Quick questions on Parametric equations: parameter elimination, sketches, and standard curves

To get a Cartesian equation, eliminate $t$ from the two equations. Strategies:

As $t$ increases, the point $(x(t), y(t))$ moves in a definite direction along the curve. For a circle parametrised by $t \in [0, 2\pi)$ with $x = \cos t$, $y = \sin t$, motion is anticlockwise starting at $(1, 0)$.

The dependable way to sketch a parametric curve, and to see what eliminating the parameter is doing, is to plot a few points and then join them in order of increasing $t$. Below, the curve $x = 2t$, $y = t^2 - 1$ (the curve from the first exam question above) is built one stage at a time.

$x = \cos t$, $y = \sin t$ goes anticlockwise. $x = \cos t$, $y = -\sin t$ goes clockwise.