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.
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.
All the functions here were either odd or even, so we were able to use symmetry as a way to look at these curves. For more general parametric equations, we can think about the ranges of \(x\)- and \(y\)-values, find coordinates of points for specific values of \(t\), and think about how a point on the curve moves as \(t\) increases. In some cases, eliminating \(t\) and working with a Cartesian equation can be helpful.