information:highlighting, structural:expansion
Geometric interpretation

The function f (x) = ax + bx + c is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. If a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x -coordinate of the vertex will be located at x = − b 2 a {\displaystyle \scriptstyle x={\tfrac {-b}{2a}}}, and the y -coordinate of the vertex may be found by substituting this x -value into the function. The y -intercept is located at the point (0, c).

The solutions of the quadratic equation ax + bx + c = 0 correspond to the roots of the function f (x) = ax + bx + c, since they are the values of x for which f (x) = 0. If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x - coordinates of the points where the graph touches the x -axis. If the discriminant is positive, the graph touches the x -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x -axis.