Reflect Polygon Ee370F

1. **State the problem:** We need to reflect the polygon with vertices $(-8, 6)$, $(-7, 9)$, $(-3, 6)$, $(-4, 2)$, and $(-7, 4)$ over the line $y = x + 3$. 2. **Formula and approach:** To reflect a point $(x, y)$ over the line $y = x + 3$, we first translate the line to $y = x$ by shifting down by 3 units, reflect over $y = x$, then shift back up by 3 units. 3. **Step 1: Translate points down by 3:** $$ (x, y) \to (x, y - 3) $$ 4. **Step 2: Reflect over $y = x$:** Reflection over $y = x$ swaps coordinates: $$ (x, y - 3) \to (y - 3, x) $$ 5. **Step 3: Translate points up by 3:** $$ (y - 3, x) \to (y - 3, x + 3) $$ 6. **Apply to each vertex:** - For $(-8, 6)$: - Translate down: $(-8, 6 - 3) = (-8, 3)$ - Reflect: $(3, -8)$ - Translate up: $(3, -8 + 3) = (3, -5)$ - For $(-7, 9)$: - Translate down: $(-7, 6)$ - Reflect: $(6, -7)$ - Translate up: $(6, -7 + 3) = (6, -4)$ - For $(-3, 6)$: - Translate down: $(-3, 3)$ - Reflect: $(3, -3)$ - Translate up: $(3, -3 + 3) = (3, 0)$ - For $(-4, 2)$: - Translate down: $(-4, -1)$ - Reflect: $(-1, -4)$ - Translate up: $(-1, -4 + 3) = (-1, -1)$ - For $(-7, 4)$: - Translate down: $(-7, 1)$ - Reflect: $(1, -7)$ - Translate up: $(1, -7 + 3) = (1, -4)$ 7. **Final reflected vertices:** $$ (3, -5), (6, -4), (3, 0), (-1, -1), (1, -4) $$ These are the points of the reflected polygon over the line $y = x + 3$.