Scheitelwert

Scheitelwert is a term used in German mathematics to denote the value of the vertex of a quadratic function, i.e., the y-coordinate of the parabola’s vertex. For a quadratic function f(x) = ax^2 + bx + c with a ≠ 0, the vertex has x-coordinate x_v = -b/(2a) and y-coordinate y_v = f(x_v). The Scheitelwert is given by y_v = f(-b/(2a)) = c - b^2/(4a) = (4ac - b^2)/(4a). If a > 0, the vertex corresponds to a minimum; if a < 0, it corresponds to a maximum.

In many contexts the term Scheitelpunkt refers to the vertex point itself, while Scheitelwert refers specifically to the y-coordinate of that point. The concept is primarily used for quadratic functions, where the extremum occurs at the vertex. For higher-degree polynomials, extremal values are typically discussed as extrema or extremwerte and often require calculus to determine.

Example: Consider f(x) = 2x^2 + 3x + 1. Then x_v = -3/4 = -0.75, and y_v = f(-0.75) = -1/8 = -0.125. Thus the vertex is at (-0.75, -0.125) and the Scheitelwert (the minimum value) is -1/8.

Notes: If a = 0, the function is linear and has no vertex, so the Scheitelwert concept does not apply. The computation above assumes real coefficients and real-valued functions.