Question: 4. Point A has coordinates $(3,4)$. After a translation 4 units left, a reflection across the $x$-axis, and a translation 2 units down, what are the coordinates of the image?
1. **State the problem:**
We start with point $A$ at coordinates $(3,4)$.
We apply three transformations in order:
- Translate 4 units left
- Reflect across the $x$-axis
- Translate 2 units down
2. **Apply the first transformation: translation 4 units left.**
Translation left means subtracting 4 from the $x$-coordinate.
$$ (x,y) \to (x-4,y) $$
So,
$$ (3,4) \to (3-4,4) = (-1,4) $$
3. **Apply the second transformation: reflection across the $x$-axis.**
Reflection across the $x$-axis changes the $y$-coordinate to its opposite.
$$ (x,y) \to (x,-y) $$
So,
$$ (-1,4) \to (-1,-4) $$
4. **Apply the third transformation: translation 2 units down.**
Translation down means subtracting 2 from the $y$-coordinate.
$$ (x,y) \to (x,y-2) $$
So,
$$ (-1,-4) \to (-1,-4-2) = (-1,-6) $$
**Final answer:** The coordinates of the image after all transformations are $$\boxed{(-1,-6)}$$.