The function given is $u(x) = -\sqrt{-x} - 2$, and the graph is plotted accordingly. Let's break down the transformations to understand how the graph is derived:

Step-by-Step Transformations:

1. Start with the base function $y = \sqrt{x}$:

   • This is the basic square root graph, which starts at the origin and curves upwards.

2. Reflect over the $y$-axis:

   • The term $-x$ inside the square root causes a reflection of $\sqrt{x}$ over the $y$-axis. The graph now exists only for $x \leq 0$.

3. Reflect over the $x$-axis:

   • The negative sign in front of $\sqrt{-x}$ reflects the graph over the $x$-axis. This inverts the curve to point downward.

4. Vertical shift down by 2 units:

   • The subtraction of 2 at the end of the function $-\sqrt{-x} - 2$ shifts the graph downward by 2 units.

Final Graph:

The graph starts at $(-2, -2)$, which is the vertex, and curves downward to the left due to the combined reflections and shifts.

Would you like a detailed breakdown of any step or further assistance? 😊

Related Questions:

1. What is the domain and range of the function $u(x) = -\sqrt{-x} - 2$?
2. How do reflections over axes affect the graph of $\sqrt{x}$?
3. Can the function $u(x)$ ever take positive values? Why or why not?
4. What is the impact of a vertical shift on the range of a function?
5. How would the graph change if $-2$ were replaced with $+2$ in $u(x)$?

Tip:

Always apply transformations systematically, starting with inner transformations (reflections, stretches) before outer ones (shifts).