Page 2: Properties and Graphing of Quadratic Functions

This page delves deeper into the properties of quadratic functions and techniques for graphing them.

Key points:

• The canonical form y=a$x-p$^2+q is useful for identifying the vertex of the parabola
• Any quadratic function can be converted between general and canonical forms
• The axis of symmetry of the parabola is given by x=p
• The vertex of the parabola is the point (p,q)

Vocabulary: Postać kanoniczna funkcji kwadratowej refers to the canonical form of a quadratic function, expressed as y=a$x-p$^2+q.

Example: For the function f$x$=3x^2-42x, we can find the vertex by determining p=$0+14$/2=7 and q=f(7)=-147, resulting in the vertex $7,-147$.

The page also covers techniques for graphing quadratic functions, including identifying key points such as the y-intercept, vertex, and axis of symmetry.

Highlight: To graph a quadratic function, it's essential to identify the vertex, y-intercept, and the direction of the parabola's opening.

Page 3: Applications and Inverse Proportionality

This page discusses applications of quadratic functions and introduces the concept of inverse proportionality.

Key points:

• Inverse proportionality is defined by the equation y=a/x, where x≠0 and a≠0
• The graph of an inverse proportion is a hyperbola
• Inverse proportions have specific properties regarding their domain and range

Definition: Proporcjonalność odwrotna (inverse proportionality) is a relationship defined by the equation y=a/x, where x≠0 and a≠0. The constant 'a' is called the coefficient of inverse proportionality.

Highlight: The graph of an inverse proportion is a hyperbola, which has two separate branches in different quadrants of the coordinate plane.

The page also covers properties of inverse proportions:

• If a>0, the graph lies in the first and third quadrants
• If a<0, the graph lies in the second and fourth quadrants
• The domain and range exclude zero

Example: In the equation xy=x-2$$y+3, we can determine if it represents an inverse proportion by solving for x and y.

This page emphasizes the practical applications of quadratic functions and inverse proportionality in real-world scenarios.

Page 3: Applications and Inverse Proportionality

This page covers practical applications of quadratic functions and introduces inverse proportionality. The Funkcja kwadratowa zastosowania demonstrates real-world uses.

Definition: Inverse proportionality is defined by y=a/x, where x≠0 and a≠0.

Highlight: The graph of an inverse proportion is a hyperbola.

Example: When a>0, the hyperbola lies in quadrants I and III; when a<0, it lies in quadrants II and IV.

Page 4: Exponential Functions

This section introduces exponential functions and their properties. The exponential function is defined as y=aˣ, where a>0 and a≠1.

Definition: An exponential function has the form y=aˣ, where a is positive and not equal to 1.

Highlight: The domain of exponential functions includes all real numbers.

Example: For y=2ˣ, the function increases exponentially when x increases.

Page 1: Introduction to Quadratic Functions

This page introduces the concept of funkcja kwadratowa and its basic properties.

Key points:

• A quadratic function is defined as y=ax^2+bx+c, where a≠0
• The domain of a quadratic function is all real numbers
• The graph of a quadratic function is a parabola
• The parabola's orientation depends on the sign of 'a'
• The y-intercept of the parabola is at point (0,c)

Definition: A funkcja kwadratowa is a function that can be expressed in the form y=ax^2+bx+c, where a, b, and c are real numbers and a≠0.

Highlight: The shape and orientation of the parabola are determined by the coefficient 'a'. When a>0, the parabola opens upward, and when a<0, it opens downward.

Example: In the function y=x^2-2√2x^2-8, we can identify a=1-2√2, b=0, and c=-8.

The page also introduces the canonical form of a quadratic function, y=a$x-p$^2+q, where (p,q) represents the vertex of the parabola.