The Standard Equation of a Parabola
Simple Explanation
A parabola with its vertex at the origin, opening upward or downward, has the standard equation x² = 4py — where p is the distance from the vertex to the focus (0,p), and the directrix is the line y=−p.
Why Do We Need It?
This is the direct algebraic translation of the focus-directrix definition, giving a clean equation to work with instead of a distance condition.
Formula
The Standard Equation of a Parabola
x² = 4py, with focus (0,p) and directrix y=−p
A parabola with vertex at the origin, opening upward (p>0) or downward (p<0), described directly by this equation.
- p
- — the distance from the vertex to the focus (and also to the directrix)
When to use it: Whenever a parabola's vertex is at the origin and its focus/directrix need to be found from the equation, or vice versa.
Worked Example
Find a parabola's focus and directrix from its equation
Find the focus and directrix of the parabola x²=8y.
Why Does This Work?
Setting the focus-directrix distance condition equal for a general point (x,y) — √(x²+(y−p)²) = |y+p| — and squaring both sides to clear the square root, then simplifying, reduces directly to x²=4py.
Real-Life Example
Designing a parabolic solar cooker
A solar cooker uses a parabolic dish to focus sunlight onto a single cooking point.
The standard equation lets engineers calculate exactly where to place the cooking surface (the focus), given the dish's shape.
Practice
For the parabola x²=12y, find p.
MediumCommon mistake
Forgetting to divide by 4 when solving for p from the y-coefficient — treating the coefficient itself as p, rather than 4p.
Quick Review
- x² = 4py, vertex at the origin.
- Focus is (0,p); directrix is y=−p.
- Derived directly from the focus-directrix distance condition.