Answer
$x^{2}-y^{2}=1, \quad x\geq 1,\qquad y\geq 0$
Work Step by Step
From the parametric equations,
$x\geq 1,\qquad y\geq 0$
Square both equations :$\left\{\begin{array}{l}
x^{2}=t+1\\
y^{2}=t
\end{array}\right.$
Substitute:
$x^{2}=y^{2}+1, \quad x\geq 1,\qquad y\geq 0$
$x^{2}-y^{2}=1, \quad x\geq 1,\qquad y\geq 0$
(upper right wing of a hyperbola).
To graph, create a table using several values for t, and then calculate $(x(t), y(t)),$ plotting the points as you go. Join with a smooth curve, noting the direction in which $t$ increases.
