How do you determine the range of a quadratic function?

Assuming you mean a polynomial of degree #2# of the form

#ax^2+bx+c#

they will always represent a parabola. Depending on the sign of #a#, there are two possibilities:

Case 1: #a>0#

In this case, the parabola "points upwards" This means that the vertex of the parabola is its global minimum, and so the graph can't reach any point lower than the vertex. On the other hand, the parabola stretches towards infinity as #x\to\pm\infty#. Remembering that the vertex is located at #x=-b/a#, the range is

#[f(-b/(2a)),\infty)#

Here's an example: the vertex is the point #(-3/2,-13/4)#, and the parabola has no upper bound. So, the range is

#[-13/4,\infty)#

graph{x^2+3x-1 [-7 7 -4 9]}

Case 2: #a>0#

This case is the opposite of the previous one. In this case, in fact, the vertex is the global maximum of the parabola, and there is no lower bound. Everything else remains the same: the range is thus given by

#(-infty, f(-b/(2a))]#

If you consider the example #-3x^2 + 15x - 4#, the vertex is the point #(5/2, 59/4)#, so the range is

#(-infty, 59/4]#

graph{-3x^2 + 15x - 4 [-1 6 -5 19]}