https://file.aiquickdraw.com/mathbot/file/1727316191449c8i2ebc7.jpg

The problem asks to graph the piecewise function:

\[
f(x) = 
\begin{cases} 
6 & \text{for } -6 \leq x < 3 \\
-x + 8 & \text{for } 3 < x \leq 6 
\end{cases}
\]

### Step 1: Understanding the Intervals
- For \( -6 \leq x < 3 \), the function is a constant \( f(x) = 6 \). 
- For \( 3 < x \leq 6 \), the function is \( f(x) = -x + 8 \), which is a linear equation with a slope of \(-1\) and a y-intercept at \(8\).

### Step 2: Graphing the Function

#### Part 1: \( f(x) = 6 \) for \( -6 \leq x < 3 \)
- This is a horizontal line at \( y = 6 \) from \( x = -6 \) to \( x = 3 \).
- At \( x = -6 \), the point is \((-6, 6)\), and since the interval includes \( -6 \), this is a **closed circle**.
- At \( x = 3 \), the point is \((3, 6)\), but since the inequality is strict (\( x < 3 \)), we use an **open circle** at this point.

#### Part 2: \( f(x) = -x + 8 \) for \( 3 < x \leq 6 \)
- This is a line with a negative slope of \(-1\) and y-intercept \(8\).
- The graph starts just after \( x = 3 \) and continues to \( x = 6 \).
- At \( x = 6 \), \( f(6) = -6 + 8 = 2 \), so we plot the point \( (6, 2) \) with a **closed circle**.
- To find another point on this line, evaluate \( f(4) \): \( f(4) = -4 + 8 = 4 \), so \( (4, 4) \) is another point on the line.

### Graphing Summary
- Draw a horizontal line at \( y = 6 \) from \( x = -6 \) to \( x = 3 \), using a closed circle at \((-6, 6)\) and an open circle at \((3, 6)\).
- Draw a sloped line from just above \( x = 3 \) to \( x = 6 \), connecting points such as \( (4, 4) \) and \( (6, 2) \), with a closed circle at \( (6, 2) \).

Let me know if you have any questions or would like further clarification!

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### Related Questions
1. How do you handle piecewise functions with more than two intervals?
2. What happens if a function has discontinuities?
3. How do you graph absolute value functions?
4. How does the slope affect the shape of a linear function?
5. How do you determine whether to use an open or closed circle at a boundary?

### Tip: 
Always check the interval boundaries carefully to decide between open and closed circles when graphing piecewise functions!

Graph the following function on the axes provided: f(x) = { 6 for -6 <= x < 3; -x + 8 for 3 < x <= 6 }

Learn how to graph the piecewise function: f(x) = 6 for -6 <= x < 3, and f(x) = -x + 8 for 3 < x <= 6. This problem covers key concepts in graphing piecewise functions, including horizontal and sloped lines, and the use of open and closed circles to indicate boundary values.

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