Graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x^2)/16 - y^2 = 1

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^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas, where 'a' and 'b' are constants that determine the shape and size of the hyperbola.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line that passes through the center and the vertices of the hyperbola. For the hyperbola given by (x^2/a^2) - (y^2/b^2) = 1, the distance from the center to each focus is given by c = √(a^2 + b^2). The foci play a crucial role in defining the hyperbola's shape and properties.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For the hyperbola in the form (x^2/a^2) - (y^2/b^2) = 1, the equations of the asymptotes are given by y = ±(b/a)x. These lines provide a framework for understanding the behavior of the hyperbola at extreme values and are essential for accurately graphing the hyperbola.