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Hyperbolas quiz

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  • What is the most distinct characteristic of a hyperbola's graph?
    A hyperbola has two curved branches that open away from each other.
  • How can you distinguish a hyperbola from an ellipse by its equation?
    A hyperbola's standard equation has a minus sign between the squared terms, while an ellipse has a plus sign.
  • What are the standard forms of the equations for horizontal and vertical hyperbolas?
    Horizontal: x²/a² - y²/b² = 1; Vertical: y²/b² - x²/a² = 1.
  • What does the leading squared term in a hyperbola's equation determine?
    It determines whether the transverse axis is horizontal (x² first) or vertical (y² first).
  • What is the center of a hyperbola in the standard form x²/a² - y²/b² = 1?
    The center is at the origin, (0, 0).
  • How do you find the values of a and b in a hyperbola's equation?
    Take the square root of the denominators under x² and y² to get a and b, respectively.
  • What is the fundamental rectangle in the context of graphing a hyperbola?
    It is a rectangle centered at the hyperbola's center, extending a units left/right and b units up/down.
  • How are the asymptotes of a hyperbola related to the fundamental rectangle?
    The asymptotes pass through the diagonals of the fundamental rectangle.
  • Where do the vertices of a hyperbola lie?
    The vertices lie on the transverse axis, at a distance of a units from the center.
  • What is the role of the asymptotes in the graph of a hyperbola?
    Asymptotes guide the branches of the hyperbola but are never touched by them.
  • How do you shift the center of a hyperbola from the origin to (h, k) in its equation?
    Replace x with (x-h) and y with (y-k) in the equation.
  • What is the constant property involving the foci (focus points) of a hyperbola?
    For any point on the hyperbola, the difference in distances to the two foci is constant.
  • How do you graph a hyperbola given its equation in standard form?
    Identify the center, find a and b, draw the fundamental rectangle, sketch asymptotes through its diagonals, and draw the branches through the vertices.
  • If the equation is (x-h)²/a² - (y-k)²/b² = 1, where is the center of the hyperbola?
    The center is at the point (h, k).
  • What is the main difference in graphing a vertical versus a horizontal hyperbola?
    For a vertical hyperbola, the y² term comes first and the vertices are on the vertical axis; for a horizontal hyperbola, the x² term comes first and the vertices are on the horizontal axis.