Which function is positive for the entire interval [–3, –2]?

On a coordinate plane, a curved line with a minimum value of (0, negative 3) crosses the x-axis at (negative 3, 0) and (3, 0), and crosses the y-axis at (0, negative 3).

On a coordinate plane, a curved line with a minimum value of (2, negative 3) crosses the x-axis at (negative 1, 0) and (5, 0), and crosses the y-axis at (0, negative 1.5).

On a coordinate plane, a curved line with a minimum value of (2, 4) and a maximum value of (0.5, 6), crosses the x-axis at (negative 1.5, 0) and crosses the y-axis at (0, 5).

On a coordinate plane, a curved line with a minimum value of (negative 1.75, negative 3.9) and a maximum value of (0, 2), crosses the x-axis at (negative 2.2, 0), (negative 0.75, 0), and (0.75, 0), and crosses the y-axis at (0, 2).

Let's break down each function and determine which one is positive for the entire interval \([-3, -2]\).

### 1. **First Function**:
- Minimum value at \((0, -3)\).
- Crosses the x-axis at \((-3, 0)\) and \((3, 0)\).

This is a symmetric curve (probably a quadratic) with roots at \(-3\) and \(3\). Since it crosses the x-axis at \(-3\), the function is zero at this point and negative between \(-3\) and \(0\). Therefore, it is not positive over the entire interval \([-3, -2]\).

### 2. **Second Function**:
- Minimum value at \((2, -3)\).
- Crosses the x-axis at \((-1, 0)\) and \((5, 0)\).

Since the x-axis crossings are outside the interval \([-3, -2]\), we can infer that the function does not cross or approach zero in this interval. However, the curve’s minimum is at \((2, -3)\), meaning the function is likely negative in the interval \([-3, -2]\).

### 3. **Third Function**:
- Minimum at \((2, 4)\) and maximum at \((0.5, 6)\).
- Crosses the x-axis at \((-1.5, 0)\).

Since the x-axis crossing at \((-1.5, 0)\) is outside the interval \([-3, -2]\), and the minimum is positive at \((2, 4)\), this function is likely positive between \([-3, -2]\). Thus, this function appears to be **positive for the entire interval \([-3, -2]\)**.

### 4. **Fourth Function**:
- Minimum at \((-1.75, -3.9)\) and maximum at \((0, 2)\).
- Crosses the x-axis at \((-2.2, 0)\), \((-0.75, 0)\), and \((0.75, 0)\).

This function has a root at \(-2.2\), and the minimum at \((-1.75, -3.9)\) shows it is negative in the interval. Therefore, it is not positive over \([-3, -2]\).

### Conclusion:
The third function is the one that is positive for the entire interval \([-3, -2]\).

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Do you want more details or have any questions?  
Here are 5 related questions to explore this concept further:

1. How can we determine the positivity of a function over an interval?
2. What is the general shape of a quadratic function with given roots?
3. How can we find the points where a function crosses the x-axis and y-axis?
4. How does the behavior of a function at its maximum and minimum points affect its sign over an interval?
5. Can a cubic function be positive for an entire interval?

**Tip**: Always check the roots and critical points of a function to understand its behavior over specific intervals.

Learn how to determine which function is positive over the interval [–3, –2] by analyzing quadratic and cubic curves. This problem includes curve behavior, critical points, and positivity over intervals, suitable for Grades 9-11.

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