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Hyperbolas definitions

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  • Hyperbola
    A conic section with two curved branches opening away from each other, distinguished by a minus sign in its standard equation.
  • Conic Section
    A shape formed by the intersection of a plane and a cone, including circles, ellipses, parabolas, and hyperbolas.
  • Branch
    One of the two separate curved parts of a hyperbola, each extending away from the center along the transverse axis.
  • Standard Form
    An equation format for hyperbolas showing squared terms divided by constants, separated by a minus sign and set equal to one.
  • Transverse Axis
    The axis along which the vertices of a hyperbola lie, determined by the leading squared term in the equation.
  • Horizontal Hyperbola
    A hyperbola with branches opening left and right, indicated by the x-squared term leading in its equation.
  • Vertical Hyperbola
    A hyperbola with branches opening up and down, indicated by the y-squared term leading in its equation.
  • Asymptote
    A line that guides the direction of hyperbola branches, approaching but never touching them, drawn through the rectangle's diagonals.
  • Vertex
    A point where a hyperbola's branch crosses the transverse axis, marking the closest approach to the center.
  • Center
    The midpoint of a hyperbola, found at (h, k) in shifted equations, serving as the reference for graphing.
  • Fundamental Rectangle
    A rectangle constructed from the center, using a and b values, whose diagonals determine the asymptotes for graphing.
  • Orientation
    The direction in which a hyperbola opens, determined by whether x-squared or y-squared leads in the equation.
  • Equation
    A mathematical expression representing a hyperbola, typically involving squared terms and a minus sign.
  • Foci
    Two fixed points inside a hyperbola; the difference in distances from any point on the branches to these points remains constant.
  • Difference of Distances
    A constant value for all points on a hyperbola, calculated between the two foci.
  • a Value
    The distance from the center to each vertex along the transverse axis, found by taking the square root of the denominator under the leading squared term.
  • b Value
    The distance from the center to the rectangle's sides perpendicular to the transverse axis, found by taking the square root of the other denominator.
  • Ellipse
    A conic section similar to a hyperbola but with a plus sign in its standard equation, resulting in a closed curve.
  • Shifted Equation
    A hyperbola equation where x and y are replaced by (x-h) and (y-k), indicating the center is not at the origin.
  • Origin
    The point (0,0) on a graph, often used as the default center for hyperbolas in unshifted equations.