Introduction to Quadratic Functions
A quadratic function is a polynomial function of degree 2. It has the general form:
f(x) = ax² + bx + c
where a, b, and c are real numbers and a ≠ 0.
Key Characteristics:
- Degree: The highest power of x is 2
- Graph: Forms a parabola (U-shaped curve)
- Vertex: The turning point of the parabola
- Axis of Symmetry: Vertical line through the vertex
- Direction: Opens up if a > 0, down if a < 0
Examples:
- f(x) = x² + 2x + 1 (a = 1, b = 2, c = 1)
- f(x) = -2x² + 4x - 3 (a = -2, b = 4, c = -3)
- f(x) = 3x² - 6x (a = 3, b = -6, c = 0)
Important Notes:
- The coefficient 'a' determines the direction and width of the parabola
- The coefficient 'b' affects the position of the vertex
- The constant 'c' is the y-intercept
- Every quadratic function has exactly one vertex
Standard Form: f(x) = ax² + bx + c
This is the most common form of a quadratic function, where a, b, and c are constants.