Page 2: Translations Along X-Axis and Y-Axis

This page delves deeper into specific types of translations and their effects on various functions.

For x-axis translations:

Formula: g(x) = f$x-p$ represents the new function after a translation by vector [p,0] along the x-axis.

Several examples are provided, showing how different functions change when translated along the x-axis.

The page then introduces y-axis translations:

Highlight: For a translation along the y-axis by vector [0,q], the new function g(x) is related to the original function f(x) by: g(x) = f(x) + q.

Visual examples illustrate how graphs shift vertically when translated along the y-axis.

Example: The function f(x) = x² translated by vector [0,2] becomes g(x) = x² + 2.

The page concludes with a table summarizing various function transformations, including both x and y translations.

Vocabulary: Przesunięcie równoległe względem osi OY refers to parallel translation along the y-axis.

Page 3: General Vector Translations

The final page covers translations by any vector [p,q], combining both horizontal and vertical shifts.

Formula: For a translation by vector [p,q], the new function g(x) is related to the original function f(x) by: g(x) = f$x-p$ + q.

This general formula encompasses both x and y translations in a single operation.

Example: The function f(x) = x² - 2x translated by vector [3,4] becomes g(x) = $x-3$² - 2$x-3$ + 4.

The page provides a detailed example of how to apply this transformation, showing the step-by-step process of simplifying the resulting function.

Highlight: Przesunięcie funkcji o wektor [p,q] combines horizontal and vertical shifts, allowing for more complex transformations of function graphs.

This comprehensive guide equips students with the knowledge to understand and apply parallel translations to various functions, enhancing their grasp of function transformations and graph manipulations.

Page 1: Introduction to Parallel Translation

This page introduces the concept of parallel translation, also known as vector translation. It explains the fundamental principles and provides visual representations of the transformation.

Definition: Parallel translation (or translacja) is a geometric transformation that moves every point by the same distance in a given direction.

The page illustrates how parallel translation affects points and shapes in a coordinate system:

Example: A point A(7,-3) translated by vector v = [-2,2] becomes A₁(5,-1).

It then focuses on translations along the x-axis:

Highlight: For a translation along the x-axis by a vector [p,0], the new function g(x) is related to the original function f(x) by the formula: g(x) = f$x+p$.

Visual examples are provided to demonstrate how the graph of a function shifts horizontally when translated along the x-axis.

Vocabulary: Przesunięcie równoległe wzdłuż osi OX refers to parallel translation along the x-axis.