Is the closure of a quasi-affine variety affine?

Question

some definitions

We work over a fixed algebraically closed field $k$.And $A^{n}$ is an affine n-space over $k$,i.e. $A^{n}=k^{n}$.

An affine variety is an irreducible algebraic set in $A^{n}$.An open subset of an affine variety is called a quasi-affine variety.

My question:

Let $Y$ be a quasi-affine variety,is its closure $\bar Y$ an affine variety?

Here is my argument: Since Y is irreducible,so is $\bar Y$.Also,$\bar Y$ is closed,hence $\bar Y$ is an affine variety.

Is my argument correct? Am I missing something?

Any help will be greatly appreciated,thanks!