Task B-2.2. If the graph of the quadratic function $f(x)=x^{2}-(m-n) x+n, m, n \in \mathbb{R}$, has its vertex at point $T(2,3)$, determine the value of $f(m-n)$.
## Solution.

The $x$-coordinate of the vertex of the quadratic function $f$ is $\frac{m-n}{2}$. Then $\frac{m-n}{2}=2$, so

$$
m-n=4
$$

(1 point)

Furthermore, the $y$-coordinate of the vertex is $\frac{4 n-(m-n)^{2}}{4}=3$. From this and (1), we get

$$
4 n-(m-n)^{2}=12
$$

or $4 n-16=12$, i.e., $n=7$.

Then $f(m-n)=f(4)=16-16+7=7$.