(a) sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases (b) Eliminate the parameter to find the Cartesian equation of the curve x=t^2, y=t^3

The parametric variable t may take any real values. As t increases from -oo to 0, x decreases from +oo to 0 and y increases from -oo to 0. For positive t x increases from 0 to +oo and y, too. Note that x is always non-negative.
a) please look at the attached picture. It shows points for t=0, +-1/2, +-7/10, +-9/10, +-1, +-1.5. The direction on the curve is also shown.
b) express y from the equation y=t^3, it is t=root(3)(y). Then substitute it to x=t^2 and obtain x=root(3)(y^2). This is the answer. We can also express this as  y=+-x^(3/2).