For all quadratic functions, the domain is always ##RR## (double R). In interval notation, we write: ##(-infty,infty)##. This means that any real number can be used as an input value.
If the quadratic has a positive lead coefficient, like y = ##3x^2- 4##, that 3 tells us that the parabola (graph shape) is opening upward and will have a vertex that is a minimum. Once we find that minimum y-value, that is where our Range begins. From low to high, the y-values will be from the minimum, to infinity. We write it like this in interval notation: ##[min, infty)##. This particular parabola has a vertex at (0, -4). Its range is: ##[-4,infty)##. See graph below.
If a quadratic has a negative lead coefficient, like y = ##-1/2x^2-4x+8##, its graph will open downward, with a vertex that is a maximum. The range is always reported as lowest value to highest value. In this case, negative infinity up to and including that maximum. The range of this function is: ##(-infty,16]##.
If you are not sure how to find the vertex of a parabola, that’s another story for another time…there are several ways! I hope this helped with domain and range in two different situations.
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