Final answer:
Given three points, one can determine if one point is the vertex of a parabola and locate this vertex as well by defining a unique parabolic curve through geometric or algebraic methods. With only two points, one cannot determine the axis of symmetry of a parabola, as there are infinitely many parabolas with various vertices and axes that could pass through these points.
Step-by-step explanation:
To answer the questions posed, we must understand the properties of a parabola and its vertex. With three points, it is generally possible to determine if one of the points is the vertex of a parabola. If those points are not collinear and can be used to define a unique parabola, then by using quadratic regression or geometric methods, one can find the vertex which is the maximum or minimum point of the parabola.
Moreover, with three non-collinear points, it is also possible to determine the location of the vertex of the parabola. Through these points, a parabolic curve can be defined, and its vertex, the point where the first derivative is zero (representing the peak or trough of the curve) can be found mathematically.
However, with just two points, one cannot determine the location of the axis of symmetry of a parabola unambiguously, because there are infinitely many parabolas that pass through any two points with different vertices and axes of symmetry. Thus without additional information, the position of the axis of symmetry cannot be determined.