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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).