Which point would map onto itself after a reflection across the line $$ y=-x $$ ? $$ (-4,-4) $$ $$ (-4,0) $$ $$ (0,-4) $$ $$ (4,-4) $$

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To determine which point maps onto itself after a reflection across the line $$ y = -x $$, let's analyze the reflection process.

### Reflection Across $$ y = -x $$

When a point $$ (x, y) $$ is reflected across the line $$ y = -x $$, its coordinates transform as follows:
$$
(x, y) \rightarrow (-y, -x)
$$

For a point to map onto itself after the reflection, it must satisfy:
$$
(x, y) = (-y, -x)
$$
This gives us the system of equations:
$$
x = -y \quad \text{and} \quad y = -x
$$
Combining these, we get:
$$
y = -x \quad \text{and} \quad x = -y \implies y = -(-y) \implies y = y
$$
This means any point on the line $$ y = -x $$ will map to itself under this reflection.

### Checking the Given Points

Let's check which of the provided points lie on the line $$ y = -x $$:

1. **$$(-4, -4)$$**
   $$
   y = -x \implies -4 = -(-4) \implies -4 = 4 \quad \text{(False)}
   $$
   
2. **$$(-4, 0)$$**
   $$
   y = -x \implies 0 = -(-4) \implies 0 = 4 \quad \text{(False)}
   $$
   
3. **$$(0, -4)$$**
   $$
   y = -x \implies -4 = -(0) \implies -4 = 0 \quad \text{(False)}
   $$
   
4. **$$(4, -4)$$**
   $$
   y = -x \implies -4 = -(4) \implies -4 = -4 \quad \text{(True)}
   $$

Only the point $$(4, -4)$$ satisfies the condition $$ y = -x $$, meaning it lies on the line of reflection and thus maps onto itself.

**Answer:** $$ \boxed{(4,-4)} $$
The point $$(4, -4)$$ maps onto itself after a reflection across the line $$ y = -x $$.

Which point would map onto itself after a reflection across the line \( y=-x \) ? \( (-4,-4) \) \( (-4,0) \) \( (0,-4) \) \( (4,-4) \)

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