Question: Given the graph of $y = f(x)$, shown as a red dashed curve, drag the movable blue point to obtain the graph of $y = f(x - 3) + 1$.
1. The problem asks to transform the graph of $y = f(x)$ to the graph of $y = f(x - 3) + 1$.
2. The formula for horizontal and vertical shifts is:
- Horizontal shift: $y = f(x - h)$ shifts the graph $h$ units to the right.
- Vertical shift: $y = f(x) + k$ shifts the graph $k$ units up.
3. Here, $y = f(x - 3) + 1$ means:
- Shift the graph of $f(x)$ horizontally to the right by $3$ units.
- Then shift it vertically up by $1$ unit.
4. The original red dashed graph has its vertex at the origin $(0,0)$.
5. After shifting right by $3$, the vertex moves to $(3,0)$.
6. After shifting up by $1$, the vertex moves to $(3,1)$.
7. Therefore, the blue solid graph should have its vertex at $(3,1)$.
8. The shape of the graph remains the same (V-shaped, opening upward).
Final answer: The blue graph is the red graph shifted right by $3$ units and up by $1$ unit, with vertex at $(3,1)$.