Parabola

A parabola is a conic, which is a curve that has a constant distance from a given line and from a given point that is not on that line.

What a parabola looks like

A parabola is defined by one point F and one line d. All points X of this parabola are then said to have the same distance from this point F and from the line d. See the figure:

• The point F is called the focus of the parabola.
• The line d is called the control line of the parabola.
• The line FD is called the axis of the parabola, is perpendicular to the control line, and passes through the focus.
• The point V is called the vertex of the parabola and is located at the centre of the line FD.
• The length of the line segment FD is called the parabola parameter. It is the distance of the focal point from the control line.

Also note in the figure that it is indeed true that the distance of the parabola point from the line and from the focus is always the same. For example, for the vertex V, the distance from the focus |VF| is the same as the distance from the straight line |VD|. Similarly, this is true for the point X, which is marked in the figure. The distance |XF| is the same as the distance |XE|.

A parabola is the graph of a quadratic function.

The equation of the parabola

There are four different cases for the parabola. How the axis of the parabola is oriented, i.e., whether the axis is vertical (parallel to the axis y), as in the first figure, or whether the axis is horizontal (parallel to the axis x). Then we distinguish the case where the parabola is bounded from below or above and "from the left" or "from the right". Let the parabola have a vertex V with coordinates [m, n].

• First case: The parabola has an axis parallel to y and is bounded from below. The following equation holds for it:

  $$(x-m)^2=2p(y-n)$$

  The focus has coordinates:

  $$F\left[m, n+\frac{p}{2}\right]$$

• Second case: The parabola has an axis parallel to the axis y and is bounded from above. The following equation holds for it:

  $$(x-m)^2=-2p(y-n)$$

  The focus has coordinates:

  $$F\left[m,n-\frac{p}{2}\right]$$

• Third case: The parabola has an axis parallel to the axis x and is bounded "from the left". For it, the following equation holds:

  $$(y-n)^2=2p(x-m)$$

  The focus has coordinates:

  $$F\left[m+\frac{p}{2},n\right]$$

• Fourth case: The parabola has an axis parallel to the axis x and is bounded "from the right". For it, the following equation holds:

  $$(y-n)^2=-2p(x-m)$$

  The focus has coordinates:

  $$F\left[m-\frac{p}{2},n\right]$$