Final answer:
The question pertains to the properties of a quadratic function displayed by a parabola. It specifies the vertex, axis of symmetry, direction of opening, domain, and range. Key features like the negative 'a' value, domain, and range are determined based on the given characteristics.
Step-by-step explanation:
The subject of this question is mathematics, specifically focusing on the characteristics of a quadratic function in vertex form. The student has provided information about a parabola, including its vertex (2, 5), axis of symmetry (x = 2), direction of opening (downwards), width (same), domain ((-inf, inf)), and range ((-inf, 5]).
Since the parabola opens down, we can deduce that the a value in the vertex form equation y = a(x - h)^2 + k is negative. The domain of any parabola is always all real numbers, hence (-inf, inf), but the range is limited by the vertex.
Since this parabola opens down and the vertex has a y-coordinate of 5, the range of the function must be all real numbers less than or equal to 5, which is written as (-inf, 5].