For a parametric equation f)(x,y)=(x(t),y(t))f)(x,y)=(x(t),y(t)), you put different values of tt, ofcourse where t>=0t≥0, to get different pairs of values for xx and yy to get sets of points, joining which, we get the desired curve.
Here we have x=sqrtt+4x=√t+4 and y=sqrtt-4y=√t−4. Let us consider t=0,1,4,9,16,25,36t=0,1,4,9,16,25,36 - note that we have intentionally selected square numbers, so that getting pair of values of xx and yy is easy.
We get (4,-4),(5,-3),(6,-2),(7,-1),(8,0),(9,1),(10,2)(4,−4),(5,−3),(6,−2),(7,−1),(8,0),(9,1),(10,2) and joining them we get the following graph nand this is a striaght line, shown below. Also observe that given x=sqrtt+4x=√t+4 and y=sqrtt-4y=√t−4, subtracting latter from former eliminates tt and we get equation of line x-y=8x−y=8. What does sqrtt√t does to this? It just restricts line to x>=4x≥4 or y>=-4y≥−4.
graph{x-y=8 [4, 24, -7, 3]}
