Even and Odd Numbers: Conceptual parallel to even and odd functions.
Key Idea: Explore the symmetry of functions with respect to the y-axis and both axes.
Even Functions
Definition: A function is even if it is symmetric when flipped over the y-axis.
Example: The classic parabola with a vertex on the y-axis.
Function: ( f(x) = x^2 ).
Mathematical Representation: ( f(x) = f(-x) ).
This indicates the function is equivalent to its y-axis reflection.
Odd Functions
Definition: A function is odd if it is symmetric when flipped over both the y and x-axes.
Example: Cubic function ( f(x) = x^3 ).
Process:
Flip over the y-axis.
Flip over the x-axis to return to the original function.
Mathematical Representation: ( f(-x) = -f(x) ).
Indicates the function is its own reflection over both axes.
Patterns in Even and Odd Functions
Exponent Connection:
Even functions often have an even exponent (e.g., ( x^n ) with even ( n )).
Odd functions often have an odd exponent (e.g., ( x^n ) with odd ( n )).
Exceptions:
Not all functions are purely even or odd.
Example of neither: ( (x - 2)^2 ) is not even, lacks symmetry over y-axis.
Shifting functions, such as ( x^3 + 3 ), breaks odd symmetry.
Unique Cases
Function Both Even and Odd:
Example: Constant function ( f(x) = 0 ).
Symmetrical about both axes due to its constant nature.
Represents a horizontal line at ( y = 0 ).
Highlights the unusual instance where a function can be both even and odd.
Conclusion
Challenge: Consider transformations and shifts that may affect function symmetry.
Reflection: The exploration of symmetry provides insight into the nature of mathematical functions and aids in understanding their geometric properties.