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Hyperbolas at the Origin quiz

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  • What is the key difference between the standard equations of an ellipse and a hyperbola?
    The ellipse uses a plus sign between the terms, while the hyperbola uses a minus sign.
  • How does the visual appearance of a hyperbola differ from an ellipse?
    A hyperbola looks like two parabolas facing away from each other, while an ellipse is a closed oval shape.
  • In the equation of a hyperbola, what does the 'a' value represent?
    The 'a' value is the distance from the center to each vertex of the hyperbola.
  • How is the 'b' value used when graphing a hyperbola?
    The 'b' value helps determine the height of the hyperbola and is critical for finding the asymptotes.
  • For a vertical hyperbola, where is the 'a' value located in the equation?
    The 'a' value is under the y squared term in the denominator.
  • How do you determine if a hyperbola is horizontal or vertical from its equation?
    If the x squared term comes first, it's horizontal; if the y squared term comes first, it's vertical.
  • Where are the vertices and foci located for a horizontal hyperbola centered at the origin?
    They are on the x-axis at (±a, 0) for vertices and (±c, 0) for foci.
  • What is the formula for finding the foci distance 'c' in a hyperbola?
    c² = a² + b²; c is the square root of the sum of a² and b².
  • How is the foci of a hyperbola unique compared to other conic sections?
    For any point on the hyperbola, the difference of the distances to the two foci is constant.
  • What is the process for graphing the asymptotes of a hyperbola?
    Draw a box using the a and b values, then draw diagonal lines through the corners of the box.
  • What is the equation for the asymptotes of a vertical hyperbola centered at the origin?
    y = ±(a/b)x, where a is the vertical distance and b is the horizontal distance.
  • How do you find the equations for the asymptotes of a horizontal hyperbola?
    The equations are y = ±(b/a)x, flipping the a and b values compared to the vertical case.
  • What are the steps to graph a hyperbola from its equation?
    Determine orientation, find vertices, find b points, draw the box, draw asymptotes, and sketch the hyperbola branches.
  • How do you find the vertices of a hyperbola given its equation?
    Take the square root of the first denominator term (a²), then plot points at (±a, 0) or (0, ±a) depending on orientation.
  • What is the main difference in the sign usage between hyperbola and ellipse equations?
    Hyperbolas use a minus sign in their standard equation and a plus sign in the foci formula, while ellipses use a plus sign in the equation and a minus sign in the foci formula.