Vector Operations and Graph Transformations
This page provides a comprehensive overview of vector operations and graph transformations, essential topics in mathematics and physics.
Definition: A vector is defined as an ordered pair of points, represented as AB = [x₂-x₁, y₂-y₁].
The length of a vector is calculated using the formula: √(x₂-x₁)² + (y₂-y₁)².
Vector operations are crucial in Działania na wektorach Matematyka and include:
- Addition: a + b = [a₁+b₁, a₂+b₂]
- Subtraction: a - b = [a₁-b₁, a₂-b₂]
- Scalar multiplication: k·a = [k·a₁, k·a₂]
- Equality: a = b (if and only if a₁=b₁ and a₂=b₂)
Highlight: Understanding vector operations is fundamental for solving problems in Działania na wektorach Fizyka and Wektory w układzie współrzędnych.
The guide also covers graph transformations, which are essential for Przekształcenia wykresów funkcji:
- Translation by a vector: y = f(x-p) + q, where [p,q] are the vector coordinates
- Reflection over x-axis: y = -f(x)
- Reflection over y-axis: y = f(-x)
- Reflection over origin: y = -f(-x)
Example: For absolute value transformations, y = |f(x)|, the part of the graph below the x-axis is reflected above it.
The document also mentions the midpoint formula for a line segment: S = ((x₁+x₂)/2, (y₁+y₂)/2).
Vocabulary: Przekształcenia wykresów funkcji trygonometrycznych refers to transformations of trigonometric function graphs, which follow similar principles.
Understanding these concepts is crucial for tackling Przekształcenia wykresów funkcji zadania and preparing for Przekształcenia wykresów funkcji Sprawdzian. The Algorytm przekształcania wykresów funkcji provided here serves as a valuable tool for solving complex problems in mathematics and physics.
