Hyperbolas NOT at the Origin definitions Flashcards
Hyperbolas NOT at the Origin definitions
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Hyperbola A conic section with two separate branches, defined by a specific quadratic equation, often centered at a point other than the origin. Center The point (h, k) representing the midpoint between the vertices and foci, determined by values subtracted from x and y in the equation. Vertex A point on the hyperbola closest to the center, found by adding and subtracting 'a' from the center's coordinates. Asymptote A straight line that the branches of the hyperbola approach but never touch, found by drawing lines through the corners of a guiding box. Branch One of the two separate curves of a hyperbola, each extending outward and approaching the asymptotes. Foci Two fixed points inside each branch, located using the relationship c² = a² + b², which help define the hyperbola's shape. Standard Equation The general form of a hyperbola's equation, modified by shifting the center to (h, k) using subtracted values. Horizontal Hyperbola A hyperbola with branches opening left and right, identified when the x-term appears first in the equation. Vertical Hyperbola A hyperbola with branches opening up and down, identified when the y-term appears first in the equation. a-value The distance from the center to each vertex, found by taking the square root of the first denominator in the equation. b-value The distance from the center to the sides of the guiding box, found by taking the square root of the second denominator. c-value The distance from the center to each focus, calculated using the formula c² = a² + b². Guiding Box A rectangle drawn using the a and b distances from the center, used to help locate asymptotes and sketch the hyperbola. Conic Section A curve formed by the intersection of a plane and a double-napped cone, including hyperbolas, ellipses, circles, and parabolas. Transformation A shift or change in the position of a graph, such as moving the center of a hyperbola from the origin to (h, k).
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