In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. Maximum = -6 at x = 10

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for analyzing their domains and ranges.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For quadratic functions, the domain is typically all real numbers, as there are no restrictions on the values that x can take. Identifying the domain is crucial for understanding the behavior of the function across its entire graph.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For a quadratic function that opens downwards, like the one described with a maximum value, the range will be all values less than or equal to the maximum point. In this case, since the maximum is -6, the range would be (-∞, -6]. Understanding the range helps in determining the vertical extent of the graph.