Page 2: Practical Example and Application

This page demonstrates a complete worked example of converting f(x) = 2(x+3)(x-4) through different forms, showcasing zamiana postaci kanonicznej na ogólną przykłady.

Example: Converting f(x) = 2(x+3)(x-4):

1. Expand to general form: f(x) = 2x²-2x-24
2. Calculate vertex coordinates using p = -b/(2a) and q = -Δ/(4a)
3. Final canonical form: f(x) = 2(x-2)²-49

Highlight: The discriminant (Δ = b²-4ac) equals 196 in this example, which is crucial for finding the vertex coordinates.

Definition: The canonical form f(x) = a(x-p)²+q represents the quadratic function in terms of its vertex (p,q).

Vocabulary: The coefficients a, b, and c from the general form are essential for calculating the discriminant and vertex coordinates.

Page 1: Converting Between Function Forms

This page introduces the fundamental concepts of converting quadratic functions between different forms, particularly focusing on the postać iloczynowa funkcji kwadratowej and its transformations.

Definition: The factored form of a quadratic function is expressed as f(x) = a(x-x₁)(x-x₂), where x₁ and x₂ are zeros of the function.

Highlight: To convert from factored to general form, multiply out the brackets following the distributive property.

Example: Converting from factored to general form: f(x) = a(x²-x₂x-x₁x+x₁x₂) → f(x) = ax²-a(x₁+x₂)x+ax₁x₂

Vocabulary: The vertex coordinates (p,q) are essential for the canonical form, where p is the x-coordinate halfway between the zeros, and q is the y-coordinate calculated by evaluating the function at p.