The problem asks to find the coordinates of $P'$ after applying a transformation $T^{(0,4)} \circ f_{y-\text{axis}}$ on point $P$. Let's break this down:

1. Initial point $P$: From the graph, the point $P$ has coordinates $(2,2)$.

2. Reflection across the y-axis $f_{y-\text{axis}}$: Reflecting a point $(x, y)$ over the y-axis changes the sign of the x-coordinate. So, after reflecting $(2,2)$ over the y-axis, we get $(-2, 2)$.

3. Translation $T^{(0,4)}$: A translation by $(0, 4)$ means adding 0 to the x-coordinate and 4 to the y-coordinate. So, translating $(-2, 2)$ by $(0, 4)$ results in: $(-2 + 0, 2 + 4) = (-2, 6).$

Thus, the coordinates of $P'$ are $(-2, 6)$.

Would you like further details or have any questions about this? Here are some related questions:

1. How do you reflect a point across the x-axis or other lines?
2. What are the general rules for translations in geometry?
3. How do you compose transformations, such as reflections and translations?
4. What would happen if the point was reflected over the x-axis instead?
5. How does this apply to more complex geometric shapes?

Tip: When performing multiple transformations, always start from the innermost transformation and work outward!