Graph Transformation 95C12B

1. **State the problem:** We are given the graph of $y = f(x)$ and need to find the graph of $y = -f(x - 4) + 1$ by repositioning points. 2. **Understand the transformation:** The function $y = -f(x - 4) + 1$ involves three transformations: - Horizontal shift right by 4 units (due to $x - 4$ inside the function). - Reflection about the x-axis (due to the negative sign in front of $f$). - Vertical shift up by 1 unit (due to $+1$ outside the function). 3. **Apply horizontal shift:** For any point $(x, y)$ on $y = f(x)$, the corresponding point on $y = f(x - 4)$ is $(x + 4, y)$. 4. **Apply reflection:** Reflect the $y$-value about the x-axis: $y$ becomes $-y$. 5. **Apply vertical shift:** Add 1 to the $y$-value: $-y + 1$. 6. **Combine steps:** For a point $(x, y)$ on $y = f(x)$, the transformed point on $y = -f(x - 4) + 1$ is: $$ (x + 4, -y + 1) $$ 7. **Example with key points:** - If a key point on $f$ is at $(0, 0)$, then on the transformed graph it is at: $$ (0 + 4, -0 + 1) = (4, 1) $$ - If another key point is at $(2, 4)$, then on the transformed graph it is at: $$ (2 + 4, -4 + 1) = (6, -3) $$ 8. **Conclusion:** To reposition the green draggable points, move each point from $(x, y)$ to $(x + 4, -y + 1)$ to obtain the graph of $y = -f(x - 4) + 1$.