For a quadratic function that decreases and then increases, it has a maximum point (vertex) and opens downwards. This means the range of the function will be from negative infinity up to and including the maximum value at the vertex.
In your options:
- \( -\infty < y \leq 3 \)
- \( -\infty < y \leq 4 \)
- \( -\infty < y < 5 \)
- None of these are correct.
The first two options suggest that the maximum value is 3 and 4, respectively, which are valid ranges for a max point. The third option suggests a maximum value of less than 5, which is not correct if we consider the value at the maximum point.
The appropriate choice for the range of a quadratic that decreases and then increases will include values up to the maximum point, and it could be either \( -\infty < y \leq 3 \) or \( -\infty < y \leq 4 \).
If we assume that the maximum of the quadratic is \( 4 \) for the sake of correctness, the best answer would be:
−∞<y≤4
So the possible range is \( -\infty < y \leq 4 \).