The general form of a quadratic function is:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola. Since you mentioned that the vertex of the function is $(1, -2)$ and that the parabola opens upwards, we know that $h = 1$, $k = -2$, and the coefficient $a > 0$ (since the parabola opens upwards).

Thus, the quadratic function can be written as:

$f(x) = a(x - 1)^2 - 2$

Now, since the parabola opens upwards, the value of $a$ is positive. If you have more information, such as a point on the parabola or the value of $a$, I could help further refine the function.

Would you like to provide additional details, or do you have any questions?

Here are some related questions to help you further explore quadratic functions:

1. How do we find the axis of symmetry of a quadratic function?
2. How do we calculate the roots or x-intercepts of a quadratic function?
3. How do we determine the equation of a parabola given its focus and directrix?
4. How can the vertex form be converted to standard form?
5. What is the significance of the value of $a$ in determining the "width" of a parabola?

Tip: To find the axis of symmetry for a parabola in vertex form $f(x) = a(x - h)^2 + k$, simply use $x = h$, which gives you the line of symmetry of the parabola.