Quadratic Function Forms and Properties

This page provides a comprehensive overview of the quadratic function, focusing on its various forms and key characteristics. The information is presented in a clear, visual manner, making it easier for students to understand the relationships between different representations of quadratic functions.

The page begins by introducing the general form of a quadratic function: y = ax² + bx + c. This form is fundamental to understanding the basic structure of quadratic equations.

Definition: The general form of a quadratic function is y = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' ≠ 0.

The direction of the parabola's opening is determined by the value of 'a':

Highlight: When a > 0, the parabola opens upward; when a < 0, it opens downward.

Next, the page introduces the canonical form of the quadratic function: y = a(x-p)² + q.

Definition: The canonical form of a quadratic function is y = a(x-p)² + q, where (p,q) represents the vertex of the parabola.

This form is particularly useful for identifying the vertex of the parabola, which is a crucial point in understanding the function's behavior.

Example: In the canonical form, p = -b/(2a) and q = -Δ/(4a), where Δ is the discriminant.

The page also presents the factored form of the quadratic function: y = a(x-x₁)(x-x₂).

Definition: The factored form of a quadratic function is y = a(x-x₁)(x-x₂), where x₁ and x₂ are the roots of the function.

This form is only applicable when the discriminant (Δ) is greater than zero, indicating that the function has two real roots.

Highlight: When Δ = 0, the function has one real root, and the factored form becomes y = a(x-x₀)².

The page concludes with a visual representation of the parabola, illustrating the relationship between the different forms and the graph's key features, such as the vertex and roots.

Vocabulary:

• Funkcja kwadratowa: Quadratic function
• Postać ogólna funkcji kwadratowej: General form of quadratic function
• Postać kanoniczna funkcji kwadratowej: Canonical form of quadratic function
• Postać iloczynowa: Factored form
• Miejsca zerowe: Roots or zeros of the function

This comprehensive overview provides students with a solid foundation for understanding and working with quadratic functions, emphasizing the connections between algebraic representations and graphical interpretations.