What are the asymptotes of a horizontal hyperbola in terms of y?

In the case of a horizontal hyperbola, the standard form of its equation is given as ((y - k)^2/a^2 - (x - h)^2/b^2 = 1). To derive the equations for the asymptotes, we need to look at the relationship expressed by the vertices and the slopes related to 'a' and 'b'.

The equations of the asymptotes for a horizontal hyperbola can be derived from the equations of the hyperbola itself. Specifically, the asymptotes are the lines that approach the hyperbola as it extends toward infinity. For a horizontally oriented hyperbola centered at the point (h, k), the equations of the asymptotes take the form:

(y - k = \pm \frac{a}{b}(x - h))

Transforming this equation, we can write it in slope-intercept form:

1. For the positive slope:

(y = \frac{a}{b}(x - h) + k)

2. For the negative slope:

(y = -\frac{a}{b}(x - h) + k)

Thus, combining these expressions, we can express both asymptotes compactly as