One. (20 points) The graph of the quadratic function $y=x^{2}+p x+q$ passes through the point $(2,-1)$ and intersects the $x$-axis at two distinct points $A(a, 0)$ and $B(b, 0)$. Let the vertex of the graph be $M$. Find the analytical expression of the quadratic function that minimizes the area of $\triangle A M B$.

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The graph of the quadratic function $y=x^{2}+p x+q$ passes through the point $(2,-1)$ and intersects the $x$-axis at two distinct points $A(a, 0)$ and $B(b, 0)$. Let the vertex of the graph be $M$. Find the analytical expression of the quadratic function that minimizes the area of $\triangle A M B$.
From the problem, we know that $4+2 p+q=-1$, which means $2 p+q=-5$.
Also, $|a-b|=\sqrt{(a+b)^{2}-4 a b}=\sqrt{p^{2}-4 q}$, and the point $M\left(-\frac{p}{2}, \frac{4 q-p^{2}}{4}\right)$, then
$$
\begin{array}{l}
S_{-A Y B}=\frac{1}{2}|A B| \cdot\left|\frac{4 q-p^{2}}{4}\right| \\
=\frac{1}{8}|a-b| \cdot\left(p^{2}-4 q\right)=\frac{1}{8}\left(p^{2}-4 q\right)^{\frac{3}{2}} .
\end{array}
$$

To make $S_{\text {_NUA }}$ the smallest, we only need to make $p^{2}-4 q$ the smallest.
And $p^{2}-4 q=p^{2}+8 p+20=(p+4)^{2}+4$, its minimum value is 4, and the minimum value of $S_{\text {_NUA }}$ is $\frac{1}{8} \times(4)^{\frac{3}{2}}=1$.
Therefore, the quadratic function that meets the requirement is $y=x^{2}-4 x+3$.