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Types of Parabola

Parabola: Axis Parallel to y-Axis
Parabola: Axis Parallel to y-Axis

By translating the parabola x 2 = 2py its vertex is moved from the origin to the point A (x 0, y 0) so that its equation: transforms to (x-x 0) 2 = 2p(y-y 0). The axis of symmetry of this parabola is parallel to the y-axis.

source: nabla.hr
Parabola: Definition, p: Semi Latus Rectum
Parabola: Definition, p: Semi Latus Rectum

According to the definition of a parabola as a conic section, the boundary of this pink cross-section, EPD, is a parabola. A horizontal cross-section of the cone passes through the vertex, P, of the parabola.

Parabola: General Case
Parabola: General Case

I'm working on understanding how to find the equation of a parabola using directrix and focus. I understand the formulas that use the pythagorean theorem/distance formula to equate the distance from a point of the parabola $P = (x, y)$ and the directrix to the distance of said point to the focus.

image: elsenaju.eu
Parabolas
Parabolas

We also get a parabola when we slice through a cone (the slice must be parallel to the side of the cone). So the parabola is a conic section (a section of a cone).

Parabolic Compass Designed by Leonardo da Vinci
Parabolic Compass Designed by Leonardo da Vinci

Design for a parabolic compass - by Leonardo da Vinci Courtesy of LeonardoDaVinci.net: MOST POPULAR PAINTINGS

Part of a Parabola
Part of a Parabola

It has following important parts: Axis of symmetry -Axis of symmetry of a parabola is the line about which it is symmetric. It is the line at which vertex and focus is located. Axis of symmetry is either X or Y axis or it is parallel to either X or Y axis. Vertex - Vertex is the point where parabola and its axis of symmetry intersect each other.

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