GOZUNTA
Try it now
All courses Math II · G-GPE.2 42 of 73
Derive the equation of a parabola from a focus and directrix.

Derive separate parabola features from focus and directrix

Problem
For a parabola with focus (0, 2) and directrix y = −2, project the focus perpendicularly to the directrix and take the midpoint for the vertex. Report the axis, signed focal parameter p, opening direction, and equal-distance check.
Coordinate plane with focus (0,2) and horizontal directrix y=−2; vertex, axis, p-value, and parabola are not shown. Open full size
Your answer
Show answer choicesAnswer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
With a free account

Find a specific skill faster.

Search the curriculum instead of wandering through random problems until the right kind appears.

Create a free account
With paid access

Open the complete course.

A subscription unlocks every problem type and variant in your selected course, not only the starter set.

Compare plans

Hint

Identify whether the directrix is horizontal or vertical.

Project the focus perpendicularly to the directrix; the vertex is their midpoint and the axis joins them.

Solution walkthrough

01

Project the focus to the directrix

\[F=(0,2);~y=-2~->~Q=(0,-2)\]

The directrix is horizontal, so its perpendicular direction is vertical. Keeping the focus's x-coordinate and moving to y=-2 gives the perpendicular foot Q.

02

Take the midpoint for the vertex

\[V=((0+0)/2,(2+(-2))/2)=(0,0)\]

A parabola's vertex lies halfway between the focus and directrix along the perpendicular axis because it is equally distant from both.

03

Determine axis, p, and opening

\[\text{axis}:~x=0;~p=2-0=2;~\text{opening}:~\text{up}\]

The axis is the vertical line through F and V. Signed p is the directed vertex-to-focus displacement; it is positive because the focus is above the vertex, so the parabola opens upward toward the focus.

04

Check the defining equality

\[VF=|2-0|=2;~\text{dist}(V,y=-2)=|0-(-2)|=2\]

The vertex is two units from the focus and two perpendicular units from the directrix, confirming the midpoint construction and all reported features.

+

Another way

  1. Average the focus y-coordinate 2 and the directrix level -2 to get the vertex level 0, then use the shared x-coordinate 0 for the axis.

!

Common mistake

The horizontal directrix y=-2 is not the axis. The axis must be perpendicular to it, so the axis is the vertical line x=0.

Answer features: focus (0,2) and directrix y=−2 have midpoint vertex (0,0), vertical axis x=0, p=2, and an upward-opening parabola.
The vertex is the midpoint between focus and directrix along the symmetry axis.