Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = -1/2 (x + 1)^2 - 3

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'. Understanding the general shape and properties of parabolas is essential for analyzing their characteristics.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens upwards or downwards. For the function f(x) = -1/2 (x + 1)^2 - 3, the vertex can be found directly from the vertex form of a quadratic equation, which is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values). For quadratic functions, the domain is typically all real numbers, while the range depends on the vertex and the direction the parabola opens. Understanding these concepts is crucial for accurately describing the behavior of the function.