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For all quadratic functions, the domain is always R? (double R)

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For all quadratic functions, the domain is always R? (double R). In interval notation, we write: (−∞,∞)(-∞,∞). This means that any real number can be used as an input value.

If the quadratic has a positive lead coefficient, like y = 3x2−43x2-4, 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,∞)[min,∞). This particular parabola has a vertex at (0, -4). Its range is: [−4,∞)[-4,∞). See graph below.

If a quadratic has a negative lead coefficient, like y = −12x2−4x+8-12x2-4x+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: (−∞,16](-∞,16]. (see graph) 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.