The reflection of \( (\boldsymbol{x},-\boldsymbol{y}) \) across the \( \boldsymbol{y} \)-axis

To find the reflection of the point \( (\boldsymbol{x}, -\boldsymbol{y}) \) across the \( \boldsymbol{y} \)-axis, we need to understand how reflections work in the Cartesian coordinate system. The reflection of a point \( (a, b) \) across the \( y \)-axis is given by the point \( (-a, b) \). This means that the \( x \)-coordinate changes sign, while the \( y \)-coordinate remains the same. Given the point \( (\boldsymbol{x}, -\boldsymbol{y}) \): 1. The \( x \)-coordinate is \( \boldsymbol{x} \). 2. The \( y \)-coordinate is \( -\boldsymbol{y} \). To reflect this point across the \( y \)-axis, we change the sign of the \( x \)-coordinate: \[ \text{Reflection across the } y\text{-axis: } (-\boldsymbol{x}, -\boldsymbol{y}). \] Thus, the reflection of \( (\boldsymbol{x}, -\boldsymbol{y}) \) across the \( y \)-axis is \[ \boxed{(-\boldsymbol{x}, -\boldsymbol{y})}. \]