1. **State the problem:** We need to find the composition of transformations that maps triangle $\triangle ABC$ with vertices $A(-4,1)$, $B(-3,4)$, and $C(-2,1)$ to triangle $\triangle A'B'C'$ with vertices $A'(5,1)$, $B'(4,4)$, and $C'(3,1)$. The transformations are: first reflect over the $y$-axis, then translate by $(x + ?, y + ?)$. 2. **Reflection over the $y$-axis:** The reflection rule over the $y$-axis is $$ (x,y) \to (-x,y) $$ Applying this to each vertex of $\triangle ABC$: - $A(-4,1) \to A'(4,1)$ - $B(-3,4) \to B'(3,4)$ - $C(-2,1) \to C'(2,1)$ 3. **Translation:** After reflection, the points are at $A''(4,1)$, $B''(3,4)$, and $C''(2,1)$. We want to translate these points to $A'(5,1)$, $B'(4,4)$, and $C'(3,1)$ respectively. The translation rule is: $$ (x,y) \to (x + h, y + k) $$ where $h$ and $k$ are constants to be found. 4. **Find translation values:** Using point $A''(4,1)$ to $A'(5,1)$: $$ 4 + h = 5 \implies h = 1 $$ $$ 1 + k = 1 \implies k = 0 $$ Check with point $B''(3,4)$ to $B'(4,4)$: $$ 3 + 1 = 4 $$ $$ 4 + 0 = 4 $$ Correct. Check with point $C''(2,1)$ to $C'(3,1)$: $$ 2 + 1 = 3 $$ $$ 1 + 0 = 1 $$ Correct. 5. **Final composition of transformations:** - Reflect over the $y$-axis: $(x,y) \to (-x,y)$ - Translate by $(x + 1, y + 0)$ **Answer:** The composition is reflect over the $y$-axis, then translate by $(x + 1, y)$.