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)=3x^2−12x−1

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function $f\left(x\right)=3x^2-12x-1$. Since the coefficient of $x^2$ is positive ($3$), the parabola opens upward, indicating the function has a minimum value.
2. To find the vertex, use the vertex formula for the x-coordinate: $x=\; -\backslash frac\{b\}\{2a\}$, where $a\; =\; 3$ and $b\; =\; -12$. Substitute these values to find the x-coordinate of the vertex.
3. Substitute the x-coordinate of the vertex back into the function $f(x)$ to find the minimum value (the y-coordinate of the vertex). This gives the minimum value of the function and the point where it occurs.
4. The domain of any quadratic function is all real numbers, so the domain is $(-\backslash infty,\; \backslash infty)$.
5. Since the parabola opens upward and the vertex represents the minimum value, the range is all real numbers greater than or equal to the minimum value found in step 3. Express the range in interval notation accordingly.

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function f(x) = ax^2 + bx + c is the point where the function attains its minimum or maximum value. It can be found using the formula x = -b/(2a). The y-coordinate of the vertex is the function value at this x, representing the minimum if a > 0 or maximum if a < 0.

Determining Minimum or Maximum Values

The sign of the leading coefficient 'a' in a quadratic function determines whether the parabola opens upward or downward. If a > 0, the parabola opens upward, and the vertex is a minimum point. If a < 0, it opens downward, and the vertex is a maximum point.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward, the range is all y-values greater than or equal to the vertex's y-coordinate; if downward, all y-values less than or equal to the vertex's y-coordinate.