1. **Problem:** Use transformation to sketch the graph of $$y = 3\sqrt{-(x + 5)} - 10$$ and state the domain and range. 2. **Base function:** The base function is $$y = \sqrt{x}$$, which has domain $$[0, \infty)$$ and range $$[0, \infty)$$. 3. **Transformations:** - Inside the square root, $$-(x + 5)$$ means a reflection about the y-axis (due to the negative sign) and a horizontal shift left by 5 units. - The coefficient 3 outside the root stretches the graph vertically by a factor of 3. - The $$-10$$ shifts the graph down by 10 units. 4. **Domain:** For the expression inside the root to be non-negative: $$-(x + 5) \geq 0 \implies x + 5 \leq 0 \implies x \leq -5$$ So, domain is $$(-\infty, -5]$$. 5. **Range:** The base $$\sqrt{u}$$ outputs values $$\geq 0$$. After vertical stretch by 3 and downward shift by 10: $$y = 3\sqrt{-(x+5)} - 10 \geq 3 \times 0 - 10 = -10$$ So, range is $$[-10, \infty)$$. 6. **Sketching:** - Start with $$y=\sqrt{x}$$. - Reflect horizontally: $$y=\sqrt{-x}$$. - Shift left 5 units: $$y=\sqrt{-(x+5)}$$. - Stretch vertically by 3: $$y=3\sqrt{-(x+5)}$$. - Shift down 10 units: $$y=3\sqrt{-(x+5)} - 10$$. 7. **Final answer:** - Domain: $$(-\infty, -5]$$ - Range: $$[-10, \infty)$$ This completes the transformation and domain/range analysis.