Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = (x - 5)^2 - 4

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.

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For the function f(x) = (x - 5)^2 - 4, 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.

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 and the direction the parabola opens, which can be determined from the vertex and the value of 'a'.