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Bot GPT-4o To analyze which statement is true about the function's graph, let's break down each statement and examine what it implies:
1. **Over the interval [–2.5, 0.5], the local maximum is 2.**
- This statement suggests that within the specified interval, the highest value the function reaches is 2. To verify this, we would need to look at the function's behavior on this interval and see if the maximum point indeed has a value of 2.
2. **As the x-values go to positive infinity, the function’s values go to negative infinity.**
- This statement addresses the end behavior of the function. If true, as x approaches positive infinity, the function's values should decrease without bound.
3. **The function is decreasing over the interval (–1, 0.75).**
- This statement implies that the function's slope is negative between x = -1 and x = 0.75, meaning the function is consistently going down over this interval.
4. **The function is negative for the interval [–2, 0].**
- This means that the function's values are below zero for all x-values within the interval [-2, 0].
To determine which statement is true, let’s summarize the criteria required for each statement and compare them against the assumed characteristics of a graph:
- **Local maximum on [–2.5, 0.5]**:
- The highest point within this interval is 2.
- **End behavior (positive to negative infinity)**:
- Function decreases steadily as x becomes very large.
- **Decreasing function over (–1, 0.75)**:
- The function's slope is negative over this specific interval.
- **Function is negative over [–2, 0]**:
- Entire segment should be below the x-axis.
Now with these observations, let’s tentatively conclude:
- If the function has any point reaching 2 on the interval [–2.5, 0.5], it fits the first statement.
- The second statement talks about end behavior, which would be evident if the graph trends downward as x goes to positive infinity.
- The third statement needs a consistently negative slope over (–1, 0.75).
- The fourth statement requires the entire interval [–2, 0] to have function values below zero.
Without the actual graph to provide exact values, the safest option is to validate by patterns often found in commonly studied functions. Assuming typical behavior often seen in quadratic, cubic, and polynomial functions:
- **Statement 1**: Having a specific local maximum is a distinct feature often seen.
- **Statement 2**: Such end behavior indicates a polynomial function with a negative leading coefficient.
- **Statement 3**: Decrease over an interval shows a specific decreasing trend usually present.
- **Statement 4**: Strictly negative over an interval indicates intercepts and specific value ranges.
**Given these considerations** and for accuracy without further details typically omissions, **statement 1** is often more common and distinct resulting in likely validation.
**Final suggestion**: The true statement is likely:
**Over the interval [–2.5, 0.5], the local maximum is 2.**