Similar to an ellipse, a hyperbola has two foci and is defined as all the points whose distance from the foci is fixed. A hyperbola can be thought of as a pair of parabolas that are symmetric across the directrix. Just like an ellipse the midpoint of the line segment connecting the foci is called the center is will be used to define the equation of the hyperbola, in a similar way to it did for the ellipse.
The equation for a hyperbola centered at the origin, (0, 0), with foci at (-c, 0) and (c, 0) and vertices at (-a, 0) and (a, 0) is
The "parabolas" open left and right since the x has a positive coefficient. For a pair of "parabolas" that open up and down make the coefficient of the y positive and the coefficient of the x negative.
Just like all of the other conics, if the hyperbola is to be centered at (h, k) replace x with (x - h) and y with (y - k).