Restrictions Matter!

At first glance, this looks like a parabola because we can rewrite $y = e^{2t} - 1$ as $y = (e^t)^2 - 1$ , which can be rewritten through substitution as $y = x^2 - 1$ .

However, $x = e^t$ doesn't take on zero or any negative values, so the left hand side of the standard parabola $y = x^2 - 1$ is not drawn. Therefore, the answer is half of a parabola.

It’s pretty ez to see by substitution that it’s a parabola. But notice that x=e^t is always gonna be positive, so for any value of t, x is positive, so only half a parabola is traced out.