\( \mathrm{J}(-7,7) \) Write an algebraic representation for the reflec

To create an algebraic representation for the reflection of the point \( J(-7, 7) \), we first need to specify **across which axis or line** the reflection is taking place. Common reflections include: 1. **Reflection Across the x-axis:** - **Rule:** \( (x, y) \rightarrow (x, -y) \) - **Application to \( J(-7, 7) \):** \[ J' = (-7, -7) \] - **Explanation:** Only the y-coordinate changes its sign, flipping the point over the x-axis. 2. **Reflection Across the y-axis:** - **Rule:** \( (x, y) \rightarrow (-x, y) \) - **Application to \( J(-7, 7) \):** \[ J' = (7, 7) \] - **Explanation:** Only the x-coordinate changes its sign, flipping the point over the y-axis. 3. **Reflection Across the Line \( y = x \):** - **Rule:** \( (x, y) \rightarrow (y, x) \) - **Application to \( J(-7, 7) \):** \[ J' = (7, -7) \] - **Explanation:** The x and y coordinates are swapped, reflecting the point over the line \( y = x \). 4. **Reflection Across the Origin (Point Reflection):** - **Rule:** \( (x, y) \rightarrow (-x, -y) \) - **Application to \( J(-7, 7) \):** \[ J' = (7, -7) \] - **Explanation:** Both coordinates change their signs, effectively rotating the point 180 degrees around the origin. **Example: Reflection Across the x-axis** Given the point \( J(-7, 7) \), reflecting it across the x-axis: \[ J' = (-7, -7) \] **Algebraic Representation:** \[ (x, y) \rightarrow (x, -y) \] This represents the general rule for reflecting any point across the x-axis. --- **Choose the appropriate reflection rule based on the desired axis or line of reflection.**