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)=−4x²+8x−3

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function \(f(x) = -4x^2 + 8x - 3\). Here, the coefficient \(a = -4\).
2. Determine whether the function has a minimum or maximum value by looking at the sign of \(a\). Since \(a = -4\) is negative, the parabola opens downward, so the function has a maximum value.
3. Find the vertex of the parabola, which gives the maximum value and where it occurs. Use the vertex formula for the \(x\)-coordinate: \(x = \frac{-b}{2a}\), where \(b = 8\) and \(a = -4\).
4. Calculate the \(y\)-coordinate of the vertex by substituting the \(x\)-value back into the function: \(f\left(\frac{-b}{2a}\right) = -4\left(\frac{-b}{2a}\right)^2 + 8\left(\frac{-b}{2a}\right) - 3\).
5. Identify the domain and range of the function. The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range depends on the maximum value found at the vertex; since the parabola opens downward, the range is \((-\infty, \text{maximum value}]\).

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 all values greater than or equal to the minimum; if downward, all values less than or equal to the maximum.