To address this question, let's analyze each transformation step-by-step.

The given function is: $g(x) = -\sqrt{x + 5} - 3$ This function can be transformed from a basic square root function, $y = \sqrt{x}$, through the following steps:

1. Horizontal Shift: The term $x + 5$ inside the square root indicates a horizontal shift left by 5 units. So, $y = \sqrt{x}$ shifts left to $y = \sqrt{x + 5}$.

2. Reflection: The negative sign in front of the square root, $-\sqrt{x + 5}$, indicates a reflection over the x-axis.

3. Vertical Shift: The $-3$ at the end of the function indicates a downward shift by 3 units.

Therefore, the transformations are as follows:

• Reflect across the x-axis
• Shift downward by 3 units

From these transformations, we can determine that the correct graph will be a downward curve, starting from a point shifted left by 5 units and then down by 3 units from the origin.

Answer to Multiple Choice

The graph of $g(x)$ will look like a square root graph reflected over the x-axis and shifted down by 3. Based on the provided graphs, select the one that matches this description.

Would you like more detailed steps, or do you have additional questions?

Here are five related questions that could expand your understanding:

1. What are the effects of horizontal shifts on the graph of a function?
2. How does reflecting a function over the x-axis change its graph?
3. Why does adding or subtracting a constant outside the square root cause a vertical shift?
4. What are the differences in transformations when constants are inside vs. outside the square root?
5. How would the graph look if the function was $g(x) = \sqrt{x + 5} - 3$ (without the reflection)?

Tip: When analyzing transformations, always work from inside to outside for terms inside the function (horizontal transformations first, then vertical).