In the the two-dimensional Cartesian $xy$-plane, any non-vertical line is of the form $$y=m(x-h)+k\tag A$$ where $y$ varies with $x$, where $m$ is the slope of the line, and where $(h,k)\in\Bbb{R}^2$ is a fixed point on the line.
It is also possible to define lines (including vertical lines) by ‘tracing’ them out with a position vector:
$$\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} \\ \pmatrix{x\\y} = \pmatrix{h\\k} + \lambda\pmatrix{b_x\\b_y}\tag B$$
where $\mathbf r$ varies with $\lambda$ for all $\lambda\in(-\infty,\infty)$, where $(h,k)\in\Bbb{R}^2$ is also a point on the line, and where $\mathbf b$ is a “difference vector” such that $$\frac{b_y}{b_x}=m$$
The two loci are identical:
$$\bigcup_{x\in\Bbb{R}}\bigl\{ (x,y) : y = m(x-h)+k \bigr\} = \bigcup_{\lambda\in\Bbb{R}} \bigl\{ (x,y) : \mathbf{r}=\mathbf{a}+\lambda\mathbf{b}\bigr\}$$
It is certainly no coincidence that form $\text{(B)}$ so closely resembles the slope-intercept form of a line nor that $$\mathbf{b}=\pmatrix{\partial x/\partial x \\ \partial y/\partial x}$$
My question is: Given a polynomial $$P_n(x)=\sum_{i=0}^{n}c_ix^i$$ what is the general formula for a vector $\mathbf{r}(\lambda)$ such that
$$\bigcup_{x\in\Bbb{R}}\bigl\{ (x,y) : y = P_n(x) \bigr\} = \bigcup_{\lambda\in\Bbb{R}} \left\{ (x,y) : \mathbf{r}(\lambda) = \pmatrix{x\\y} \right\}\quad?$$
I have tried trying to reverse-engineer a formula using parametric representations of loci of the form $$\cases{x=f(t)\\y=g(t)}$$ but have had no luck.