Let's break this problem down step-by-step.

The base function is:

$f(x) = \sqrt{x}$

The given function to transform is:

$h(x) = -\sqrt{x + 9} + 9$

We need to identify the correct transformations applied to $f(x) = \sqrt{x}$ to result in $h(x)$.

Step 1: Horizontal Transformation

• Inside the square root, the expression is $\sqrt{x + 9}$.
• This indicates a horizontal shift to the left by 9 units (since $x + 9$ shifts left, not right).

Step 2: Reflection about the x-axis

• The negative sign in front of the square root, $-\sqrt{x + 9}$, means the graph is reflected about the x-axis (flips the graph vertically).

Step 3: Vertical Shift

• The +9 added outside the square root shifts the graph up by 9 units. However, this vertical shift is not needed to answer the multiple-choice question, which focuses on horizontal shifts and reflections.

Answer Choice

From the above analysis:

• Horizontal shift to the left by 9 units
• Reflection about the x-axis

The correct answer is:

$\boxed{C.} \text{The graph of } \sqrt{x} \text{ should be horizontally shifted to the left by 9 units and reflected about the x-axis.}$

Do you need further details on this solution, or have any additional questions?

Related Questions

1. What happens when a negative sign is placed inside the square root, like $\sqrt{-x}$?
2. How does a vertical stretch or compression affect a square root graph?
3. What is the effect of shifting $\sqrt{x}$ vertically instead of horizontally?
4. Can you describe how both x-axis and y-axis reflections change the graph's shape?
5. How would you plot the function $h(x) = -\sqrt{x + 9} + 9$ step-by-step?

Tip: A transformation inside the square root always affects the graph horizontally and behaves oppositely to the sign seen inside the function (e.g., $+9$ shifts left).