In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (y+2)^2/4−(x−1)^2/16=1

Key Concepts

Hyperbola

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's equation can be expressed as (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center, and a and b determine the distances to the vertices and asymptotes.

Asymptotes

Asymptotes are lines that a curve approaches as it heads towards infinity. For hyperbolas, there are two asymptotes that intersect at the center of the hyperbola. The equations of the asymptotes can be derived from the standard form of the hyperbola and are crucial for sketching the graph accurately, as they guide the direction and shape of the branches.

Foci

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 denoted by 'c', where c² = a² + b². The foci play a significant role in defining the hyperbola's shape and are essential for understanding its geometric properties, including how the branches relate to these points.