Let's break down the transformation step by step:

1. Translation $T_{(-1, 3)}$:

The translation $T_{(-1, 3)}$ means shifting the point by $-1$ unit along the $x$-axis and $+3$ units along the $y$-axis.

For the point $(-9, -2)$:

• The new $x$-coordinate will be $-9 + (-1) = -10$.
• The new $y$-coordinate will be $-2 + 3 = 1$.

So after the translation, the point becomes $(-10, 1)$.

2. Reflection across the $x$-axis $r_{x-axis}$:

To reflect a point across the $x$-axis, the $x$-coordinate remains the same, while the $y$-coordinate changes its sign.

For the point $(-10, 1)$:

• The $x$-coordinate remains $-10$.
• The $y$-coordinate becomes $-1$.

Thus, after the reflection, the point is $(-10, -1)$.

Final Answer:

The image of the point $(-9, -2)$ after the transformation $r_{x-axis} \circ T_{(-1, 3)}$ is $(-10, -1)$.

Would you like further details or have any questions? Here are five related questions for further exploration:

1. How would the transformation change if the reflection was across the $y$-axis instead?
2. What is the matrix representation of a reflection across the $x$-axis?
3. How do you calculate the inverse of a translation transformation?
4. What happens if you apply the transformations in reverse order?
5. How does the slope of a line connecting the original and final points change after these transformations?

Tip: When reflecting across the $x$-axis, only the $y$-coordinate changes, while the $x$-coordinate remains unchanged.