Properties of Quadratic Functions in Canonical Form
This page delves deeper into the postać kanoniczna funkcji kwadratowej and its properties.
The canonical form of a quadratic function is given by y = ax−p² + q. The page provides exercises to practice converting quadratic functions to canonical form.
Example:
a) y = -3x² + −1,3 becomes y = -3x+1² + 3
b) y = 2x² + −0,−5 becomes y = 2x² - 5
c) y = -x² + 5,0 becomes y = -x−5²
An important theorem states that the graph of y = ax−p² + q is obtained from the graph of f = ax² by a parallel translation by the vector p,q.
Vocabulary: The vertex of a quadratic function in canonical form is located at the point Wp,q. The axis of symmetry has the equation x = p.
The page concludes with exercises to identify the vertex and axis of symmetry for given quadratic functions in canonical form.
Example:
For gx = x² - 8x, the vertex is W4,−16 and the axis of symmetry is x = 4.