Given the vertex, (2, -4) as the minimum point, and that it is an upward-facing parabola that passes point (1, -2):

We could substitute these values into the vertex form of the quadratic function,

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

where:

(h,k) is the vertex

The axis of symmetry is the vertical line x = 2

The value of “a” determines whether the graph opens up or down, and makes the parent function wider or narrower.

The value of “h” determines how far left or right the parent function is translated.

The value of “k” determines how far up or down the parent function is translated.

Now that we’ve established these definitions, we can substitute the given values into the vertex form:

f(x) = a(x - 2)^2 - 4

In order to determine the value of “a”, we must substitute the values of the other given point, (1, -2) into the equation:

f(x) = a(x - 2)^2 - 4

-2 = a(1 - 2)^2 - 4

-2 = a(-1)^2 - 4
-2 = a - 4

Add 4 to both sides to solve for “a”

-2 + 4 = a - 4 + 4
2 = a

Now that we have the value of a = 2, we can establish our quadratic function in vertex form:

f(x) = 2(x - 2)^2 - 4

I’m also including a screenshot of the graph of the function we came up with.

Please mark my answers as the Brainliest if you find my explanations helpful :)