Funktionskurva

Funktionskurva, or the function curve, is the graph of a function y = f(x) in the Cartesian plane. It consists of all points (x, f(x)) where x lies in the function’s domain, the set of input values for which the function is defined. The corresponding outputs form the range. The funktionskurva provides a visual representation of how the dependent variable y changes with the input x.

The curve helps analyze the behavior of the function. For continuous functions, the graph is a connected line or smooth curve, while discrete or piecewise functions may appear as separate points or line segments. The slope of the curve at a point relates to the derivative, indicating the instantaneous rate of change. Properties such as monotonicity, extrema, symmetry, and periodicity are read directly from the shape and position of the curve. Second-order information, like convexity or concavity, comes from the second derivative.

Common examples include linear functions y = ax + b, quadratic functions y = ax^2 + bx + c, exponential functions y = a·e^(kx), logarithmic functions y = a·log_b(x), and trigonometric functions such as y = sin(x). Transformations such as horizontal or vertical shifts, stretching, compression, and reflections alter the funktionskurva in predictable ways.

In practice, the funktionskurva is a fundamental tool in mathematics and applied fields. It supports reasoning about limits, continuity, and approximation, and it is essential in data visualization, modeling, and teaching the concepts of function behavior. Graphs can be constructed analytically from formulas or plotted using graphing software and calculators.