Here are the essential concepts you must grasp in order to answer the question correctly.
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 (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas, indicating that the transverse axis is vertical.
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Vertices and Foci
The vertices of a hyperbola are the points where the branches are closest to each other, located at (0, ±a) for the equation (y^2/a^2) - (x^2/b^2) = 1. The foci are points located along the transverse axis, at a distance of c from the center, where c is calculated using the formula c = √(a^2 + b^2). These points are crucial for understanding the shape and orientation of the hyperbola.
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Graphing Hyperbolas
Graphing a hyperbola involves plotting its vertices and foci, and then sketching the asymptotes, which guide the shape of the branches. The asymptotes for the hyperbola in the given equation can be found using the equations y = ±(a/b)x, which help in determining the direction and spread of the branches. Matching the equation to a graph requires recognizing these features and their relationships.
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