In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. f(x) = -x^2 + 14x - 106

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 properties of quadratic functions is essential for determining their maximum or minimum values.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point of the graph, depending on its orientation. For a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a). This point is crucial for identifying the maximum or minimum value of the function, as it occurs at the vertex when the parabola opens downwards or upwards, respectively.

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 (f(x)-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, but the range is determined by the vertex and the direction the parabola opens, which affects the minimum or maximum output value.