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Hyperbolas NOT at the Origin quiz

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  • What is the main difference in the equation of a hyperbola centered at the origin versus one not at the origin?
    The equation for a shifted hyperbola includes (x - h) and (y - k) instead of just x and y, where (h, k) is the new center.
  • How do you identify the center (h, k) of a hyperbola from its equation?
    The center (h, k) is found by looking at the values subtracted from x and y in the equation: (x - h) and (y - k).
  • How can you tell if a hyperbola is horizontal or vertical from its equation?
    If the y-term comes first and is positive, the hyperbola is vertical; if the x-term comes first and is positive, it is horizontal.
  • What does the value 'a' represent in the equation of a hyperbola?
    'a' is the square root of the first denominator and determines the distance from the center to each vertex along the transverse axis.
  • How do you find the vertices of a vertical hyperbola centered at (h, k)?
    The vertices are at (h, k + a) and (h, k - a), where 'a' is the square root of the first denominator.
  • How do you find the vertices of a horizontal hyperbola centered at (h, k)?
    The vertices are at (h + a, k) and (h - a, k).
  • What does the value 'b' represent in the equation of a hyperbola?
    'b' is the square root of the second denominator and is used to find the 'b points' for drawing the rectangle that helps locate the asymptotes.
  • How do you find the 'b points' for a vertical hyperbola?
    The 'b points' are at (h + b, k) and (h - b, k), where 'b' is the square root of the second denominator.
  • What is the purpose of drawing a rectangle (box) when graphing a hyperbola?
    The rectangle connects the vertices and b points, and its diagonals are used to draw the asymptotes.
  • How do you find the equations of the asymptotes for a hyperbola not centered at the origin?
    Draw lines through the corners of the rectangle (box) formed by the vertices and b points; these lines are the asymptotes.
  • What is the relationship between a, b, and c in a hyperbola?
    The relationship is c² = a² + b², where 'c' is the distance from the center to each focus.
  • How do you find the coordinates of the foci for a vertical hyperbola centered at (h, k)?
    The foci are at (h, k + c) and (h, k - c), where c = sqrt(a² + b²).
  • How do you find the coordinates of the foci for a horizontal hyperbola centered at (h, k)?
    The foci are at (h + c, k) and (h - c, k).
  • What do the branches of a hyperbola do in relation to the asymptotes?
    The branches approach the asymptotes but never cross them.
  • Why is it important to determine the orientation (horizontal or vertical) of a hyperbola before graphing?
    The orientation determines the direction in which the vertices, b points, and foci are placed relative to the center.