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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.

Let f(x) = 2sqrt(x). If g(x) is the graph of f(x) shifted up 6 units and right 1 unit, write a formula for g(x).

Learn how to shift the graph of f(x) = 2sqrt(x) by 6 units up and 1 unit to the right to get the new function g(x). This covers function transformations for square root functions, suitable for high school students.

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