1. **State the problem:** We have a rectangle ABCD with vertex A at coordinates $(1,4)$. 2. **Transformation rule:** The transformation is a reflection across the line $y = x$. 3. **Formula for reflection across $y = x$:** When a point $(x,y)$ is reflected across the line $y = x$, its image is $(y,x)$. 4. **Apply the rule to point A:** $$A = (1,4) \implies A' = (4,1)$$ 5. **Interpretation:** The reflected point $A'$ has coordinates $(4,1)$. **Final answer:** $A' = (4,1)$