Consider the following function.

f(x)=x−2‾‾‾‾‾√−4

Step 1 of 2:

Determine the more basic function that has been shifted, reflected, stretched, or compressed.

The function provided is:

\[
f(x) = \sqrt{x - 2} - 4
\]

### Step 1: Identify the Basic Function

The core of this function is the square root, \(\sqrt{x}\). The more basic function is:

\[
g(x) = \sqrt{x}
\]

Now, let's analyze the transformations applied to this basic function to get \(f(x)\).

### Transformations:

1. **Horizontal Shift**: 
   - Inside the square root, we have \(x - 2\). This represents a horizontal shift to the right by 2 units.
   - The transformation is \(g(x) = \sqrt{x - 2}\).

2. **Vertical Shift**:
   - The constant \(-4\) outside the square root represents a vertical shift downward by 4 units.
   - The transformation is \(f(x) = \sqrt{x - 2} - 4\).

Thus, the basic function \(g(x) = \sqrt{x}\) has been shifted to the right by 2 units and shifted downward by 4 units to obtain \(f(x)\).

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

Here are 5 related questions to expand on this topic:
1. What is the domain of the function \(f(x)\)?
2. How would you reflect this function across the x-axis?
3. What effect does multiplying the square root by a constant have?
4. How would the graph change if the function was \(f(x) = -\sqrt{x - 2} - 4\)?
5. What would the function look like if the vertical shift was +4 instead of -4?

**Tip**: A horizontal shift is always the opposite of the sign inside the function. For example, \(x - 2\) means shifting right by 2, not left!

Consider the following function. f(x) = sqrt(x - 2) - 4. Determine the more basic function that has been shifted, reflected, stretched, or compressed.

Explore the transformation of the square root function f(x) = sqrt(x - 2) - 4, including horizontal and vertical shifts. Understand the step-by-step analysis of function transformations for students in Grades 9-11.

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