by 4 p times the quantity x minus h squared

p > 0: parabola opens upward
p < 0: parabola opens downward
focus: (h, k + p)
directrix: y = k – p
axis of symmetry: x = h

Graph of parabola with a vertex at h comma k, an axis of symmetry x is equal to h, a focus at h comma k plus p and a directrix y is equal to k minus p.

Vertical Directrix

x minus h equals 1 divided by 4 p times the quantity y minus k squared

p > 0: parabola opens right
p < 0: parabola opens left
focus: (h + p, k)
directrix: x = h – p
axis of symmetry: y = k

Graph of parabola with a vertex at h comma k, an axis of symmetry y is equal to k, a focus at h plus p comma k and a directrix x is equal to h minus p.