Parabolas

Parabola Focus and Directrix

Find the focus and directrix of an axis-aligned parabola from vertex form.

parabola-focus-directrix For y = a(x-h)^2 + k, p = 1/(4a), the focus is (h, k+p), and the directrix is y = k-p.

Example

Use p to locate the focus and directrix.

highlighted = computed this step

Step 1 — Vertex form

Read the vertex form.

y=14(x−2)2−3y= \hlmath{\frac{1}{4}} (x- \hl{2} )^{2} - \hl{3}

Step 2 — Find p

Compute p from the coefficient.

p=1/(4⋅14)=1p= 1 /( 4 \cdot \frac{1}{4} )= \hl{1}

Step 3 — Focus

Move p units to the focus.

F=(2,-2)F=( \hl{2} , \hl{-2} )

Step 4 — Directrix

Move p units to the directrix.

y=-4y= \hl{-4}

Step 5 — Plot focus and directrix

Plot the parabola, focus, and directrix.

Coordinate plot\text{Coordinate plot}
Parabola focus-directrix plotParabola with focus F(2, -2) and directrix y=-4.directrixVF

Step 6 — Focus and directrix

State the focus and directrix.

F=(2,-2)y=-4F=( \hl{2} , \hl{-2} ) \quad y= \hl{-4}