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Hyperbolas NOT at the Origin quiz

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  • What is the standard form equation for a hyperbola not centered at the origin?
    The standard form is ((y - k)^2)/a^2 - ((x - h)^2)/b^2 = 1 for a vertical hyperbola, where (h, k) is the center.
  • How do you determine the center of a hyperbola from its equation?
    The center is (h, k), where h is subtracted from x and k is subtracted from y in the equation.
  • What does the 'h' value represent in the hyperbola's equation?
    The 'h' value represents the horizontal shift of the center from the origin.
  • What does the 'k' value represent in the hyperbola's equation?
    The 'k' value represents the vertical shift of the center from the origin.
  • How do you identify if a hyperbola is vertical or horizontal from its equation?
    If the y-term is positive and comes first, the hyperbola is vertical; if the x-term is positive and comes first, it is horizontal.
  • How do you find the vertices of a vertical hyperbola not at the origin?
    Keep h constant and add and subtract 'a' from k to get the vertices at (h, k + a) and (h, k - a).
  • How do you calculate the value of 'a' in the hyperbola's equation?
    'a' is the square root of the denominator under the positive term in the equation.
  • How do you find the 'b' points for a vertical hyperbola?
    Keep k constant and add and subtract 'b' from h to get the points (h + b, k) and (h - b, k).
  • How do you calculate the value of 'b' in the hyperbola's equation?
    'b' is the square root of the denominator under the negative term in the equation.
  • What is the relationship between a, b, and c in a hyperbola?
    The relationship is c^2 = a^2 + b^2, where c is the distance from the center to each focus.
  • How do you find the coordinates of the foci for a vertical hyperbola?
    Keep h constant and add and subtract 'c' from k to get (h, k + c) and (h, k - c).
  • How do you graph the asymptotes of a hyperbola not at the origin?
    Draw a rectangle using the vertices and b points, then draw lines through the diagonals of the rectangle.
  • What is the purpose of drawing a box when graphing a hyperbola?
    The box helps locate the vertices and b points and guides the drawing of the asymptotes.
  • How do the branches of a hyperbola relate to the asymptotes?
    The branches start at the vertices and approach the asymptotes but never cross them.
  • What changes in the equation of a hyperbola when it is shifted from the origin?
    The variables x and y are replaced with (x - h) and (y - k), shifting the center to (h, k).