https://file.aiquickdraw.com/mathbot/file/1726721706811va1zqmpd.jpg

https://file.aiquickdraw.com/mathbot/file/1726721706812n5uiov26.jpg

Let's break down and answer each part of the problem from the images you've uploaded.

### Original Function:
\[ f(x) = \sqrt{x} \]

#### 1. Reflection across the x-axis:
When reflecting a function across the x-axis, we multiply the entire function by \(-1\). This gives:
\[ f(x) = -\sqrt{x} \]

So, the correct choice is:
\[ f(x) = -\sqrt{x} \]

#### 2. Shift left by 2 units:
To shift a function left by \(k\) units, replace \(x\) with \(x + k\). In this case, shifting left by 2 units means replacing \(x\) with \(x + 2\), giving:
\[ f(x) = \sqrt{x + 2} \]

So, the correct choice is:
\[ f(x) = \sqrt{x + 2} \]

#### 3. Shift down by 1 unit:
To shift a function down by \(k\) units, subtract \(k\) from the entire function. For a downward shift of 1 unit:
\[ f(x) = \sqrt{x} - 1 \]

So, the correct choice is:
\[ f(x) = \sqrt{x} - 1 \]

#### 4. Shift right by 2 units and up by 1 unit:
To shift a function right by \(k\) units, replace \(x\) with \(x - k\), and to shift up by \(m\) units, add \(m\) to the entire function. So, shifting right by 2 and up by 1 gives:
\[ f(x) = \sqrt{x - 2} + 1 \]

So, the correct choice is:
\[ f(x) = \sqrt{x - 2} + 1 \]

#### 5. Vertical stretch by a factor of 3:
To stretch a function vertically by a factor of \(a\), multiply the entire function by \(a\). For a vertical stretch by 3:
\[ f(x) = 3\sqrt{x} \]

So, the correct choice is:
\[ f(x) = 3\sqrt{x} \]

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Would you like further details on these transformations or any other questions about the problems? Here are some additional practice questions you can consider:

1. How do you reflect a function across the y-axis?
2. What is the effect of a horizontal stretch on \( f(x) \)?
3. How would you describe the transformation \( f(x) = -2\sqrt{x - 3} + 4 \)?
4. What happens to the domain and range after a vertical stretch of \( f(x) \)?
5. How do transformations affect the intercepts of a function?

**Tip:** Always apply horizontal transformations (like shifts or stretches) inside the function's argument and vertical ones outside.

Suppose f(x) = √x. Provide the function for each of the following transformations: reflection across the x-axis, left shift, down shift, right shift and up shift, and vertical stretch.

Learn how to apply transformations to square root functions, including reflections, horizontal and vertical shifts, and vertical stretches. This guide covers the essential formulas and steps needed to transform f(x) = √x.

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