To determine the domain of the parabola based on the information given, let's analyze the situation:
- The top of the parabola, or its vertex, is at the point \((0, 4)\), which means it reaches a maximum value of \(4\) on the y-axis.
- The parabola also passes through the point \((2, 0)\) on the x-axis, indicating that \(x = 2\) is where the parabola intersects the x-axis (where \(y = 0\)).
Given that the parabola opens downwards (since it has a maximum point at the vertex \((0, 4)\)), it will extend to the left of the vertex, reaching the x-axis at points less than \(2\).
However, as we analyze the choices for the domain:
- A. -2 ≤ x ≤ 2: This suggests that the parabola exists between -2 and 2 on the x-axis.
- B. 0 ≤ x ≤ 2: This suggests that the parabola only exists for positive values of \(x\), from 0 to 2.
- C. x is all real numbers less than or equal to 4: This has no direct correlation to the x-values since it discusses the y-values instead.
- D. x is all real numbers: This suggests the parabola extends infinitely in both the negative and positive directions.
Since the parabola opens downward with the right intersection at \(x = 2\) and the vertex at \(x = 0\), it likely spans from the left indefinitely until it touches the x-axis at \((2, 0)\). Therefore, this suggests that the left side might be smaller than \(2\).
Thus, the correct domain is A. -2 ≤ x ≤ 2 since it includes all points where the parabola exists from -2 to 2, accessing the full range of the parabola between these intersections.