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Eliminate the Parameter quiz

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  • What does it mean to eliminate the parameter in parametric equations?
    It means to remove the variable t from the equations, leaving an equation that only involves x and y.
  • What is the typical process for eliminating the parameter in parametric equations?
    Solve one equation for t and substitute it into the other equation.
  • Why is it usually better to solve the x equation for t and substitute into the y equation?
    This often results in a more familiar equation, such as a line or parabola, involving only x and y.
  • Given x = √t and y = t - 3, what is the rectangular equation after eliminating the parameter?
    The rectangular equation is y = x² - 3.
  • Why are t values sometimes restricted in parametric equations?
    Restrictions on t prevent undefined or imaginary values, such as negative values under a square root.
  • What effect do restrictions on t have on the graph of a parametric equation?
    They limit the portion of the graph represented, so only part of the full rectangular equation is shown.
  • What familiar shape does the equation y = x² - 3 represent?
    It represents a parabola.
  • What is a rectangular equation?
    A rectangular equation is an equation involving only x and y, with no parameter t.
  • What is the main challenge when eliminating the parameter in parametric equations involving trigonometric functions?
    Direct substitution can lead to complicated expressions, so a different strategy is needed.
  • What identity is used to eliminate the parameter in parametric equations with sine and cosine?
    The Pythagorean identity sin²(t) + cos²(t) = 1 is used.
  • Given x = cos(t) and y = 3sin(t), what are the expressions for cos(t) and sin(t)?
    cos(t) = x and sin(t) = y/3.
  • How do you use the Pythagorean identity to eliminate the parameter in x = cos(t) and y = 3sin(t)?
    Substitute x for cos(t) and y/3 for sin(t) into the identity, resulting in x² + (y/3)² = 1.
  • What type of conic section does the equation x² + (y/3)² = 1 represent?
    It represents an ellipse.
  • What is the general method for eliminating the parameter in parametric equations with trig functions?
    Solve each equation for the trig function, square them, and use a Pythagorean identity to relate them.
  • Why does the graph of a parametric equation sometimes only show part of the full rectangular graph?
    Because the range of t is restricted, so only a segment of the full graph is plotted.