The quadratic function

A quadratic function is a function that can be described by an equation of the form f(x) = ax² + bx + c, where a ≠ 0.
The highest degree of the polynomial is two, hence it’s called quadratic.
The coefficients a, b and c are constants and x represents an unknown variable.

The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
By completing the square of the general form, you can express a quadratic function in standard form.
The vertex of the parabola is the point at which the parabola’s curvature is at a minimum (if the parabola opens up) or maximum (if the parabola opens down).

The graph of a quadratic function is called a parabola.
If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.
The vertex represents the maximum or minimum point of the parabola.
The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides it into two symmetric halves.

The roots of a quadratic equation are the x-values where the parabola intersects the x-axis.
These can be found using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / (2a).
The term in the square root, b² - 4ac, is known as the discriminant. It can determine the types of roots of the equation:
- If the discriminant > 0, the function has two distinct real roots.
- If the discriminant = 0, the function has one real root, also known as a repeated root.
- If the discriminant < 0, the function has two complex roots.

Quadratic equations can be solved by various methods including factorisation, completing the square, using the quadratic formula, or graphically.
The method chosen often depends on the specific form of the quadratic equation. All methods should give the same roots.

Quadratic functions are used in various fields including physics, engineering, and business for solving real-world problems such as projectile motion, product pricing, and profit maximisation.