Math101 · Independent Ontario learning library
Math101Parabola Equation
A parabola is the set of points equidistant from a focus and a directrix. Standard form $(x-h)^2=4p(y-k)$ opens vertically, while $(y-k)^2=4p(x-h)$ opens horizontally.
Focus-directrix equations explain reflectors, satellite dishes, headlights, and projectile graph shapes while linking analytic and synthetic geometry.
Intuition and core definition
A parabola is the set of points equidistant from a focus and a directrix. Standard form $(x-h)^2=4p(y-k)$ opens vertically, while $(y-k)^2=4p(x-h)$ opens horizontally. The vertex is $(h,k)$ and signed $p$ gives focus direction and distance.
Notation, language, and conditions
For vertical form, focus is $(h,k+p)$ and directrix $y=k-p$; $p>0$ opens up and $p<0$ down. For horizontal form, focus is $(h+p,k)$ and directrix $x=h-p$. The coefficient is $4p$, not $p$.
Why this idea matters
A parabola is equidistant from a focus and directrix, and its standard equation reveals vertex, orientation, and focal parameter.
A dependable method
- Identify which variable is squared to determine axis orientation.
- Read vertex from shifted coordinates.
- Set the coefficient of the unsquared displacement equal to $4p$.
- Use signed $p$ to locate focus and directrix.
- Check that the vertex is midway between focus and directrix and test a point.
