Consider a separated morpshism $f:X\longrightarrow Y$ between two $K$-schemes ($K$ is a field). The graph of $f$ is the image of the morphism $(Id_X,f):X\longrightarrow X\times_{\operatorname {spec}K} Y$ and it is indicated as $\Gamma_f$; one can show that $\Gamma_f$ is a closed subscheme of $X\times_{\operatorname {spec}K} Y$, namely $(Id_X,f)$ is a closed immersion.
I need to write down explicitly an affine open covering for $\Gamma_f$ and then identify each chart of this affine covering as the common zeros set of some polynomials, but I have problems to perform this task.
Remark:
If $X$ and $Y$ where only topological space and $f$ a continous function, I think that an open covering of $\Gamma_f$ (with the induced topology in the product ) can be found in the following way: let be $\{V_i\}$ an open covering of $Y$, then
$$\Gamma_f=\bigcup_if^{-1}(V_i)\times (V_i\cap f(X))$$
I need a similar expression in the case of fibered product of schemes but I can't found it.
Hoping that my question is quite clear, I would to thank you in advance for your answers.
For $p \in X$ you can find affine neighborhoods $U$ of $p$ and $V$ of $f(p)$ such that $f(U) \subset V$. The product $U \times_K V$ is an affine open subscheme of $X \times_K Y$, and so its intersection with the closed subscheme $\Gamma_f$ is also affine. Moreover, such products cover $\Gamma_f$.
In explicit examples one could be more, well, explicit.