1. **State the problem:** We are given the function $$y = (x - 5)^2 + 1$$ and asked to find the coordinates of its vertex. 2. **Recall the vertex form of a parabola:** The function is already in vertex form $$y = a(x - h)^2 + k$$ where $ (h, k) $ is the vertex. 3. **Identify the vertex:** Comparing, we see $ h = 5 $ and $ k = 1 $. 4. **Write the vertex coordinates:** The vertex is at $$\boxed{(5, 1)}$$. 5. **Explain the graph shape:** Since $a = 1 > 0$, the parabola opens upwards. 6. **Graph window hint:** To include the vertex, choose an x-range around 5 and y-range around 1.