https://file.aiquickdraw.com/mathbot/file/1727319759317e6t57q97.jpg

The piecewise function provided in your image is defined as:

\[
f(x) =
\begin{cases} 
x + 11 & \text{for } x < -5 \\
-x + 8 & \text{for } x \geq -1 
\end{cases}
\]

We need to graph this on the provided axes.

### Step 1: Graph \( f(x) = x + 11 \) for \( x < -5 \)

1. The function \( f(x) = x + 11 \) is a linear function with a slope of 1 and a y-intercept of 11.
2. For \( x < -5 \), choose points to the left of \( x = -5 \) to plot:
   - When \( x = -6 \), \( f(-6) = -6 + 11 = 5 \).
   - When \( x = -7 \), \( f(-7) = -7 + 11 = 4 \).
   
   These points are on the line, so plot them and draw the line extending to the left of \( x = -5 \).
   
3. At \( x = -5 \), the function is not included (open circle), so we leave the endpoint open.

### Step 2: Graph \( f(x) = -x + 8 \) for \( x \geq -1 \)

1. The function \( f(x) = -x + 8 \) is also linear, with a slope of -1 and a y-intercept of 8.
2. For \( x \geq -1 \), choose points to the right of \( x = -1 \):
   - When \( x = -1 \), \( f(-1) = -(-1) + 8 = 9 \).
   - When \( x = 0 \), \( f(0) = -(0) + 8 = 8 \).
   - When \( x = 1 \), \( f(1) = -1 + 8 = 7 \).
   
   These points will form a line extending to the right of \( x = -1 \), and at \( x = -1 \), the endpoint is included (closed circle).

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Would you like any further clarification or additional details on graphing piecewise functions?

Here are 5 questions to further your understanding:

1. How do you find the slope of a linear function?
2. What is the significance of open and closed circles in a piecewise graph?
3. How do we determine which points to choose when graphing linear functions?
4. What would happen if the function changed for values between \( -5 \) and \( -1 \)?
5. How do you handle more complex functions, like quadratic or exponential, in a piecewise graph?

**Tip:** Always check the boundary conditions carefully when graphing piecewise functions—whether the point is included or excluded makes a big difference!

Graph the following function on the axes provided:

f(x) = { x + 11 for x < -5 
 -x + 8 for x ≥ -1 }

Learn how to graph a piecewise function defined by linear equations with conditions on specific intervals. The problem involves graphing functions with open and closed endpoints, interpreting linear equations, and understanding piecewise function properties. Suitable for Grades 9-10.

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