Final answer:
None of the provided options correctly identify the vertex, axis of symmetry, maximum value, and range of the parabola y = -x² - x. The vertex is (-1/2, 1/4), the axis of symmetry is x = -1/2, the maximum value is 1/4, and the range is y ≤ 1/4.
Step-by-step explanation:
To identify the vertex, axis of symmetry, the maximum or minimum value, and the range of the parabola y = -x² - x, we first complete the square to find the vertex form of the equation. The given equation can be rewritten as y = -(x² + x + 1/4 - 1/4). Completing the square gives us y = -(x + 1/2)² + 1/4. The vertex form of the equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Comparing our completed square form, we get that the vertex is (-1/2, 1/4).
The axis of symmetry is the vertical line that passes through the vertex, hence it is x = -1/2. Since a = -1 in our equation and it is negative, the parabola opens downwards, indicating a maximum value, which is the y-coordinate of the vertex, 1/4. Lastly, the range of a downward-opening parabola is all real numbers less than or equal to its maximum value, which in this case is y ≤ 1/4.
Therefore, the correct statement from the given options would be none of them, because the correct vertex is (-1/2, 1/4), the axis of symmetry is x = -1/2, the maximum value is 1/4, and the range is y ≤ 1/4, which is not listed in any of the choices.