In this example, we explore how to graph a hyperbola, focusing on its orientation, vertices, b points, asymptotes, and foci. The first step is to determine the orientation of the hyperbola by examining the equation. Since the term with \(x^2\) appears first, we conclude that the hyperbola is oriented horizontally along the x-axis.

Next, we identify the vertices. The value of \(a^2\) is derived from the first denominator, which is 16. Taking the square root gives us \(a = \sqrt{16} = 4\). Therefore, the vertices are located at the points (4, 0) and (-4, 0) on the graph.

Following this, we calculate the b points, which are positioned vertically opposite the vertices. Here, \(b^2\) corresponds to the second number in the denominator, which is 20. Thus, \(b = \sqrt{20}\), which can be simplified to \(2\sqrt{5}\) or approximately 4.47. The b points are then at (0, \( \sqrt{20} \)) and (0, \(-\sqrt{20}\)), or approximately (0, 4.47) and (0, -4.47).

To find the asymptotes, we draw a rectangle (or box) that connects the vertices and b points. The asymptotes are represented by lines that extend through the corners of this box. These lines help define the behavior of the hyperbola as it approaches these asymptotes.

Next, we sketch the branches of the hyperbola, which start at the vertices and approach the asymptotes. One branch extends from (4, 0) towards the upper asymptote and downwards, while the other branch starts at (-4, 0) and approaches the lower asymptote.

Finally, we determine the foci of the hyperbola. For a horizontal hyperbola, the foci are located along the x-axis. We use the relationship \(c^2 = a^2 + b^2\) to find \(c\). Here, \(a^2 = 16\) and \(b^2 = 20\), leading to \(c^2 = 16 + 20 = 36\). Taking the square root gives us \(c = \sqrt{36} = 6\). Thus, the foci are positioned at (6, 0) and (-6, 0).

In summary, we have successfully graphed the hyperbola, identified its vertices at (4, 0) and (-4, 0), b points at (0, \( \sqrt{20} \)) and (0, \(-\sqrt{20}\)), drawn the asymptotes, and located the foci at (6, 0) and (-6, 0).