To solve this problem, let's start by analyzing the function transformations.

The original function given is: $f(x) = 2\sqrt{x}$

We need to find $g(x)$, which is the graph of $f(x)$ shifted up 2 units and left 6 units.

Step-by-Step Solution

1. Shift Left 6 Units: Shifting a function to the left by 6 units involves replacing $x$ with $x + 6$ in the function. So, $f(x+6) = 2\sqrt{x+6}$

2. Shift Up 2 Units: Shifting a function up by 2 units involves adding 2 to the function. Therefore, $g(x) = 2\sqrt{x+6} + 2$

Thus, the formula for $g(x)$ is: $g(x) = 2\sqrt{x+6} + 2$

This is the expression you can enter in the box provided.

Would you like more details on transformations, or do you have any other questions?

Here are five related questions to expand your understanding:

1. What would be the formula if the graph was shifted down 3 units and right 4 units?
2. How would the function change if it were reflected across the x-axis?
3. How does adding a constant inside versus outside the square root affect the graph?
4. How can you generalize transformations for any function $f(x)$?
5. What happens if you apply both vertical and horizontal stretching to the function?

Tip: Remember that horizontal shifts affect the input $x$, while vertical shifts affect the entire function output.