Advanced Properties of Quadratic Functions

The wyznaczanie wzoru funkcji kwadratowej na podstawie jej własności (determining the quadratic function formula based on its properties) often involves analyzing its discriminant and roots.

The discriminant (Δ) of a quadratic function is given by: Δ = b² - 4ac

Vocabulary: The discriminant (Δ) determines the nature and number of roots of a quadratic function.

Miejsca zerowe funkcji kwadratowej (roots of quadratic function) depend on the value of Δ:

• Δ < 0: No real roots
• Δ = 0: One real root (x₀ = -b / 2a)
• Δ > 0: Two real roots (x₁ = (-b - √Δ) / 2a and x₂ = (-b + √Δ) / 2a)

Example: For Δ = 0, the quadratic function has one root, and its factored form is f(x) = a(x-x₀)².

The postać iloczynowa funkcji kwadratowej (factored form of quadratic function) depends on its roots:

• For Δ > 0: f(x) = a(x-x₁)(x-x₂)

Highlight: The maximum or minimum value of a quadratic function occurs at its vertex. For a > 0, it's the minimum; for a < 0, it's the maximum.

Understanding these properties is crucial for badanie funkcji kwadratowej (analyzing quadratic functions) and solving related problems.