The function given is $h(x) = \sqrt{x + 2} + 1$, and we need to graph this using transformations of the standard square root function $f(x) = \sqrt{x}$.

Steps to graph the transformation:

1. Start with the graph of $f(x) = \sqrt{x}$. This is the basic square root function, which begins at the origin (0, 0) and increases to the right.

2. Horizontal shift: The term $+2$ inside the square root function means a shift 2 units to the left. So, the starting point for $\sqrt{x + 2}$ will be at $(-2, 0)$.

3. Vertical shift: The $+1$ outside the square root indicates a shift 1 unit upwards. This will move the entire graph of $\sqrt{x + 2}$ one unit higher.

Thus, the graph of $h(x) = \sqrt{x + 2} + 1$ will start at $(-2, 1)$, and will follow the typical shape of the square root graph, shifted accordingly.

Now, let's compare this with the provided options in the image. Based on the transformations, the graph that best matches this description is the one that starts at $(-2, 1)$, with the familiar square root shape.

Do you need further help analyzing the specific options provided in the image? Feel free to ask!

Related questions:

1. How do horizontal and vertical shifts affect the graph of functions?
2. What would happen if the transformation were $h(x) = \sqrt{x - 3} - 2$?
3. How can we apply this method to graph other transformations of basic functions?
4. Can the graph of $h(x)$ have any other transformations, such as reflections?
5. What are the key points that help identify transformations in a graph?

Tip: When identifying transformations, always look for changes inside the function (affecting $x$) for horizontal shifts and changes outside the function (affecting the whole graph vertically).