We are given the function $f(x) = 2\sqrt{x}$, and we need to find a new function $g(x)$ based on shifts to the graph of $f(x)$.

The graph of $f(x)$ is: $f(x) = 2\sqrt{x}$

Now, we are told that $g(x)$ is obtained by shifting $f(x)$ up by 6 units and right by 1 unit.

Step 1: Shifting Right by 1 Unit

Shifting a graph to the right by 1 unit means replacing $x$ with $x - 1$. So, the function becomes: $f(x - 1) = 2\sqrt{x - 1}$

Step 2: Shifting Up by 6 Units

Shifting a graph up by 6 units means adding 6 to the function. Therefore, we add 6 to the previous expression: $g(x) = 2\sqrt{x - 1} + 6$

Final Answer:

$g(x) = 2\sqrt{x - 1} + 6$

Let me know if you would like further details or have any questions.

Here are five related questions for further practice:

1. What is the effect of shifting a graph down by 4 units instead of up?
2. How would the formula change if the graph was shifted left by 2 units?
3. What happens to the graph of $g(x)$ if it is reflected over the x-axis?
4. How would you shift $f(x)$ left by 3 units and down by 5 units?
5. Can you describe the transformations that occur if $f(x)$ is scaled vertically by a factor of 3?

Tip: To shift a graph horizontally, you change the input $x$ by adding or subtracting inside the function, while vertical shifts involve adding or subtracting directly from the function itself.