The problem asks for the coordinates of point A' after two transformations: a 90-degree rotation (denoted as $R_{90^\circ}$) and a reflection over the line $y = x$.

Given the point $A(-4, 1)$, let's break down the transformations step by step:

1. Rotation of 90 degrees counterclockwise: A 90-degree counterclockwise rotation around the origin transforms a point $(x, y)$ to $(-y, x)$. So, applying this to the point $A(-4, 1)$, we get: $A' = ( -1, -4 )$

2. Reflection over the line $y = x$: A reflection over the line $y = x$ swaps the coordinates of the point $(x, y)$ to $(y, x)$. So applying this to $A'(-1, -4)$, we get: $A'' = (-4, -1)$

Thus, the coordinates of the final image of the point are $(-4, -1)$, which corresponds to answer B.

Let me know if you'd like a more detailed explanation!

Further questions:

1. What is the general formula for a 90-degree counterclockwise rotation?
2. How do different transformations (like translations or scaling) affect a point?
3. Can a rotation and a reflection be combined in other ways to form different transformations?
4. How does reflecting a point over other lines (like $y = -x$) change its coordinates?
5. What other types of symmetry can a geometric shape have?

Tip: When performing transformations, it's helpful to visualize the steps by plotting points on a graph!