To solve the transformation of the point $(-3, -2)$ under the combined transformations of reflection across the $x$-axis $(r_{x-axis})$ and a rotation of $270^\circ$ counterclockwise $(R_{270^\circ})$, we need to break down the process into two steps:

1. Reflection across the $x$-axis: The reflection across the $x$-axis transforms a point $(x, y)$ to $(x, -y)$.

   • For the point $(-3, -2)$, after reflecting across the $x$-axis: $r_{x-axis}(-3, -2) = (-3, 2)$

2. Rotation by $270^\circ$ counterclockwise: A $270^\circ$ counterclockwise rotation transforms a point $(x, y)$ to $(y, -x)$.

   • Applying this rotation to the point $(-3, 2)$ (the result of the reflection): $R_{270^\circ}(-3, 2) = (2, 3)$

Thus, the image of the point $(-3, -2)$ after the combined transformation $r_{x-axis} \circ R_{270^\circ}$ is (2, 3).

Would you like further details on this process?

Here are 5 related questions to deepen your understanding:

1. How does a rotation of $90^\circ$ counterclockwise differ from $270^\circ$?
2. What is the general formula for reflecting a point across the $y$-axis?
3. What happens when a point is rotated $180^\circ$ counterclockwise?
4. How does the combination of transformations affect the overall result?
5. What would the point be after only applying the reflection without the rotation?

Tip: Always apply transformations in the correct order, as reversing them can yield different results.