1. **State the problem:** We are given a point $(4, -6)$ on the graph of $y = f(x)$ and need to find the corresponding coordinates on the graph of $$y = \frac{2}{\sqrt{x + 3}} - 4.$$ 2. **Understand the function:** The new function is $$y = \frac{2}{\sqrt{x + 3}} - 4.$$ This means for any $x$, we calculate $\sqrt{x + 3}$, then divide 2 by that value, and finally subtract 4. 3. **Calculate the new $y$ value for $x=4$:** Substitute $x=4$ into the new function: $$y = \frac{2}{\sqrt{4 + 3}} - 4 = \frac{2}{\sqrt{7}} - 4.$$ 4. **Simplify the expression:** $$y = \frac{2}{\sqrt{7}} - 4.$$ We can leave it as is or rationalize the denominator: $$y = \frac{2}{\sqrt{7}} - 4 = \frac{2\sqrt{7}}{7} - 4.$$ 5. **Final coordinates:** The $x$-coordinate remains $4$, and the new $y$-coordinate is $$\frac{2\sqrt{7}}{7} - 4.$$ **Answer:** The point on the graph of $$y = \frac{2}{\sqrt{x + 3}} - 4$$ corresponding to $x=4$ is $$\left(4, \frac{2\sqrt{7}}{7} - 4\right).$$