Range of Polynomial Functions (F 509)

A parabola has a floor or ceiling because both ends of its graph point the same direction. x² grows large and positive whether x is large and positive or large and negative, so an upward parabola can't ever go below its lowest point. A cubic has no such limit: as x goes to negative infinity, x³ goes to negative infinity too, and as x goes to positive infinity, x³ does the same. One end heads down, the other up, so the graph sweeps through every real number on its way.

For a quadratic f(x) = ax² + bx + c, the vertex is the single turning point, and its y-coordinate is the boundary of the range. If a is positive, the parabola opens upward and the vertex gives the minimum; every other output is equal to or greater than that value. If a is negative, the parabola opens downward and the vertex gives the maximum. The x-coordinate of the vertex is -b/(2a); plug that back into f(x) to get the boundary value itself.

This sits right next to domain, which for any polynomial is all real numbers, since you can plug in anything. Range is the harder half because it depends on shape: degree, leading coefficient, and turning points. As even degree climbs from two to six, the same floor-or-ceiling behavior holds, just with more wiggles in between, while any odd-degree polynomial keeps producing every real number.