Parameterizing $f(x, y) = z$ then what does $f(x(t), y(t)) = z$?

Question

Let's say we have a function $f(x, y) = z$ and we parameterize it as follows: $$\{x(t), y(t)\} = \{0.1, 2\} + t \nabla f(0.1, 2)$$ $$f(x(t), y(t)) = z$$ As I understand it, we've essentially transformed it into a function $f: \mathbb R \rightarrow \mathbb R$, which has the highest intitial ($t = 0$) $\frac{\partial z}{\partial t}$ of all possible parameterizations at the point $\{0.1, 2, f(0.1, 2)\}$. So if we graph this parameterized $f$, what will it represent?