https://file.aiquickdraw.com/mathbot/file/17266241743925nimxg44.jpg

The problem asks to graph the piecewise function:

\[
f(x) = \begin{cases} 
-3x + 9 & \text{for } x < 1 \\
-x + 14 & \text{for } x > 5 
\end{cases}
\]

### Analysis of the Function:

1. **For \( x < 1 \):**
   - The function is \( f(x) = -3x + 9 \), which is a straight line with a slope of -3 and a y-intercept of 9. 
   - When \( x = 1 \), \( f(1) = -3(1) + 9 = 6 \), so the point is (1, 6). Since \( x = 1 \) is not included in this domain (the condition is \( x < 1 \)), the point at (1, 6) will be an open circle.

2. **For \( x > 5 \):**
   - The function is \( f(x) = -x + 14 \), which is a straight line with a slope of -1 and a y-intercept of 14.
   - When \( x = 5 \), \( f(5) = -(5) + 14 = 9 \), so the point is (5, 9). Since \( x = 5 \) is not included in this domain (the condition is \( x > 5 \)), the point at (5, 9) will be an open circle.

### Verification of Your Graph:
- From the image, the line segment for the function \( -3x + 9 \) appears to be graphed correctly for values of \( x < 1 \). The graph shows a line extending with a correct slope and a point near (1, 6) marked as an open circle.
- The second segment for \( -x + 14 \) for \( x > 5 \) is not yet graphed.

### Steps to Complete the Graph:
1. **Graph the first segment** \( f(x) = -3x + 9 \) for \( x < 1 \). Ensure there is an open circle at \( (1, 6) \) and the line has a slope of -3.
2. **Graph the second segment** \( f(x) = -x + 14 \) for \( x > 5 \). Ensure there is an open circle at \( (5, 9) \) and the line has a slope of -1.
   
Would you like any more details or clarifications?

Here are five questions to expand your understanding:
1. How do you calculate the slope of a line from its equation?
2. What is the significance of an open circle in graphing piecewise functions?
3. How can you check the continuity of a piecewise function?
4. What other tools can help in graphing functions more efficiently?
5. How does the slope affect the steepness of a line?

**Tip:** When graphing piecewise functions, always ensure you mark endpoints with open or closed circles depending on whether the inequalities include equality (≤ or ≥).

Graph the piecewise function: f(x) = { -3x + 9 for x < 1, -x + 14 for x > 5 }.

This problem involves graphing a piecewise function with two linear equations: f(x) = -3x + 9 for x < 1 and f(x) = -x + 14 for x > 5. Learn how to properly represent each segment with correct slopes and endpoints using open circles for non-inclusive points. Suitable for students in Grades 9-12.

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