1. **State the problem:** We need to find the coordinates of point $R(4,5)$ after reflecting it once across the $x$-axis and once across the $y$-axis. 2. **Reflection in the $x$-axis:** - The rule for reflecting a point $(x,y)$ across the $x$-axis is to keep the $x$-coordinate the same and change the sign of the $y$-coordinate. - Formula: $$ (x,y) \to (x,-y) $$ - Applying this to $R(4,5)$: $$ (4,5) \to (4,-5) $$ 3. **Reflection in the $y$-axis:** - The rule for reflecting a point $(x,y)$ across the $y$-axis is to keep the $y$-coordinate the same and change the sign of the $x$-coordinate. - Formula: $$ (x,y) \to (-x,y) $$ - Applying this to $R(4,5)$: $$ (4,5) \to (-4,5) $$ 4. **Explanation of the relationship:** - Reflecting across the $x$-axis changes the vertical position by negating the $y$-coordinate. - Reflecting across the $y$-axis changes the horizontal position by negating the $x$-coordinate. - Both reflections keep one coordinate the same and negate the other, effectively flipping the point over the respective axis. **Final answers:** - After reflection in the $x$-axis: $R' = (4,-5)$ - After reflection in the $y$-axis: $R'' = (-4,5)$