Parametric equations for nine curves are defined by combining the row and column headings in the table below. The domain of \(t\) for each curve is the set of values of \(t\) for which the \(x\) and \(y\) coordinates are defined.

Sketches of some of these curves are available on cards.

Place each sketch in the cell with the appropriate row and column headings for the parametric equations of the curve and sketch the missing curves.

\(x=\cos t\)

\(t\in \mathbb{R}\)


\(t\in \mathbb{R}\)

\(x=\tan t\)

\(-4\pi\leq t\leq 4\pi\) \(t\neq \pm\tfrac{\pi}{2}, \pm \tfrac{3\pi}{2}, \pm \tfrac{5\pi}{2}, \pm \tfrac{7\pi}{2}\)

\(y=2\sin t\)

\(t\in \mathbb{R}\)


\(t\in \mathbb{R}\)


\(-4\pi\leq t \leq 4\pi\) and \(t\neq 0\)
  • What are the domains and ranges of the functions which give the parametric equations for the curve?

  • What is the range of values of the parameter \(t\) for the curve?

  • In which direction does the point \(P(x,y)\) move along the curve as \(t\) increases?

  • What are the values of \(t\) at some important points on the curve?

As you look along the rows or down the columns in the table, what features of the curves stay the same and which change? Explain how this is related to the parametric equations of the curves.