Mistakes regarding parametric curves

• Sometimes people go on with rules without thinking and do this: [math]\displaystyle{ y = x^2 \to y(t) = t^2 }[/math] and [math]\displaystyle{ x = \sqrt{y} \to x(t) = \sqrt{t} }[/math] which is to randomly do computations without knowing what it means. The graph of [math]\displaystyle{ (\sqrt{t}, \ t^2) }[/math] is half a parabola because of the square root and is also a distorted parabola because the horizontal coordinate does not have a constant rate of change.