To graph the function $f(x) = -\sqrt{x + 5} + 1$, let's break it down into transformations step-by-step:

1. Start with the basic function $y = \sqrt{x}$.
2. Horizontal shift: The term $x + 5$ indicates a shift 5 units to the left.
3. Vertical reflection: The negative sign in front of the square root, $-\sqrt{x}$, reflects the graph across the x-axis.
4. Vertical shift: The $+1$ moves the graph up by 1 unit.

Domain and Range:

• Domain: The expression under the square root must be non-negative, so $x + 5 \geq 0$, which means $x \geq -5$.
• Range: Since the graph is reflected across the x-axis and then shifted up by 1, the maximum value occurs at $x = -5$ and is $y = 1$. The range is $(-\infty, 1]$.

Steps to plot:

1. Begin with the point at $x = -5$, where $y = 1$.

2. Calculate a few points such as for $x = -4$, $x = -1$, and $x = 0$.

   • For $x = -4$, $f(-4) = -\sqrt{-4 + 5} + 1 = -1 + 1 = 0$.
   • For $x = -1$, $f(-1) = -\sqrt{-1 + 5} + 1 = -2 + 1 = -1$.
   • For $x = 0$, $f(0) = -\sqrt{0 + 5} + 1 = -\sqrt{5} + 1 \approx -1.24$.

3. Plot these points and sketch the curve that reflects the transformations.

Would you like further details or clarification on any step? Here are some related questions to consider:

1. How would the graph change if there were no reflection, i.e., if the function were $\sqrt{x + 5} + 1$?
2. What is the vertex form of a square root function, and how is it used to find transformations?
3. How do different coefficients affect the graph of the square root function?
4. How would you determine the range if the function were $f(x) = \sqrt{x + 5} - 1$?
5. How can we verify the graph using a table of values?

Tip: When graphing transformations, always consider the order of transformations: horizontal shifts, reflections, stretches/compressions, and vertical shifts.