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Writing Parametric Equations quiz

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  • What is the first step in writing parametric equations from a rectangular equation?
    The first step is to choose an expression for the parameter t, often involving x or y.
  • Why is it common to set t equal to x when parameterizing an equation?
    Setting t equal to x is simple and avoids domain restrictions, making it a reliable choice for most problems.
  • After defining t, what is the next step in finding parametric equations?
    Solve for x in terms of t to get x(t), then substitute this into the original equation to find y(t).
  • Why should you avoid choosing t as an even power of x, like x^2, when parameterizing?
    Choosing t as an even power of x can restrict the domain and may lead to imaginary numbers if t is negative.
  • How do you parameterize equations involving x^2 + y^2?
    Use the Pythagorean identity by setting x and y in terms of cosine and sine functions of t.
  • What should you do if the equation is not in the form f(x)^2 + g(y)^2 = 1?
    Rewrite the equation, often by dividing both sides by a constant, to get it into the form f(x)^2 + g(y)^2 = 1.
  • How do you find x(t) for an equation like x^2/4 + y^2 = 1?
    Set x/2 = cos(t), so x(t) = 2cos(t).
  • How do you find y(t) for an equation like x^2/4 + y^2 = 1?
    Set y = sin(t), so y(t) = sin(t).
  • What is a good way to check if your parametric equations are correct?
    Eliminate the parameter t from your parametric equations to see if you return to the original rectangular equation.
  • If t = x + 1, how do you express x in terms of t?
    Solve for x to get x(t) = t - 1.
  • Given y = 2x + 5 and t = x + 1, what is y(t)?
    Substitute x(t) into y to get y(t) = 2(t - 1) + 5 = 2t + 3.
  • For y = (x + 2)^2 - 3, what is a good choice for t?
    Set t equal to the expression inside the parentheses, so t = x + 2.
  • Given t = x + 2, how do you write x(t) and y(t) for y = (x + 2)^2 - 3?
    x(t) = t - 2 and y(t) = t^2 - 3.
  • How do you parameterize 9x^2 + y^2 = 9?
    Rewrite as x^2 + (y^2)/9 = 1, then set x = cos(t) and y = 3sin(t).
  • Why is it important to avoid domain restrictions when choosing t?
    Domain restrictions can limit the possible values of t and may result in undefined or imaginary values for x or y.