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)=3x²−12x−1

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function \(f(x) = 3x^2 - 12x - 1\). Here, the coefficient \(a = 3\). Since \(a > 0\), the parabola opens upward, which means the function has a minimum value.
2. Find the vertex of the parabola, which gives the minimum value and where it occurs. Use the vertex formula for the \(x\)-coordinate: \(x = -\frac{b}{2a}\). Here, \(b = -12\) and \(a = 3\), so calculate \(x = -\frac{-12}{2 \times 3}\).
3. Substitute the \(x\)-value of the vertex back into the function to find the minimum value: \(f\left(-\frac{b}{2a}\right) = 3\left(-\frac{b}{2a}\right)^2 - 12\left(-\frac{b}{2a}\right) - 1\).
4. Determine the domain of the function. Since \(f(x)\) is a quadratic function, its domain is all real numbers, which can be written as \((-\infty, \infty)\).
5. Determine the range of the function. Since the parabola opens upward and has a minimum value at the vertex, the range is all real numbers greater than or equal to the minimum value found in step 3. Express the range as \([\text{minimum value}, \infty)\).

Key Concepts

Properties of Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. This determines whether the function has a minimum (a > 0) or maximum (a < 0) value.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its minimum or maximum value. It can be found using the formula x = -b/(2a). Substituting this x-value back into the function gives the corresponding minimum or maximum value.

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 and the direction the parabola opens: if it opens upward, the range is from the minimum value to infinity; if downward, from negative infinity to the maximum value.