The question in the image appears to ask about a series of geometric transformations applied to point $L(-2, -1)$. Let's break down the process step by step:

1. Dilation:

   • The point $L(-2, -1)$ is dilated with respect to the origin $O(0, 0)$ by a scale factor of $-2$.
   • To dilate a point $(x, y)$ by a factor of $k$ with respect to the origin, the formula is: $L' = (k \cdot x, k \cdot y)$ For a scale factor of $-2$, the new point becomes: $L' = (-2 \cdot -2, -2 \cdot -1) = (4, 2)$

2. Reflection over the line $y = x$:

   • When reflecting a point $(x, y)$ over the line $y = x$, the coordinates are swapped: $(x, y) \to (y, x)$ So, reflecting $(4, 2)$ over the line $y = x$ gives: $L'' = (2, 4)$

3. Rotation by $90^\circ$ counterclockwise around the origin:

   • To rotate a point $(x, y)$ by $90^\circ$ counterclockwise around the origin, the transformation is: $(x, y) \to (-y, x)$ Applying this to $(2, 4)$: $L''' = (-4, 2)$

Thus, the final image of the point after all the transformations is $(-4, 2)$.

Would you like any additional details or clarifications on this solution?

Here are 5 related questions to expand your understanding:

1. How does the dilation factor affect the direction of the point's movement?
2. What would happen if the dilation scale factor was positive instead of negative?
3. How would the transformation change if the reflection was over the y-axis instead of the line $y = x$?
4. What is the general formula for rotating a point by $180^\circ$ around the origin?
5. Can you explain the difference between counterclockwise and clockwise rotations in transformation geometry?

Tip: When working with transformations, always apply them step-by-step to avoid confusion, especially with sequences of reflections, rotations, and dilations.