What are the equations of the asymptotes?

The equations of the asymptotes depend on the type of curve you are dealing with. I will explain the most common types: lines and hyperbolas.

For a line, specifically a straight line, it does not have asymptotes as it extends to infinity without any defined horizontal or vertical asymptotes.

For a hyperbola, which is an open curve that has two separate branches, the asymptotes are straight lines that the hyperbola approaches but never touches. The standard form of the equation of a hyperbola centered at the origin with horizontal and vertical asymptotes is:

(x^2/a^2) - (y^2/b^2) = 1 for a horizontal hyperbola, or (y^2/a^2) - (x^2/b^2) = 1 for a vertical hyperbola.

For a horizontal hyperbola, the asymptotes are given by the equations y = ±(b/a)x. For a vertical hyperbola, the asymptotes are x = ±(a/b)y.

If the center of the hyperbola is not at the origin and is at (h,k), the equations of the asymptotes would be y - k = ±(b/a)(x - h) for a horizontal hyperbola, or x - h = ±(a/b)(y - k) for a vertical hyperbola.