How to differentiate between "shapes" when given parametric equations? Having a hard time understanding how to figure out how a function looks like when it's in parametric form.
Here are two examples, just wondering if someone could help me develop some sort of intuition as to how to look at it.
\begin{pmatrix}
\cosh s \\
t+\cosh s \\
3+2t \\
\end{pmatrix}
and 
\begin{pmatrix}
2e^ {-t} \\
3+e^t \\
3+2t \\
\end{pmatrix}
 A: The first vector function has two parameters, $s$ and $t$. Therefore the graph of that function is a surface with two dimensions. It may be curved, but it will look like a (perhaps twisted) sheet.
To think better how the surface will look, not that the parameter $s$ appears only in the expression $\cosh s$. We can replace that expression with parameter $u$, remembering that since $u=\cosh s$ it will take the values $1\le u$. We then get
$$\begin{pmatrix} u \\t+u \\3+2t \end{pmatrix}=
\begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}+
t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}+
(u-1)\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$$
Either expression shows that the surface is (part of) a plane and that the point $(1,1,3)$ is on the surface. Since $u$ is limited, that means the surface is a half-plane, and $(1,1,3)$ is on the edge of the plane.
That probably is enough for intuition: playing with the expression just a little shows that we have a half-plane. Here is the graph:

Your second vector function has only one parameter, $t$. That means it is a curve in space, with one dimension. It will look like a (perhaps twisted) line.
To see what kind of curve, we see that removing the constant term $(0,3,3)$ from the function, the $x$ and $y$ coordinates are near-reciprocals of each other. Actually, we would get $xy=2$. The graph of that curve in the $xy$ plane is a hyperbola, but since both $x$ and $y$ must be positive we would get only one branch of the hyperbola.
The $z$ linear term means the hyperbola branch is turned out of the $xy$ plane, and the constant $(0,3,3)$ means the hyperbola apparently does not have its center at the origin. Again, by playing with the expression a little we figure out its basic shape. Here is its graph:

A: In 3 space ( 3 coordinates) if there is one  independent parameter it is some curved line. If there are two independent parameters, then the two curved line mesh together making up a surface. The plot in either case is parametric in 2D and 3D.
