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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, either horizontally or vertically.
  • What are asymptotes in the context of a hyperbola?
    Asymptotes are lines that the branches of a hyperbola approach but never touch, guiding the shape of the branches.
  • Where do the vertices of a hyperbola lie?
    The vertices lie on the transverse axis, which is the axis along which the branches open.
  • How can you distinguish between a horizontal and a vertical hyperbola using its equation?
    If the x² term comes first, it's a horizontal hyperbola; if the y² term comes first, it's a vertical hyperbola.
  • What is the standard form equation for a horizontal hyperbola centered at the origin?
    The standard form is x²/a² - y²/b² = 1.
  • What is the standard form equation for a vertical hyperbola centered at the origin?
    The standard form is y²/b² - x²/a² = 1.
  • How do you find the values of a and b from the equation of a hyperbola?
    a is the square root of the denominator under the leading squared term, and b is the square root of the other denominator.
  • What is the fundamental rectangle in graphing a hyperbola?
    It is a rectangle centered at the hyperbola's center, with sides of length 2a and 2b, used to help draw the asymptotes and branches.
  • How do you graph the asymptotes of a hyperbola?
    Draw straight lines through the diagonals of the fundamental rectangle, extending in both directions.
  • What is the relationship between the foci and any point on a hyperbola?
    For any point on the hyperbola, the absolute difference in distances to the two foci is constant.
  • How does the equation of a hyperbola change if the center is not at the origin?
    The equation becomes (x-h)²/a² - (y-k)²/b² = 1 for horizontal, or (y-k)²/b² - (x-h)²/a² = 1 for vertical, where (h, k) is the center.
  • How do you determine the center of a hyperbola from its equation?
    The center is at (h, k), which are the values subtracted from x and y in the squared terms.
  • What is the purpose of the fundamental rectangle when graphing a hyperbola?
    It helps locate the vertices, draw the asymptotes, and guides the sketching of the branches.
  • Where are the vertices located for a horizontal hyperbola centered at (h, k)?
    The vertices are at (h+a, k) and (h−a, k).
  • What is the main difference between the equations of an ellipse and a hyperbola?
    A hyperbola's equation has a minus sign between the squared terms, while an ellipse's equation has a plus sign.