Which function is characterized by having a maximum or minimum value that represents the highest or lowest points of a dataset?

The function that is characterized by having a maximum or minimum value, which represents the highest or lowest points of a dataset, is the quadratic function. Quadratic functions take the form of a polynomial expression with the highest exponent of 2, typically written as ( f(x) = ax^2 + bx + c ). These graphs form parabolic shapes, which either open upwards or downwards depending on the coefficient of the quadratic term (( a )).

When the parabola opens upwards, the vertex represents the minimum value of the function, while if it opens downwards, the vertex marks the maximum value. Therefore, the vertex of a quadratic function is crucial as it identifies these extreme values of the dataset, which is foundational in various applications such as optimization problems.

In contrast, logistic, exponential, and cubic functions do not have the same characteristics regarding maximum or minimum points. Logistic functions are defined for growth and saturation and do not have a single maximum or minimum but approach horizontal asymptotes. Exponential functions typically increase or decrease without bound, lacking a maximum or minimum value. Cubic functions can have local maxima or minima but do not possess a single highest or lowest point across the entire dataset as clearly defined as in quadratics.