*d*is the directrix of the parabola, F is its focus, and S is its vertex.

### Properties

- The axis of the parabola is an axis of symmetry. This axis contains the vertex and the focus. It is called the
**focal axis**of the parabola.

- We can say that a point P is on the parabola if and only if: d(P, F) = d(P,
*d*),

which means if: \(\textrm{m}\overline{\textrm{PH}}\) = \(\textrm{m}\overline{\textrm{PF}}\). - The vertex of a parabola is the point of intersection between the axis of the parabola and the parabola.
- All chords that pass through the focus of the parabola and that connect two points on this parabola are called
**focal chords**. Those that are perpendicular to the focal axis are called the latus rectum of the parabola. - All parabolas boil down, by translation or rotation, to a parabola in which the focus is on one of the axes of the Cartesian plane and in which the vertex is at the origin. The parabola of the equation \(y^{2} = 2px\) is called the basic parabola.

(1) *y*\(^{2}\) = 2*px* and *p* > 0 :

(2) *y*\(^{2}\) = 2*px* and *p* < 0 :

(3) *x*\(^{2}\) = 2*py* and *p* > 0 :

(4) *x*\(^{2}\) = 2*py* et *p* < 0 :

If, after a translation of the coordinate system, the coordinates of the vertex are (*h*, *k*), the equation of the parabola takes one of these standard forms:

(i) The equation is (*y* – *k*)\(^{2}\) = 2*p*(*x* – *h*), its vertex is S(*h*, *k*), is focus is F(*h *+ \(\frac{p}{2}\), *k*) and the equation of its directrix is *x* = *h* – \(\frac{p}{2}\).

It is open to the right if *p* > 0 and to the left if *p* < 0.

The equation of its line of symmetry is *y* = *k*.

(ii) The equation is (*x* – *h*)\(^{2}\) = 2*p*(*y* – *k*), its vertex is S(*h*, *k*), is focus is F(*h*, *k + *\(\frac{p}{2}\)) and the equation of its directrix is *y* = *k* – \(\frac{p}{2}\).

It opens toward the top if *p* > 0 and toward the base if *p* < 0.

The equation of its line of symmetry is *x* = *h*.