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 (x²/a²) - (y²/b²) = 1 or (y²/b²) - (x²/a²) = 1. Here, 'a' represents the distance from the center to the vertices along the x-axis or y-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 in standard form, the equations of the asymptotes can be derived from the standard form equations. They are given by y = (b/a)x and y = -(b/a)x for hyperbolas centered at the origin. Recognizing the asymptotes helps in sketching the hyperbola accurately and understanding its behavior.

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis, while the co-vertices are associated with the conjugate axis. For a hyperbola in standard form, the vertices are located at (±a, 0) or (0, ±b) depending on the orientation. Identifying these points is essential for graphing the hyperbola and understanding its geometric properties.