Consider the graph of each quadratic function.(a) Give the domain and range.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. In this case, the function f(x) = -7(x + 5)^2 + 7 opens downwards because 'a' is negative.

Domain and Range

The domain of a function refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (y-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, while the range depends on the vertex of the parabola. In this case, the vertex indicates the maximum value of the function, which helps determine the range.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. For the function f(x) = -7(x + 5)^2 + 7, the vertex is located at the point (-5, 7). This point is crucial for determining the range of the function, as it represents the maximum value when the parabola opens downwards.