Final answer:
The given equation x² - y² = 100 represents a hyperbola. The vertices are (-10, 0) and (10, 0), the foci are (-10, 0) and (10, 0), and the asymptotes have equations y = 0. The graph is symmetric with respect to the y-axis and the origin is the center.
Step-by-step explanation:
The given equation x² - y² = 100 represents a hyperbola. To find the vertices, foci, and asymptotes, we can rewrite the equation in standard form as (x/h)² - (y/k)² = 1, where (h, k) is the center of the hyperbola.
Comparing the equation to the standard form, we have (x/0)² - (y/0)² = 1. So the center of the hyperbola is (0, 0).
The vertices are located at (±a, 0) where a is the distance from the center to the vertices. In this case, a = √100 = 10. So the vertices are (-10, 0) and (10, 0).
The foci are located at (±c, 0) where c is the distance from the center to the foci. In this case, c = √(a² + b²) = √(10² + 0²) = 10. So the foci are (-10, 0) and (10, 0).
The asymptotes have equations y = ±(b/a)x, where b is the distance from the center to the asymptotes. In this case, b = 0. So the asymptotes have equations y = 0.
Sketching the graph, we have a hyperbola centered at the origin (0, 0) with vertices at (-10, 0) and (10, 0), and the foci also at (-10, 0) and (10, 0). The asymptotes are the x and y axes. The graph is symmetric with respect to the y-axis and the origin is the center.