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 the orientation and dimensions of the graph.
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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 given equation. 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 spread of the hyperbola.
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Asymptotes
Asymptotes are lines that the branches of a hyperbola approach but never touch. For the hyperbola in the given equation, the equations of the asymptotes can be derived from the relationship y = ±(a/b)x, where a and b are the denominators of the y^2 and x^2 terms, respectively. These lines help in sketching the hyperbola accurately by indicating its direction and growth.
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