The graph shown seems to represent a square root function, specifically a transformation of the basic square root graph $y = \sqrt{x}$. Based on the visual, we can assess which transformation matches the graph.

In the given options:

1. $y = \sqrt{x+1}$: This would shift the basic square root function $y = \sqrt{x}$ to the left by 1 unit. The graph would pass through (-1, 0) and rise upward from there.

2. $y = -\sqrt{x+1}$: This would shift the graph left by 1 unit and reflect it across the x-axis, making it a downward-facing curve starting from (-1, 0).

3. $y = -\sqrt{x} + 1$: This reflects the basic square root function $y = \sqrt{x}$ over the x-axis, shifts it upwards by 1, but does not move it horizontally. The curve starts from the origin (0,1).

4. $y = -\sqrt{x} - 1$: This reflects $y = \sqrt{x}$ over the x-axis, shifts it down by 1 unit, and the curve begins at the origin but lower.

The graph shown in the image appears to be a downward-facing curve starting at approximately (-1, 0), indicating it has been reflected across the x-axis and shifted to the left.

Thus, the correct equation would be $y = -\sqrt{x + 1}$.

Do you have any further questions, or would you like additional details?

Here are some related questions to explore:

1. What transformations occur with $y = \sqrt{x}$ when adding or subtracting inside the square root?
2. How does reflecting a graph over the x-axis affect its equation?
3. What impact does a vertical shift have on the graph of $y = \sqrt{x}$?
4. Can transformations apply to other types of functions (e.g., cubic or exponential)?
5. How does shifting a graph horizontally affect its intercepts?

Tip: Reflections across the x-axis change the sign of the output values (i.e., they multiply the function by -1).