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)=5x²−5x

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function \(f(x) = 5x^{2} - 5x\). Here, the coefficient \(a = 5\).
2. Since \(a > 0\), the parabola opens upward, which means the function has a minimum value (not a maximum).
3. To find the vertex (where the minimum occurs), use the vertex formula for the \(x\)-coordinate: \(x = -\frac{b}{2a}\). Here, \(b = -5\), so calculate \(x = -\frac{-5}{2 \times 5}\).
4. Substitute the \(x\)-value found into the function \(f(x)\) to find the minimum value: \(f\left(-\frac{b}{2a}\right) = 5\left(-\frac{b}{2a}\right)^{2} - 5\left(-\frac{b}{2a}\right)\).
5. Determine the domain and range: The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range starts from the minimum value found and goes to infinity, so it is \([\text{minimum value}, \infty)\).

Key Concepts

Properties of Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax² + 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: if the parabola opens upward, the range is all values greater than or equal to the minimum; if downward, all values less than or equal to the maximum.