The rule for reflection across the x-axis is (x, y) (x, -y).

The coordinates of the image are $\boxed{(-3, -7)}$.

2. Reflect the point B $(6, -2)$ in the line $y = x$.

The coordinates of the image are $\boxed{(-2, 6)}$.

3. Rotate the point C $(-4, 5)$ by 180 degrees about the origin.

The coordinates of the image are $\boxed{(4, -5)}$.

4. Rotate the point D $(2, -3)$ by 90 degrees counter-clockwise about the origin.

The coordinates of the image are $\boxed{(3, 2)}$.

5. A square has vertices at $(0, 0)$, $(2, 0)$, $(2, 2)$, and $(0, 2)$. Find the image after reflecting in the y-axis.

For $(0, 0)$: $(-0, 0) = (0, 0)$
For $(2, 0)$: $(-2, 0)$
For $(2, 2)$: $(-2, 2)$
For $(0, 2)$: $(-0, 2) = (0, 2)$
The coordinates of the image are $\boxed{(0, 0), (-2, 0), (-2, 2), (0, 2)}$.

6. Reflect the point E $(-5, -8)$ in the line $y = -x$.

The coordinates of the image are $\boxed{(8, 5)}$.

7. Reflect the point F $(7, -3)$ in the y-axis. Then reflect that image in the x-axis.

The rule is $(x, y) \to (-x, y)$.

The rule is $(x, y) \to (x, -y)$.

The final coordinates are $\boxed{(-7, 3)}$.

8. Rotate the point G $(6, 2)$ by 270 degrees counter-clockwise about the origin.

The coordinates of the image are $\boxed{(2, -6)}$.

9. A triangle has vertices at P $(-2, 1)$, Q $(0, 4)$, and R $(3, -1)$. Rotate the triangle by 90 degrees counter-clockwise about the origin.

For $P(-2, 1)$: $P'(-1, -2)$
For $Q(0, 4)$: $Q'(-4, 0)$
For $R(3, -1)$: $R'(-(-1), 3) = R'(1, 3)$
The coordinates of the image triangle P'Q'R' are $\boxed{P'(-1, -2), Q'(-4, 0), R'(1, 3)}$.

10. A rectangle has vertices at $(-4, 1)$, $(-4, 5)$, $(3, 5)$, and $(3, 1)$. Reflect the rectangle in the line $y = x$.

For $(-4, 1)$: $(1, -4)$
For $(-4, 5)$: $(5, -4)$
For $(3, 5)$: $(5, 3)$
For $(3, 1)$: $(1, 3)$
The coordinates of the image rectangle are $\boxed{(1, -4), (5, -4), (5, 3), (1, 3)}$.