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Parametric Equations quiz

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  • What is a parametric equation?
    A parametric equation expresses x and y in terms of a third variable, usually called t, the parameter.
  • How do you graph parametric equations?
    Create a table of values for t, x(t), and y(t), then plot the corresponding (x, y) points on the plane.
  • What is the main difference between parametric equations and standard two-variable equations?
    Parametric equations involve two equations for x and y in terms of t, while standard equations relate y directly to x.
  • What is a 'plane curve' in the context of parametric equations?
    A plane curve is the graph of parametric equations, which can be a line, parabola, or other shapes.
  • How is the direction or orientation of a parametric curve indicated?
    The direction is shown with arrows along the curve, indicating increasing t values.
  • What does 'eliminating the parameter' mean?
    It means removing t from the equations to obtain a single equation involving only x and y.
  • What is the typical process for eliminating the parameter?
    Solve one equation for t and substitute it into the other equation to get a rectangular equation.
  • Why might you only get part of a curve when graphing parametric equations?
    Domain restrictions on t can limit the portion of the curve represented by the parametric equations.
  • How do you eliminate the parameter in parametric equations involving trigonometric functions?
    Solve for the trig functions in terms of x and y, then use a Pythagorean identity to relate them and eliminate t.
  • What Pythagorean identity is commonly used to eliminate the parameter in trig-based parametric equations?
    The identity sin²(t) + cos²(t) = 1 is used to relate sine and cosine terms.
  • How do you parameterize a rectangular equation?
    Choose a simple expression for t, solve for x(t), and substitute into the original equation to find y(t).
  • What should you avoid when choosing t for parameterization?
    Avoid expressions that introduce domain restrictions, such as even powers or square roots of x.
  • How can you check if your parameterization is correct?
    Eliminate the parameter from your parametric equations; you should recover the original rectangular equation.
  • How do you parameterize equations of circles or ellipses?
    Rewrite the equation in the form f(x)² + g(y)² = 1, then set f(x) = cos(t) and g(y) = sin(t), solving for x(t) and y(t).
  • What is the result of eliminating the parameter from x = cos(t), y = 3sin(t)?
    You get x² + (y/3)² = 1, which is the equation of an ellipse.