Quadratic Functions and Equations
Quadratic functions are polynomial functions of degree 2, forming the foundation for modeling many real-world phenomena that involve acceleration, area relationships, and optimization problems.
Definition and Standard Form
A quadratic function can be written in the standard form:
f(x) = ax² + bx + c
where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is called a parabola.
Key Features of Quadratic Functions
Vertex Form
A quadratic function can be rewritten in vertex form:
f(x) = a(x - h)² + k
where (h, k) represents the vertex of the parabola.
- If a > 0, the parabola opens upward, and the vertex is a minimum point.
- If a < 0, the parabola opens downward, and the vertex is a maximum point.
The vertex form is particularly useful for identifying the maximum or minimum value of a quadratic function.
Axis of Symmetry
Every parabola has an axis of symmetry that passes through the vertex. For a quadratic function in standard form, the axis of symmetry is given by:
x = -b/(2a)
This vertical line divides the parabola into two mirror images.