To graph a hyperbola from its equation, it is essential to first determine its orientation—whether it is horizontal or vertical. This can be identified by examining the equation; if the term with \(x^2\) appears first in the denominator, the hyperbola is horizontal. For example, if the equation is of the form \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\), it indicates a vertical hyperbola, while \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) indicates a horizontal hyperbola.

Next, identify the vertices of the hyperbola. For a horizontal hyperbola, the vertices are located at \((h \pm a, k)\), where \(a\) is derived from \(a^2\) in the equation. For instance, if \(a^2 = 9\), then \(a = 3\), placing the vertices at \((h+3, k)\) and \((h-3, k)\).

Following this, determine the \(b\) values, which are found from \(b^2\) in the equation. If \(b^2 = 64\), then \(b = 8\). The \(b\) points for a horizontal hyperbola are located at \((h, k \pm b)\), resulting in points at \((h, k+8)\) and \((h, k-8)\).

To find the asymptotes, draw a rectangle using the vertices and \(b\) points. The asymptotes are then represented by lines that pass through the corners of this rectangle, following the equations \(y - k = \pm \frac{b}{a}(x - h)\).

Finally, sketch the branches of the hyperbola, which approach the asymptotes but never intersect them. The general shape of the hyperbola will open towards the direction of the vertices.

To locate the foci, use the relationship \(c^2 = a^2 + b^2\). For example, if \(a^2 = 9\) and \(b^2 = 64\), then \(c^2 = 9 + 64 = 73\), leading to \(c = \sqrt{73} \approx 8.54\). The foci for a horizontal hyperbola are positioned at \((h \pm c, k)\), which means they will be at approximately \((h+8.54, k)\) and \((h-8.54, k)\).

By following these steps, one can effectively graph a hyperbola and identify its key features, including vertices, asymptotes, and foci, all derived from the equation.