In Exercises 27–32, find the standard form of the equation of each hyperbola.

The standard form of a hyperbola centered at the origin is given by the equations (x^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas and (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas. Here, 'a' represents the distance from the center to the vertices along the transverse axis, while 'b' represents the distance to the asymptotes. Understanding this form is crucial for identifying the hyperbola's orientation and dimensions.

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin, the equations of the asymptotes can be derived from the standard form. They are given by y = (b/a)x and y = -(b/a)x for horizontal hyperbolas, and x = (a/b)y and x = -(a/b)y for vertical hyperbolas. These lines help in sketching the hyperbola and understanding its behavior at infinity.

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis, located at (±a, 0) for horizontal hyperbolas and (0, ±a) for vertical hyperbolas. The foci, which are points that help define the hyperbola's shape, are located at (±c, 0) for horizontal and (0, ±c) for vertical hyperbolas, where c = √(a^2 + b^2). Knowing the positions of the vertices and foci is essential for graphing the hyperbola accurately.