https://file.aiquickdraw.com/mathbot/file/17280644941232q6icbja.jpg

Let's analyze the piecewise function \( f(x) \) given in the image:

\[
f(x) =
\begin{cases}
-\frac{1}{4}x - 3, & \text{for } x < -4 \\
-4, & \text{for } -4 \leq x \leq 3 \\
-x, & \text{for } x > 3
\end{cases}
\]

We will break down each part and match it with the appropriate graph:

1. **For \( x < -4 \)**: The function is \( f(x) = -\frac{1}{4}x - 3 \).
   - This is a linear function with a slope of \( -\frac{1}{4} \) and a y-intercept at \( -3 \).
   - The graph should show a line with a negative slope for values of \( x \) less than \(-4\).

2. **For \( -4 \leq x \leq 3 \)**: The function is \( f(x) = -4 \).
   - This represents a horizontal line at \( y = -4 \) for the interval from \( x = -4 \) to \( x = 3 \).
   - The graph should show a flat line at \( y = -4 \) between these values.

3. **For \( x > 3 \)**: The function is \( f(x) = -x \).
   - This is a linear function with a slope of \( -1 \), passing through the origin if extended.
   - The graph should display a line with a slope of \( -1 \) for \( x > 3 \).

Now, looking at the four graph options:

- **Option A** shows a correct negative slope for \( x < -4 \), a horizontal line at \( y = -4 \), and a correct slope of \( -1 \) for \( x > 3 \). This option matches the piecewise function description.
  
- **Option B** shows a different pattern for the slope, so it's incorrect.

- **Option C** also differs in how it handles the lines, making it incorrect.

- **Option D** is similarly inconsistent with the given piecewise function.

Thus, **Option A** is the correct graph for this piecewise function.

Would you like more detailed explanations for any of the steps? Here are some related questions to consider:

1. How do you find the slope and y-intercept of a linear equation?
2. What is the significance of the domain restrictions in a piecewise function?
3. How do you graph a constant function like \( f(x) = -4 \)?
4. Can you explain how the slope of a line affects the steepness and direction?
5. What are some strategies for graphing piecewise functions efficiently?

**Tip:** Always carefully check the boundaries (open or closed circles) in piecewise function graphs to ensure they match the function definition exactly.

Graph the piecewise function f(x) = { -1/4x - 3, for x < -4; -4, for -4 ≤ x ≤ 3; -x, for x > 3 }.

Learn how to graph a piecewise function with three parts: f(x) = -1/4x - 3 for x < -4, f(x) = -4 for -4 ≤ x ≤ 3, and f(x) = -x for x > 3. This problem covers key concepts in piecewise functions, linear equations, and graphing, suitable for students in Grades 9-12.

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