It looks like you're describing a transformation of the function $\sqrt{x}$ based on a pattern or set of coordinates given in the sequence "123456-1-2-3-4-5-6123456-1-2-3-4-5-6." However, without a visual of the graph, I'll interpret that the sequence might represent a shift, reflection, or stretch/compression of the graph of the square root function.

Here’s a general approach to find the equation of the transformed function $g(x) = a \sqrt{b(x - h)} + k$:

1. Vertical Stretch/Compression ($a$): If the graph is stretched vertically, $a > 1$; if compressed, $0 < a < 1$; if reflected across the x-axis, $a < 0$.

2. Horizontal Stretch/Compression ($b$): If the graph is stretched horizontally, $0 < b < 1$; if compressed, $b > 1$; if reflected across the y-axis, $b < 0$.

3. Horizontal Shift ($h$): If the graph shifts to the right, $h > 0$; if to the left, $h < 0$.

4. Vertical Shift ($k$): If the graph shifts up, $k > 0$; if down, $k < 0$.

To find the exact equation $g(x)$, we need more specific information about the transformations applied to $\sqrt{x}$. If you can provide details on how the graph is transformed—such as specific shifts, stretches, or reflections—I can give you the exact function.

If you have a graph, you can describe it further, or if there are particular values from a table of points, that would help clarify the transformation!

Would you like more detailed steps on how to identify these transformations, or is there a specific transformation pattern that applies to your scenario?

Here are some related questions that could help:

1. What happens to $g(x)$ if the function $\sqrt{x}$ is reflected over the x-axis?
2. How would you determine $g(x)$ if the graph is shifted 3 units to the right and 2 units up?
3. What changes occur if the graph of $\sqrt{x}$ is compressed vertically by a factor of 0.5?
4. How does a horizontal stretch affect the domain of the function $\sqrt{x}$?
5. Can you identify the function transformation from a given set of points on a graph?

Tip: Analyzing transformations systematically—shifts, stretches/compressions, and reflections—can help you write the function equation accurately!