1. **Problem Statement:** Sketch the graph of the function $$y = \sqrt{-2(x - 4)} - 5$$ and understand its transformations. 2. **Formula and Important Rules:** The base function is $$y = \sqrt{x}$$. Transformations include: - Horizontal shift by 4 units to the right (due to $$x - 4$$ inside the square root). - Horizontal reflection and stretch by factor 2 (due to the coefficient $$-2$$ inside the square root). - Vertical shift down by 5 units (due to $$-5$$ outside the square root). 3. **Rewrite the function to understand the domain:** $$y = \sqrt{-2(x - 4)} - 5 = \sqrt{-2x + 8} - 5$$ 4. **Find the domain:** The expression inside the square root must be non-negative: $$-2(x - 4) \geq 0$$ $$-2x + 8 \geq 0$$ Divide both sides by $$-2$$ (remember to reverse inequality): $$\cancel{-2}(x - 4) \leq \cancel{-2}0$$ $$x - 4 \leq 0$$ $$x \leq 4$$ 5. **Find the starting point (vertex) of the graph:** At $$x = 4$$, $$y = \sqrt{-2(4 - 4)} - 5 = \sqrt{0} - 5 = -5$$ So the graph starts at point $$(4, -5)$$. 6. **Behavior of the graph:** As $$x$$ decreases from 4, the value inside the square root increases, so $$y$$ increases. 7. **Summary of transformations:** - The graph is reflected horizontally because of the negative inside the square root. - It is horizontally stretched by factor $$\frac{1}{\sqrt{2}}$$ due to the coefficient 2. - Shifted right by 4 units. - Shifted down by 5 units. **Final answer:** The graph of $$y = \sqrt{-2(x - 4)} - 5$$ starts at $$(4, -5)$$ and extends leftwards, increasing as $$x$$ decreases, with the described transformations applied to the base $$y=\sqrt{x}$$ function.