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The given function \( f(x) \) is a piecewise function defined as follows:

\[
f(x) =
\begin{cases}
-4 & \text{if } -6 \leq x < -2 \\
-\frac{1}{2}x + 3 & \text{if } -2 \leq x < 6
\end{cases}
\]

### Step-by-Step Breakdown:
1. **For \( -6 \leq x < -2 \)**:
   The function is a constant: \( f(x) = -4 \). This means that for all values of \( x \) in this interval, the function has a value of -4. So, you would draw a horizontal line at \( y = -4 \) for \( x \) from -6 to -2 (excluding \( x = -2 \)).

2. **For \( -2 \leq x < 6 \)**:
   The function is linear: \( f(x) = -\frac{1}{2}x + 3 \). This is a line with a slope of -1/2 and a y-intercept of 3. For this interval, you plot the line starting from \( x = -2 \) (including this point) up to \( x = 6 \) (excluding \( x = 6 \)).

### Key Points to Plot:
- For \( x = -6 \) to \( x = -2 \), \( f(x) = -4 \). So, plot points like (-6, -4) and (-2, -4).
- For \( x = -2 \), \( f(-2) = -\frac{1}{2}(-2) + 3 = 4 \). This gives the point (-2, 4).
- For \( x = 6 \), \( f(6) = -\frac{1}{2}(6) + 3 = 0 \). The point is (6, 0), but it is not included in the graph (open circle).

### Plot:
- The first segment is a horizontal line from \( (-6, -4) \) to \( (-2, -4) \).
- The second segment is a straight line from \( (-2, 4) \) to just before \( (6, 0) \), with an open circle at \( (6, 0) \).

Would you like a graph plotted for this? Let me know if you'd like more details or explanations.

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Here are 5 related questions for further understanding:
1. How do you find the slope of a line?
2. What is the difference between open and closed intervals on a graph?
3. How do you determine whether a point is included or excluded in a graph of a function?
4. Can you graph other piecewise functions with different intervals and slopes?
5. How does the slope of a line affect its steepness on a graph?

**Tip:** When graphing piecewise functions, always pay close attention to the interval boundaries to ensure you correctly include or exclude points.

Graph this function: f(x) = -4 if -6 ≤ x < -2; f(x) = -(1/2)x + 3 if -2 ≤ x < 6.

Learn how to graph a piecewise function where f(x) is constant at -4 for -6 ≤ x < -2 and linear as -(1/2)x + 3 for -2 ≤ x < 6. This solution includes detailed explanations of the steps involved, making it ideal for high school students (Grades 9-12).

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