Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = -3x^2 + 24x - 46

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + 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 characteristics.

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a) to determine the x-coordinate, and then substituting this value back into the function to find the corresponding y-coordinate. The vertex is crucial for identifying the maximum or minimum value of the function.

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, but the range is determined by the vertex; if the parabola opens upwards, the range starts from the vertex's y-coordinate to positive infinity, and if it opens downwards, it extends from negative infinity to the vertex's y-coordinate.