Properties of Quadratic Functions Based on Parameter 'a'

This section examines how the coefficient 'a' affects the shape and behavior of quadratic functions.

Definition: The coefficient 'a' determines both the opening direction and the "width" of the parabola.

Example:

• When a > 0: Parabola opens upward
• When a < 0: Parabola opens downward
• Larger |a| values create narrower parabolas

Highlight: The value of 'a' never affects the domain of the function, which remains R.

Forms of Quadratic Functions

A detailed exploration of the three main forms of quadratic functions.

Definition:

• Postać ogólna (Standard form): f$x$ = ax² + bx + c
• Postać kanoniczna (Vertex form): f$x$ = a$x-p$² + q
• Wzór funkcji kwadratowej w postaci iloczynowej (Factored form): f$x$ = ax-x₁$$x-x₂

Highlight: Each form highlights different properties of the quadratic function:

• Standard form shows the y-intercept
• Vertex form shows the vertex coordinates
• Factored form shows the zeros

Converting Between Forms

This section covers the methods for converting between different forms of quadratic functions.

Example: To convert from standard to canonical form:

1. Find p = -b/(2a)
2. Calculate q by substituting x = p into the original function

Vocabulary:

• Delta (Δ): The discriminant b² - 4ac
• Zeros: The x-values where f$x$ = 0

Maximum and Minimum Values

Analysis of extreme values of quadratic functions.

Definition: The vertex represents either the maximum value (if a < 0) or minimum value (if a > 0) of the function.

Example: For a closed interval, find:

1. Calculate the discriminant
2. Find the vertex
3. Compare values at endpoints and vertex

Quadratic Inequalities

Solving quadratic inequalities through systematic steps.

Highlight: The solution process involves:

1. Rearranging to standard form
2. Finding zeros
3. Creating a sign chart
4. Determining intervals

Example: For 3x² - 6x - 16 ≥ 0:

1. Find zeros: x = -4 or x = 2
2. Solution: x ∈ $-∞, -4] ∪ [2, ∞$

Word Problems and Applications

Practical applications of quadratic functions in real-world scenarios.

Example: Rectangle optimization problem:

• Given a 42m perimeter
• Find dimensions for maximum area
• Solution: 10.5m × 10.5m gives maximum area of 110.25m²

Highlight: Real-world applications often involve:

• Area optimization
• Motion problems
• Economic models

Introduction to Quadratic Functions

This section introduces the fundamental concepts of quadratic functions and their basic properties.

Definition: A quadratic function is a polynomial function of degree 2, typically written as f$x$ = ax².

Highlight: The domain of a quadratic function is always all real numbers (R), while the range depends on the direction of the parabola.

Example: For f$x$ = x², the vertex is at (0,0), and the parabola opens upward.

Vocabulary:

• Vertex (wierzchołek): The highest or lowest point of the parabola
• Axis of symmetry (oś symetrii): The vertical line passing through the vertex