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Parabolas

Description And History of This Conic: A parabola is a figure composed of all points in a plane that are the same distance from a fixed point, called the focus, as they are from a fixed line, called the directrix. In geometric terms, a parabola is a tilted plane that intersects only one nappe of a cone and is parallel to only one generator. For a clear picture of a parabola, refer to the diagram on the homepage.

The Basics

Here is a diagram of basic parabola terms to help you better understand its description. The golden line, called the axis of symmetry, is a line that goes through the focus and is perpendicular to the directrix. The point where the parabola intersects the axis of symmetry is called the vertex. As you see in the diagram, though not to scale, the directrix is "p" distance from the vertex, while the focus is also "p" distance from the vertex. This means the directrix and focus are "2p" distance apart. Another interesting characteristic of a parabola is its latus rectum. The latus rectum is a line segment that goes through the focus, but is perpendicular to the axis of symmetry. The two points at which it intersects the parabola are the points that define the latus rectum.

The Equations

While the basic equation for all conics is "Ax^2 + Cy^2 + Dx + Ey + F = 0", a parabola has only one squared variable, thus making a parabola defined as AC=0. Since only one variable can be squared, we are left with two situations: a parabola that opens along the y-axis or x-axis. Let's start with positive parabolas! The basic equation for a parabola that opens upwards along the y-axis is y = 1/4p (x - h)^2 + k, as represented by parabola #1. This is where the variable "p" makes its appearance, and in a parabola equation, 1/4p= the distance from the vertex to either the directrix or focus. For a parabola such as #2 that opens rightwards along the x-axis, just switch x and y and their respective vertex variables, h and k : x = 1/4p(y - k)^2 + h. In order for your parabola to be negative, the only step you must do to either equation is make "a" negative: For parabola #3, your equation is y = - 1/4p(x - h)^2 + k, and for #4, the equation is x = - 1/4p (y - k)^2 + h. It's that easy!!!

Special Cases: Parabola

Conics can express certain special characteristics given the right circumstances. One of these cases is when there is rotation of axes. When there is rotation of axes, the origin of the conic remains the same, while the x and y axes are rotated through a given angle to a new position, with new variables x' and y'. In this case, the normal equation for a conic changes from "Ax^2 + Cy^2 + Dx + Ey + F = 0" to "Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0", thus adding "Bxy". What you must know is how to identify which conic you are exploring by just looking at this new equation. To do this we must use the discriminant, "B^2 - 4AC", and the parabola is defined as "B^2 - 4AC = 0." *************************************************************************************** Another property we must explore is that of eccentricity. Eccentricity is defined as the ratio of the distance (d) from the focus (F) to a given point (P) in a conic to the distance (d) from the directrix (D) to the point (P). Look to the diagram on the right to see this definition written as an equation. Eccentricity is also defined in simpler terms as "c/a." In the case of this conic, a parabola is defined as "eccentricity = 1" (e = 1). WOW, we finished parabolas! Take a minute to let all this info soak in, and feel free to read over this information as many times until you are comfortable. Once you're ready, hit the "Back" button in your browser and enjoy the parabola question to test how well you're doing!