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The piecewise function given is:

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

### To graph this:

1. **For the first piece** \( f(x) = x + 6 \) on the interval \( -4 \leq x \leq -1 \):
   - At \( x = -4 \), \( f(x) = (-4) + 6 = 2 \).
   - At \( x = -1 \), \( f(x) = (-1) + 6 = 5 \).
   - So, plot points at \( (-4, 2) \) and \( (-1, 5) \) and draw a straight line connecting these two points.

2. **For the second piece** \( f(x) = 6 \) on the interval \( -1 < x < 6 \):
   - This is a horizontal line at \( y = 6 \) from \( x = -1 \) to \( x = 6 \), excluding \( x = -1 \) (open circle at \( x = -1 \)) and including \( x = 6 \) (closed circle at \( x = 6 \)).

### Key Points:
- From \( x = -4 \) to \( x = -1 \), the graph is a line with slope 1.
- From \( x = -1 \) to \( x = 6 \), the graph is a constant horizontal line at \( y = 6 \).

Would you like a visual representation of this graph?

### Follow-up questions:
1. How does the domain affect the graph of piecewise functions?
2. What happens to the function at the boundaries where the pieces meet?
3. How do you handle open and closed circles in piecewise functions when graphing?
4. Can you interpret the slope of the linear part of the function?
5. How would the graph change if the interval for \( f(x) = x+6 \) were extended?

**Tip**: When graphing piecewise functions, always carefully check whether the boundary points are included or excluded by using open or closed circles.

Graph the following function on the axes provided: f(x) = { x + 6 for -4 ≤ x ≤ -1, 6 for -1 < x < 6 }

Learn how to graph a piecewise function with both a linear and a constant segment. The function is f(x) = {x + 6 for -4 ≤ x ≤ -1, 6 for -1 < x < 6}, covering key concepts such as piecewise functions and linear equations. Suitable for students in Grades 9-11.

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