x=-t^2 y=2t^2-2

Question

Eliminate the parameter

Answer 1

For this parametric equation we have to be careful because of the squares on the t variable.  We will start with x = -t².  Multiplying by negative 1, we get -x = t².  We substitute this into the other equation (for the t²) and get y = 2(-x) - 2 or y = -2x-2.  Now we have to make sure that we restrict the domain appropriately.  It's clear that no matter what value of t, we will get a -x value. so we must limit  our domain to: (-∞, 0].  For the range, the t² ensures that the lowest y will get is -2.  Therefore, our range is limited to [-2, ∞).  Fortunately, the domain fits this range perfectly (when x is zero y is -2 and when x is -∞ y is ∞.) and we don't have any areas where x exists but y doesn't or vice versa.