Question

How do you graph the curve whose parametric equations are given and show its orientation given `x = sqrt{t} + 4`, `y = sqrt{t} - 4`, where `t>=0`?

Answer 1

Pleasesee below.

Explanation:

For a parametric equation `f)(x,y)=(x(t),y(t))`, you put different values of `t`, ofcourse where `t>=0`, to get different pairs of values for `x` and `y` to get sets of points, joining which, we get the desired curve.

Here we have `x=sqrtt+4` and `y=sqrtt-4`. Let us consider `t=0,1,4,9,16,25,36` - note that we have intentionally selected square numbers, so that getting pair of values of `x` and `y` is easy.

We get `(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+4` and `y=sqrtt-4`, subtracting latter from former eliminates `t` and we get equation of line `x-y=8`. What does `sqrtt` does to this? It just restricts line to `x>=4` or `y>=-4`.

graph{x-y=8 [4, 24, -7, 3]}