Parabola (Focus-Directrix Definition) Formula
The Formula
Horizontal axis: (y - k)^2 = 4p(x - h) with vertex (h, k), focus (h+p, k), directrix x = h - p.
When to use: Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.
Quick Example
If p = 2: focus at (0, 2), directrix at y = -2, equation y = \frac{1}{8}x^2.
Notation
What This Formula Means
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.
Formal View
Worked Examples
Example 1
easySolution
- 1 Rewrite in standard form: x^2 = 8y. This matches x^2 = 4py where 4p = 8, so p = 2.
- 2 The parabola opens upward. The focus is at (0, p) = (0, 2).
- 3 The directrix is y = -p = -2.
Answer
Example 2
mediumExample 3
mediumCommon Mistakes
- Confusing the 4p form with the y = ax^2 form: if y = ax^2, then a = \frac{1}{4p}, so p = \frac{1}{4a}. A small a means a wide parabola with a far-away focus.
- Putting the directrix on the same side as the focus: the directrix is ALWAYS on the opposite side of the vertex from the focus.
- Forgetting that p can be negative: if p < 0, the parabola opens downward (or leftward), and the focus is below (or to the left of) the vertex.
Why This Formula Matters
The reflective property of parabolas (all rays from the focus reflect parallel to the axis) is used in satellite dishes, car headlights, telescopes, and solar concentrators. Understanding the focus-directrix form connects the algebraic y = ax^2 to geometry.
Frequently Asked Questions
What is the Parabola (Focus-Directrix Definition) formula?
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
How do you use the Parabola (Focus-Directrix Definition) formula?
Every point on a parabola is exactly the same distance from the focus as it is from the directrix line. This geometric property is why satellite dishes and flashlight reflectors are parabolic—signals from the focus reflect off the curve in parallel lines.
What do the symbols mean in the Parabola (Focus-Directrix Definition) formula?
p = directed distance from vertex to focus (positive means focus is above/right of vertex; negative means below/left).
Why is the Parabola (Focus-Directrix Definition) formula important in Math?
The reflective property of parabolas (all rays from the focus reflect parallel to the axis) is used in satellite dishes, car headlights, telescopes, and solar concentrators. Understanding the focus-directrix form connects the algebraic y = ax^2 to geometry.
What do students get wrong about Parabola (Focus-Directrix Definition)?
The vertex is halfway between the focus and directrix. If you know any two of {vertex, focus, directrix}, you can find the third.
What should I learn before the Parabola (Focus-Directrix Definition) formula?
Before studying the Parabola (Focus-Directrix Definition) formula, you should understand: quadratic functions.