Final answer:
A quadratic function with a vertex at (1, -2) and opening upwards can be written in vertex form as f(x) = a(x - 1)^2 - 2, where 'a' is a positive constant. This form simplifies graphing and understanding the parabola's properties, though additional information is needed to find a unique solution.
Step-by-step explanation:
Based on the information provided, the quadratic function in question is one that opens upwards and has a vertex at the point (1, -2). In mathematical terms, such functions are represented in the vertex form of the quadratic equation, which is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and 'a' determines the opening of the parabola and its width. Since the parabola opens upwards, we know that 'a' is positive. The vertex form makes it easy to graph the function and understand its properties. It also simplifies the process of finding the function's zeros and its axis of symmetry, which in this case is the line x = 1.
When you're given scenarios such as a quadratic function exhibiting specific behavior, such as having a positive value at certain points or having a slope that is changing in magnitude, this is indicative of the nature of the quadratic curve it represents. Understanding the vertex and direction of opening of the quadratic function allows one to determine where the function will have its maximum or minimum values, and how it behaves as x assumes values greater or less than the vertex's x-coordinate.
To actually find the specific function that matches a vertex of (1, -2) and opens upwards, one must consider additional points the function passes through or further restrictions on 'a'. Only then can a unique quadratic function be determined. Without additional information, multiple quadratic functions could satisfy the given vertex and direction criteria. Nevertheless, they all would exhibit the graph of a parabola that opens upward and has a vertex at (1, -2).