A quadratic function has a nonzero coefficient on the squared term. Its graph is a parabola. When that coefficient is positive, the parabola opens upward and the vertex is a minimum. When it is negative, the parabola opens downward and the vertex is a maximum. The domain of a polynomial rule is all real numbers unless the problem sets a narrower one.
Use the roots and the vertical intercept for different tasks
For the function below, factoring gives roots 1 and 5. Those inputs give output zero. At x equal to zero the output is 5, which is the vertical intercept. The horizontal intercepts solve a quadratic equation, while the function also assigns outputs to inputs between and beyond those intercepts.
Locate the vertex on the axis of symmetry
The midpoint of roots 1 and 5 is 3. Substituting 3 gives negative 4. Completing the square gives the same information and also works when there are no real roots. In this example the graph is the square of x minus 3, shifted down by 4. The range starts at negative 4 and includes every greater real value.
Decide whether a value is above or below the axis
Between 1 and 5, one factor is positive and the other is negative, so the output is negative. Outside that interval the factors have the same sign, so the output is positive. At the endpoints it is zero. This sign pattern uses the upward opening and two distinct roots. A downward-opening parabola with two distinct roots reverses the signs.
