This problem asks for the image of the point $(2, 8)$ after a reflection over the line $y = x$. Here's how to solve it step by step:

Step-by-step solution:

1. Understand Reflection over $y = x$:

   • When a point $(a, b)$ is reflected over the line $y = x$, the coordinates of the reflected point are swapped. The new coordinates become $(b, a)$.

2. Apply the Transformation:

   • The original point is $(2, 8)$.
   • After swapping the coordinates (reflecting over $y = x$), the new point becomes: $(8, 2)$

Final Answer:

The image of the point $(2, 8)$ after reflection over $y = x$ is: $(8, 2)$

Let me know if you need further clarification!

Related Questions:

1. What is the general rule for reflecting points over the $x$-axis or $y$-axis?
2. How do you reflect a point over the line $y = -x$?
3. What happens when a point lies on the line of reflection $y = x$?
4. How do you reflect a geometric shape (e.g., triangle) over the line $y = x$?
5. Can you derive the reflection rule using linear algebra (matrices)?

Tip:

Always verify reflections by considering the symmetry of the line of reflection—points should be equidistant from the line.