Fluency exercise

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Problem

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.

 $x=\cos t$ $t\in \mathbb{R}$ $x=2t$ $t\in \mathbb{R}$ $x=\tan t$ $-4\pi\leq t\leq 4\pi$ $t\neq \pm\tfrac{\pi}{2}, \pm \tfrac{3\pi}{2}, \pm \tfrac{5\pi}{2}, \pm \tfrac{7\pi}{2}$ $y=2\sin t$ $t\in \mathbb{R}$ $y=t^2$ $t\in \mathbb{R}$ $y=\dfrac{2}{t}$ $-4\pi\leq t \leq 4\pi$ and $t\neq 0$

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.