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To determine the interval where the function is both increasing and linear, we need to carefully analyze the graph.

From the graph:
1. A **linear** function has a constant slope, meaning it appears as a straight line.
2. An **increasing** function has a positive slope (the graph rises as you move from left to right).

### Let's examine the intervals:
- **Between \( x = -9 \) and \( x = -4 \)**: The function is decreasing (the graph slopes downward), so this is not the correct interval.
- **Between \( x = -4 \) and \( x = -1 \)**: The graph is non-linear, curving up and down. So, this is also incorrect.
- **Between \( x = -1 \) and \( x = 5 \)**: The function is increasing from \( x = -1 \) to \( x = 1 \), but only a portion is linear.
- **Between \( x = 5 \) and \( x = 9 \)**: The function is increasing, but the graph is not linear here. The curve shows a slope change, indicating it's non-linear.

The correct interval is **between \( x = -1 \) and \( x = 5 \)**, as this contains a portion that is increasing and linear (a straight, rising line).

Would you like more details on this? Here are 5 related questions you might explore:
1. What distinguishes a linear from a non-linear function in general?
2. How can we identify increasing vs decreasing intervals on a graph?
3. Can a function be increasing but non-linear?
4. How does one calculate the slope of a linear function from a graph?
5. What does it mean if a function is decreasing and linear?

**Tip**: Always check both the slope and the shape of the graph to determine if a function is increasing/decreasing and linear/non-linear!

The graph of a function is shown below. On which interval is the function increasing and linear?

Explore how to determine the interval where a function is both increasing and linear on a graph. This problem involves identifying the characteristics of linear and increasing functions, analyzing slopes, and applying the definition of linear functions. Ideal for Grades 8-10.

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