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To determine the intervals where the function \( f(x) \) is increasing, we need to identify the segments of the graph where the slope is positive (i.e., the graph is moving upwards as we move from left to right).

### Step-by-Step Solution:
1. **Identify the Turning Points:**  
   - The graph rises up to a peak near \( x = -6 \) and \( x = -1 \).
   - It also has a trough near \( x = 4 \).

2. **Determine Increasing Intervals:**
   - The graph is increasing from:
     - **Left of \( x = -6 \)**: The graph rises up towards \( x = -6 \).
     - **Between \( x = -1 \) and \( x = 4 \)**: The graph falls after \( x = -6 \), reaches a low point near \( x = -1 \), then rises again until it reaches \( x = 4 \).
     - **Right of \( x = 4 \)**: After decreasing at \( x = 4 \), the graph starts to rise again and continues to increase towards \( x = 9 \).

Therefore, the intervals where the function \( f(x) \) is increasing are:
\[
(-\infty, -6) \cup (-1, 4) \cup (4, \infty)
\]

### Conclusion:
The function \( f(x) \) is increasing on the intervals \((- \infty, -6)\), \((-1, 4)\), and \((4, \infty)\).

Would you like further details or have any questions about this solution?

Here are some related questions for further exploration:
1. What are the intervals where the function \( f(x) \) is decreasing?
2. How can you determine the local maxima and minima from the graph?
3. What is the difference between increasing and strictly increasing functions?
4. How does one find intervals of concavity for a function?
5. What are critical points, and why are they important in determining increasing/decreasing intervals?

**Tip:** For more complex functions, you can use the first derivative to determine increasing and decreasing intervals by finding where \( f'(x) > 0 \) or \( f'(x) < 0 \).

Determine all intervals on which the graph of f is increasing.

Learn how to find the intervals where a function is increasing by analyzing its graph. Understand the steps to identify turning points and determine behavior based on slope.

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