In Exercises 39–44, an equation of a quadratic function is...

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. $$ f(x)=6 x^{2}-6 x $$

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.
$$
f(x)=6 x^{2}-6 x
$$

Key Concepts

Completing the Square

Completing the square is a method used to rewrite a quadratic function from its standard form into vertex form. This form explicitly shows the vertex of the parabola and makes it easier to identify the function’s minimum or maximum value and understand its graph.

Vertex

The vertex of a quadratic function is the point where the function attains its extreme value (minimum or maximum). It is found using the formula x = -b/(2a) for the x-coordinate, with the y-coordinate determined by substituting this value back into the function.

Domain and Range

The domain of any quadratic function is all real numbers, while the range is determined by the vertex and the direction in which the parabola opens. If the parabola opens upward, the range starts at the y-coordinate of the vertex and extends to infinity; if it opens downward, the range extends from negative infinity up to the y-coordinate of the vertex.

Quadratic Functions

Quadratic functions are second-degree polynomial functions typically expressed in the standard form ax² + bx + c. They graph as parabolas, which are symmetric and exhibit a distinct vertex where the function reaches either its maximum or minimum value.

Leading Coefficient

The leading coefficient, represented by 'a' in the standard form, determines the direction in which the parabola opens. A positive value of 'a' means the parabola opens upward, resulting in a minimum value at the vertex, whereas a negative value indicates it opens downward, yielding a maximum value at the vertex.

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