In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x^2+8x−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 coefficient 'a' determines the direction of the parabola: if 'a' is positive, the parabola opens upwards, indicating a minimum value; if 'a' is negative, it opens downwards, indicating a maximum value.

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 a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a). The y-coordinate of the vertex gives the maximum or minimum value of the function.

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, and the range depends on the vertex's position, either extending to positive or negative infinity based on whether the parabola opens upwards or downwards.