https://file.aiquickdraw.com/mathbot/file/17300475202999nor3h79.jpg

To sketch the graph of \( f(x) \) based on the piecewise function provided, let's examine each part of the function and plot accordingly on the graph.

The function is defined as:
\[
f(x) = 
\begin{cases} 
-0.5x - 7 & \text{if } x \leq -2 \\
2 & \text{if } -2 < x \leq 3 \\
-2x + 7 & \text{if } x > 3 
\end{cases}
\]

### Step-by-Step Analysis

1. **For \( x \leq -2 \):** The function is \( f(x) = -0.5x - 7 \).
   - Calculate \( f(-2) \):
     \[
     f(-2) = -0.5(-2) - 7 = 1 - 7 = -6
     \]
   - Plot the point \((-2, -6)\) with a **closed dot** because \(x = -2\) is included.
   - This part of the function is a straight line with a slope of \(-0.5\) and y-intercept of \(-7\). Continue the line to the left.

2. **For \( -2 < x \leq 3 \):** The function is \( f(x) = 2 \).
   - This is a horizontal line from \(x = -2\) to \(x = 3\).
   - At \( x = -2 \), mark an **open dot** at \((-2, 2)\) since \(x = -2\) is not included here.
   - At \( x = 3 \), mark a **closed dot** at \((3, 2)\) because \( x = 3 \) is included.

3. **For \( x > 3 \):** The function is \( f(x) = -2x + 7 \).
   - Calculate \( f(3) \):
     \[
     f(3) = -2(3) + 7 = -6 + 7 = 1
     \]
   - Plot the point \((3, 1)\) with an **open dot** because \(x = 3\) is not included.
   - This part of the function is a straight line with a slope of \(-2\) and y-intercept of \(7\). Extend the line to the right from this point.

### Plot Summary:
- **Left Section ( \( x \leq -2 \) ):** Line with equation \( f(x) = -0.5x - 7 \), starting at \((-2, -6)\) and extending left.
- **Middle Section ( \( -2 < x \leq 3 \) ):** Horizontal line at \( y = 2 \), with an open dot at \((-2, 2)\) and a closed dot at \((3, 2)\).
- **Right Section ( \( x > 3 \) ):** Line with equation \( f(x) = -2x + 7 \), starting from an open dot at \((3, 1)\) and extending right.

Would you like further details on how to interpret or draw specific parts of this graph?

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#### Follow-Up Questions
1. What is the purpose of open and closed dots in piecewise graphs?
2. How can you determine the slope and intercept of a line segment in a piecewise function?
3. Why does the function have different rules for different intervals of \(x\)?
4. What would happen if two intervals in a piecewise function overlap?
5. How can piecewise functions represent real-world situations?

#### Tip
In piecewise functions, always check the endpoints of each interval to determine whether they should have open or closed dots, depending on the inequality signs used.

Sketch a graph of f(x) based on the piecewise function provided.

Learn how to sketch the graph of a piecewise function with three linear segments: f(x) = -0.5x - 7 for x ≤ -2, f(x) = 2 for -2 < x ≤ 3, and f(x) = -2x + 7 for x > 3. This guide includes step-by-step plotting of open and closed dots based on interval endpoints, suitable for students in Grades 9-12.

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