Answer
$y^{2}-x^{2}=4,\quad y \gt 0$
.
Work Step by Step
Restrictions on x and y, due to the parametric equations:
$x\in(-\infty,\infty),\quad y \gt 0$
From the identity $\cosh^{2}t-\sinh^{2}t=1,$
multiply by $ 2^{2}$:
$2^{2}\cosh^{2}t-2^{2}\sinh^{2}t=4$
substitute,
$y^{2}-x^{2}=4,\quad y \gt 0$
This is the hyperbola, vertical axis, upper wing.
To graph, create a function value table using values for t, in ascending order of t, calculating the x- and y-coordinates of points on the graph.
Plot and join the points obtained with a smooth curve, noting the direction in which t increases.
