To find the image of the point $(0, -9)$ under the given transformation $T_{2,-4} \circ r_{y=x}$, let us break the transformation into two parts:

1. Reflection $r_{y=x}$:
   The reflection over the line $y = x$ swaps the $x$- and $y$-coordinates of a point. Applying this transformation to $(0, -9)$, we get:
   $r_{y=x}(0, -9) = (-9, 0).$

2. Translation $T_{2,-4}$:
   The translation $T_{2,-4}$ adds 2 to the $x$-coordinate and subtracts 4 from the $y$-coordinate. Applying this transformation to the reflected point $(-9, 0)$, we get:
   $T_{2,-4}(-9, 0) = (-9 + 2, 0 - 4) = (-7, -4).$

Final Result:

The image of the point $(0, -9)$ after the transformation $T_{2,-4} \circ r_{y=x}$ is: $\boxed{(-7, -4)}.$

Let me know if you'd like further clarification or step-by-step visualization of this process.

---

Related Questions:

1. What is the result of reflecting a general point $(a, b)$ over the line $y = x$?
2. How do you represent translations as matrix transformations?
3. What is the effect of combining reflections and translations geometrically?
4. How would the transformation change if the translation was $T_{-3,5}$?
5. How do you find the inverse of the transformation $T_{2,-4} \circ r_{y=x}$?

Tip: Always apply reflections and translations step-by-step, respecting the order of transformations to avoid errors.