Symmetry of Logarithmic Functions
This page discusses the symmetry and transformations of funkcja logarytmiczna logarithmicfunctions. It explains how to modify the basic logarithmic function to achieve different symmetrical properties.
The basic form of a logarithmic function is presented as fx = log₁ x. This serves as the foundation for understanding various transformations.
Definition: A funkcja logarytmiczna is a function of the form fx = logₐ x, where a is the base of the logarithm and x is the input variable.
To reflect the logarithmic function over the x-axis, a negative sign is placed before the entire function. For example, fx = -log₂ x is the reflection of fx = log₂ x over the x-axis.
Example: The function fx = log₂ x becomes fx = -log₂ x when reflected over the x-axis.
For symmetry with respect to the y-axis, a negative sign is placed before the x variable inside the logarithm. This transformation is represented as fx = log₂−x.
Highlight: The dziedzina funkcji logarytmicznej domainofthelogarithmicfunction changes when applying the transformation fx = log₂−x, as the input must be negative for the function to be defined.
The page also illustrates these transformations graphically, showing how the original function's shape changes with each modification.
Vocabulary: Przesunięcie funkcji logarytmicznej refers to the shifting of the logarithmic function graph along the coordinate axes.
These transformations are crucial in understanding the behavior of logarithmic functions and their applications in various mathematical and real-world scenarios.