x = t^2 + 1
y = t^2 - 1.
Subtract the second equation from the first. You will get
x - y = 1 - (-1) = 2.
So, you eliminated the variable " t " and obtained the equation of the straight line x - y = 2 in (x,y)- coordinate plane.
The image of the map x ----> (t^2+1,t^2-1) is the ray: half of the straight line in (x,y)-coordinate plane.
