A parabola is one of the conic sections, formed when a plane cuts a cone parallel to one of the cone's sides. It is also the set of all points that are the same distance from a fixed point called the focus and a fixed line called the directrix. This definition makes the curve more than just a graph from algebra because it connects geometry, distance, and equations.
Parabolas matter because they model projectiles, satellite dishes, headlights, bridges, and many optimization problems.
Key Facts
- Focus-directrix definition: for every point P on a parabola, distance(P, focus) = distance(P, directrix).
- Standard vertical form: (x - h)^2 = 4p(y - k), with vertex (h, k) and focus (h, k + p).
- Standard horizontal form: (y - k)^2 = 4p(x - h), with vertex (h, k) and focus (h + p, k).
- For y = ax^2, the focal length is p = 1/(4a), so the focus is (0, p) and the directrix is y = -p.
- The axis of symmetry passes through the vertex and focus and is perpendicular to the directrix.
- Reflective property: a ray parallel to the axis of symmetry reflects through the focus, and a ray from the focus reflects parallel to the axis.
Vocabulary
- Parabola
- A parabola is the set of all points in a plane that are equally distant from a fixed focus and a fixed directrix.
- Focus
- The focus is the fixed point used in the geometric definition of a parabola.
- Directrix
- The directrix is the fixed line used in the geometric definition of a parabola.
- Vertex
- The vertex is the point where the parabola turns and lies halfway between the focus and directrix.
- Axis of symmetry
- The axis of symmetry is the line through the vertex and focus that divides the parabola into two mirror-image halves.
Common Mistakes to Avoid
- Confusing the focus with the vertex. The vertex is halfway between the focus and directrix, while the focus is inside the opening of the parabola.
- Using 2p instead of 4p in the standard equation. The correct conic form is (x - h)^2 = 4p(y - k) or (y - k)^2 = 4p(x - h).
- Forgetting that the sign of p controls direction. If p is positive the parabola opens toward the positive axis direction, and if p is negative it opens toward the negative axis direction.
- Treating the directrix as parallel to the axis of symmetry. The directrix is perpendicular to the axis of symmetry, not parallel to it.
Practice Questions
- 1 A parabola has vertex (0, 0) and focus (0, 3). Write its standard equation and the equation of its directrix.
- 2 Find the focus and directrix of the parabola (x - 2)^2 = 12(y + 1).
- 3 Explain why a parabolic satellite dish sends incoming rays that are parallel to its axis toward the receiver placed at the focus.