3B. Given the formula

$$
y=x^{2}-2(k-1) x+k^{2}, k \in \mathbb{R}
$$

a set of parabolas is defined. Determine the geometric locus of the vertices of the parabolas.
The solution. The vertex of the parabola $y=a x^{2}+b x+c$ is given by $T\left(-\frac{b}{2 a}, \frac{4 a c-b^{2}}{4 a}\right)$. Substituting $a=1, b=2(k-1)$, and $c=k^{2}$, we obtain that the vertex is $T(k-1,2 k-1)$ for a given real number $k$. Since $2 k-1=2(k-1)+1$, the required geometric locus is the line $y=2 x+1$