The function $g(x) = -\sqrt{x} + 2$ can be understood as a transformation of the basic square root function $\sqrt{x}$.

Key features of the graph:

1. Basic shape: The graph of $g(x) = -\sqrt{x} + 2$ is based on the square root function $\sqrt{x}$, but it has been transformed.
2. Reflection: The negative sign in front of the square root reflects the graph of $\sqrt{x}$ over the x-axis, so it opens downwards instead of upwards.
3. Vertical shift: The "+2" shifts the entire graph vertically upward by 2 units.
4. Domain: The square root function is only defined for non-negative values of $x$, so the domain of $g(x)$ is $x \geq 0$.
5. Range: Since the graph is reflected downward and then shifted up, the range is $(-\infty, 2]$, meaning the graph will never exceed 2 but will approach negative infinity as $x$ increases.

Step-by-step description of transformations:

• Start with the graph of $\sqrt{x}$, which looks like a curve starting at the origin and increasing to the right.
• Reflect the graph across the x-axis due to the negative sign, so it now opens downwards.
• Shift the entire graph upward by 2 units.

Plot points:

• When $x = 0$, $g(x) = -\sqrt{0} + 2 = 2$.
• When $x = 1$, $g(x) = -\sqrt{1} + 2 = 1$.
• When $x = 4$, $g(x) = -\sqrt{4} + 2 = 0$.
• When $x = 9$, $g(x) = -\sqrt{9} + 2 = -1$.

Shape:

• The curve starts at the point $(0, 2)$ and slopes downwards as $x$ increases, approaching $-\infty$.

Would you like me to plot this graph visually?

Here are some follow-up questions you might be interested in:

1. How does shifting the graph vertically or horizontally affect its shape?
2. What would happen if you change the sign of the square root function again?
3. How do transformations work on other types of functions?
4. Can you predict the domain and range of more complicated functions?
5. What happens to the graph if the constant "+2" is replaced by another number?

Tip: Reflecting a function across the x-axis changes its orientation, but not its domain or general shape.