In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−3)^2−4(y+3)^2=4

Key Concepts

Hyperbola Definition

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (x-h)²/a² - (y-k)²/b² = 1, where (h, k) is the center, and 'a' and 'b' determine the distance from the center to the vertices and co-vertices, respectively.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in standard form, the equations of the asymptotes can be derived from the center and the values of 'a' and 'b'. Specifically, they are given by the equations y - k = ±(b/a)(x - h), where (h, k) is the center of the hyperbola, providing a guide for sketching the graph.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The distance from the center to each focus is determined by the formula c = √(a² + b²), where 'c' represents the distance to the foci. The foci play a crucial role in defining the shape of the hyperbola and are essential for understanding its geometric properties.