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 $f$.
$f(x)=3 x^2+6 x-1$
(GRAPH CAN'T COPY)

Key Concepts

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers because there are no restrictions on the values that x can take. The range, however, depends on the direction in which the parabola opens. For an upward opening parabola, the range starts from the y-coordinate of the vertex and extends to infinity, whereas for a downward opening parabola, the range extends from negative infinity to the vertex’s y-coordinate.

Vertex of a Parabola

The vertex of the parabola is the point where the function reaches its maximum or minimum value. It can be found using the formula x = -b/(2a) for the x-coordinate, and substituting this value back into the function to get the y-coordinate. The vertex represents the turning point of the parabola.

Parabola's Orientation and Extremum

The coefficient a in the quadratic function determines the orientation of the parabola. If a > 0, the parabola opens upward and has a minimum value at the vertex; if a < 0, it opens downward and the vertex is the maximum point. This property directly relates to the determination of the function's extremum (minimum or maximum value).

Standard Form of a Quadratic Function

The standard form (ax² + bx + c) is a convenient way to express a quadratic function that easily reveals its coefficients, which are used in determining the vertex, axis of symmetry, and the direction in which the parabola opens. This form makes it straightforward to apply formulas, such as the vertex formula, to analyze the graph.

Quadratic Function

A quadratic function is a second-degree polynomial function typically written in the form ax² + bx + c, where a, b, and c are constants with a ? 0. Its graph is a parabola, which is a symmetric curve that either opens upward or downward depending on the sign of the coefficient a.

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