Definition: A quadratic function is a function whose rule may be written in the form
f(x) = ax² + bx + c where a, b and c are real numbers and a is not 0.

The graph of a quadratic function is a parabola which opens up if a > 0 and opens down if a < 0.

The y-intercept of the graph of a quadratic function is (0, c).

The zeros of a quadratic function may be found with the quadratic formula.

Note that if b² - 4ac < 0 then is a complex number and if b² - 4ac > 0 then is a real number. Of course if b² - 4ac = 0 then = 0.

Therefore, the quadratic formula may yield either real zeros of the quadratic function or complex zeros of the quadratic function.

The expression b² - 4ac is called the discriminant of the quadratic function.

Using that terminology we may observe the following:

If the discriminant of a quadratic function is positive, the quadratic function has two real zeros. They represent two distinct x-intercepts of the graph of the quadratic function.

If the discriminant of a quadratic function is zero, the quadratic function has one real zero namely . It represents the single x-intercept of the graph of the quadratic function. Since it is the only x-intercept of the graph, it must be the vertex of the parabola.

If the discriminant of a quadratic function is negative, the quadratic function has two complex zeros. They are conjugates of one another. Since only real numbers are represented on the x-axis, these complex zeros cannot represent x-intercepts of the graph of the quadratic function. Therefore the graph of the quadratic function has no x-intercepts. Therefore it is either entirely above the x-axis or entirely below the x-axis.

The vertex of the graph of a quadratic function is .

The first coordinate of the vertex of the graph of a quadratic function is midway between the x-intercepts of that quadratic function.