What is the standard form equation for the asymptotes of a vertical hyperbola?

For a vertical hyperbola, the standard form of its equation is expressed as ((y - k)²/a² - (x - h)²/b² = 1). In this equation, the quantities ((h, k)) represent the coordinates of the center of the hyperbola, while (a) and (b) are related to the distances that define the shape of the hyperbola.

To understand why this equation is relevant for vertical hyperbolas, consider that a vertical hyperbola opens up and down. The vertical orientation means that the (y) part of the equation is positive, indicating that as the absolute distance from the center increases in the vertical direction, the values of (y) will dominate, leading to the creation of the hyperbola's arms extending vertically.

On the other hand, the asymptotes of a hyperbola can also be derived from the standard form. The asymptotes represent lines that the hyperbola approaches but never intersects. For the equation given, the asymptotes are described by the equations approximately (y - k = \pm \frac{a}{b}(x - h)). This further confirms that the structure of the hyperbola is dictated by