https://file.aiquickdraw.com/mathbot/file/1727923929662izo4uunt.jpg

The given function to graph is:

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

### Analysis and Key Points:
1. **Domain**: The expression under the square root, \( x - 4 \), must be non-negative. Thus, \( x - 4 \geq 0 \), which implies \( x \geq 4 \). The domain of the function is \( [4, \infty) \).

2. **Leftmost Point**: The leftmost point is when \( x = 4 \). Plugging \( x = 4 \) into \( f(x) \):

   \[
   f(4) = -2\sqrt{4 - 4} = -2\sqrt{0} = 0
   \]

   So, the point is \( (4, 0) \).

3. **Additional Points**:
   Let's choose three additional points in the domain \( x \geq 4 \).

   - **Point 1**: Let \( x = 5 \).
     
     \[
     f(5) = -2\sqrt{5 - 4} = -2\sqrt{1} = -2
     \]
     Point: \( (5, -2) \).

   - **Point 2**: Let \( x = 8 \).
     
     \[
     f(8) = -2\sqrt{8 - 4} = -2\sqrt{4} = -4
     \]
     Point: \( (8, -4) \).

   - **Point 3**: Let \( x = 12 \).
     
     \[
     f(12) = -2\sqrt{12 - 4} = -2\sqrt{8} = -2\sqrt{4 \cdot 2} = -4\sqrt{2} \approx -5.66
     \]
     Point: \( (12, -5.66) \).

### Points to Plot:
- \( (4, 0) \)
- \( (5, -2) \)
- \( (8, -4) \)
- \( (12, -5.66) \)

You can now use these points to plot the function on the graph.

Would you like more details or have any questions?

### Related Questions:
1. What is the range of the function \( f(x) = -2\sqrt{x - 4} \)?
2. How does the graph of \( f(x) = -2\sqrt{x - 4} \) compare to \( g(x) = \sqrt{x} \)?
3. What are the x-intercepts and y-intercepts of the function?
4. How does changing the coefficient \( -2 \) affect the shape of the graph?
5. What transformations are applied to the parent function \( y = \sqrt{x} \)?

**Tip**: When graphing square root functions, always consider the domain and transformations such as vertical stretching or reflecting.

Graph the function f(x) = -2√(x - 4). Plot four points on the graph of the function: the leftmost point and three additional points.

Learn how to graph the function f(x) = -2√(x - 4) by plotting four key points, including the leftmost point, and understand the domain and range of the function. Suitable for high school students in Grades 9-12.

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