Which transformations of RST will result in the same image? Check all that apply. a reflection over the \( y \)-axis and then a translation 2 units right a \( 180^{\circ} \) rotation about the origin, then a translation 2 units right, and then a reflection over the \( x \)-axis a reflection over the \( y \)-axis and then a translation 2 units left a \( 180^{\circ} \) rotation about the origin, then a translation 2 units left, and then a reflection over the \( x \)-axis a reflection over the \( x \)-axis and then a translation 2 units right a reflection over the \( x \)-axis and then a translation 2 units left Save and Exit

To determine which sequences of transformations result in the same image, we'll analyze each option step by step. We'll represent each transformation mathematically to see if different sequences yield identical final transformations. ### Transformation Breakdown: 1. **Reflection over the \( y \)-axis and then a translation 2 units right** - **Reflection over \( y \)-axis:** \( (x, y) \rightarrow (-x, y) \) - **Translation 2 units right:** \( (-x, y) \rightarrow (-x + 2, y) \) - **Final Transformation:** \( (-x + 2, y) \) 2. **\( 180^{\circ} \) rotation about the origin, then a translation 2 units right, and then a reflection over the \( x \)-axis** - **\( 180^{\circ} \) rotation:** \( (x, y) \rightarrow (-x, -y) \) - **Translation 2 units right:** \( (-x, -y) \rightarrow (-x + 2, -y) \) - **Reflection over \( x \)-axis:** \( (-x + 2, -y) \rightarrow (-x + 2, y) \) - **Final Transformation:** \( (-x + 2, y) \) 3. **Reflection over the \( y \)-axis and then a translation 2 units left** - **Reflection over \( y \)-axis:** \( (x, y) \rightarrow (-x, y) \) - **Translation 2 units left:** \( (-x, y) \rightarrow (-x - 2, y) \) - **Final Transformation:** \( (-x - 2, y) \) 4. **\( 180^{\circ} \) rotation about the origin, then a translation 2 units left, and then a reflection over the \( x \)-axis** - **\( 180^{\circ} \) rotation:** \( (x, y) \rightarrow (-x, -y) \) - **Translation 2 units left:** \( (-x, -y) \rightarrow (-x - 2, -y) \) - **Reflection over \( x \)-axis:** \( (-x - 2, -y) \rightarrow (-x - 2, y) \) - **Final Transformation:** \( (-x - 2, y) \) 5. **Reflection over the \( x \)-axis and then a translation 2 units right** - **Reflection over \( x \)-axis:** \( (x, y) \rightarrow (x, -y) \) - **Translation 2 units right:** \( (x, -y) \rightarrow (x + 2, -y) \) - **Final Transformation:** \( (x + 2, -y) \) 6. **Reflection over the \( x \)-axis and then a translation 2 units left** - **Reflection over \( x \)-axis:** \( (x, y) \rightarrow (x, -y) \) - **Translation 2 units left:** \( (x, -y) \rightarrow (x - 2, -y) \) - **Final Transformation:** \( (x - 2, -y) \) ### Analysis of Final Transformations: - **Options 1 and 2:** Both result in \( (-x + 2, y) \) → **Same Image** - **Options 3 and 4:** Both result in \( (-x - 2, y) \) → **Same Image** - **Options 5 and 6:** Result in \( (x + 2, -y) \) and \( (x - 2, -y) \) respectively → **Different Images** ### Conclusion: The sequences of transformations that result in the **same image** are: - **a.** Reflection over the \( y \)-axis and then a translation 2 units right - **b.** \( 180^{\circ} \) rotation about the origin, then a translation 2 units right, and then a reflection over the \( x \)-axis - **c.** Reflection over the \( y \)-axis and then a translation 2 units left - **d.** \( 180^{\circ} \) rotation about the origin, then a translation 2 units left, and then a reflection over the \( x \)-axis **Options e and f** produce different images and should **not** be selected. **Final Answer:** All transformations labeled a, b, c, and d result in the same image.