4B. Given $y=x^{2}-2(k-1) x+k^{2}$, where $k$ is a real number, is a set of parabolas (each number $k$ defines one parabola). Determine the geometric place of the points which are the vertices of the given parabolas.
Solution. The abscissa of the vertex of the parabola

$$
y=x^{2}-2(k-1) x+k^{2}
$$

is $x=-\frac{b}{2 a}=k-1$, and the corresponding ordinate is $y=(k-1)^{2}-2(k-1)(k+1)+k^{2}=2 k-1$.

By eliminating the parameter $k$, we obtain the equation of the desired geometric locus of points:

$$
y=2 x+1
$$

Thus, the desired geometric locus of points is a straight line.