1. The problem is to analyze the function $f(x) = x^2 - 2x + 1$. 2. This is a quadratic function in standard form. We can rewrite it to identify its properties. 3. Notice that $x^2 - 2x + 1$ can be factored as $$(x - 1)^2$$. 4. This means the function is a perfect square trinomial, which is always non-negative and has its minimum value at $x = 1$. 5. The vertex form of a quadratic is $f(x) = a(x - h)^2 + k$, where $(h,k)$ is the vertex. 6. Here, $a = 1$, $h = 1$, and $k = 0$, so the vertex is at $(1,0)$. 7. The graph is a parabola opening upwards with its minimum point at $(1,0)$. 8. This matches the function $f(x) = (x - 1)^2$. 9. The function $f(x) = (x - 1)^4 + 1$ mentioned earlier is similar but has a minimum at $(1,1)$ and is steeper near the vertex. 10. For $f(x) = x^2 - 2x + 1$, the minimum value is $0$ at $x=1$. Final answer: The function $f(x) = x^2 - 2x + 1$ can be rewritten as $f(x) = (x - 1)^2$ with vertex at $(1,0)$ and minimum value $0$.