Parabola

1. Definition of a Parabola
   A parabola is the set of all points on a plane equidistant from a fixed point (called the focus) and a fixed line (called the directrix) that does not pass through the focus. The line that passes through the focus and is perpendicular to the directrix is called the axis of the parabola, and the point where the parabola intersects the axis is called the vertex of the parabola.
   
   
   
2. Let be the foot of the perpendicular from the point on the parabola whose focus is to the directrix ,
   
3. Equation of a Parabola
1. Equation of a Parabola with the Focus on the -axis
   When the focus is at and the directrix is the line , the equation of the parabola is:
   
   
   
   
   
   
   
   
   
2. Equation of a Parabola with the Focus on the -axis
   When the focus is at and the directrix is the line , the equation of the parabola is:
   
   
   
   
   
   


4. Translation of a Parabola
5. i. For the equation , if the parabola is translated by units horizontally and units vertically, the new equation becomes:

   Focus
   Directrix

6. ii. For the equation , if the parabola is translated by units horizontally and units vertically, the new equation becomes:

   Focus
   Directrix