Parabola

A parabola is a conic section, a curve in which every point is the same distance from a given line and from a given point that does not lie on that line.

What a parabola looks like

A parabola is defined by one point F and one line d. Every point X of the parabola is the same distance from the point F as from the line d. Have a look at the picture:

• The point F is called the focus of the parabola.
• The line d is called the directrix of the parabola.
• The line FD is called the axis of the parabola; it is perpendicular to the directrix and passes through the focus.
• The point V is called the vertex (turning point) of the parabola, and it lies at the midpoint of the segment FD.
• We call the length of the segment FD the parameter of the parabola. It is the distance from the focus to the directrix.

Notice in the picture that the distance of a point on the parabola from the line and from the focus really is always the same. For example, for the vertex V, the distance from the focus |VF| equals the distance from the line |VD|. The same holds for the point X marked in the picture: the distance |XF| equals the distance |XE|.

A parabola is the graph of a quadratic function.

The equation of a parabola

For a parabola we distinguish four different cases. First, how is the axis oriented: is it vertical (parallel to the y-axis), as in the first picture, or horizontal (parallel to the x-axis)? Second, which way does the parabola open: upwards or downwards, to the right or to the left. Let the parabola have vertex V with coordinates (m, n).

• First case: The parabola has its axis parallel to the y-axis and opens upwards. Its equation is:

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

  The focus has coordinates:

  $$F(m,n+\frac{p}{2})$$

• Second case: The parabola has its axis parallel to the y-axis and opens downwards. Its equation is:

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

  The focus has coordinates:

  $$F(m,n-\frac{p}{2})$$

• Third case: The parabola has its axis parallel to the x-axis and opens to the right. Its equation is:

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

  The focus has coordinates:

  $$F(m+\frac{p}{2},n)$$

• Fourth case: The parabola has its axis parallel to the x-axis and opens to the left. Its equation is:

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

  The focus has coordinates:

  $$F(m-\frac{p}{2},n)$$