1. **State the problem:** We are given the function $f(x) = -4\sqrt{-x}$ and want to understand its properties and graph. 2. **Recall the square root function:** The basic square root function is $y = \sqrt{x}$, which is defined for $x \geq 0$ and produces non-negative outputs. 3. **Analyze transformations:** - Inside the square root, we have $-x$, which reflects the graph of $\sqrt{x}$ over the y-axis, so the domain becomes $x \leq 0$. - The coefficient $-4$ outside the square root means the graph is vertically stretched by a factor of 4 and reflected over the x-axis (because of the negative sign). 4. **Domain:** Since the expression inside the square root must be non-negative, we have: $$-x \geq 0 \implies x \leq 0$$ 5. **Range:** Because of the negative sign outside and the stretch factor 4, the output values are non-positive and scaled: $$f(x) = -4\sqrt{-x} \leq 0$$ 6. **Summary:** The graph is the reflection of $\sqrt{x}$ over the y-axis and x-axis, stretched vertically by 4, defined for $x \leq 0$. 7. **Desmos function:** $y = -4\sqrt{-x}$