The graph of a quadratic function f is given. (a) Find the...

The graph of a quadratic function $f$ is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of $f$. (c) Find the domain and range of $f$.
$f(x)=2 x^2-4 x-1$
(GRAPH CAN'T COPY)

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial expressed in the standard form f(x) = ax² + bx + c. It graphs as a parabola, and its shape and orientation are determined by the coefficient a. This function serves as the foundation for analyzing properties such as the vertex, axis of symmetry, and the function's extremum (maximum or minimum).

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the function reaches its minimum (if the parabola opens upward) or maximum (if it opens downward). It is found using the formula x = -b/(2a) for the x-coordinate, and substituting this back into the function gives the corresponding y-coordinate. The vertex is essential for graphing the function and understanding its extremum value.

Completing the Square

Completing the square is a method used to convert a quadratic function from standard form into vertex form, which makes the vertex apparent. By rearranging and factoring the quadratic expression, one can rewrite the function as f(x) = a(x - h)² + k, where (h, k) is the vertex. This technique is useful for analyzing the properties of the quadratic function including its vertex and axis of symmetry.

Parabola Direction and Extremum

The coefficient a in a quadratic function indicates the direction in which the parabola opens. If a > 0, the parabola opens upward, meaning it has a minimum value at the vertex. Conversely, if a < 0, it opens downward and has a maximum value at the vertex. This property helps determine whether the vertex represents a minimum or maximum value.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers because any real x-value can be substituted into the function. The range, however, depends on the direction of the parabola's opening. For a parabola opening upward (a > 0), the range is from the y-coordinate of the vertex to infinity, while for one opening downward (a < 0), the range extends from negative infinity to the y-coordinate of the vertex.

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