To find the coordinates of $P'$ under the transformation $T_{(-4,0)} \circ r_m$, where $m$ is the line $y = -2$, let's break down the steps:

1. Reflection across the line $m$:

   • The line $m$ has the equation $y = -2$.
   • The point $P$ appears to be at $(1, -3)$ on the grid.
   • Reflecting $P$ across $y = -2$ requires finding its symmetrical point with respect to this line.
   • The distance from $P$ to $y = -2$ is 1 unit (since $-3 - (-2) = -1$), so reflecting it across will place it 1 unit above $y = -2$ at $(1, -1)$.

2. Translation $T_{(-4,0)}$:

   • The translation vector $(-4,0)$ means moving the point 4 units to the left without any change in the $y$-coordinate.
   • Applying this translation to the reflected point $(1, -1)$, we get: $(1 - 4, -1) = (-3, -1)$

Therefore, the coordinates of $P'$ are: $(-3, -1)$

Would you like more details or have any questions on this transformation?

Here are some related questions to explore:

1. How do you reflect a point across a line other than the x-axis or y-axis?
2. How is the distance calculated between a point and a line?
3. What changes in the transformation if we reflect across a different line?
4. How do translation transformations affect coordinates in general?
5. What are the properties of compositions of transformations?

Tip: For reflections, always find the perpendicular distance to the line to determine the reflected point’s position accurately.