Graph Transformation Acc24C

1. **State the problem:** We are given the graph of $y = f(x)$ and need to transform it to the graph of $y = -f(x + 3) + 2$. 2. **Understand the transformations:** The function $y = -f(x + 3) + 2$ involves three transformations applied to $f(x)$: - Horizontal shift left by 3 units due to $x + 3$ inside the function. - Reflection about the x-axis due to the negative sign in front of $f$. - Vertical shift upward by 2 units due to the $+2$ outside the function. 3. **Apply the horizontal shift:** Replace every $x$ in $f(x)$ with $x + 3$, shifting the graph 3 units to the left. 4. **Apply the reflection:** Multiply the output of $f(x + 3)$ by $-1$, flipping the graph upside down. 5. **Apply the vertical shift:** Add 2 to the entire function, moving the graph up by 2 units. 6. **Summary of transformations:** $$y = -f(x + 3) + 2 = -\bigl(f(x + 3)\bigr) + 2$$ 7. **Interpretation on the graph:** - The vertex of the original $f(x)$ is at $(0,0)$. - After shifting left by 3, the vertex moves to $(-3,0)$. - Reflecting about the x-axis, the vertex remains at $(-3,0)$ but the V opens downward. - Shifting up by 2 moves the vertex to $(-3,2)$. 8. **Check draggable points:** - The vertex of the solid curve should be at $(-3,2)$. - Other points on the graph should be transformed accordingly by applying the same transformations. **Final answer:** The graph of $y = -f(x + 3) + 2$ is obtained by shifting the original graph 3 units left, reflecting it about the x-axis, and then shifting it 2 units up.