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 = a$x-p$² + q. The page provides exercises to practice converting quadratic functions to canonical form.

Example: a) y = -3x² + $-1, 3$ becomes y = -3$x+1$² + 3 b) y = 2x² + $-0, -5$ becomes y = 2x² - 5 c) y = -x² + [5, 0] becomes y = -$x-5$²

An important theorem states that the graph of y = a$x-p$² + 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 W(p, 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 g$x$ = x² - 8x, the vertex is W$4, -16$ and the axis of symmetry is x = 4.

Forms and Properties of Quadratic Functions

This page focuses on the different forms of funkcja kwadratowa and their properties.

Quadratic functions can be expressed in three forms:

1. General form: f$x$ = ax² + bx + c
2. Canonical form: f$x$ = a$x-p$² + q
3. Factored form (if it exists)

Example: Converting between forms y = -3$x+1$² - 2 = -3$x² + 2x + 1$ - 2 = -3x² - 6x - 3 - 2 = -3x² - 6x - 5

The page outlines key properties of quadratic functions based on the sign of the coefficient 'a':

For a > 0:

1. Domain is ℝ
2. Range is [q, +∞)
3. f$x$ decreases in the interval $-∞, p$ and increases in $p, +∞$
4. Axis of symmetry: x = p
5. Vertex: W(p, q)
6. Minimum: f_min$p$ = q

For a < 0, similar properties are listed with appropriate changes.

Highlight: The vertex of a quadratic function can be found using the formula p = -b / (2a).

Transformations and Forms of Quadratic Functions

This final page covers transformations between different forms of funkcja kwadratowa and introduces the concept of trójmian kwadratowy (quadratic trinomial).

The page demonstrates how to convert from general form to canonical form using the completing the square method:

y = ax² + bx + c = a$x² + (b/a)x$ + c = a$x² + (b/a)x + (b/(2a))² - (b/(2a))²$ + c = a$x + b/(2a)$² - $b²/(4a)$ + c

Definition: The quadratic trinomial has the form ax² + bx + c, which is equivalent to a quadratic function.

The page provides examples of converting quadratic functions to canonical form:

Example: f$x$ = 2x² - 12x + 19 can be transformed to 2$x-3$² + 1

The concept of discriminant $Δ = b² - 4ac$ is introduced, which is useful in determining the nature of the roots of a quadratic equation.

Highlight: The canonical form of a quadratic function provides valuable information about its graph, including the vertex and axis of symmetry.

The page concludes with exercises to practice converting quadratic functions to canonical form and identifying key properties.

Introduction to Quadratic Functions

This page introduces the concept of funkcja kwadratowa (quadratic function) and its various forms.

A quadratic function is defined as a function that assigns to each real number x a value determined by the formula y = ax² + bx + c, where a ≠ 0 and a, b, c are real numbers. The page outlines different cases of quadratic functions based on the values of coefficients.

Definition: A quadratic function is described by the formula f$x$ = ax² + bx + c, where a ≠ 0 and a, b, c ∈ ℝ.

Example:

1. f$x$ = ax² (when b = 0 and c = 0)
2. f$x$ = ax² + bx $when c = 0$
3. f$x$ = ax² + c $when b = 0$
4. f$x$ = ax² + bx + c (full quadratic function)

The page also introduces the concept of postać kanoniczna funkcji kwadratowej (canonical form of a quadratic function), which is expressed as y = a$x-p$² + q.

Highlight: The canonical form provides valuable insights into the properties of the quadratic function, including its vertex and axis of symmetry.